Coherent transport through nanoelectromechanical systems · Die Untersuchung von kohärentem Transport durch nanoelektromechanische Sys-teme (NEMS) steht im Fokus dieser Arbeit. NEMS

Coherent transport through

nanoelectromechanical systems

vorgelegt von

Diplom-Physikerin

Anja Metelmann

Dresden

Von der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktorin der Naturwissenschaften

Dr. rer. nat

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. Michael Lehmann

1. Gutachter: Prof. Dr. rer. nat. Tobias Brandes

2. Gutachter: Dr. Andrew Armour, Associate Professor and Reader

Tag der wissenschaftlichen Aussprache: 27.07.2012

Berlin 2012

D 83

2

Zusammenfassung

Die Untersuchung von kohärentem Transport durch nanoelektromechanische Sys-teme (NEMS) steht im Fokus dieser Arbeit. NEMS stellen Bauteile dar, bei denenein quantenmechanisches Transportsystem an die Freiheitsgrade eines mechanischenSystems koppelt. Damit ist eine Untersuchung der Elektron-Phonon Wechselwirkungin einer Nicht-Gleichgewichtsumgebung möglich.

Im Allgemeinen wird bei der semi-klassischen Betrachtung dieser mechanisch-elektrischen Systeme eine adiabatische Näherung durchgeführt. In dieser Näherunggeht man davon aus, dass sich die Elektronen die durch das System tunneln deutlichschneller bewegen als der Oszillator, d.h. der Oszillator spürt nur ein effektives vonden Elektronen verursachtes Potential, während er seine Position im Gegensatz zuden Elektronen nur langsam verändert. Die interessantesten Phänomene zeigen sichjedoch, wenn sich die Elektronen und der Oszillator sich auf einer ähnlichen Zeitskalabewegen.

Im Rahmen dieser Arbeit wird die Dynamik des Oszillators in adiabatischer und innicht-adiabatischer Näherung untersucht. Um Zugang zu den mechanischen Eigen-schaften zu erlangen kann die Methodik der Feynman-Vernon Influenz Funktionalegenutzt werden. Wir konzentrieren uns auf zwei elektronische Modellsysteme – dasEin- und Zwei-Level System – die linear an eine einzige bosonische Mode koppeln.Wir nutzen den Greenschen Formalismus zur Berechnung der elektronischen Eigen-schaften, dieser bietet, in seiner Erweiterung durch Keldysh, Möglichkeiten Nicht-gleichgewichts - Prozesse zu beschreiben.

Die Dynamik für einen Oszillator im elektronischen Zwei-Level System zeigt einnicht-triviales Verhalten, es treten beispielsweise Grenzzyklen und Bistabilitäten auf.Wir untersuchen ausführlich die auftretenden Effekte mit Methoden für nichtlinearedynamische Syteme. Zudem erfolgt die Berechnung von Strom und Rauschen, bei-des Eigenschaften die experimentell zugänglich sind. Eine weitere Besonderheit derverwendeten Methode ist, dass keine störungstheoretische Behandlung der Kopplungzwischen dem Quantensystem und den elektronischen Anschlüssen erfolgt.

Zudem zeigen wir, dass sich unsere nicht-adiabatische Methodik ohne weiteres aufbestimmte Systeme übertragen lässt. Dazu nutzen wir das Modell eines elektron-ischen Transportsystems mit einem elektronischen Level, dass an einen großes En-semble von Spins koppelt, und dem Einfluss eines externen Magnetfeldes unterliegt.Das Spin Ensemble kann durch einen großen effektiven Spin beschrieben werden, fürden eine semi-klassische Beschreibung möglich ist. Dabei werden die Quantenfluktu-ationen des Systems als klein angesehen und die Wechselwirkung zwischen den Spin-Systemen im Rahmen einer Meanfield-Näherung beschrieben.

5

Abstract

In this thesis we compare the semiclassical description of nanoelectromechanical sys-tems (NEMS) within and beyond the Born-Oppenheimer approximation. NEMSenable the detailed study of the interaction between electrons, tunneling through anano-scale device, and the degrees of freedom of a mechanical system.

We work in the semiclassical regime where an expansion around the classical path isperformed. The advantage of this method is, that it is nonperturbative in the system-leads coupling, because the exact electronic solutions are included. We develop anonadiabatic approach, where we can treat the oscillator and the electrons on thesame time-scale without further constrains.

The considered NEMS models contain a single phonon (oscillator) mode linearlycoupled to an electronic few-level system in contact with external particle reservoirs(leads). Using Feynman-Vernon influence functional theory, we derive a Langevinequation for the oscillator’s trajectories. A stationary electronic current through thesystem generates nontrivial dynamical behaviour of the oscillator, even in the adia-batic regime. We present a detailed prescription of the oscillator’s phase space andinvestigate the observed dynamical features with methods for nonlinear dynamicalsystems.

The backaction of the oscillator onto the electronic properties is studied as well.For the cases of one and two coupled electronic levels, we discuss the differencesbetween the adiabatic and the nonadiabatic regime of the oscillator dynamics.

Furthermore, we apply the developed methods to a single-level system which isanisotropically coupled to a large spin under the influence of an external magneticfield. Here, the semiclassical treatment of the large spin’s dynamics is included withina mean-field approach for the spin-spin interaction.

This system possess rich dynamical properties, like self-sustained and chaotic os-cillations. We investigate the system in the nonequilibrium regime for high externalbias, where we can compare our nonadiabatic method to a rate equation approach,which is perturbative in coupling to the leads. Additionally, we study the system inthe low bias case, where the dynamics are even richer as in the infinite bias case.

7

Acknowledgement

First of all, I would like to thank Tobias Brandes for his marvellous support in the lastthree years. I am thankful for all the interesting discussions and for the opportunitieshe opened for me. His enthusiasm in physics was quite inspiring for me.

Moreover, I am thankful for all the amazing people I was able to met during thelast years and with whom I had fruitful discussions about physics. I want to thankAndrew Armour for the possibility to visit his group in Nottingham, which stronglymotivated me further in my work.

Furthermore, I like to thank all the people of my group at the TU Berlin. Espe-cially, the former members Philipp Zedler and Robert Hussein, for the collaborativework on the NEMS system. Philip Zedler, sharing the office with we, had alwaystime for discussions about Green’s functions, numerics and the remaining topics ofthis world.

Not to forget Christian Nietner, who corresponds to my social-unit almost everyday in the environment of a small cup of espresso. I thank Mathias Hayn, KlemensMosshammer and Malte Vogl for proof-reading of this work and Claudius Hubig,who was hunting with me for fixed points.

For financial funding I thank the the GRK 1558 and especially the Rosa Luxem-bourg Foundation.

I thank my family and friends for their patience and their support, Saskja Schadefor keeping me in the team and Super-G-star Balindt for being my friend.

Finally, a thanks goes to the confederates who crossed my way, all the people whoimagined Kathryn, Jean-Luc and Benjamin and last, but not least, Jonathan Davis.

9

Contents

1. Introduction 13

1.1. A brief history of mechanical systems . . . . . . . . . . . . . . . . . . 151.2. A route map for this work . . . . . . . . . . . . . . . . . . . . . . . . 19

2. Mechanical system coupled to a single electronic level 21

2.1. Adiabatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.1. Hamiltonian and model . . . . . . . . . . . . . . . . . . . . . 222.1.2. Langevin equation: stochastic equation of motion . . . . . . . 232.1.3. Electronic forces and friction in the adiabatic regime . . . . . 262.1.4. Effective potential and phase space portraits . . . . . . . . . . 31

2.2. Nonadiabatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.1. Stochastic equation of motion with nonadiabatic ansatz . . . 352.2.2. Time-dependent occupation of the single-level . . . . . . . . . 362.2.3. An alternative way of deriving the occupation . . . . . . . . . 392.2.4. Phase space in the nonadiabatic regime . . . . . . . . . . . . 42

2.3. Electronic properties influenced by the mechanical system . . . . . . 442.3.1. Current for the single-level system . . . . . . . . . . . . . . . 442.3.2. Noise for the single-level system . . . . . . . . . . . . . . . . . 47

3. Mechanical system coupled to an electronic two-level system 53

3.1. Dynamics: adiabatic vs. nonadiabatic . . . . . . . . . . . . . . . . . 533.1.1. Langevin equation for the two-level system . . . . . . . . . . 543.1.2. Time-dependent occupation difference . . . . . . . . . . . . . 553.1.3. Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2. Analysis using the adiabatic approach . . . . . . . . . . . . . . . . . 623.2.1. Why limit cycles ? . . . . . . . . . . . . . . . . . . . . . . . . 623.2.2. Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.3. Fixed point analysis . . . . . . . . . . . . . . . . . . . . . . . 67

3.3. Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.1. Current and noise for the two-level system . . . . . . . . . . . 733.3.2. Lead- and level-transition functions . . . . . . . . . . . . . . . 783.3.3. A brief summary for the two-level system . . . . . . . . . . . 80

4. Large external spin coupled to a single electronic level 81

4.1. Single-level with spin and linear potential . . . . . . . . . . . . . . . 814.1.1. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.1.2. Nonadiabatic approach . . . . . . . . . . . . . . . . . . . . . . 84

11

Contents

4.1.3. From Berlin to Madrid: rate equation approach . . . . . . . . 854.2. Abundant dynamics in the finite bias regime . . . . . . . . . . . . . . 94

4.2.1. Finite bias: adiabatic approach . . . . . . . . . . . . . . . . . 944.2.2. Dynamical analysis . . . . . . . . . . . . . . . . . . . . . . . . 974.2.3. Results for the nonadiabatic approach in the finite bias regime 1094.2.4. Which approach works best? . . . . . . . . . . . . . . . . . . 115

5. Conclusion 119

A. Theoretical concepts 121

A.1. Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.1.1. Feynman-Vernon influence functional . . . . . . . . . . . . . . 121A.1.2. Classical action and Langevin equation . . . . . . . . . . . . . 123

A.2. Green’s functions in the adiabatic approach . . . . . . . . . . . . . . 124A.2.1. Undisturbed Green’s functions of the leads . . . . . . . . . . . 124A.2.2. Green’s functions of the two-level system . . . . . . . . . . . . 125A.2.3. Green’s functions of the spin system . . . . . . . . . . . . . . 125

A.3. Basics for a fixed point analysis . . . . . . . . . . . . . . . . . . . . . 127

B. Miscellaneous calculations 129

B.1. Single-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129B.1.1. Once more: time-dependent single-level occupation . . . . . . 129B.1.2. Check: Analytic result for the occupation . . . . . . . . . . . 130

B.2. Two-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B.2.1. Occupation difference . . . . . . . . . . . . . . . . . . . . . . 132B.2.2. Friction and occupation in the infinite bias case . . . . . . . . 133B.2.3. Transition functions in the adiabatic limit . . . . . . . . . . . 134

B.3. Spin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136B.3.1. Nonadiabatic equations for the coupling to a large spin . . . . 136B.3.2. SDS: infinite bias results for the adiabatic approximation . . 138B.3.3. Adiabatic corrections . . . . . . . . . . . . . . . . . . . . . . . 139

12

1. Introduction

Transport spectroscopy is a powerful tool to investigate objects on the atomic scale,enabling one to take a deep look inside electronic structures. It allows the study offundamental interactions, which originate from the quantum nature of matter, as wellas the observation of the interplay between electronic devices with other quantities,like phonons or photons.

Especially, nanoelectromechanical systems (NEMS) enable the detailed study ofthe interaction between electrons, tunnelling through a nano-scale device, and thedegrees of freedom of a mechanical system. The electronic current affects the me-chanical system and vice versa. The length scale of the mechanical object can behuge compared to the scale of individual atoms. The dimensions of systems usedin recent experiments range down to scales, where the observation of fundamentalquantum behaviour for a comparatively macroscopic object is possible [1, 2].

The influence of strong electron-phonon coupling in molecules or suspended carbonnanotubes yields highly interesting effects [3, 4, 5], like the Franck-Condon blockade,where the influence of the mechanical system suppresses the electronic current [4,5], or switching in molecular junctions [6]. Furthermore, the interaction with theelectronic current can drive the mechanical system in the nonlinear regime, whereeffects like self-oscillations appear [7, 8].

In an experiment, an electronic device, which can act as a detector of the oscil-lator’s displacement [1, 9], always exhibits backaction onto the mechanical system.This effect can be utilised to cool the resonator, preparing ultracold and quantumstates of mechanical structures [9]. Additionally, the backaction of the electronic de-vice can be used to control the mechanical system. For instance, to generate quantumsuperposition states by using Cooper-pair boxes as detectors [10, 11, 12, 13].

A standard approach to solve the dynamics of NEMS is to apply a perturbationtheory in the tunnel Hamiltonian and thus treating the coupling of the detector withthe external electronic reservoirs perturbatively. This has been successfully done upto the co-tunnelling regime [14] and is a well-explored path, where the dynamics ofthe system is described by master equations or generalisations thereof in Liouvillespace. Many interesting physical results can be obtained via this approach, e.g.avalanche-type molecular transport [15, 16, 17] or laser-like instabilities [18]. Thismethod was useful to describe NEMS coupled single electron transistors (SETS)[18, 19, 20] or for the discussion of quantum shuttles [21, 22, 23, 24].

All the mentioned methods produce good results in the range of high bias, whereNon-Markovian effects [25, 26], due to quantum coherences between the externalreservoirs and the electronic system, can be neglected [27]. However, they fail tocorrectly predict the system’s dynamics in the Non-Markovian regime. It is therefore

13

1. Introduction

desirable to develop tools that allow a description of NEMS beyond the Masterequation regime (weak electron-leads coupling) and at the same time are not merelyperturbative in the coupling of the oscillator to the electronic environment [28].

In the past, the coupling of electrons to a single bosonic mode has been solvedexactly for the case where only one single electron is present [29, 30], i.e. in an emptyband approximation. The inclusion of Fermi sea reservoirs at different chemicalpotentials transforms this into a difficult nonequilibrium many-body problem, andapproximations are necessary [28, 31, 32, 33]. To gain access to the small biasregime, one has to work perturbatively in the system-oscillator coupling insteadof the system-leads coupling [28]. Alternatively, if the oscillator is treated in asemiclassical regime, Feynman-Vernon influence functional techniques are suitable[34, 35, 33, 36, 37].

In general, the semiclassical treatment is combined with an adiabatic approxima-tion, where the oscillator is slow compared to the electrons which pass through theelectronic device. But the interaction between these systems is strongest, if both acton the same timescale [19]. In recent publications, different approaches, e.g. basedon scattering theory [38, 39], were used to go beyond the adiabatic approximationor to verify the range of validity for this approach [40, 41].

In this work we go one step further, leaving the adiabatic approximation behind.We develop an approach, enabling us to treat the system in a complete nonadiabaticmanner. As a consequence, we have to calculate the system’s quantities by consider-ing their full time-dependence. For this reason, the oscillator and the electrons canact on the same time-scale without further constraints. This allows us to criticallyassess the validity of the adiabatic approach.

To this end, we stay in the semiclassical regime, where we combine exact solutionsof the electronic system with a semiclassical expansion using the Feynman–Vernoninfluence functional (double path integral) theory [34]. The NEMS models we areinterested in are the single-level and the two-level system linearly coupled to a singlebosonic mode. Especially, the last one possess rich and nontrivial dynamics. Theinteraction with the stochastic current of the electronic system, drives the oscillatorinto the nonlinear regime, which is rich in bifurcations, leading to the appearance offeatures like limit cycles and bistabilities [42, 43].

In addition, we show that the nonadiabatic approach is straightforwardly transfer-able to different kinds of systems. The requirement for this transfer, is an externalsystem, treated in a semiclassical manner, which is coupled to a nonequilibrium en-vironment, as represented by an electronic transport system. In this work, we choosethe example of the Fano-Anderson model [44, 45] coupled to a large external spin,where the spin of an electron tunneling through the device interacts with a largeensemble of spins [46].

14

1.1. A brief history of mechanical systems


Omne quod movetur ab alio movetur - Everything that is in motion must be movedby something [47]. This is one of the basic principles of the Aristotelian Physicsaround 300 BC, which can be regarded as the starting point for the description ofclassical mechanical systems. In this theory a so-called motor conjunctus – a causefor movement, or change of movement – is always needed. And every moving objecthas to stay in contact with its cause.

Figure 1.1.: Aristotle, Ptolemy andCopernicus discussing,title page of [48].

In the medieval, before Newton published hisPhilosophiae Naturalis Principia Mathematica

900 years later, Arabian philosophers enhancethe theory of motion and criticise the Aris-totelian Physics. That a force is proportionalto acceleration was already signified in the 12thcentury and a lot of other concepts where intro-duced in first principles, like inertia, momen-tum and gravity [49]. These early works influ-enced the development of the impetus theoryin the 14th century, a supplement to Aristo-tle’s physical principles. It contains the con-cept, that the moving object implies its causeof movement. The mathematical definition ofthe impetus is velocity times mass, quite closeto the definition of momentum.

Within this theory, the cornerstone for theunderstanding of oscillatory motion was laid inthe framework of a thought-experiment. Theso-called tunnel-experiment, where a cannon-ball oscillates along the inside of a tunnel pass-ing straight through the earth [49]. The link to the motion of a pendulum was veryvaluable for the development of the basics of mechanical dynamics. In the 17thcentury, Galileo published his result from experiments with pendulums in Dialogue

Concerning the Two Chief World Systems, whose title page of the Latin edition isdepicted in Fig. 1.1 and in which he compared the Copernican system with the tra-ditional Ptolemaic system. Not much later, Newton developed his law’s of motionand therewith the main principles of classical mechanics were established.

With the beginning of the 19th century, the development of relativity and quantumtheory lead to a complete re-thinking in the understanding of dynamical principles.Consequently, the range of applicability for classical mechanics became limited. Ingeneral, this limitation is correlated with the length-scale of the dynamical system,whereby quantum mechanics become relevant for objects on the atomic-scale. Not toforget about macroscopic effects and properties of matter, which are based on quan-tum mechanics, e.g. superconductivity or ferromagnetism. Until now, the crossoverbetween quantum and classical mechanics is not entirely clear [50].

15

1. Introduction

In recent years, the progress in fabrication of nano-scale devices and the develop-ment of sensitive optical techniques [51], has lead to detectors which can be used forquantum-limited measurement.

For example, one important experiment has allowed a detailed study and controlof single phonons by cooling a macroscopic resonator mode close to its ground stateand coupling it to single electronic degrees of freedom [1].

Figure 1.2.: Micromechanicalresonator coupledto a qubit [1].

In Fig. 1.2 an optical micrograph of the useddevice is depicted. The resonator is capacitivelycoupled to a qubit, which can be modelled as aneffective two-level system. The qubit is used forquantum-limited measurement of the oscillatorsmotion. Although, the mechanical oscillator hasa quite short ring-time (Q ∼ 260), they were ableto detect both systems in their respective groundstate, due to the strong coupling between them.

A strong coupling is important, because the in-teraction between the two systems must be fasterthan the dissipation of energy from them [2]. Thequality factor Q of a resonator, describes the life-time of the mechanical oscillations. A high Q im-plies low dissipation and therewith a long oscilla-tion time.

Another recent experimental set-up used a so-called quantum-drum with a comparatively longlife-time (Q ∼ 3.6 × 105). On the basis of op-tomechanical concepts, they used a superconduct-ing cavity to cool the mechanical drum resonatoruntil it reaches its quantum ground state [2].

A lot of other experimental groups investi-gate mechanical resonators at the quantum limit[13, 52, 53]. Details of further cooling experiments,including a listing of the different set-ups and theirproperties, can be found in [51]. There, the au-thors discuss the differences between several mechanical systems. Denoting that forbottom-up devices, like semiconductor based beams or cantilevers, as well as theresonators used for the presented experiments [1, 2], the Q factor decreases with thesize of the resonator. Defects originating from the fabrication technique are mostlyresponsible for this. It is expected, that so-called top-down devices, like nanowires orcarbon nanotubes, own higher quality factors. But until now, the position detectionfor these nanoelectromechanical systems is not as far developed as for the bottom-updevices.

There are several theoretical works considering a nanowire in a magnetic fieldbetween two superconducting leads [55, 56, 57], which propose ground state coolingby a supercurrent flow [57], as well as the suppression of fluctuations [56] due to

16


Figure 1.3.: Two different set-ups for nanoelectromechanical systems. LEFT: Sus-pended carbon nanotube used in Delft [54], taken from [51]. RIGHT:Bottom-up nano device and measurement diagram from [9]. SiN andAl nanomechanical resonator (NR) coupled to a superconducting single-electron transistor (SSET), with simplified measurement circuit diagram.Details in [9].

tunnelling events of electrons.

As mentioned above, carbon nanotubes are expected to possess a high qualityfactor. The deposition of the tube is the last step in fabrication and therefore theprobability for the occurrence of defects is lower. But it is difficult to bring theoscillator in the linear regime. The reason for this is the coupling between thestochastic electrons and the mechanical oscillator, which is strong and enhancesbackaction effects [7, 8]. At the one side this leads to damping and therefore cooling ofthe mechanical motion and one the other side to nonlinear behaviour of the nanotubesmotion [58, 59, 60]. The development of measurement protocols [61], the control ofthe backaction effects [60] and the classification of the phonon decay rates [62, 63]are topics in the scope of recent research.

Backaction effects play an important role for experiments with mechanical oscil-lators. They can lead to cooling and amplification of the mechanical motion as wellas the increase of the coupling strength between mechanical systems and electronictransport systems. In Reference [64] a macroscopical resonator was driven by themesoscopic backaction of electrons tunnelling through a radio-frequency quantumpoint contact. The authors used noise measurement to investigate the influence ofthis backaction, using the effect that the noise of the detector is affected by themechanical system.

In Fig. 1.3 a set-up for measuring the backaction of a superconducting single-electron transistor (SSET) on a radio-frequency nanomechanical resonator is depicted[9]. Here, a capacitively coupled SSET is used to probe the resonators position. Thetransport properties for this kind of system was intensively studied [18, 65, 66, 67].In the framework of quantum master equations Armour, Rodrigues and co-workers

17

1. Introduction

observed damping effects for pure SET [19] and self-oscillations in SSET [18].

Self-oscillations can be utilised for single-electron transport, so-called quantumshuttles as proposed by [68]. These devices possibly allow to close the quantummetrology triangle [69], which links voltage, current and frequency via independentquantum effects based on the fundamental constants ~ and e [70]. In general, themeasurement of single-electron transport is limited by co-tunnelling events, whereelectrons coherently pass the system through virtual states [70, 71].

Moreover, nonlinear dynamical effects in driven nanoelectromechanical systemsare in the scope of recent research [72, 73], because they can help to investigatethe transition from classical to quantum mechanics [74, 75]. For example, one aimis to prepare a mechanical resonator in a quantum superposition state and watch itbecome classical by using a superconducting qubit to drive and probe the mechanicalresonator in a superposition state [11, 12].

A further example for nonlinear effects under an external force appears in sus-pended carbon nanotubes. There the interplay of mechanical and electric degreesof freedom in the tube changes, if the system experiences stress due to an externalforce [76]. This stress leads to a modification of the coulomb blockade and a strongenhancement of phonon blockade. The behaviour depends on the strength of the ap-plied force. Below a critical force the system acts as a harmonic oscillator and abovethe so-called Euler instability, the system behaves like a Duffing oscillator [77].

The experimental success in reaching and detecting the quantum ground stateof a mechanical object has lead to a huge number of new experiments over therecent years. Especially in the field of optomechanics [78, 79, 80]. The simplestset-up for a optomechanical system is a Fabry-Pérot cavity resonator build of tworeflecting mirrors, one of those is fixed and the other one is movable. An appliedelectromagnetic field exerts a radiation pressure force, leading to a displacement ofthe movable mirror and a modification of the cavity mode, due to backaction.

Some goals of optomechanical experiments are quite similar to the ones of elec-tromechanical systems. The best example for this is the cooling of a mechanicalsystem to the quantum ground state with optomechanical techniques [81, 2], as wellas using these devices for force sensing and quantum information processing technol-ogy [78]. Another similarity to electromechanical systems is found in the dynamicalfeatures of optomechanical systems. For instance, the observation of optically in-duced self-oscillations in mechanical resonators [82] has lead to the proposal forphoton shuttles [83]. Noise and backaction effects play an important role for theelectromechanical systems. Another goal is the investigation of photon shot noise[78, 84], which is important in optical experiments.

There are several ideas for further electro- and optomechanical systems, e.g. withoptically levitated nanodielectrics [85, 86]. The range of utilisation in present exper-iments is large. A quite exciting field is the detection of gravitational waves using anoptomechanical design [87], where mirrors are used as pendulums, which respond toa passing gravitational wave. But also terrestrial technology is expected to benefitfrom these systems, in the range of mass sensors or in signal processing [51].

18

1.2. A route map for this work

1.2. A route map for this work

We want to give the reader a quick overview about the structure of this work.In chapter 2 the single-level system, which is coupled to a single bosonic mode, is

in the scope of our investigation. This comparatively simple nanoelectromechanicalmodel is an ideal starting point to discuss our several theoretical approaches and tohighlight the effects of the interaction between the mechanical and the electronic sys-tem. We discuss in detail the derivation of the adiabatic Langevin equation obtainedwith the help of Feynmann-Vernon influence functional technique. Followed by apresentation of the result for the effective potential and the phase space trajectoriesof the oscillator.

Subsequently, we introduce our nonadiabatic approach. After the illustration ofthe necessary theoretical modifications, we discuss differences to the adiabatic resultsfor the oscillator’s dynamics. We also turn towards the electronic properties of thesystem in the end of this chapter.

In chapter 3, we use the methods developed for the single-level system and extendthem to an electronic two-level system. The latter, includes much richer dynamics asthe single-level case, but as well the theoretical effort increases, especially concerningthe nonadiabatic approach. We analyse the oscillator’s phase space results by usingmethods from nonlinear dynamical systems. Subsequently, we focus on the electronicproperties of the system, where we be able to investigate the difference between theapproaches in more detail.

In chapter 4 we turn to another interacting transport system, which can be de-scribed in a semiclassical manner. The model we focus on is a single electronic levelcoupled to a large external spin in the environment of an external magnetic field. Wetransfer our prior developed approaches to investigate this model for two differentbias situations.

Within this whole thesis, the reduced Planck constant is set to one (~ = 1).

19

2. Mechanical system coupled to a

single electronic level

The modelling of a large class of systems used for electronic transport spectroscopy,like single electron transistors (SET) or molecules, is accomplished by the single res-onant level model. This simple model contains one electronic level confined betweentwo electronic reservoirs – namely the leads.

The coupling of such a system to mechanical degrees of freedom yields highlyinteresting features, like current blockade [88], damping of the resonators motion[19] or molecular switching [35]. The theoretical investigations in the frameworkof classical approaches was performed by several authors [88, 20, 89], as well as theextension to the quantum regime by using Master equations or generalisations thereof[19, 41, 28].

The description of the system’s dynamics in a semiclassical framework is possi-ble within a Langevin equation ansatz [35, 40, 39], due to the stochastic nature ofthe electronic current flowing through the system. The Langevin equation can beobtained via linear response theory [32] or by using Feynman-Vernon influence func-tional theory which has been used in simple NEMS models by Mozyrsky, Brandbygeand co-workers [35, 36, 37].

This method was incorporated in recent works from several authors [40, 39, 90, 42].But is combined within an adiabatic approximation assuming a slow oscillator. Werevise this method to a nonadiabatic description. However, we start this section withthe adiabatic approach in order to introduce the basic concepts. Additionally, theadiabatic results are necessary for a later comparison to the ones obtained from thenonadiabatic ansatz.

2.1. Adiabatic approach

In this section we study a single bosonic mode linearly coupled to an electronic single-level system. The latter is confined between two external electronic reservoirs withdetuned chemical potentials. The electronic current running through the systemprovides a nonequilibrium environment for the oscillator. Due to the coupling tothe electronic system, the initial oscillator potential gets modified. The stochasticnature of the transport process leads to the appearance of an intrinsic friction, whichdamps the oscillator’s motion.

Within an adiabatic approach, we assume the oscillator to be slow compared tothe electrons which are jumping through the system. We describe the nonlinear dy-namical system with a semiclassical approach, where the quantum fluctuations are

21

2. Mechanical system coupled to a single electronic level

considered to be small. One advantage of the used method is, that it is nonperturba-tive in the coupling between the electronic level and the external reservoirs. Followingfrom that, we can assume strong coupling to the leads and work in arbitrary biasregimes without further constraints.

We use the adiabatic approach to introduce the basic concepts and the dynamicaleffects occurring due to the interaction between the mechanical and the electronicsystem. As we will learn later, with the introduction of the nonadiabatic approach inSec. 2.2, the analytical study of the system is rather possible within a nonadiabaticapproach.

The dynamics of the oscillator is the scope of this section. After a brief intro-duction of the single electronic level model, we derive a Langevin equation, whichdescribes the oscillator motion in phase space. We use the Feynman-Vernon influencefunctional technique [34] for this derivation. To calculate the appearing electronicforces, we need a formalism which enables us to consider the nonequilibrium proper-ties of the system. The Keldysh Green’s functions [91, 92] are ideal for this purpose,which we will see in the third part of this section. This is followed by the presentationof the results for the oscillator’s trajectories in phase space. Additionally, we definean effective potential for a broader analytic investigation of the parameter space.

2.1.1. Hamiltonian and model

L RΓL ΓR

Vbiasεd

Figure 2.1.: Sketch of a single-level system coupled to an oscillator.

Our electronic model consists of a single electronic level which is confined betweena left (L) and a right (R) electronic reservoir. These reservoirs are assumed to bein equilibrium and filled with noninteracting electrons up to the Fermi level energycorresponding to chemical potentials µα∈L,R. When we apply an external bias Vbias

we obtain an electronic current in the direction of the detuning, as illustrated inFig. 2.1. The tunnelling rate ΓL determines the strength of tunnelling from the leftlead to the electronic level, and in the same sense ΓR the tunnelling to the right lead.

This single-level system, or also called Fano-Anderson model [44, 45], is one of thesimplest transport systems. The Hamiltonian for this system is

He =∑

kα

εkαc†kαckα +

∑

kα

(Vkαc

†kαd +H.c.

)+ εdd

†d. (2.1)

22


The first term corresponds to the energy of the reservoir electrons with momentum kand energy εkα. The operators c†kα/ckα create/annihilate electrons in the left (α = L)or right (α = R) lead. The level has energy εd and creation/annihilation operatorsd†/d. The second term of the Hamiltonian describes the transitions between a statein the lead α and the level with the tunnelling amplitude Vkα.

We combine the introduced electronic system with a single bosonic mode, whichis described by a parabolic potential

Hosc =1

2mp2 +

1

2mω2

0 q2. (2.2)

Here, qt and pt denote the position and momentum operators of an oscillator withfrequency ω0 and mass m. Naturally, we can also express the oscillator Hamiltonianin second quantisation as the electronic system, by using bosonic operators a/a†.But for our later semiclassical description the used notation is more suitable.

Linear coupling between the electronic and the mechanical subsystem leads to ourfinal model for the nanoelectromechanical system. The total Hamiltonian reads

H = He +Hosc − λd†dq, (2.3)

where λ equals the coupling strength. This system is also called the Anderson-Holstein model [93]. The last term of Eq. (2.3) can be interpreted as an electronicforce F = λd†d = λn acting on the oscillator. Due to this force the initial harmonicoscillator potential gets modified and we obtain a multistable system with nonlineardynamical properties. Obviously, the backaction onto the electronic system modifiesthe electronic properties as well. But at first we concentrate on the dynamicalbehaviour of the oscillator.

2.1.2. Langevin equation: stochastic equation of motion

There are several ways for deriving an (semiclassical) equation of motion for theoscillator trajectories in a NEMS system. For instance, Rodrigues and co-workersderived a closed set of equations for the averaged dynamics of a resonator coupledto a single-electron transistor (SET) by starting from a quantum master equationand applying a mean-field approximation [65]. In their work they found a qualitativegood accordance of the semiclassical approach and the complete quantum solutions.For a similar system, the usage of Linear Response theory was also successful todescribe the dynamics of the oscillator [94].

On the basis that the mechanical system underlies a Langevin dynamic one canalso directly write down a Langevin equation and consider the electronic nonequilib-rium environment as origin for the stochastic forces. From this starting point, Bodeet. al. calculated these current forces by expanding F = λn in the framework ofGreen’s functions and a scattering matrix approach [95]. With this, they developed ageneralised description for collective modes coupled to a mesoscopic conductor withthe possibility to include multiple electronic orbitals.

23


By using Feynman-Vernon influence functional theory, we derive the Langevinequation for our NEMS system in a self-consistent manner. We revise the methodby Mozyrsky, Brandbyge and co-workers [33, 37, 36]. Within this formalism wecombine the exact solutions of the electronic system with a semiclassical expansionfor the oscillator trajectory. This method is nonperturbative in the coupling betweenthe level and the electronic reservoirs. Hence, considering the transport system, weare able to go beyond a master equation approach.

Speaking in a quantum dissipation language, we consider the electronic model,containing the leads and the electronic level, as our bath and the oscillator as oursystem. Assuming that the total density matrix χ(t) factorises at the initial time t0into a system and a bath part χ(t0) = ρosc(t0) ⊗ ρB, we trace out the bath degreesof freedom and obtain the reduced density matrix ρosc(q, q

′, t) = 〈q|ρosc(t)|q′〉 of theoscillator. Their propagation in time can be described by a double path integral[96, 97]

ρosc(q, q′, t) =

∫dq0

′dq′0 ρosc(q0′, q′0, t0

′)

q∫

q0

Dq(τ)

q′∫

q′0

D∗q′(τ) ei(Sq′−Sq′ )F [q; q′](τ),

(2.4)

where every path in phase space is weighted with the influence functional F [q, q′](τ).The classical action is denoted by Sq and the details of their evaluation are shown inthe Appendix section, see Sec.A.1.2. The influence functional can be expressed viathe time evolution operator U [q](t, t0), containing the effective bath part Heff

B [q] (t) =

He − λF qt, it yields

F [q; q′](t0, t) = trBU †[q′](t, t0)U [q](t, t0)ρB

. (2.5)

In the next step we transform to centre-of-mass and relative coordinates

xt =qt + q′t

2, yt = qt − q′t, (2.6)

to detach the classical trajectory xt from the quantum mechanical deviations yt.Within a Born-Oppenheimer approximation, this change of variables allows us tostudy a slow oscillator by an adiabatic approximation of the classical trajectory

xt ≈ x0 + tx0. (2.7)

In this adiabatic approach, the typical time-scale of the oscillator movement is slowcompared to the electronic transition rates. In the subsequent, when having intro-duced the angular oscillator frequency by ω0 and electron transition rates by ΓL, ΓR

the condition ω0 ≪ ΓL, ΓR has to be satisfied.

We now decompose the bath into an unperturbed part H0 = He− F x0 and a per-turbation part V (t) = −F (tx0+

12yt). With this, we introduce an effective interaction

24


picture with respect to the unperturbed part:

U [q](t, t0) = eiH0(t−t0)U [q](t, t0) = Te−i

t∫t0

dt′ V (t′)

,

and V (t) = eiH0(t−t0)V (t)e−iH0(t−t0). (2.8)

Here, T equals the time-ordering operator, arranging operators with later times tothe left. We remain in this interaction picture for all our further calculations in theadiabatic regime and denote this by using the tilde symbol, e.g. A.

In the next step, we expand the influence functional up to second order in theperturbation V (t). We emphasise, that this results in terms of quadratic orderconcerning both, the off-diagonal path yt and the electronic force term 〈F 〉. Dueto the semiclassical approximation, we assume the fluctuations around the classicaltrajectory yt to be small and to obey Gaussian statistics [33]. With this we canneglect higher order terms (corresponding to non-Gaussian noise) in yt. To improveour approach concerning the electronic terms, we apply a cluster expansion [98] andre-include all electronic second order terms.

Finally, we obtain for the influence functional

Fpert[qt; q′t] = e−Φ[x0;yt], (2.9)

with the influence phase

Φ[x0; yt] = −i

t∫

t0

dt′[〈F [x0](t

′)〉 − x0A[x0](t′)]yt′ +

t∫

t0

dt′t∫

t0

ds 〈ξ(t)ξ(t′)〉 yt′ys.

(2.10)

The influence phase can be regarded as a cumulant generating functional for theforce operator correlation functions. Using this language, the first cumulant equalsan electronic force which is proportional to the level occupation. The force operatorin the effective interaction picture yields

F [x0](t) = ei(He−F x0)tF e−i(He−F x0)t. (2.11)

The second cumulant contains the fluctuations of the electronic operator δF [x](t) =F [x0](t) − 〈F [x0](t)〉. Due to the adiabatic approximation, Eq. (2.7), a separationinto real and imaginary part of the second cumulant is possible. The second termtx0 of the Taylor expansion in Eq. (2.7) leads to an explicit friction term

A[x0](t) = 2

∫ t

t0

dt′ t′ Im〈δF [x0](t)δF [x0](t′)〉. (2.12)

The real part of the second cumulant corresponds to a stochastic force with thecorrelation function

〈ξ(t)ξ(t′)〉 = 2Re〈δF [x0](t)δF [x0](t′)〉. (2.13)

25


After further calculations, we obtain a stochastic equation of motion for the classicaltrajectory. A detailed derivation of it is contained in Sec. A.1. To achieve a self–consistent equation of motion, we re-insert the full time-dependence of the fixedclassical trajectory in accordance with the adiabatic approximation and end up withthe Langevin equation

mxt + V ′osc(xt)− 〈F [x](t)〉 + xtA[x](t) = ξt. (2.14)

Here, the derivation of the harmonic potential V ′osc(xt) corresponds to a classical

restoring force with the spring constant k = mω2. Following Gaussian statistics, thestochastic force has zero mean 〈ξt〉 = 0 and the correlation function 〈ξ(t)ξ(t′)〉. Asmentioned above, the electronic force corresponds to the level occupation, so it onlycontributes if the level is occupied. The friction term A[x](t) results from stochasticelectron jumps between the leads and the level and is often considered as the firstadiabatic correction term [39].

Although we replaced x0 by xt in the Langevin equation Eq. (2.14), the expectationvalue of the electronic forces are calculated for fixed x0. This is due to our adiabaticinteraction picture which only includes the fixed classical variable. When finallycalculating the Langevin equation numerically, we reconsider the time-dependenceof the electronic forces and obtain self-consistent equations for the trajectories.

2.1.3. Electronic forces and friction in the adiabatic regime

Keldysh Green’s functions are quite suitable for the description of nonequilibriumsystems [92]. There, the Green’s functions are defined on a Keldysh contour, whichruns from the past, where the system is assumed to be in equilibrium, to a certainpresent time and back to the past. Analytic continuation allows the derivation ofthe several Keldysh Green’s functions to real time space.

In the frame of this work, the knowledge of the single particle Green’s functions issufficient for the calculation of the electronic properties. For instance, consider thelevel occupation 〈n(t)〉 ∈ [0, 1], which determines the electronic force

〈F [x](t)〉 = λ〈n(t)〉 = −iλ G<(t, t). (2.15)

This force vanishes for an unoccupied level and reaches its maximum for occupationequal to unity. The single-level lesser Green’s function without coupling in energyspace reads [92]

G<(ω) = i∑

α∈L,RΓα

fα(ω)

(ω − ε)2 + Γ2/4, ε ≡ εd − λx, (2.16)

with the tunnelling rates Γα = 2π∑

k |Vkα|2δ(ω − εkα) in flat band approximationand the definition Γ ≡ ΓL + ΓR. As mentioned before and in accordance with theadiabatic approximation, we keep the centre of mass coordinate fixed, xt = x, duringthe calculation of the Green’s functions. Then, the coupling to the oscillator simplyshifts the level εd by λx and we use the abbreviation ε defined in Eq. (2.16).

26


Describing the occupation of the α-lead in Eq. (2.16), fα(ω) denotes the Fermifunction

fα(ω) =1

eβ(ω−µα) + 1, (2.17)

with the inverse temperature β = 1/kBT and the chemical potential µα of the α-lead.In the high temperature case, structures become washed out, so we mainly focus onthe low temperature case, where the signatures of the interaction are clearly visible.

In the adiabatic approach, the time-dependent Green’s functions depend only ontime-differences and are straightforwardly obtained by performing a Fourier trans-formation of Eq. (2.16). The resulting Green’s functions evolve in time with e−Γ/2t

[99, 100, 42], which vanishes for large times and Γ > 0. From this follows, that thestationary result for the Green’s functions is sufficient and we set t = 0 when per-forming the Fourier transformation. In the zero-temperature limit, we replace theFermi functions with the Heaviside theta function fα(ω) = Θ(µα − ω) and obtain

−iG<(t = 0) =

∫ ∞

−∞

dω

2πe−iω(t=0)G<(ω) =

1

2− 1

π

∑

α

Γα

Γarctan

[ 2Γ(ε− µα)

].

(2.18)

For finite temperature one has to regard Fermi functions instead of the Heavisidetheta function and instead of the arctan the Digamma function Ψ [101] appears.Note, that the stationary or frozen Green’s function Eq. (2.18) become again time-dependent, when we set x → xt at the end.

The second quantity we want to calculate is the friction term A[x](t). As mentionedin Sec. 2.1.2, it contains the imaginary part of the correlation function

〈δn(t)δn(t′)〉 = 〈d†(t)d(t′)〉〈d(t)d†(t′)〉 = G<(t′ − t)G>(t− t′), (2.19)

introducing the greater Green’s function G>(t− t′). In energy space the latter yield

G>(ω) = −i∑

α∈L,RΓα

[fα(ω)− 1]

(ω − ε)2 + Γ2/4. (2.20)

The link between the correlation function and the Green’s function is obtained byusing Wick’s theorem [100]. Inserting Eq. (2.19) into Eq. (2.12), we obtain

A[x](t) = 2λ2

t∫

t0

dt′ t′ Im[G<(t′ − t)G>(t− t′)

]=

λ2

π

∫dωG<(ω)

∂

∂ωG>(ω).

(2.21)

Here, we applied a Fourier transformation in the second step and used the definitionof the delta function

∫dωeiωτ = 2πδ(τ) [100]. In analogy to the calculation of

the occupation we first keep the centre of mass variable fixed while we calculate the

27


friction. In the zero-temperature limit we can make use of the relation ∂∂ωΘ(ω−µ) =

δ(ω − µ) and in the end we obtain

A[x](t) =λ2

2π

∑

α,β

ΓαΓβ1 + sgn(µα − µβ)[(µβ − ε)2 + Γ2

4

]2 . (2.22)

This friction is always positive and disappears in the infinite bias limit (µL,R = ±∞).However, in the finite bias case the oscillator gets damped due to the electronicsubsystem.

The real part of the second electronic cumulant corresponds to the correlationfunction of the stochastic force ξt in the Langevin equation Eq. (2.14). In the adia-batic regime, the Gaussian fluctuations are assumed to appear on a short time-scale.Following from that, the correlation function can be approximated by

〈ξ(t)ξ(t′)〉 = 2λ2 Re[G<(t′ − t)G>(t− t′)

]

= 2λ2

∫dω1

2π

∫dω2

2πRe[G<(ω1)e

−iω1(t′−t)G>(ω2)eiω2(t′−t)

]

= 2λ2 Re

[∫dε

2πe−iε(t−t′)

∫dω

2πG<(ω)G>(ω + ε)

]

≃ 2λ2δ(t − t′)∫

dω

2πG<(ω)G>(ω) ≡ δ(t− t′)D[x](t). (2.23)

In the final Langevin equation, the stochastic force is then described by a standardwhite noise term with an additional multiplicative term: ξt = ξwhite

√D[x](t). This

result coincides with [40, 102]. For T = 0 we obtain (∆αβ = µα − µβ)

D[x](t) =2λ2ΓLΓR

πΓ2

∑

α

sgn ∆αβ

(µα − ε)[

(µα − ε)2 + Γ2

4

] + 2

Γarctan

(2(µα − ε)

Γ

) .

(2.24)

The contributions A[x](t) and D[x](t) correspond to the noise produced by the back-action force, where the symmetric part D[x](t) of this quantum noise spectral densityis responsible for heating and the asymmetric part A[x](t) induces backaction damp-ing. The fluctuation-dissipation theorem links these both quantities and enables thedefinition of an effective frequency-dependent temperature Teff(ω) [38]. The latercharacterises the electronic system as an effective thermal environment. In ther-mal equilibrium, the fluctuation-dissipation theorem yields D(x) = 2TA(x) [39]. InReference [38] the backaction noise is studied within a scattering theory approach.The authors found two contributions to the backaction force noise. One due to theposition detector which scales inversely with the shot noise of the current and oneoriginating from the phase of the oscillator.

28


x [1/l0]

Feff(x) A(x) D(x)

increa

singbiasVbias

Γ=

0.7

Γ=

1.4

Γ=

2.8

Γ=

10 1

1

1

1

1

1

1

0

00

0

0

0

00

00

0

0

00

00

0

0

−1

−1

−1

2

2

22

2

2

22

2

2

22

4

4

4

3

3

6

0.5

0.5

1.5

1.5

0.06

0.04

0.02

Figure 2.2.: Electronic forces and friction for increasing external bias Vbias and sev-eral values of the tunnelling rate Γ. The first column depicts resultsfor the effective force as a function of the oscillator position x. Thesecond/third column shows the friction/fluctuation force. Parametersare εd/ω0 = 3.0 and g = 2.45 at zero temperature. The explicit biasvoltages (symmetric choice) read Vbias/ω0 = 1.0/2.5/3.0.

Here, this so-called phase backaction is not included but the backaction induced bythe measurement. The graphs in Fig. 2.2 depict results for the electronic forces as afunction of the oscillators position and several values of the bias and the tunnellingrate. In order to obtain correct physical units, we introduce the dimensionless cou-pling constant

g =λ

mω20l0

, (2.25)

whereby l0 ≡ 1/√mω0 equals the oscillator length. Additionally, we define an effec-

29


tive force

Feff [x](t) = −mω20x+ λ〈n(t)〉 = −ω0

l0

[x

l0− g〈n(t)〉

], (2.26)

where we combine the electronic force and the restoring force of the oscillator. Theeffective force has several zeros in the small bias regime and these zeros correspondto fixed points of the dynamical system. Due to the fact that the friction is alwayspositive, the oscillator gets damped and end up in these stationary solution, seeSec. 2.1.4 for details of the dynamical behaviour.

The shape of the effective force illustrates the two contributions from Eq. (2.26).The linear behaviour for large values of x originates from the restoring force and theelectronic force is responsible for a shift to positive x values. The wavy behaviour inthe regime where all parameters of the system are small, cf. Fig. 2.2, is as well causedby the occupation of the electronic level. When we consider the result for Vbias =1.0ω0, we observe two local extrema. Between these extrema, the electronic forceis larger than the restoring force. The zero in the middle of this slope correspondsto a point of balance, where both forces are equal. There, the occupation has itsaveraged value 1/2. In the case of a small tunnelling rate, e.g. Γ = 0.7ω0, we obtaintwo additional zeros near to x0 ≈ 0, λ, corresponding to zero and full occupation. IfΓ is increased these both zeros disappear and solely the middle fixed point remains.

For a certain range of bias, the effective force has up to 5 zeros, see lowest forcegraph in Fig. 2.2. These additional zeros appear due to the influence of the bias.There, the occupation has step shape and these steps appear at x = [εd ∓ Vbias/2]/λ.By further increasing the bias only the x0 ≈ λ/2 fixed point survives.

The second column of Fig. 2.2 depicts the result for the friction term A(x). Forsmall bias and small tunnelling rate we observe a peak near x0 ≈ λ/2. The peakheight drastically decreases by increasing one of the two parameters. Depend-ing on the applied bias the friction peak splits up into two maxima located atx = [εd ∓ Vbias/2]/λ. There, the shifted level energies ε are in resonance with thechemical potentials µL,R. In the regime of high bias or large tunnelling rate thefriction vanishes completely.

This is different for the fluctuation term D(x), which is plotted in the last columnof Fig. 2.2. It indeed vanishes by increasing the tunnelling rate, but it becomesconstant in the high bias limit. In this case the first term in Eq. (2.24) equals zero,but with limx→±∞ [arctan(x)] = ±π/2 we obtain for the second term

limµL,R→±∞

D[x](t) =2λ2ΓLΓR

Γ2. (2.27)

The results for all quantities which determine the interaction between the electronicand the mechanical subsystem show, that most effects appear in the range of smallbias and for small tunnelling rate. This stands in contrast to the adiabatic approach,which requires Γ/ω0 ≫ 1. In Sec. 2.2 we test the adiabatic approximation with acomplete nonadiabatic approach. But in this section, we stay in the regime wherethe adiabatic approach should be at the limit of its validity.

30


2.1.4. Effective potential and phase space portraits

The coupling to the electronic nonequilibrium environment modifies the initial har-monic oscillator potential. In the case without the friction term, we can define aneffective potential [42]

Ueff(x) =

x∫

0

dx′[V ′osc(x

′)− 〈F [x′](t)〉]. (2.28)

This potential determines the adiabatic dynamics of the oscillator. With Eq. (2.18)the potential for zero temperature reads

Ueff(x) =1

2mω2

0x2 − λ

2x+

1

π

∑

α

Γα

Γ

[(εd − µα) arctan

(2(εd − µα)

Γ

)

−(ε(x)− µα) arctan

(2(ε(x)− µα)

Γ

)+

Γ

4ln

(ε(x)− µα)

2 + Γ2

4

(εd − µα)2 +Γ2

4

].

(2.29)

U eff(x)

[1/ω0]

x [1/l0]

Vbias/ω0

0

0

0.2

0.1

−0.1 1

1 2

3

3

2.5

Figure 2.3.: Effective potential for differ-ent bias voltages Vbias.

Here, we use again the abbreviationε(x) = εd − λx. If λ = 0 the arctan-terms cancel each other and the har-monic potential result is recovered. Inthe high bias limit, or for large valuesof the tunnelling rate Γ, the last threeterms disappear and only the term lin-ear to x survives as influence of the elec-tronic environment. In Fig. 2.3 the ef-fective oscillator potential for differentbias values Vbias is depicted. We choosethe detuning of the left and right leadin a symmetric manner, this results inµL = −µR = Vbias/2. For a bias equalto the oscillator’s frequency Vbias = ω0,two distinct minima appear, they correspond to the zeros of the effective forceFeff = −mω2

0x + λ〈n(t)〉. For such a small bias value the occupation of the levelhas step shape with a jump from zero to one near to x = λ/2.

By further increasing the bias, one additional minimum appears and the systembecomes tristable. This lasts only for a small range of bias values and after thisregime solely one minimum survives. There, the occupation approaches 1/2 and theresulting potential is again harmonic with a minimum shifted to x = λ/2. The samehappens when we keep the bias fixed and instead increase the tunnelling rate Γ. Thisis clearly visible in Fig. 2.4. There the effective oscillator potential is plotted as afunction of the oscillators position x and the tunnelling rate Γ. The initial tristable

31


Γ[ω

0]

x [1/l0]

0.25

0.2

0.15

0.1

0.05

−0.05

−0.1

0.4

0.8

1.2

1.6

00

0

1

2

2 3

Figure 2.4.: Density plot of the effective oscillator potential Ueff(x) as a function oftunnelling rate Γ and oscillator position x in units of ω0. Parametersare εd/ω0 = 3.0 and g = 2.45 at zero temperature. The bias voltage(symmetric choice) read Vbias/ω0 = 2.5.

Vbias

[ω0]

x [1/l0]

0.3

0.2

0.1

−0.1

−0.2

−0.3

−0.4

−0.50

0

0

1

1

2

2

3

3

4

5

6

Figure 2.5.: Density plot of the effective oscillator potential Ueff(x) as a function ofbias voltage Vbias and oscillator position x in units of ω0. The graphdepicts the results for the tunnelling rate Γ/ω0 = 1.4. Remaining pa-rameters are εd/ω0 = 3.0 and g = 2.45 at zero temperature.

32


−0.5

00.5

11.5

2

2.53

−0.5

−0.3

−0.1

0.1

0.3

0.5

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.5 0 0.5 1 1.5 2 2.5 3−0.5

−0.3

−0.1

0.1

0.3

0.5

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.5 0 0.5 1 1.5 2 2.5 3−0.5

−0.3

−0.1

0.1

0.3

0.5

x [1/l0]

p [1/ω0 l0 ]

x [1/l0]

p[1/ω0l 0]

x [1/l0]

p[1/ω

0 l0 ]

Figure 2.6.: Results for the Langevin equation without friction plotted together withthe effective oscillator potential. Parameters are εd/ω0 = 3.0, Γ/ω0 =1.4 and g = 2.45 at zero temperature. The bias voltage (symmetricchoice) reads Vbias/ω0 = 2.5.

potential, which appears for small Γ and a bias voltage of Vbias = 2.5ω0 see Fig. 2.3,vanishes by increasing Γ and the minimum in the middle, corresponding to λ/2,survives as expected. For clarity, the graph Fig. 2.4 shows only results up to Γ = 2ω0,but even for this comparatively small range of values this behaviour is recognisable.

The density plot in Fig. 2.5 depicts the oscillator potential as a function of theapplied bias for Γ = 1.4ω0. We recognise the expected behaviour. The multistabilityof the system vanishes when the bias voltage is increased.

For a further study of the dynamical behaviour we solve the Langevin equationEq. (2.14) numerically. We neglect the fluctuations due to the stochastic force ξt,such that the equation of motion for the classical trajectory 〈xt〉 reads

m〈xt〉+ 〈xt〉A[〈x〉](t) − Feff (〈xt〉) = 0. (2.30)

As mentioned before, the electronic force and the friction now depend on the centreof mass variable 〈xt〉 and hence become time-dependent as well.

The phase diagrams without friction follow directly from the shape of the effectivepotential. Therefore, we illustrate the effective potential in three dimensions includ-ing the phase space results in Fig. 2.6. In the chosen parameter regime the potentialis tristable and the effective force has five zeros and following from that, five fixedpoints appear. With methods of dynamical stability analysis this fixed points canbe classified [103]. For a two dimensional system the trace and the determinant ofthe Jacobian of the linearised system decide about class and stability of the fixedpoints, see Sec.A.3. For the considered system the trace equals the negative frictionτ = −A(〈x∗〉), where 〈x∗〉 denotes the position coordinate of the fixed points. The

33


〈xt〉 [1/l0]

〈pt〉[1/ω0l 0]

222

111

0

0

0

0

0

0

−1−1−1

Figure 2.7.: Phase space portraits resulting from the Langevin equation Eq. (2.30).The grey (black) lines depict the results without (with) friction. Thebias voltage increases from left to right; explicit values are Vbias/ω0 =0.5/2.5/5.0. The initial conditions are chosen for each phase diagramseparately in order to make the characteristic shapes visible. Remainingparameters are the same as for Fig. 2.6.

determinant in units of ω0l0 reads

∆ =1

l0− g

∂

∂〈xt〉〈n(t)〉

∣∣∣∣∣〈x∗〉

=1

l0− g2

2π

∑

α

Γα1[

(ε(〈x∗〉)− µα)2 +Γ2

4

] . (2.31)

In the case shown in Fig. 2.6 the friction is omitted and therewith the trace is equal tozero. Three of the fixed points 〈x∗〉 ≈ 0/0.5g/g can be classified as centers in phasespace. Here the trajectories run in circles about the fixed points, which correspondto minima of the oscillator potential. Between these centers are two saddle points〈x∗〉 ≈ [εd ∓ Vbias/2]/g, where the determinant is smaller than zero and therewiththe eigenvalues are real with opposite sign.

Things change if the friction is included. Then the centers turn into stable spirals.Depending on the initial conditions the trajectories run into one of the stable fixedpoints in the long-time limit. In Fig. 2.7 results with friction for different values ofthe bias voltage are plotted. The friction is strongly peaked around the saddle points,see Fig. 2.2, and therewith a trajectory which approaches these peaked regime getsstrongly damped and a kink-like behaviour for the trajectory is produced.

The first plot in Fig. 2.7 shows the two minima case, where the left fixed pointcorresponds to the zero occupied electronic level and the right fixed point to thefully occupied level. For increasing bias voltage, we obtain a third stable spiralin between, corresponding to the average occupied state. For sufficiently high biasvoltage (large transport window) only the average occupied state survives.

34

2.2. Nonadiabatic approach


In the previous section we investigated a harmonic oscillator linearly coupled to anelectronic subsystem containing one electronic level by using an adiabatic approach[42]. This includes the assumption, that the oscillator’s movement is slow comparedto the electrons which are jumping through the system. The foregoing results wereobtained in the regime, where the adiabatic approach was at the limit of its validity.

If the electrons and the oscillator act on the same time-scale the interaction be-tween both is strongest [19]. Hence, for Γ ≈ ω0 the most interesting features appear.In the regime Γ ≫ ω0 the oscillator solely experiences a shift of its rest positioncaused by the stochastic processes initiated by the current.

Going beyond the adiabatic approximation or probing its range of validity is in thescope of recent research [38, 104, 41]. For instance, on the basis of scattering theory[38, 104]. But these methods still imply a limitation concerning the requirementson the different time-scales of the system. In our work, we go one step further anddevelop a method which enables us to work in a complete nonadiabatic manner.

Therefore, we start this section with the modification of our general path-integralansatz, obtaining an explicit time-dependent interaction picture. Following fromthat, we have to calculate the systems quantities in a full time-dependent manner.This allows us to critically assess and probe the validity of the adiabatic approach.

2.2.1. Stochastic equation of motion with nonadiabatic ansatz

In the adiabatic approximation, Sec. 2.1.2, a Taylor expansion for the centre of massvariable is performed (xt ≈ x0 + tx0), leading to an interaction picture with respectto H0 = He − F x0 and a perturbation V [q](t) = −F (tx0 +

12yt). Consequently, the

expectation value of the force operator in the adiabatic interaction picture was calcu-lated for fixed x0. Additionally, due to the second term tx0 of the Taylor expansion,an explicit friction term arises in the (adiabatic) Langevin equation Eq. (2.14).

Things change if we turn to a nonadiabatic approach, where higher terms of theTaylor expansion can not longer be neglected. The differences start with the decom-position of the bath Hamiltonian in the effective interaction picture

U [qt] = Te−i

t∫0

dt′ [He−F xt′− 1

2F yt′ ]

= U [xt] U [yt] ; U [yt] = Tei

t∫0

dt′ 1

2F (t′)yt′

, (2.32)

where the term with the off-diagonal path yt is regarded as a perturbation andF (t) = U † [xt] F U [xt]. With Eq. (2.32) the influence functional reads

F[qt; q

′t

]= trB(U

† [−yt] U† [xt] U [xt] U [yt])

= trB(U† [−yt] U [yt]). (2.33)

35


The further calculations are similar to the adiabatic case. We expand the functionalto second order, perform a cluster expansion [98] and finally obtain

Fpert[qt; q

′t

]= e−Φ[xt;yt], (2.34)

with the influence phase

Φ [xt; yt] =− i

t∫

0

dt′f(t′)yt′ +

t∫

0

dt′t∫

0

ds C(t′, s)yt′ys. (2.35)

The force correlation function yields

C(t′, s) = trB

(F (t′)− f(t′)

)(F (s)− f(s)

)≡ 〈δF (t′)δF (s)〉. (2.36)

The force term f(t) ≡ 〈F (t)〉 depends on the centre of mass path xt. The influ-ence phase, Eq. (2.35), can still be regarded as a cumulant generating functionalfor the force operator correlation functions. The term of quadratic order in yt de-scribes the Gaussian fluctuations around the classical trajectory which is determinedself-consistently in our approach. Higher order terms in yt (corresponding to non-Gaussian noise) describe higher quantum fluctuations which are neglected here.

To second order in yt the double path integral for the reduced density matrixdescribes a classical stochastic process for the diagonal path xt that is defined by a(nonadiabatic) Langevin equation

mxt + V ′osc(xt)− f [xt] = ξt, (2.37)

with V ′osc(xt) = mω2

0xt and a Gaussian stochastic force ξt that has a correlationfunction 〈ξt′ξs〉 = C(t′, s). Eq. (2.37) is the starting point for our nonadiabaticcalculations. Note that the force f [xt] = λ〈d†(t)d(t)〉 = iλG<(t, t) is a complicatedfunctional that contains the full time-dependence of the position operator xt.

In contrast, in the nonadiabatic Eq. (2.37), all higher orders of the Taylor expan-sion for xt are included and the first challenge is to calculate the force term f [xt]considering the full time-dependence of xt. The latter is done in the next section.We neglect the stochastic fluctuations (ξt = 0), this provides a direct test-bed for theadiabatic results. Furthermore, remaining on the level of a complete nonadiabaticapproach and additionally include fluctuations is hardly possible. The reason forthis is the complex time-dependence of the correlation functions as we present in thenext section.

2.2.2. Time-dependent occupation of the single-level

Until now we assumed that xt ≡ x while calculating the adiabatic electronic forceF [x](t) = λn(t). Without the adiabatic approximation the occupation of the levelhas to be calculated with full time-dependence of xt. There are several ways to derivethis time-dependent occupation, e.g. using the equation of motion technique [99] or

36


time-dependent Green’s functions. In this section we use the second method, due toits elegance and shortness. But we also present an alternative derivation in the nextsection.

Concerning the nonadiabatic approach, we denote the occupation number of thelocal level by n ≡ N as a better distinction to the adiabatic model. The calculationis performed with the time-dependent lesser Green’s function [92]

N [xt] =〈d†(t)d(t)〉 = −iG<(t, t). (2.38)

Note, that in the time-dependent regime the Green’s functions do depend separatelyon two times and not on time-differences. Following from that, we cannot benefitfrom a Fourier transformation. The lesser Green’s function is obtained with help ofthe Keldysh equation

G<(t, t) =

∫dt1

∫dt2 GR(t, t1) Σ

<(t1, t2) GA(t2, t), (2.39)

containing the advanced/retarded Green’s function

GR,A(t, t′) = gR,A(t, t′) e∓

t∫

t′

dt′′ Γ2

= ∓iΘ(±t∓ t′) e−i

t∫

t′

dt′′[ε(t′′)∓iΓ2], (2.40)

with ε(t) ≡ εd − λxt, where Θ denotes the Heaviside step function. In the time-dependent case the lesser self energy reads

Σ<(t1, t2) = i∑

α

∫dω

2πe−iω(t1−t2) fα(ω) Γα, (2.41)

whereby fα(ω) denotes the Fermi function. We assume constant tunnelling ratesΓα = 2π

∑k |Vkα|2δ(ω − εkα) = Γ/2, i.e. the left and the right tunnelling rate are

equal with Γ ≡ ΓL + ΓR. After inserting the advanced/retarded Green’s functionsand the self-energy into the Keldysh equation Eq. (2.39) and a reshaping of the limitsof integration, we finally obtain for the lesser Green’s function

G<(t, t) = i∑

α

∫dω

2πΓα fα(ω)

t∫

t0

dt1

t∫

t0

dt2 e−i

t∫t1

dt′(ε(t′)−ω− iΓ2)ei

t∫t2

dt′(ε(t′)−ω+ iΓ2).

(2.42)

With this we find as a result for the time-dependent occupation

N [xt] =∑

α∈L,RΓα

∫dω

2πfα(ω) |A(ω, t)|2, (2.43)

including the spectral function

A(ω, t) = −i

t∫

t0

dt′ e−i

t∫

t′

dt′′ (ε(t′′)−ω−iΓ2). (2.44)

37


This spectral function can be solely obtained from numerical evaluations, which isoutlined in more detail in Sec. 2.2.4. Note, that this is not a spectral function in theusual sense and as it is defined in the framework of Green’s functions. For instance,there the relation A = i[GR − GA] should be fulfilled, which is not the case here.But the shape of equation Eq. (2.42), including the square of the absolute value of A,reminds of the fluctuation dissipation theorem in equilibrium G<

eq(ω) = if(ω)Aeq(ω).The latter links the equilibrium lesser Green’s function, containing information aboutthe system’s fluctuations, to the spectral function, which describes the decay in thetime-domain and thus the dissipation.

For now, we want to check if these results for the level occupation coincide withthe adiabatic result Eq. (2.16). Once more we keep the position variable fixed xt ≡ x,therefore the integral in the exponent becomes trivial and we obtain

A(ω, t) =− i

t∫

t0

dt′ e−i(εd−λx−ω−iΓ2)(t−t′) =

e−i(εd−λx−ω−iΓ2)(t−t0) − 1(

εd − λx− ω − iΓ2) . (2.45)

For large times t − t0, the first exponential can be neglected and the spectral func-tion becomes stationary and is equivalent to the retarded Green’s function of theelectronic level. Taking the square of the absolute value we recover the adiabaticresult

Nadiabatic [x] =∑

α∈L,RΓα

∫dω

2π

fα(ω)[(εd − λx− ω)2 + Γ2

4

] . (2.46)

The correlation function of the fluctuations yields

〈ξtξs〉 = C(t, s) = λ2 G<(t, s) G>(t, s). (2.47)

For its calculation the Green’s functions for different times are necessary. Theycan be derived similarly to the occupation and we obtain for the lesser and greaterfunction

G<(t, s) = i∑

α

Γα

∫dω

2πfα(ω) A(ω, t) A

∗(ω, s) e−iω(t−s),

G>(s, t) =− i∑

α

Γα

∫dω

2π(1− fα(ω)) A(ω, s) A

∗(ω, t) eiω(t−s), (2.48)

including the spectral function from Eq. (2.44). Its easy to see, that with (t, s) →(t, t) the time-dependent lesser Green’s function is recovered. Additionally, with theknowledge of the stationary spectral function Eq. (2.45) and setting (t, s) → (t− s),we are able to recover the adiabatic correlation function corresponding to Eq. (2.23).

But there is no simple way to derive the friction term, which arises in the adiabaticLangevin equation, corresponding to the imaginary part of the correlation function.The spectral function contains the whole xt dependence, therefore, if we use a Taylorexpansion up to second order, as in the adiabatic approximation, the time integrals

38


yield expressions containing the error function of complex argument. Expanding thisexpression to lowest order in xt does not recover the expression for the friction.

This error function is related to the Gaussian cumulative distribution functionand should include higher cumulants. But the derivation of the friction includingthe correct time-ordering is challenging. The influence functional method provideda correct time-ordering in the current interaction picture. For a derivation of alladiabatic force terms, starting from the time-dependent Green’s functions, one hasto expand the retarded/advances Green’s function in a Wigner presentation as in [95].There, the authors use the Wigner presentation to distinct the time-scales, assumingthat the oscillator is slow compared to the electrons. Within their approach, theygain the same force and friction term as in our adiabatic approach.

2.2.3. An alternative way of deriving the occupation

In the last section we derived the time-dependent occupation by using KeldyshGreen’s functions. From that we obtained a spectral function which has to be trans-formed into a differential equation and could only be solved numerically. Similarequations are obtained if we use the equation of motion method. This method isinspired by the work from Pederson and Co-workers [105, 106]. They used an equa-tion of motion approach on the basis of many-particle states, which enables them toincluding coulomb interaction and level-broadening effects.

In this section, we explain how to derive the occupation using this method. Thiskind of derivation is straightforward to transfer to more complex system as the single-level model. In this work we use the nonadiabatic approach also for more complicatedsystems and therefore we use the equation of motion method. We want to use thesimplicity of the single-level model, to illustrate the principles of this method andshow its equivalence to the Green’s function ansatz.

In an interaction picture the time-evolution of operators is calculated with re-spect to the unperturbed Hamiltonian H0. The Heisenberg equation of motions foroperators of the level and the leads yield

˙d = i

[H0, d

]−

= −i(εd − λx) d− i∑

kα

V ∗kαckα,

˙ckα = i[H0, ckα

]−= −iεkαckα − i Vkαd. (2.49)

The square brackets denote the commutator and the equation of motion for d† equalsthe complex conjugate of d. We can solve the inhomogeneous differential equationfor the lead operator, this leads to

ckα(t) = e−iεkαtckα(t = 0)− iVkα

t∫

0

dt′e−iεkα(t−t′) d(t′). (2.50)

39


Hence, the differential equation for the level reads

˙d(t) = −iε(t)d(t)− i

∑

kα

Vkα e−iεkαt ckα(0) −∑

kα

V 2kα

t∫

0

dt′ e−iεkα(t−t′) d(t′).

(2.51)

The last term can be transformed according to

∫dω∑

kα

|V |2kα δ(ω − εkα)

t∫

0

dt′ e−iω(t−t′) d(t′)

=∑

α

Γα

∫dω

2π

t∫

0

dt′ e−iω(t−t′) d(t′) = Γ

t∫

0

dt′ δ(t− t′) d(t′) =Γ

2d(t),

where we used the assumption Γ = ΓL+ΓR. This simplifies the equation of motionsfor the level operators to

˙d(t) = −i

(ε(t)− i

Γ

2

)d(t)− i

∑

kα

V ∗kαe

−iεkαtckα(0),

˙d†(t) = i

(ε(t) + i

Γ

2

)d†(t) + i

∑

kα

Vkαeiεkαt c†kα(0). (2.52)

These equations can be likewise solved as the lead operators. The obtained time-integrals inserted into the equation for the occupation N(t) = 〈d†(t)d(t)〉 lead to aresult which is equal to Eq. (2.43), for details see Sec. B.1.1. But here, we continuewith the equation of motion for the occupation

N(t) = −ΓN(t) + i∑

kα

[Vkα eiεkαt〈c†kα(0)d(t)〉 − V ∗

kαe−iεkαt〈d†(t)ckα(0)〉

]

= −ΓN(t) +∑

kα

[〈C†

kα(t)d(t)〉+ 〈d†(t)Ckα(t)〉], (2.53)

where we defined the new operators Ckα(t) = −iV ∗kαe

−iεkαtckα(0) and C†kα(t) =

iVkαeiεkαtc†kα(0). The expectation values of the combination of the lead and the level

operator obey the following time-evolution

d

dt〈C†

kα(t)d(t)〉 = −i

(ε(t) − εkα − i

Γ

2

)〈C†

kα(t)d(t)〉+∑

kα

|Vkα|2 f(εkα),

d

dt〈d†(t)Ckα(t)〉 = i

(ε(t)− εkα + i

Γ

2

)〈d†(t)Ckα(t)〉+

∑

kα

|Vkα|2 f(εkα). (2.54)

40


Multiplication with δ(ω − εkα) and summing over all k states leads to a system ofcoupled equation

N(t) = −ΓN(t) +∑

α

∫dω 2 Re [Bα(ω, t)] ,

Bα(ω, t) = −i

(ε(t)− ω − i

Γ

2

)Bα(ω, t) +

Γα

2πfα(ω), (2.55)

with the function Bα(ω, t) =∑

k δ(ω − εkα)〈C†kα(t)d(t)〉. Solving Eq. (2.55) for fixed

centre of mass variable leads to the adiabatic result for the occupation, see Sec. B.1.2.We can also reconsider the nonadiabatic result obtained from the Green’s functionmethod. Therefore we transform Eq. (2.55) into the integral equation (ε = εd−λx(t))

N(t) =∑

α

Γα

2π

∫dωfα(ω)

t∫

0

dt′t′∫

0

ds 2 Re

e

is∫

t′

dt′′(ε−ω−iΓ2) eΓ(t

′−t). (2.56)

We modify the time-integral by taking the time-ordering t > t′ > s into account.With this the exponent yieldse

is∫

t′

dt′′(ε−ω−iΓ2)ei

t∫

t′

dt′′iΓ

= e

is∫

t′

dt′′(ε−ω−iΓ2)ei

t∫

t′

dt′′iΓ

ei

t∫

t′

dt′′(ε−ω−iΓ2)e−i

t∫

t′

dt′′(ε−ω−iΓ2)

= ei

t∫

t′

dt′′(ε−ω+iΓ2)e−i

t∫s

dt′′(ε−ω−iΓ2). (2.57)

Inserting the latter into the equation for the occupation we obtain

N(t) =∑

α

Γα

2π

∫dωfα(ω)

t∫

0

dt′t′∫

0

ds 2 Re

e

it∫

t′


t∫s


=∑

α

Γα

2π

∫dωfα(ω)

t∫

0

dt′t∫

0

ds ei

t∫

t′


t∫s


=∑

α

Γα

2π

∫dω fα(ω) |A(ω, t)|2. (2.58)

In the second step, we rearranged the integral limits in accordance with the as-sumed time-ordering [107]. Therewith, the Green’s function result is recovered, cf.Eq. (2.43).

For the single-level model this method seems more elaborate than the Green’sfunction calculation. But if the system becomes more complicated, the number ofcoupled equations of motion, corresponding to the additional correlations, increases.Even if we construct all Green’s functions for such a system, the reduction to a systemof coupled differential equation is challenging. And this reduction is necessary forthe numerical calculation.

41


2.2.4. Phase space in the nonadiabatic regime

The numerical effort to calculate the trajectories in phase space was low in the adi-abatic approximation. There, we had to solve two coupled equations correspondingto the centre of mass variable and the momentum. The summation over the lead en-ergies could be performed analytically or with a simple numeric integration routine.This changes for the nonadiabatic approach. There, we include the whole time-dependence of the Green’s functions and obtain an equation of motion for every leadelectron with energy εkα. The number of coupled differential equations depends nowon the chosen discretisation scheme.

The oscillator’s equation of motion reads

mxt + V ′osc(xt)− λN [xt] = 0. (2.59)

To solve the equation of motion without any further approximation and expansion,we transform Eq. (2.44) into a differential equation

A(ω, t) = −i− i

(ε(t)− ω − i

Γ

2

)A(ω, t). (2.60)

For the numerical integration a trapezoidal rule for discrete functions is applied.Thus we solve the system

xt =1

mpt,

pt =− Vosc −∑

α∈L,R

Γα

4π∆ω

N−1∑

n=0

[fα(ωn+1) |A(ωn+1, t)|2 + fα(ωn) |A(ωn, t)|2

],

(2.61)

with ∆ω = |ωN − ωmin|/N , in which N equals the number of points of the discreti-sation scheme and ωmin → −∞. We decompose the spectral function into real andimaginary part A(ω, t) = AR(ωn, t) + iAI(ωn, t) and obtain

AR(ωn, t) =− Γ

2AR(ωn, t) + (εd − λxt − ωn)A

I(ωn, t),

AI(ωn, t) =− Γ

2AI(ωn, t)− (εd − λxt − ωn)A

R(ωn, t). (2.62)

These equations do not depend on the α-lead and therewith the summation over αin Eq. (2.61) is trivial and we obtain [2N + 2] coupled equations. But even with asmart discretisation scheme the number of points should exceed N ∼ 104 and wecan state that this method is numerically much more expensive than the adiabaticapproach (N = 2).

Fig. 2.8 depicts the results in phase space for different bias values. All data areobtained for a small tunnelling rate Γ = 1.4ω0, close to the limit of validity of theadiabatic approach which requires Γ ≫ ω0. Row A shows the adiabatic results. The

42


-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

0 2

-1

0

1

-1

0

1

0 2

-1

0

1

-1

0

1

0 2

-1

0

1

position 〈xt〉 [1/l0]

momentum

〈pt〉[1/ω0l 0]

A1 A2 A3

B1 B2 B3

Figure 2.8.: Phase space portraits resulting from the (non-) adiabatic approach inunits of ω0 and l0 with the parameters Γ = 1.4ω0, εd = 3.0ω0 andg = 2.45 at zero temperature. Row A depicts the adiabatic results,additionally the case without friction is plotted (dotted lines). Row (B)depicts the nonadiabatic results. The bias voltage (symmetric choice) isincreased from left to right, explicit values are Vbias/ω0 = 0.5/2.5/5.0.

dotted lines correspond to the adiabatic case without the first adiabatic correctionterm. Here, the trajectories run about the fixed points of the system, as seen inSec. 2.1.4. By varying the applied bias, the number of fixed points changes and inthe case of high bias only one fixed point survives. Turning on the friction (firstadiabatic correction term) leads to the solid line results in the graphs of row A.Here, the centers turn into stable spirals and the trajectories end up in the fixedpoints.

In row B the nonadiabatic results are depicted. The trajectories approach the samefixed points as in the adiabatic case, but the evolution of the damped oscillationsdiffers for small times. If we choose the same initial conditions as for the adiabaticapproach, the final reached fixed point in the nonadiabatic regime can be different.By comparing both approaches we can state, that the largest differences emerge forsmall times. This follows from the fact, that the nonadiabatic approach containsthe short-time dynamics. And we can estimate, that in the long-time limit thesecontributions are not relevant for the considered system.

43


2.3. Electronic properties influenced by the mechanical

system

So far we have concentrated on the influence of the electronic environment onto theoscillator’s dynamics. In the range of small bias voltage the system is multistableand depending on the initial conditions, the oscillator trajectories evolve in timetowards one of the potential minima. Effectively, the oscillator gets shifted to newrest positions x∗, whereby only one rest position remains for higher bias voltages.

As a matter of course, the mechanical system affects the electronic properties aswell. Due to the linear coupling to the oscillator, the initial energy level εd getsshifted by λx and we obtain a new effective energy level ε = εd − λx. The electroniccurrent flowing through the system depends on the position of the energy level, ifthe bias, and therewith the transport window, is comparably small.

The electronic quantities are relevant for the measurement of the features ofelectron-phonon interaction [5, 3]. We start this section with the comparison ofthe electronic current results obtained from the adiabatic and the nonadiabatic ap-proach. The second part of this section illustrates the difficulties arise if we tryto calculate the nonadiabatic noise. Due to these drawbacks we present numericalresults for the adiabatic noise.

2.3.1. Current for the single-level system

The electronic current flowing through the system can be calculated from the time-evolution of the occupation number operator of the leads [92]

Iα = −e〈Nα〉, Nα =∑

k

c†kαckα. (2.63)

Nα does not commutes with the tunnelling Hamiltonian, therefore we obtain

Iα =2e∑

k

Vkα〈c

†kαd〉 − V ∗

kα〈d†ckα〉= 2e

∑

k

Re[VkαG

<d,kα(t, t)

], (2.64)

introducing the level-reservoir Green’s functions G<d,kα(t, t) = −[G<

d,kα(t, t)]∗, which

are obtained via

G<d,kα(t, t) =

∫dt′V ∗

kα

[GR(t, t′)g<kα(t

′, t) +G<(t, t′)gAkα(t′, t)], (2.65)

containing the level Green’s functions Eq. (2.39) and Eq. (2.40), as well as the undis-turbed lead functions g<,R,A

kα , see Sec.A.2.1. In the adiabatic approximation, allthese Green’s functions solely depend on time-differences and a Fourier transforma-tion can be performed. In the nonadiabatic approach, the time-dependencies are notthat trivial, the level-reservoir function yield

G<d,kα(t, t) = i

t′∫

t0

dt′V ∗kαe

iεkα(t−t′)[GR(t, t′)fα(εkα) +G<(t, t′)

]. (2.66)

44

2.3. Electronic properties influenced by the mechanical system

0 5 10 15

0

0.1

0.2

0.3

0.40 1

0

0.1

0.2

40 80

0

0.02

0.04

x/g [1/l0]I

L(t

→∞

)[1

/ω

0e]

IL

(x)

[1/ω

0 e]I

L(t)

[1/ω

0 e]

Vbias [ω0]

〈x∗1〉〈x∗

3〉

〈x∗2〉

g = 3.5

g = 2.45

adiabatic

non–adiabatic

non–adiabatic

infinite biaswithout coupling

Vbias = 2.0 ω0

Vbias = 1.0 ω0

t [1/ω0]

Figure 2.9.: Current results for t → ∞ as a function of Vbias for two different couplingparameters g and with Γ/ω0 = 1.4, εd/ω0 = 3.0 at zero temperature.The black circles/grey diamonds depict the nonadiabatic results. Forcomparison the infinite bias result and the current without coupling areplotted. The areas between the dotted lines mark the regime where threefixed points exist, leading to three current channels. For Vbias/ω0 = 2.0the position dependent current IL(x) is plotted (upper right graph), thethree fixed points are marked as crosses.

Inserting the latter into the equation for the current Eq. (2.64) we obtain

Iα(t) = −eΓα

[N [xt] +

∫dω

πfα(ω)Im [A(ω, t)]

]. (2.67)

where N [xt] equals the time-dependent occupation Eq. (2.43).

In general, the imaginary part of the spectral function is negative and describesthe current flowing from the leads to the single electronic level. In contrast, theoccupation describes the current flowing out of the central region. If the leads aredetuned the electrons tunnel to the lead with lower chemical potential. As we seein the right lower graph of Fig. 2.9, where the nonadiabatic current for Vbias = 1ω0

is depicted, is it possible to obtain negative values of the current, corresponding toelectrons tunnelling against the applied bias. This feature is not pronounced for thesingle-level system and appears only for small values for the bias and small times.

45


But this kind of effect can never be observed for the adiabatic current, which yield

IL = eΓ

4π

[arctan

2(µL − εd + λx)

Γ− arctan

2(µR − εd + λx)

Γ

]= −IR. (2.68)

This result is straightforwardly obtained by using the adiabatic occupation Eq. (2.46)and the stationary spectral function Eq. (2.45) from the last section. It is well knownand for the infinite bias case we obtain IIB

L,R = ±eΓ/4 as expected [99]. In theadiabatic case the values for left and right current only differ in their sign.

The left graph of Fig. 2.9 depicts the stationary left current IL(t → ∞) for in-creasing bias and for two different coupling parameters g. As explained above, theoscillator’s trajectories end up in fixed points for large times. Hence, the currentIL(t → ∞) becomes stationary and its value corresponds to a single electronic levelwhich is shifted by −gx∗/l0, whereby x∗ corresponds to a fixed point. Because thesystem owns multiple fixed points, we obtain several current channels depending onthe initial condition. The dashed-dotted line in Fig. 2.9 corresponds to the case with-out coupling to the oscillator (g = 0). For small bias, the current is low comparedto the infinite bias case (dashed line). There, the effective level ε = εd − gx∗/l0 issituated outside the transport window. The latter is also valid for the case withoutcoupling, due to ε = εd = 3.0ω0. The bias range for the current suppression is largerin the case of stronger coupling to the oscillator (g = 3.5). We obtain a hysteresis-likeshape for the current, which is due to the multistability of the system. The couplingbetween the electronic and the mechanical system leads to a modified oscillator po-tential with additional minima. Switching between these states is possible and wastheoretical proposed and studied by several authors [108, 33, 102].

The beginning and the ending of the hysteresis regime, where two current channelsexist, are denoted by vertical dotted lines in Fig. 2.9. For the nonadiabatic case theregime where two current channels exist, differs slightly from the adiabatic case.

The upper right graph of Fig. 2.9 shows the current IL(x) for the bias value Vbias =2.0ω0. Here, three fixed points occur, denoted by a cross. For 〈x∗2〉 ≈ g/2l0 theeffective level is situated in the middle of the transport window and following fromthat the current is maximal. For the two other fixed points the effective level is againsituated outside the transport window and the current is small.

By comparing the adiabatic and the nonadiabatic stationary currents, we canconclude that in the long-time limit only small differences exist. The differences areat their maximum for small times, which is clearly visible in the lower right graphof Fig. 2.9, which depicts IL(t). There, the oscillations in the nonadiabatic case aremuch larger.

Additionally, the left and right time-dependent currents differ for small times tin the nonadiabatic case. When we integrate IL and IR over all times the resultscoincide, so that there is no violation of current conservation. The spectral functiondefined in Eq. (2.44), is sensitive to small time-differences t − t0. For larger timesthe oscillator’s trajectories end up in one of the fixed points, whereas the spectralfunction becomes stationary and hence also the current.

46


2.3.2. Noise for the single-level system

The noise equals the second cumulant of the current. It is a very relevant quantityin optical and electronical experiments. There are several sources of noise in experi-ments, q.e. thermal noise and shot noise [109]. The latter appears due to the discretenature of electrons and is independent of frequency and temperature. In the regimeof low temperature, shot noise is the dominant contributor to electronic current noise[25] and therewith a theoretical understanding of the noise is quite important.

On the one side, the appearing of noise in experiments represents a benefit, be-cause it contains additional information about the system, which cannot be obtainedfrom current or conductance measurements [110]. For instance, in [66, 111, 112] theauthors investigate a nanoelectromechanical system, consisting of a superconductingelectron transistor (SSET) strongly coupled to a harmonic resonator. In this system,the resonator can be driven due to an external field or by the interaction of thesystem itself. They were able to probe the correlations appearing in this system byinvestigating the noise spectrum.

As an experimental example we can take an optomechanical systems. There thestrength of the interaction between a cavity and the mechanical system can be clas-sified with a coupling parameter g0, which is correlated with the noise. Hence, themeasurement of the noise spectrum provides access to this coupling parameter [113].

On the other side, the noise provides a limit for the accuracy of a measurement[114]. Especially, if one crosses the border to the quantum regime. It affects thepossibility to cool a resonator to its ground state [115] or to detect gravitationalwaves [116].

In this section we derive the expression for the nonadiabatic noise correspondingto our single-level system. The correlation function for the current yields

Sα(t, t′) = 〈δIα(t), δIα(t)〉 = 〈

Iα(t),Iα(t′)

〉 − 2〈Iα(t)〉〈Iα(t′)〉

= (ie)2∑

kk′

[VkαVk′α〈c†kα(t)d(t)c

†k′α(t

′)d(t′)〉− VkαV

∗k′α〈c†kα(t)d(t)d†(t′)ck′α(t′)〉

− V ∗kαVk′α〈d†(t)ckα(t)c†k′α(t′)d(t′)〉

+ V ∗kαV

∗k′α〈d(t)†ckα(t)d†(t′)ck′α(t′)〉+ h.c.− 2〈Iα(t)〉〈Iα(t′)〉

].

(2.69)

For the further calculation we follow the derivation in [92] based on nonequilibriumGreen’s functions. For the interested reader we recommend the detailed derivationincluded there, here we solely sketch the steps and continue with the explicit expres-sion for the noise in our nonadiabatic approach.

The expectation values of the operators in Eq. (2.69) correspond to two-particleGreen’s functions. These functions are not easily obtained, but it is possible toexpress them through two-particle Green’s functions involving solely operators ofthe electronic level. Using Wick’s theorem splits the two-particle functions into one-particle functions. With this the disconnected terms cancel out the last term in

47


Eq. (2.69) and the noise yields

Sα(τ, τ′) = e2

∑

k

|Vkα|2gkα(τ

′, τ)G(τ, τ ′) + gkα(τ, τ′)G(τ ′, τ)

+ e2∑

kk′

|Vkα|2|Vk′α|2∫dτ1

∫dτ2−gkα(τ1, τ)gk′α(τ2, τ

′)G(τ, τ2)G(τ ′, τ1)

+ gkα(τ2, τ)gk′α(τ′, τ1)G(τ, τ ′)G(τ1, τ2)

+ gkα(τ, τ1)gk′α(τ2, τ′)G(τ ′, τ)G(τ1, τ2)

−gkα(τ, τ1)gk′α(τ′, τ2)G(τ2, τ)G(τ1, τ

′)

+ h.c., (2.70)

containing the undisturbed lead Green’s function gkα and the electronic level func-tions G(τ, τ ′) = −i〈Tcd(τ)d

†(τ ′)〉. We consider a time-ordering on the Keldysh con-tour as τ > τ ′ and the integrations can be transformed to the real axis by applyingthe continuation rules [92]. We consider the time-ordering t′ < t1 < t2 < t, use thetime-dependent Green’s functions from Sec. 2.2.2 and obtain

Sα(t, t′) =e2Γ2

α

[i

Γα

∫dω

2π

fα(ω)e

−iω(t′−t)G>(t, t′)− [1− fα(ω)]eiω(t′−t)G<(t′, t)

+G>(t, t′)

G<(t′, t) +

∫dω

2πfα(ω)e

−iω(t′−t)A−(ω, t′)

−∫

dω′

2πfα(ω

′)e−iω′(t′−t)A∗−(ω

′, t)

+G<(t′, t)

∫dω

2π[fα(ω)− 1]e−iω(t′−t)A∗

+(ω, t′)

−∫

dω′

2π[fα(ω

′)− 1]e−iω′(t′−t)A+(ω′, t)

+

∫dω

2π

∫dω′

2πfα(ω)[fα(ω

′)− 1]A−(ω, t′)A+(ω

′, t)e−i(ω+ω′)(t′−t)

+

∫dω

2π

∫dω′

2πfα(ω

′)[fα(ω)− 1]A∗−(ω

′, t)A∗+(ω, t

′)e−i(ω+ω′)(t′−t), (2.71)

with a spectral function similar to Eq. (2.44)

A±(ω, t) = −i

t∫

t0

ds e−i

t∫s

dt′′ (ε(t′′)±ω−iΓ2). (2.72)

It is hardly possible to proceed with this calculation due to the dependence ontwo different times. In general, one performs a Fourier transformation to obtain thenoise spectrum

Sα(Ω) =

∫dτ e−iΩτ ; τ = t− t′. (2.73)

48


x/g [1/l0] x/g [1/l0] x/g [1/l0]

SL(x)[ 1/ω0e2]

Vbias[ω

0/2]

Vbias[ω

0/2]

0 00

01 1 1

2

2

2

2 2−1 −1 −1

4 4

6 6

0.1

0.1

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.020.040.060.08

0.120.140.160.18

0.22

0.05

0.1

0.15

0.2

Figure 2.10.: Density plot for the Fano factor F (left graph) and the current IL(middle graph) for increasing bias Vbias and in dependence of the posi-tion x/g in units of the coupling parameter g = 2.5. The last plot inline depicts the noise result as a function of x/g for three different biasvalues, Vbias/ω0 = 1(black); 2(red); 4(blue). Remaining parameters forall three graphs are Γ = 1ω0 and εd = 3ω0.

As long as the Green’s functions, and therewith the noise, do depend on two in-dependent times, a Fourier transformation provides no benefits in the nonadiabaticapproach.

Due to these issues, we return back to the adiabatic approach, where the analyticcalculation of the Green’s function is possible. It is customary to consider not thewhole spectrum of the noise but its zero-frequency component. The latter is thecommon determinable quantity in experiments. The zero frequency noise yields

SL(0) =e2Γ2

2

∫dω

2π

(fL(ω) [1− fL(ω)] + fR(ω)[1− fR(ω)])G

A(ω)GR(ω)

+[fL(ω)− fR(ω)]2GA(ω)GR(ω)

[1− Γ2

4GA(ω)GR(ω)]

],

(2.74)

whereby the first term corresponds to thermal noise which vanishes for zero tem-perature and the second term equals the shot noise which vanishes for zero bias.Assuming T = 0 the noise is obtained from

SL(0) = e2

[IL/e−

Γ2

8π

(µL − ε)

(µL − ε)2 + Γ2

4

− (µR − ε)

(µR − ε)2 + Γ2

4

]. (2.75)

Here, we have identified the first term as the current Eq. (2.68). This result coincideswith the Levitov-Lesovik formula result [117, 118, 119, 120, 102, 110]. The usualdefined transmission coefficient equals here T = Γ2/4 GAGR.

For infinite bias the second term in Eq. (2.75) vanishes and the noise is constantSIB = e2Γ/4 = IIB, this equality with the infinite bias current is resulting fromthe assumption of symmetric tunnelling rates ΓL = ΓR. In the infinite bias case,the effective force is as well constant, whereby the friction vanishes. Following fromthat, the oscillator performs periodic oscillations and no further backaction effects

49


onto the electronic properties can be resolved in the framework of this method. Theinfluence of the electron-phonon interaction leads solely to a shift for the oscillator’srest position.

The right graph in Fig. 2.10 depicts the zero temperature noise and different valuesof the external bias. The noise increases with the bias. The grey dashed linescorrespond to the fixed points 〈x∗〉 ≈ 0; g/2; g. As mentioned before, two saddlepoints exist between these fixed points in the regime of small bias. For Vbias = ω0

two small peaks appear due to this turning points. We can estimate, that for smallbias, where we have two or three stable fixed points, the noise is minimal if theoscillator runs into the left or right point. For higher bias the oscillators trajectoriesrun into the middle fixed point where also the noise is maximal. The triangle shape ofthe noise in Fig. 2.10 is determined by the applied bias and by the chosen tunnellingrate. If we further increase the size of the transport window the noise increases upto the constant value SIB = e21/4ω0.

The middle graph in Fig. 2.10 depicts the current. The size of the transport windowdetermines the triangle-like shape, as for the noise. The condition εd = 3ω0 leads toa shift to positive values of x/g. As discussed in the last section, for small bias theeffective level ε = εd − gx is situated outside of the transport window and therewiththe current is suppressed and hence also the noise.

It is common to consider the ratio between the correlation function and its meanvalue. Here, this so-called Fano factor reads

F(x) =SL(x)

2IL(x). (2.76)

If F = 1 the stochastic process underlies Poissonian statistics. A deviation fromthe latter is characterised by the Fano factor [109]. For super-Poissonian statisticsthe Fano factor is F > 1 and vice versa for sub-Poissonian noise. Additionally, theFano factor gives a hint to indicate bunching (F > 1) and anti-bunching (F < 1).Whereby bunching means, that the electrons tunnel in packages and not irregularlythrough the device. In contrast when anti-bunching is present, the electrons passthe system regularly and therewith the noise is smaller. But as recently shown, thisrelation is only valid for Markovian systems with no appearance of delay processes[121]. The authors of [121] compared the Fano factor with the time-dependent g2-function, which indicates bunching and anti-bunching in optical systems. Theirresults show, that the Fano factor, as a long-time quantity, cannot describe theseeffects in a general manner.

The left graph in Fig. 2.10 depicts the Fano factor for our single-level systemcoupled to an oscillator. For a comparably small range of bias (blue area) the Fanofactor is minimal (F ≈ 0.2). Inside the triangle, the Fano factor equals F ≈ 1/2 =F IB, which is expected for a single-level system with symmetric chosen leads ΓL =ΓR. A high symmetry minimises the Fano factor [99].

Unfortunately, we obtain no additional information about the system’s correlationfrom the presented adiabatic noise results. The reason lies in the adiabatic approx-imation. Although, we observed a qualitative accordance of the adiabatic and the

50


nonadiabatic current in the long-time limit, we do not expect this accordance in thenoise. As discussed in [25], the stationary averaged current is unaffected by shorttime-dynamics and includes no Non-Markovian corrections. But for the noise Non-Markovian corrections are relevant, even the stationary noise includes informationabout the past [122]. One reason why the noise includes additional information.

The nonadiabatic approach includes the short-time dynamics and therefore weestimate, that the nonadiabatic noise will differ from the adiabatic result. A way ofimprovement could be the calculation of the current distribution function. This wasperformed by Pistolesi and co-Workers in [102]. From the Langevin equation theauthors derive a Fokker-Planck equation corresponding to a probability distribution.The expectation value of the current as well as the shot noise is computed withthe stationary distribution. The noise caused by the fluctuations over the positionis obtained by the time-dependent solution of the Fokker-Planck equation. Theseparation, and afterwards addition, of the noise contributions is possible, becauseof the different time-scales in an adiabatic approach. For instance, the positionfluctuations occur on a long time-scale, following from that, this part can than bemore important as the shot noise at low frequencies. Fluctuations occurring on thesame time-scale require the inclusion of correlation between the sources of the noise[123].

Due to the strong feedback in nanoelelectromechanical systems current noise [8]and phonon noise [41, 124] are important measurement quantities and therefor animprovement of the adiabatic approach is desirable. Another possibility could be theinclusion of adiabatic correction terms also for the noise [39]. Although this is notin the scope of this thesis but should be accomplished in future works.

51

3. Mechanical system coupled to an

electronic two-level system

One possible extension of a SET is realised by the so-called superconducting elec-tron transistor (SSET). For this device transport occurs via resonance processes andit can be modelled with a two-level system. This system coupled to a mechanicalsystem induces two most relevant backaction effects. Firstly, the detector backactiondamps the oscillator as it also has been observed for the SET. This feature is used forcooling experiments [9] and has been theoretical described by several authors for theSSET’s [125, 126] and other nanoelectromechanical systems based on an electronictwo-level device [127, 128]. Secondly, the coupling to the electronic nonequilibriumenvironment gives rise to nonlinear friction coefficients leading to both – positiveand negative damping [94]. The SSET close to the Josephson quasi-particle reso-nance can induce self-oscillations and regions of multistability of the oscillator [18].Hence, the SSET can possess laser-like instabilities [94] and a laser-like pumping wasexperimentally observed [129].

The coupling of a two-level system to a mechanical system was theoretically studiedin the framework of master equations [111, 18, 130] or adiabatic Langevin equations[94, 39, 40] and other semiclassical methods [65, 131]. In this chapter, we use ournonadiabatic method, which is beyond the master equation and the adiabatic ap-proximation, to probe the adiabatic approach and investigate the influence of a lowexternal bias.

3.1. Dynamics: adiabatic vs. nonadiabatic

The investigation of the interaction between a more complex electronic system anda mechanical system is in the scope of this section. Instead of one electronic levelwe now consider two electronic levels in series. The transition amplitude Tc be-tween these two energy levels becomes a new important quantity. Again we usethe Feynman-Vernon influence functional theory and derive the nonadiabatic andadiabatic Langevin equation for the oscillators trajectory. Already in the simpleadiabatic approach, we obtain nontrivial dynamical behaviour of the oscillator, likebistabilities and self-sustained oscillations. Additionally, we observe more quantita-tive differences between the adiabatic and nonadiabatic approach.

53

3. Mechanical system coupled to an electronic two-level system

L RΓL ΓR

Tc

Vbias

νL

νR

Figure 3.1.: Sketch of a two-level system coupled to an oscillator.

3.1.1. Langevin equation for the two-level system

The model we are treating in this section consists of two single electronic levels whichare coupled by a tunnel barrier. Again we assume a linear coupling to a single bosonicmode. The total Hamiltonian is composed of the oscillator part Hosc, cf. Eq. (2.2),the electronic part He and an interaction part which describes the coupling betweenthe oscillator and the two levels. In contrast to the single-level model, the oscillatorcouples to the difference of the occupation numbers with the coupling strength λ.The total Hamiltonian therefore reads

H = He +Hosc − λq(d†LdL − d†RdR), (3.1)

containing the electronic part

He =∑

kα

εkαc†kαckα +

∑

kα

[Vkα c†kαdα + V ∗

kα d†αckα

]

+∑

α

ναd†αdα + Tc d†L dR + T ∗

c d†R dL, (3.2)

where να∈L,R designates the left/right energy levels and Tc denotes the tunnel cou-

pling matrix element between them. d†α/dα denotes the creation/annihilation op-erator of the αth (α ∈ L,R) level. All other quantities are equivalent to thesingle-level system, c.f. Sec. 2.1.1. In the case of one electronic level we obtained anelectronic force which was proportional to the occupation. This is different for thetwo-level system, where the population difference σz corresponds to the electronicforce operator

F = λ(d†LdL − d†RdR

)= λ σz. (3.3)

The derivation of the equation of motion is similar to the single-level case. Therefore,we just sketch this calculation here as a reminder for the reader.

The oscillator’s dynamics is described by the reduced density matrix, whose prop-agation in time can be written as a double path integral over qt (forward) and q′t(backward) path weighted by the Feynman-Vernon influence functional [97, 96]. Ex-panding this functional to second order and performing a cluster expansion leads to a

54


Langevin equation. During this calculation we transform to relative (yt) and centre-of-mass (xt) path variables and introduce an effective interaction picture. With thiseffective interaction picture the differences between the adiabatic and nonadiabaticapproach start to appear.

In the nonadiabatic case the interaction picture with respect to H0 = He − F xt,contains an explicit time-dependent unperturbed part and thus all system quanti-ties are calculated with respect to this time-dependence. To second order in thequantum fluctuations yt the double path integral for the reduced density matrix de-scribes a classical stochastic process for the diagonal path xt that is defined by thenonadiabatic Langevin equation

mxt + V ′osc(xt)− λ 〈σz [xt]〉 = ξt, (3.4)

with V ′osc(xt) = mω2

0xt and a Gaussian stochastic force ξt that has a correlationfunction 〈ξtξs〉 = C(t, s) = 〈δσz [xt]δσz [xs]〉. Eq. (3.4) is the starting point for ournonadiabatic calculations. Note that the force f [xt] = λ 〈σz [xt]〉 is a complicatedfunctional that contains the full time-dependence of the position operator xt.

In the adiabatic approximation a Taylor expansion for the centre-of-mass variableis performed (xt ≈ x0 + tx0), leading to an interaction picture with respect toH0 = He − F x0 and a perturbation V [q](t) = −F (tx0 +

12yt). Consequently, the

expectation value of the force operator in the adiabatic interaction picture can becalculated for fixed x0. Additionally, due to the second term of the Taylor expansiontx0, an explicit friction term arises in the adiabatic Langevin equation,

mxt + V ′osc(xt) + xtA [xt]− λ〈σ0

z [xt]〉 = ξt. (3.5)

The friction term A [xt] can be interpreted as the first adiabatic correction term.

In contrast, in the nonadiabatic Eq. (3.4), all higher orders of the Taylor expansionare included and the first challenge is to calculate the force term f [xt] consideringthe full time-dependence of xt. Note, that in all further calculations we neglect thestochastic fluctuations (ξt = 0) like in the calculations for the single-level model.

3.1.2. Time-dependent occupation difference

In this section we derive the occupation difference for the adiabatic and the nona-diabatic case. Therefore, we revert to the equation of motion method introducedin Sec. 2.2.3. Calculating the time-dependent occupations for the two-level systemusing Green’s functions is a challenge due to the complex dependencies and couplingsof the system’s operators. We choose a more direct way by using the equations ofmotion technique, leading to a large system of coupled differential equations whichhave to be solved numerically.

Starting from the Heisenberg equations of motion for the operators of the electronic

55


levels and the leads we obtain

˙dL(t) =− i

(νL(t)− i

Γ

4

)dL(t)− iT ∗

c dR(t) +∑

k

CkL(t),

˙dR(t) =− i

(νR(t)− i

Γ

4

)dR(t)− iTcdL(t) +

∑

k

CkR(t), (3.6)

with the abbreviations να(t) = να ∓ λxt and Ckα(t) = −iV ∗kαe

−iεkαtckα(0) as newoperators. Again, we assume constant tunnelling rates ΓL = ΓR = Γ/2, wherebyΓα ≡ 2π

∑k |Vkα|2δ(ω − εkα). The equations Eq. (3.6) already include the solution

for the inhomogeneous differential equation for the lead operator ckα, see Eq. (2.50).The tilde denotes the interaction picture introduced above, cf. Eq. (2.32). Hence,the effective time-dependent energy level να(t) contains only the classical variablext. The differential equations for the corresponding electronic level annihilationoperators are derived in a similar manner.

With the equation for the level operators Eq. (3.6) we obtain for the propagationin time of the population difference

d

dt〈σz(t)〉 =− Γ

2〈σz(t)〉+ 2i

[Tc〈d†R(t)dL(t)〉 − T ∗

c 〈d†L(t)dR(t)〉]

+∑

k

[〈C†

kL(t)dL(t)〉+ 〈d†L(t)CkL(t)〉]

−∑

k′

[〈C†

k′R(t)dR(t)〉+ 〈d†R(t)Ck′R(t)〉]. (3.7)

This inhomogeneous differential equation contains multiple expectation values ofoperator combinations and for every combination another equation of motion can bederived. We can transform Eq. (3.7) into an inhomogeneous integral equation, whichyields in an abbreviated form

〈σz [xt]〉 = Blead(t) − 4 |Tc|2t∫

t0

dt′ eΓ

2(t′−t) Re

t′∫

t0

dt′′ 〈σz[xt′′ ]〉eit′′∫

t′

ds(∆ν−2λxs−iΓ2)

,

(3.8)

with the energy level difference ∆ν = νL − νR. The Blead function contains fur-ther contributions from the leads and the transitions between the electronic levels.We present this integral form to emphasise what this nonadiabatic approach accom-plishes. For calculating the occupation difference at one time t we have to integrateover all times before. Hence we are able to include memory effects into our system.For a final calculation we use the differential form and numerical support.

Deriving all equations of motion for the expectation values of operator combina-tions in Eq. (3.7) we finally obtain an inhomogeneous system of coupled differential

56


equations with time-dependent coefficients. Multiplication with δ(ω−εk,α) and sum-ming over all k states leads to

〈 ˙σz(t)〉 = − Γ

2〈σz(t)〉+ 2 Re

[2 DRL(t) +

∫dωBLL(ω, t)−

∫dω′BRR(ω

′, t)],

DRL(t) = i

(νR(t)− νL(t) + i

Γ

2

)DRL(t)− |Tc|2 〈σz(t)〉

+

∫dω′BRL(ω

′, t)−∫

dωB∗LR(ω, t),

Bαα(ω, t) = − i

(να(t)− ω − i

Γ

4

)Bαα(ω, t) −Bαβ(ω, t) +

Γ

4πfα(ω),

Bαβ(ω, t) = − i

(νβ(t)− ω − i

Γ

4

)Bαβ(ω, t) + |Tc|2Bαα(ω, t), α 6= β, (3.9)

with the definitions:

Bαα(ω, t) = i Vkα δ(ω − εkα) eiεkαt〈c†kα(0)dα(t)〉,

BRL(ω, t) = −Tc VkR δ(ω − εkR) eiεkRt〈c†kR(0)dL(t)〉,

BLR(ω, t) = −T ∗c VkL δ(ω − εkL) e

iεkLt〈c†kL(0)dR(t)〉,DRL(ω, t) = iTc 〈d†R(t)dL(t)〉. (3.10)

We name the Bαβ functions as lead-transition functions, describing the tunnellingof electrons from the electronic levels into the leads and vice versa. The DRL func-tion corresponds to the transitions between the levels. Therefore, we name it level-transition function.

The adiabatic result for the occupation can directly be obtained from Eq. (3.9) byomitting the time-dependence of xt ≡ x. Then the lead-transition functions, as wellas the occupation difference and level-transition function, decouple from the equationof motion for the oscillator’s variables. Additionally, the lead-transition functions donot depend on the occupation difference or the level-transition function, therewitha pair of coupled equations remains. With a Laplace transformation these systemsare straightforwardly solvable, see Sec. B.2.3. Here, the time-dependence drops downexponentially and therefore we consider solely the long-time limit. The stationarylead-transition functions read

Bαα(ω) =− Γ

4πfα(ω)

i(ω − νβ + iΓ4 )[(ω − νL + iΓ4 )(ω − νR + iΓ4 )− |Tc|2

] = −iΓ

4πfα(ω)G

Rαα(ω),

Bαβ(ω) =− Γ

4πfα(ω)

|Tc|2[(ω − νL + iΓ4 )(ω − νR + iΓ4 )− |Tc|2

] = − Γ

4πfα(ω)

GRαβ(ω)

T(∗)c

.

(3.11)

We recognise the two-level Green’s function in the adiabatic regime, cf. Sec.A.2.2.Setting the time-derivatives for the occupation difference and the transition func-tion equal to zero, cf. Eq. (3.9), we get rid of the level-transition function and the

57


stationary occupation difference solely depends on the lead-transition functions

〈σz(t → ∞)〉 =2

Γ

[(νR − νL)

2 + Γ2

4

]

[(νR − νL)2 +

Γ2

4 4|Tc|2]

×∫

dω Re

[2 [BLL(ω)−BRR(ω)] +

4 [BRL(ω)−B∗LR(ω)][

Γ2 − i(νR − νL)

]].

(3.12)

Inserting the lead-transition functions from Eq. (3.11) we obtain

〈σz[x]〉 =Γ

4π

∞∫

−∞

dω fL(ω)(ω − νR − λx)2 +

(Γ4

)2 − |TC |2∣∣∣[ω − νL + λx− iΓ2

] [ω − νR − λx− iΓ2

]− |Tc|2

∣∣∣2

−∞∫

−∞

dω fR(ω)(ω − νL + λx)2 +

(Γ4

)2 − |TC |2∣∣∣[ω − νL + λx− iΓ2

] [ω − νR − λx− iΓ2

]− |Tc|2

∣∣∣2

.

(3.13)

The integration intervals are clearly defined through the Fermi functions included inthe lead-transition functions. Therefore, we neglect the prime for the ω integrationover the right lead’s energies. The upper result coincides with the Green’s functionsresult, see Sec. B.2.1, where we also present the solutions of the integrals for T = 0.

For zero temperature and infinite bias the integrals in Eq. (3.13) can easily becalculated, see Sec. B.2.2. For symmetric detuned energy levels νR = −νL we obtain

〈σz[x]〉IB =(νL − λx)2 + Γ2

16[(νL − λx)2 + |Tc|2 + Γ2

16

] . (3.14)

Note that the derivation of the first adiabatic correction term, the so-called friction,cannot be directly derived from the nonadiabatic ansatz. Like in the single-level case,one has to expand the time-dependent Green’s functions as in [95]. For the Fano-Anderson model coupled to a large (semiclassical) spin, which we discuss in the nextchapter, we performed this kind of expansion as a complement. The correspondingderivation is sketched in Sec. B.3.3. The latter is straightforwardly transferable tonanomechanical systems, for more details see [95].

3.1.3. Phase space

To obtain the phase space portraits for the oscillators trajectories we numericallysolve the equation of motion Eq. (3.4) together with the time-dependent occupationdifference derived in the last section. We apply a trapezoidal rule for discrete func-tions to overcome the integrals over the discrete lead energies. Then the system for

58


the coupled differential equations reads

xt =1

mpt,

pt = −mω20xt + λ〈σz(t)〉,

〈 ˙σz(t)〉 = −Γ

2〈σz(t)〉+ 4DR

RL(t)

+ ∆ωN−1∑

n=0

[BR

LL(ωn+1, t) +BRLL(ωn, t)

]−∆ω′

N ′−1∑

j=0

[BR

RR(ωj+1, t) +BRRR(ωj , t)

],

DRRL(t) = −Γ

2DR

RL(t)− (νR − νL + 2λxt)DIRL(t)− |Tc|2 〈σz(t)〉

+∆ω′N ′−1∑

j=0

[BR

RL(ωj+1, t) +BRRL(ωj, t)

]−∆ω

N−1∑

n=0

[B∗R

LR(ωn+1, t) +B∗RLR(ωn, t)

],

DIRL(t) = −Γ

2DI

RL(t) + (νR − νL + 2λxt)DRRL(t)

+ ∆ω′N ′−1∑

j=0

[BI

RL(ωj+1, t) +BIRL(ωj, t)

]+∆ω

N−1∑

n=0

[B∗I

LR(ωn+1, t) +B∗ILR(ωn, t)

].

(3.15)

The discretisation scheme includes N/N ′ points and the step sizes ∆ω/∆ω′. Addi-tional to this equations we obtain 4[N+1]+4[N ′+1] equations for the lead-transitionfunctions Bαβ and therewith 13 + 4[N ′ + N ] coupled equations. We optimise thediscretisation schemes by defining multiple intervals with different step sizes. Forzero temperature, 2.4× 104 equations are sufficient for numerical stable results. Theintegration over time is performed with adaptive step-size control and an embedded8th order Runge-Kutta Prince-Dormand method with 9th order error estimate [132].This helps us to minimise the computation time.

For the adiabatic approach we solely solve a system of two coupled equationscorresponding to the Langevin equation Eq. (3.5). There an explicit friction termA[xt] appears as for the adiabatic single-level system. An explicit derivation anddiscussion of these friction is given in Sec. 3.2.2. Here, we just want to highlight oneimportant property on the basis of the infinite bias result namely

AIB[xt] = 4 |Tc|2λ2

Γ(νR − νL)

(νR − νL)2 + |Tc|2 + 5Γ2

16[(νR − νL)2 + |Tc|2 + Γ2

16

]3 , (3.16)

with the abbreviations νL,R = νL,R∓xt (for the derivation see Sec. B.2.2). Regardingthe prefactor (νR − νL) we recognise that we obtain regions where the friction isnegative and regions where it is positive. The latter holds also for the finite biascase. One requirement for a damped dynamical system to exhibit stable periodicoscillations is the additional appearance of anti-damping (positive friction in our

59


-1

0

1

-1

0

1

-1

0

1

-1

0

1

0

2

0

2

-2

0

-2

0

-2

0

2

-4 -2 0 2 4

-2

0

2

-2

0

2

-4 -2 0 2 4

-2

0

2

-2

0

2

-4 -2 0 2 4

-2

0

2

-2

0

-4 -2 0 2 4

-2

0

position 〈xt〉 [1/l0]

mom

entu

m〈p

t〉[1

/ω

0l 0

]

A1 A2 A3 A4

B1 B2 B3 B4

LC

LC

Figure 3.2.: Phase space portraits for various tunnel couplings |Tc|2 increasing fromleft to right. Upper row A: adiabatic results. Lower row B: nonadiabaticresults. In graphs A2 and B3 limit cycles (LC) appear. Explicit parame-ters are |Tc|2 = 0.2; 0.4; 1; 4 ω2

0. The tunnelling rate equals Γ = 2ω0 andfor the chemical potentials we assumed µL = ω0 and µR = −5ω0. Thedimensionless coupling constant is chosen as g = 2.5 and the internalbias voltage as Vint = 5ω0, whereas νL = −νR = eVint/2.

depiction). For the two-level system this means, that energy transfer processes occurwhich lead to the acceleration of the oscillator. The occurrence of positive frictionis possible when νL > νR and the electrons can transfer energy to the oscillator, cf.Sec. 3.2.1.

In Figure 3.2 results for the phase space trajectories are plotted. As initial condi-tion the momentum was set to zero and the positions were chosen in order to showthe various shapes of the trajectories. The tunnel coupling |Tc|2 increases from leftto right.

The upper row depicts the results for the adiabatic case including the first adiabaticcorrection term A[xt] resulting from the nonequilibrium electronic environment. Ifthe friction is turned off, periodic cycles appear which are stable and run aroundone or more fixed points. Certain states of the two-level system correspond to thesefixed points. If the left level is occupied the rest position of the oscillator is shiftedto the right and correspondingly to the left for an occupied right level. The stateswhen both levels are occupied or empty correspond to the fixed point in the middle.

When the friction is turned on the characteristics of the fixed points changes. Forexample, three fixed points appear in the range of small tunnel coupling (A1). There,the trajectories form stable spirals and run into the fixed points. For |Tc|2 = 0.4ω2

0

a limit cycle appears in the middle (A2). By further increasing |Tc|2 the limit cycleturns into an unstable spiral (A3) and in the end only the left fixed point remains as

60


stable solution (A4). As mentioned before, the appearance of the limit cycle in thissystem is possible due to energy transfer processes between the electrons and theoscillator. In the stable spiral case, the influence of the electrons leads to dampingof the oscillator. For example for νL < νR, the electrons need energy to pass throughthe system.

In these calculations Γ = 2ω0 is chosen because we want to probe the adiabaticapproach. As discussed for the single-level system, this low value for the tunnellingrate stands in contrast to the range of validity of the adiabatic approach, whichimplies a slow oscillator (Γ ≫ ω0). If Γ ∼ ω0 the interaction between the current andthe oscillator is strongest [19], because both act on the same time-scale. Interestingeffects still appear with a slightly enlarged Γ, but for Γ ≫ ω0 only one stable spiralremains. This means, that the oscillators rest position is shifted from its groundposition caused by the stochastic processes initiated by the current.

The second row (B) shows the results for the nonadiabatic approach. Qualitativelythe same features emerge, as the appearance of the limit cycle and the stable spi-rals. But we obtain quantitative differences, when comparing the adiabatic and thenonadiabatic approach, like the change of the middle fixed point into a limit cyclewhich happens at higher values of |Tc|2 as in the adiabatic case. The results differmost for small times, similar to the single-level case in Sec. 2.2. By further decreasingthe tunnelling rate Γ the differences between the approaches increase. Results fora smaller tunnelling rate are presented in Sec. 3.3.1, where we discuss the electroniccurrent for Γ = ω0. Furthermore, the differences between the adiabatic and thenonadiabatic results become much sharper when we consider the current and we areable to discuss the advantages and disadvantages of both approaches in more detail.

61


3.2. Analysis using the adiabatic approach

A mechanical mode coupled to an electronic two-level system includes interestingdynamical effects as we had seen in the last section. There, the adiabatic and thenonadiabatic results coincide in a qualitative manner. Doing analytics using thenonadiabatic approach is not really possible due to the complex hierarchy of coupledequations. But in the adiabatic case we obtain an explicit friction term and wecan define an effective force term leading to an effective potential. In this sectionwe use the adiabatic approximation for further investigations and a characterisationof the dynamics of the system. Therefore we start this section with an intuitiveinterpretation for what happens in our system. In the second section we calculatethe intrinsic friction term with the help of Keldysh Green’s functions. The latteris needed for the last section part where we accomplish a fixed point analysis withanalytical methods for nonlinear dynamical systems [103]. There, the differentialequation of the system is examined by studying its fixed points [133, 65].

3.2.1. Why limit cycles ?

Before we try to answer this question, we should reconsider what a limit cycle isand which requirements this nonlinear dynamical feature has to obey. Following thedefinition in [103], a limit cycle is an isolated closed trajectory in phase space. All tra-jectories in the neighbourhood of it are not closed and if the limit cycle is stable, thesesurrounding trajectories approach the limit cycle. In this case, the dynamical systemexhibits self-sustained oscillations and if such a system undergoes a small disturbanceit returns back to the initial cycle afterwards. This is a real nonlinear effect withits origin in the structure of the system. As a matter of course periodic oscillationsappear in linear systems, but these oscillations are not isolated. Their amplitude,and therewith the radius of the cycle in phase space, depend on the initial conditions.

2

-1

-2

2

-3 -1 1

-3

-2

1

3

x

x

Figure 3.3.: Van der Pol oscilla-tor.

One famous example for a dynamical systemwho’s dynamics imply a limit cycle, is the vander Pol oscillator [134]

x+ ǫ(x2 − 1)x+ x = 0.

For ǫ > 0 this system performs self-sustained os-cillations around the unstable fixed point x = 0.In phase space all trajectories outside and insideof the limit cycle move towards the cycle, seeFig. 3.3. The nonlinear damping term ǫ(x2 − 1)can be positive (|x| < 0) and negative (|x| > 0)and along with it the oscillator gets deceleratedand accelerated. One special thing about the vander Pol oscillator is that for every ǫ > 0 a unique limit cycle appear. In general thiskind of stability or continuity does not appear in dynamical systems.

62


LLL RRR

A B C

VbiasVbiasVbias

νLνLνL

νRνRνR

νL

νL

νL

νR

νRνR

∆

Figure 3.4.: Illustration of three processes that transfer energy from or to the oscil-lator. Graph A depicts the emission of energy by the electronic system,resulting in negative damping of the oscillator. In contrast, graph Band C show processes of energy absorption causing positive damping. ∆denotes the energy that electrons must expend to enter an empty statein the right lead.

Therewith we have some ingredients for the appearance of limit cycles: positiveand negative friction combined with an unstable fixed point. But the predictionof limit cycles in a dynamical system is a challenging task. One way is to searchfor bifurcations in the system, which we try to accomplish in Sec. 3.2.3 for our two-level system. As mentioned before in Sec. 3.1.3, this system fulfils the requirementof positive and negative friction. From Eq. (3.16) we learned that the sign of thefriction depends on the position of the effective levels νL,R = νL,R ∓ xt.

In a classical picture this can be understood intuitively: if νL < νR, the electronneeds energy to pass from the left to the right level which it takes away from theoscillator. If in contrast νL > νR, the passage of the electron from the left to theright level releases energy which accelerates the oscillator.With finite bias, see Fig. 3.4, the situation becomes more complicated as the levels

spectral functions are effected by the lead occupations. If νL < νR the friction isstill positive and for νL > νR a bounded range exists where the friction is negative.The latter region is limited by the Fermi level of the leads, for details see Sec. 3.2.2.When the effective level energy νR passes the value of the chemical potential µR theelectron needs energy to cross over from the right level to the right lead which itreceives from the oscillator, so the friction is again positive.

That negative damping occurs in nanoelectromechanical systems was also observedby Clerk and Benett [125]. They investigate the double Josephson quasi-particle(DJQP) process, which is somehow comparable with our DQD, by setting up aLangevin equation in linear response. For small detuning from the centre of resonancethey found that energy is transferred from or to the oscillator when the system isbelow or above the DJQP resonance.

The transfer of energy between the electrons and the oscillator in both directionswas also observed in a work by Armour and co-workers [18] for a nanoelectromechan-ical system. There, the authors consider a resonator coupled to a superconductingsingle electron transistor (SSET). As a result of the interplay of positive and negativedamping in a certain parameter range they observed limit cycles and bistability inthe phase plane. Here a likewise behaviour for the oscillator is obtained. In contrast

63


to the semiclassical approximation they investigate the Wigner function of the sys-tem with numerical master equations. They compare these results to a mean fieldevaluation of the expectation value of the oscillator position [65]. For weak couplingthe mean field approach gives quantitatively correct results, and for higher couplingit still describes the dynamics qualitatively correct. These results suggest that theuse of average oscillator positions and momenta is qualitatively correct for a descrip-tion of the oscillator instabilities.

3.2.2. Friction

In an adiabatic approach the centre-of-mass path variable is not time-dependentand therewith we can straightforwardly calculate the friction with Keldysh Green’sfunctions. In a two-level system the nonequilibrium Green’s functions correspondingto the level operators dα/d

†α acquires a matrix structure

G2 =

(GLL GLR

GRL GRR

).

Note that the standard relations obtained from the analytic continuation rules, likethe Keldysh equation G< = GRΣ<GA, include now matrix multiplications. InSec.A.2.2 we derive several Green’s functions of the two-level system in frequencyspace. As for the single-level case the friction corresponds to the imaginary part ofthe correlation function of the occupation difference

A[xt] = 2λ2

t∫

t0

dt′ t′ Im[〈δσz(t)δσz(t′)〉

]

=λ2

π

∫dω

[G<

LL(ω)∂

∂ωG>

LL(ω) +G<RR(ω)

∂

∂ωG>

RR(ω)

− G<LR(ω)

∂

∂ωG>

RL(ω)−G<RL(ω)

∂

∂ωG>

LR(ω)

]. (3.17)

For the second step we used a Fourier transformation and the time-integration isperformed using the definition of the derivative of Dirac’s delta. The lesser/greaterGreen’s function Keldysh equation follow from the Keldysh equation and the tun-nelling rates are considered to be frequency independent. Therefore, the self energiesare (T = 0)

Σ<,>γ (ω) = ±iΓγ Θ(±µγ ∓ ω). (3.18)

Because of the lengthy calculation, we just outline the calculation for the LL-termTLL. The other terms (TRL,TLR and TRR) can be derived in a similar way. Here, theproduct of retarded and advanced Green’s functions equates the squared modulus of

64


Gαβ , and

GRLL(ω) G

ALL(ω) = |GR

LL(ω)|2 = |GALL(ω)|2,

GRLR(ω) G

ARL(ω) = |GR

LR(ω)|2 = |GARL(ω)|2. (3.19)

For simplicity we omit the superscript of the retarded Green’s function and abbre-viate the moduli by

|GLL(ω)|2 ≡ |GRLL(ω)|2, |GLR(ω)|2 ≡ |GR

LR(ω)|2.

The first term of the friction reads

TLL =Γ2

4

∫dω

[|GLL(ω)|2Θ(µL − ω) + |GLR(ω)|2Θ(µR − ω)

]

×[[

∂

∂ω|GLL(ω)|2

]Θ(ω − µL) + |GLL(ω)|2δ(ω − µL)

+

[∂

∂ω|GLR(ω)|2

]Θ(ω − µR)|GLR(ω)|2δ(ω − µR)

]. (3.20)

Some parts of the integration terms can directly be evaluated with the help of thedelta and Heavyside functions. By choosing the condition µL > µR a case distinctionis avoided which would be necessary for two product terms which included a delta-function. After performing also an integration by parts, we arrive at

TLL =Γ2

8

[|GLL(µL)|4 + |GLR(µR)|4 + 4

µL∫

µR

dω |GLL(ω)|2[∂

∂ω|GLR(ω)|2

]

− 2 |GLL(µL)|2 |GLR(µL)|2 + 4 |GLL(µR)|2 |GLR(µR)|2]. (3.21)

In the following the Green’s functions, derived in Sec.A.2.2, are inserted, whereasequal tunnelling rates are assumed for the left and the right side, ΓL = ΓR = Γ

2 .Then a number of integrations by parts is performed to dispose the derivations inthe integral term. Afterwards a closed expression is obtained

TLL =Γ2

8

[[(µL − νR)

2 + Γ2

16

]2

N(µL)2+

|Tc|4N(µR)2

+ 2 |Tc|2[(µR − νR)

2 + Γ2

16

]

N(µR)2− 4 |Tc|2

µL∫

µR

dω(ω − νR)

N(ω)2

], (3.22)

with the denominator N(ω) =∣∣∣[ω − νL − iΓ2

] [ω − νR − iΓ2

]− |Tc|2

∣∣∣2

of the Green’s

function. In an analogue way the other terms in (3.17) can be derived, and finallythe

65


x [1/ω0]

A(x)

[l0/ω0]

νR < µR

νL < µL

νR < νL νR > νL

νR = νL

µL = 1ω0 µR = −5ω0

µL = 1ω0 µR = −6ω0

µL = 4ω0 µR = −5ω0

µL = 4ω0 µR = −6ω0

µL = ∞ µR = −∞

0

0

6

3

−3

−6−2 2

Figure 3.5.: Results for the friction for different bias values. The vertical dashedlines separates the regimes where the left level is above/below the rightlevel. With the chosen parameters the effective energy levels readνL,R = ±2.5(1− x)ω0. Therefore, the crossing position is near to x = 1.Remaining parameters are Γ = 3ω0 and |Tc|2 = ω0.

solution for the friction is

A[xt] =λ2

π

Γ2

8

[[(µL − νR)

2 + Γ2

16

]2

N(µL)2+

[(µR − νL)

2 + Γ2

16

]2

N(µR)2+

|Tc|4N(µL)2

+|Tc|4

N(µR)2

+ 2 |Tc|2[(µR − νR)

2 + Γ2

16

]

N(µR)2− 2 |Tc|2

[(µL − νR)

2 + Γ2

16

]

N(µL)2

− 4 |Tc|2[(µR − νR)(µR − νL)− Γ2

16

]

N(µR)2− 8 |Tc|2 (νL − νR)

µL∫

µR

dω1

N(ω)2

].

(3.23)

The last term, containing the integral, is responsible for the negative regime of thefriction. The prefactor (νL − νR) defines the crossing for the case of infinite bias,see Eq. (3.16), and for finite bias the crossing also lays near to νL = νR. This isclearly visible in Fig. 3.5 where the friction for different bias values is depicted. The

66


orange dotted line describes the infinite bias case where the friction is positive forνR < νL (shaded area above zero axis) and negative for νR > νL (shaded area belowzero axis). This corresponds to the processes where energy is transferred from orto the oscillator, see graphs A and B in Fig. 3.4. With finite bias the range of thenegative friction is limited. By increasing the transport window the positive peakon the right hand side breaks up and the influence of the right and the left leadbecomes visible. The left peak corresponds to the process in graph C in Fig. 3.4where the right effective energy level is positioned below the right chemical potentialand therefore an electron needs energy to occupy an empty state in the lead.

3.2.3. Fixed point analysis

The prediction of the dynamical behaviour in phase space of the oscillators trajec-tories is possible by using methods from nonlinear dynamics [103]. The key pointin a dynamical analysis is the characterisation of its fixed points by studying theirstability and how they react if a small disturbance appears. For the reader who isnot familiar with this kind of methods we sketch their basics in Sec.A.3.

The dynamical system for the two-level system reads

xt =1

mpt,

pt =ω0

l0

[−xtl0

+ g〈σz(t)〉 −ptω0l0

l20A[xt]

]. (3.24)

Fixed points occur under the condition pt = xt = 0, i.e. pt = 0 and following fromthat, the fixed points position coordinates are equal to the zeros of the effective force

Feff (xt) = −xtl0

+ g〈σz(t)〉 = 0. (3.25)

Depending on the parameters, we have found constellations with zeros of the effectiveforce ranging from 1 to 7. Like in the single-level case we can reach a huge numberof zeros only at small bias. That is an advantage over the pure Master equationapproach, which is not reliable in this low–bias regime where Non-Markovian effectsare important. The Jacobian matrix is obtained from

J∗ =

(0 1

ω0

l0

[− 1

l0+ g ∂

∂xt〈σz(t)〉

∣∣x∗

]−A[x∗]

),

evaluated at the fixed point x∗, whereby p∗ = 0. For a two dimensional system theeigenvalues of the Jacobian matrix are determined by their trace and their determi-nant, cf. Eq. (A.32). The determinant ∆ and the trace τ from the upper Jacobianbecome

∆ =ω0

l0

[1

l0− g

∂

∂xt〈σz(t)〉

∣∣x∗

], τ = −A[x∗]. (3.26)

The trace decides about the stability of a fixed point and is equal to the negativefriction here. For the case without friction the trace is equal to zero. Therefore, only

67


x [1/l0]x [1/l0]

p[1/ω0l 0]

p[1/ω0l 0]

τ ∆Feff

0

0

0

0

0

0

0.4 0.4

−0.4 −0.4

1

1

1

12 23 3

−1

−1

−1

−1−2 −2−3 −3

Figure 3.6.: Examination of the two-level system for two different tunnel couplingsTc. The left/right graphs correspond to |Tc|2 = 0.43/0.49ω2

0 . The uppergraphs show the results of a fixed point analysis. ∆ corresponds tothe determinant (grey line) and τ to the trace of the Jacobian matrix(dashed line). The dotted lines running through all graphs representthe positions of the fixed points. These points equal the zeros of theeffective force Feff (black line). The middle row depicts sketches for thephase space portraits predicted by the upper fixed point analysis. Bothsketches contain two stable spirals on the left and right side. In directionto the middle two saddle points follow. The only difference is in thecentral region, where the left graph predicts a stable spiral and the rightone an unstable spiral. The third row displays the actual results for thephase space obtained from numerical calculations. We used equal ratesΓL = ΓR = 1.5ω0. For the chemical potentials µL = 1ω0 and µR = −5ω0

are assumed. The dimensionless coupling constant is chosen as g = 2.5,and the internal bias voltage as Vint = 5ω0, whereas νL = −νR = eVint/2.

68


centers occur in the phase plane. In the case with friction the trace can be eitherpositive or negative leading to both, stable and unstable fixed points.

For comparison see Fig. 3.6, where the upper row depicts the results of a fixedpoint analysis for two different values of tunnelling amplitude |Tc|2. The effectiveforce Feff is plotted together with the trace and the determinant. Thus the fixedpoints can be characterised and it is possible to predict the shape of the phase spaceportrait. In the middle graphs these predictions are illustrated followed by the actualresults depicted in the third row.

For both tunnel couplings we obtain five fixed points. The vertical dotted lines inthe graphs of Fig. 3.6 denote their position. Considering the position and the numberof these points the predictions of the analysis coincide with the results obtained fromthe Langevin equation. Likewise the direction of evolution for the trajectories equalsthe forecasts of the fixed point analysis.

The analytic classification of the fixed points is almost correct. For the left graph,corresponding to |Tc|2 = 0.43ω2

0 , we obtain a stable spiral in the middle enclosed bytwo saddle points and followed by a stable spiral on each side. In the right diagramthe mean point changes to an unstable spiral. Around these unstable spiral runtwo limit cycles. The appearance of the small cycle is due the change of stabilityof the centre fixed point from stable to unstable for |Tc|/ω2

0 ≃ 0.49. There, thesystem undergoes a Hopf bifurcation [103], which happens when a pair of complexeigenvalues of the dynamical system cross the imaginary axis. These eigenvaluesdetermine the evolution in the phase plane, see section Sec. 3.2.3. In other words,the trace τ changes its sign and at the bifurcation point the eigenvalues are purelyimaginary λ1,2 = ∓i2

√∆, see equation Eq. (A.32).

The limit cycle with the small radius exists in the range of 0.49 ≤ |Tc|2/ω20 ≤ 0.54.

Its prediction was not directly possible from the foregoing analysis. The crossingof the eigenvalues is a strong indication for its appearance, but its range is notpredictable. For example, if we increase the coupling |Tc|2/ω2

0 ≥ 0.54 the middlefixed point stays unstable but no stable limit cycles appear.

The same insecurity occurs for the second limit cycle which appears in both phasespace diagrams. It exists in the range of 0.42 ≤ |Tc|2/ω2

0 ≤ 0.5 and origins from thechange in numbers of the fixed points. If |Tc|2/ω2

0 ≤ 0.42 the system owns seven fixedpoints, one additional saddle point and an unstable node. When the latter disappearthe limit cycle with the larger radius comes up. This leads to the centre bistability,with a stable spiral/limit cycle in the middle and a surrounding stable limit cycle.As far as our investigations show, these kinds of bistabilities do not appear in thenonadiabatic approach.

The bifurcation analysis depicted in Eq. (3.7) illustrates the explained behaviour.There, the fixed points of the system are depicted as a function of the parameter|Tc|2. The dashed lines represent saddle points, where the eigenvalues are real andhave opposite sign. In this plot four bifurcations are visible. The first one, a saddle-node bifurcation, appears at |Tc|2 ≈ 0.3ω2

0 where a saddle point and an unstable nodewere created. Therefore the number of fixed points changes from 3 to 5. The nextbifurcation happens when the stable spiral in the middle comes up together with a

69


x[1/l 0]

|Tc|2[ω20

]

bistable

stable spiral

stable spiral

unstable node

saddle point

stable spiral unstable spiral

limitlimit

cyclecycle

0

1

2

−1

−2

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Figure 3.7.: Bifurcation graph corresponding to the parameter regime in Fig. 3.6.

saddle point (|Tc|2 ≈ 0.35ω20). This bifurcation can be interpreted as a saddle-node

bifurcation on invariant circle (SNIC) [135], where two fixed points were destroyedand a limit cycle is born. The system’s fixed point number decreases again for|Tc|2 ≈ 0.42ω2

0 from 7 to 5, and, as mentioned above, the limit cycle with the largeradius is created. Finally, the Hopf bifurcation appears when the eigenvalue of themiddle fixed point changes its sign and the stable spiral becomes unstable.

For the single-level system the dynamics were quite simple and the knowledge ofthe potential minima and the positive friction leads always to stable spirals embeddedby saddle points. For the two-level system things are more complicated. The numberof fixed points can be large as depicted in Fig. 3.8. There, we plotted the number offixed points as a function of the tunnelling amplitude between the two energy levelsand the tunnelling rate into the two-level system.

The maximum of 7 fixed points exists only for a small range (red area), requiringsmall values of |Tc|2 and Γ. The number of fixed points is always odd, because ofso-called saddle-node or fold bifurcations [103]. For every appearing or disappear-ing fixed point with positive determinant an additional saddle point comes up ordisappears as well. This bifurcation can easily be explained.

70


7

5

3

1

Γ[ω

0]

|Tc|2[ω20

]1

2

2 3

4

4 5

6

6 7

1

3

5

7

8

9

Figure 3.8.: Density plot obtained from the fixed point analysis. The graph depictsthe number of fixed points as a function of tunnel coupling |Tc|2 andtunnelling rate Γ. The remaining parameters are the same as for Fig. 3.6.

Γ[ω

0]

|Tc|2[ω20

]

stable

unstable

nonad. LC1

1

2

2

3

3

4

4

5

5

6

6

7

7

Figure 3.9.: Stability analysis for the middle fixed point. The left border betweenthe stable and unstable region is highlighted with a red line, there a Hopfbifurcation appears. Remaining parameters as for Fig. 3.6.

71


Imagine a parabola f(x) = x2+a, where, if we vary the parameter a, the number ofzeros change. For a > 0 the functions graph situated above the x-axis and f(x) = 0has no real solution. If a = 0 one zero appears at x = 0. For a < 0 we obtain two zerosx1,2 = ±x∗ and these two points have deviations with opposite sign f ′(±x∗) = ±2x∗.

We can transform this simple example to the fold bifurcation. The system changesits dynamical behaviour by varying the parameter a. As long a > 0 no fixed pointsappear. The so-called bifurcation point is a = 0, where one half-stable fixed pointcame up. By further increasing the parameter two fixed points are obtained, onestable and one unstable. These kind of bifurcations can lead to the appearance oflimit cycles. In Fig. 3.6 a limit cycle came up, after two fixed points were destroyed.The latter feature appears along the right border of the 7-to-5 bifurcation in Fig. 3.8.At every border bifurcations appear and the system changes its dynamical behaviour.But for the borders corresponding to higher values of |Tc|2 or Γ, we do not observeany limit cycles. In the end, the remaining fixed point refers to the stable spiral onthe left side 〈x∗ ≈ −g〉, cf. Fig. 3.6.

Additionally, the appearing fixed points can pass a modification of their eigenval-ues. The best example is the middle fixed point (〈x∗ ≈ 0〉). In Fig. 3.9 the results forthe stability analysis of the latter are depicted. This fixed point is mainly unstable,but the corresponding area in the |Tc|2 − Γ plane is surrounded by a stable edge. Inthe light grey area in Fig. 3.9 the middle fixed point does not exist. For small valuesof the tunnel coupling |Tc|2 a Hopf bifurcation appears along the left edge denotedas a red line in Fig. 3.9.

In the next section we discuss the results for the adiabatic and nonadiabaticcurrent for Γ = ω0. In the adiabatic regime the limit cycle appear in the region0.04 < |Tc|2/ω2

0 < 0.48 and the creation point coincides with the result depicted inFig. 3.9. But its life-time is quite short although the fixed point stays unstable. Inthe nonadiabatic approach the existence of the limit cycle lasts until |Tc|2 ≈ 6.5ω2

0

and therefore almost over the whole unstable range in the |Tc|2−Γ plane. The latterholds also for other values of the tunnelling rate Γ. In the area below the yellowline in Fig. 3.9 limit cycles appear in the nonadiabatic approach. Above this line themiddle fixed point is stable and for higher values of the tunneling rate it vanishes asin the adiabatic approach.

72

3.3. Electronic properties

U eff(x)

[1/ω0]

I L(t)

[1/ω0e]

x [1/l0] t [1/ω0]

λ = 0LC

0

0

0 0.02

0.04

0.06

0.08

−0.5

0.5

1

1.5

−4 4−2 2 960 980 1000

Figure 3.10.: Adiabatic effective potential and electronic current corresponding toits minima. The green rectangle highlights the limit cycle region (LC).


The coupling to the electronic nonequilibrium environment modifies the oscillatordynamics and vice versa. In consequence the properties of the electronic systemare affected by the mechanical system. This backaction is relevant for an experi-mental investigation, where the cumulants of the electronic current are the relevantmeasurement parameters.

This section starts with the discussion of the current results obtained from theadiabatic and nonadiabatic approach. The comparison of the results leads to abetter understanding of the several benefits and disadvantages of these methods. Asa completion, we also present results for the current noise in the adiabatic regime.Further analytic studies using the nonadiabatic approach are rarely possible, butin the second part of this section we take a deeper look at the current- and thelevel-transition function. These quantities provide more information about the innerelectronic processes.

3.3.1. Current and noise for the two-level system

At first, we briefly present which kind of features show up in the current, if wecouple a two-level system to a single bosonic mode. Therefore, we make use of theadiabatic approach. There, similar to the single-level case, an effective potential forthe oscillator can be derived, see Sec. 2.1.4. This effective potential contains severalminima corresponding to the zeros of the effective force. In Fig. 3.10 the potential isdepicted. The lines running through the potential minima correspond to fixed pointsof the system. The left and the right minima equate to the stable spiral situation.There, the trajectories perform damped oscillations and in the long-time limit theyend up in one of the potentials minima. This is different for the central minima,corresponding to a limit cycle case, where we have stable oscillations for all times.

73


I L(t)

[1/ω0e]

I L,R

(t)

[1/ω0e]

t [1/ω0] t [1/ω0]

adiabatic nonadiabatic LEFT RIGHT

0

0

0.05

0.05

−0.05

0.1

985 990 995960 980 1000

Figure 3.11.: Current in the limit cycle case. LEFT: Adiabatic vs. nonadiabatic.RIGHT: Left and right current of the nonadiabatic case.

These dynamical situations are reflected in the electronic current, which is clearlyvisible in the current graph of Fig. 3.10. We obtain three current channels. Twoof them become stationary in the long-time limit. Likewise to the oscillator, thecurrent performs self-sustained oscillations. In this regime the system acts as aDC/AC transformer. The current belonging to the left stable spiral is suppressed.This is easily explained: We are in a finite bias situation where a small transportwindow appears. Effectively, the electronic levels get shifted due to the coupling tothe mechanical system and thus if they are situated outside of the transport windowthe electrons cannot pass through the system. Thus the current is suppressed.

All these features also appear if we use the nonadiabatic method. But the firstdifference occurs, if we look at the current through the left and the right lead. Inthe adiabatic case, the time-resolved current for the right lead is equal to the currentthrough the left lead with opposite sign:

IadiabaticL =eΓLΓR

∫dω

2π[fL(ω)− fR(ω)]G

RLR(ω)G

ARL(ω)

=eΓLΓR

µL∫

µR

dω

2π

|Tc|2∣∣∣[ω − νL − iΓ2

] [ω − νR − iΓ2

]− |Tc|2

∣∣∣2 = −Iadiabatic

R .

(3.27)

For the nonadiabatic case, right and left time-resolved currents are different buttheir time-averages coincide. If we are in the long-time limit and the system performsno oscillations, left and right current are equal. In contrast, in the limit cycle case weobtain a driven system leading to currents IL,R whose time-dependence differ sincecharge temporarily accumulates in the electronic levels. This behaviour is depicted inFig. 3.11. There we also recognise an additional feature appearing in the nonadiabatic

74


current. In contrast to the adiabatic result where the current direction is determinedby the applied bias, we obtain current in both directions. Considering the result forthe left current in Fig. 3.11. In our denotation a positive current describes electronspassing the system from left to right. For the chosen detuning this should be theexpected direction. But, also negative ranges appear corresponding to a currentflowing in the opposite direction. The latter is quite intuitive if we imagine varyingenergy levels confined in a finite transport window. here appear constellations wherethe left energy level is above the left chemical potential and the electrons can occupyempty states in the left lead. If we consider the relation for the current through leadα

Iα(t) = −e

[ΓαNα [xt]−

∫dω 2 Re [Bαα(ω, t)]

], (3.28)

derived via the Heisenberg equations of motion, the second term describes the tran-sitions of electrons between the α-level and the α-lead. For example, the left currentflows against the applied bias, if this term is smaller than the occupation of the leftlevel NL [xt], which is always positive.

The left graphs of Figure Fig. 3.12 depict the stationary current IL(t → ∞) asa function of the tunnel coupling |Tc|2. In the adiabatic case and for small valuesof |Tc|2 (A1) we observe two fixed points and one limit cycle leading to a tristablecurrent. In the limit cycle case the current oscillates in time. The correspondingaveraged current (dotted line) is not completely shown in (A1), due to the largevalues, since the current increases further until |Tc|2 = 0.48ω2

0 . There, the limit cycledisappears and two fixed points remain until |Tc|2 = 1.8ω2

0 (A2). We also obtain twofixed points in the small range of |Tc|2 ≤ 0.03ω2

0 , which is not resolved in graph A1of Fig. 3.12.

The current corresponding to the fixed point x∗ ≃ 2/l0 (solid line below g = 0case) increases approximately in the same fashion as in the case without coupling.In this regime the left effective level is situated inside the transport window (νL,R ≃∓2.5ω0). For large tunnel coupling one fixed point persists, x∗ ≃ −2.3/l0 and thecorresponding current (lowest solid line) is strongly suppressed compared to the casewithout coupling. There, both effective levels νL,R ≃ ±7.5ω0 are clearly situatedoutside the transport window. Therefore, tunnelling through the two-level system israrely possible.

In the left graphs of Fig. 3.12 the symbols denote the nonadiabatic current in thelong-time limit. Here, the system has also two fixed points, but the limit cycle rangeis much larger, 0.35 ≤ |Tc|2/ω2

0 ≤ 6.5. For |Tc|2 ≤ 0.35ω20 we observe two stable fixed

points and by increasing the tunnel coupling the middle stable spiral turns into alimit cycle and the mechanical system performs periodic oscillations. For the lattercase, the circles in Fig. 3.12 denote the averaged current.

The nonadiabatic current corresponding to the middle fixed point/limit cycle fol-lows the result without coupling. As long as the fixed point x∗ ≃ 0.03/l0 is stable theresulting effective level is approximately νL,R ≃ νL,R as in the case without coupling.In graph B1 of Figure 3.12, the phase space trajectories are plotted for the case

75


0 0.1 0.2 0.3 0.4 0.5

0

0.01

-1 0 1

-1

0

1

-1 0 1

-1

0

1

970 980 990

0

0.1

0

0.1

0.5 1 1.5 2

0

0.02

|Tc|2

[

ω20

]

|Tc|2

[

ω20

]

IL

(t→

∞)

[1/ω

0e]

IL

(t)[1

/ω

0 e]〈p

t 〉[1

/ω

0 l0 e]

〈xt〉 [1/l0e]

t [1/ω0]

A1

A2

B1

B2

|Tc|2

= 0.45 ω20

|Tc|2

= 0.45 ω20stable spiral

stable spiral

limit cycle

limit cycle

with

out

couplin

gadia

batic

adia

batic

LC

non–adiabatic

non–adiabatic

non–adiabatic

non–adiabatic

adiabatic

adiabatic

Figure 3.12.: LEFT: Current for t → ∞ as a function of tunnel coupling |Tc|2. A1displays the results for |Tc|2 ≤ 0.5ω2

0 and A2 the results in the rangeof 0.5 ≤ |Tc|2/ω2

0 ≤ 2.0. The symbols denote the nonadiabatic results.The diamonds correspond to the stable spiral situations. There theoscillation of the dynamical system disappears in the long-time limitand the current becomes stationary. Circles denote averaged currentvalues for the limit cycle case when the system performs periodic os-cillations. The indigo solid (dotted) lines depict the adiabatic resultsfor the stable spiral (limit cycle) case. The dashed-dotted line depictsthe current without coupling. RIGHT: The graph B1 shows the phasespace results for |Tc|2 = 0.45ω2

0 . Here the radius for the adiabatic limitcycle is much larger than in the nonadiabatic case. The correspondingtime-dependent left current is depicted in graph B2, which oscillates aswell. Explicit parameters are Γ = ω0, µL = ω0 and µR = −5ω0. Thedimensionless coupling constant is chosen as g = 2.5 and the internalbias voltage as Vint = 5ω0, whereas νL = −νR = eVint/2.

76


x [1/l0] x [1/l0]x [1/l0]

SL(x) IL(x)F(x)

F(x)

adiabatic

adiabatic LC

g = 0

|Tc|2[ω20

]

|Tc|2[ ω

2 0

]

|Tc|2[ ω

2 0

]

|Tc|2[ ω

2 0

]

000

000

0 00

11

1

1

1

111 2 222

1.51.51.5

0.8

0.90.10.1

0.2

0.40.50.50.5

0.6

0.8

Figure 3.13.: Results for the Fano factor F , the Noise SL and the current IL asa function of the oscillator position and tunnel coupling |Tc|2. Theright plot depict the Fano factor corresponding to numerical resultsof the Langevin equation. The dashed-dotted line represents the casewithout coupling to the oscillator (g = 0). Remaining parameters asfor Fig. 3.12.

when the system performs periodic oscillations. The limit cycle corresponding to thenonadiabatic results runs in small cycles about the origin and, following from that,the averaged current is similar to the case without coupling. In the adiabatic casethe radius is much larger and the shape of the limit cycle is not smoothly circular.Hence, the current is much larger then in the nonadiabatic case (A1). In graphB2 the related time-dependent current is depicted. Frequency and amplitude differstrongly in both cases. The frequency of the current oscillations is equal to theoscillator frequency (nonadiabatic: ω ≈ 0.86ω0).

The calculation of the noise in the framework of the nonadiabatic approach israrely possible without further approximations as in the single-level case. The noiseS(t, t′) depends on two different times and not solely on time-differences. Therefore,we directly go to the adiabatic approach where we can easily calculate the zerofrequency noise

Sadia.L (0) = 2e2

1

2eIadiabaticL − Γ4

16

µL∫

µR

dω

2π

|Tc|4∣∣∣[ω − νL − iΓ2

] [ω − νR − iΓ2

]− |Tc|2

∣∣∣4

.

(3.29)

Here, we assumed zero temperature where the thermal noise term vanishes, cf.Sec. 2.3.2. In Fig. 3.13 the noise is depicted together with the current and the Fanofactor as a function of the oscillator position and the tunnel coupling. The first thingwe recognise by comparing the density plots with our results from Fig. 3.12 is, thatall current channels corresponding to the fixed points of the system are in the rangeof very low bias, cf 〈x∗〉 ≈ −2.5; 0; 2.5/l0 . The electronic current is maximal in thisparameter regime for x/l0 ≈ 1. There the effective level energy vanishes.

The last graph of Fig. 3.13 represents the Fano factor for fixed point values cor-responding to Fig. 3.12. When the oscillators trajectories run into the left stablefixed point (upper red line) the current is strongly suppressed and the Fano factorequals ≈1. For the right spiral case the current is higher and the Fano factor slightly

77


increases for higher tunneling amplitude (lower red line). The appearing limit cycleruns around the middle fixed point 〈x∗ ≈ 0〉 and due to the large radius the currentis clearly higher as for the stable spiral cases. Additionally, the Fano factor dropsdown, which is denoted by the blue line in the graph.

3.3.2. Lead- and level-transition functions

In this section we concentrate on the lead- and the level-transition functions definedearlier, cf. Eq. (3.10). We consider the limit cycle case where all quantities oscillatein time. As mentioned before, the lead-transition functions Bαβ(t) describe thetransition between the electronic levels and the leads. For instance, the BLL functiondetermines the electronic transfer from the left lead to the left energy level and viceversa. This function contributes directly to the left electronic current, cf. Eq. (3.28),whereas the BLR function does not directly appear. The latter describes transitionsbetween the left lead and the right energy level.

Likewise, the DRL function does not couple directly to the current but to theoccupation difference, cf. Eq. (3.9). This function describes transitions between theleft and the right energy level. Its equation of motion also contains a coupling to theBLR/BRL functions.

The current reaches its maximum when the distance between the left (right) effec-tive level and the left (right) chemical potential is minimal (maximal). This is clearlyvisible in Fig. 3.14 where the time-evolution for current and the different lead- andlevel-transition functions are depicted. The evolution in time of all functions is notsmooth sinusoidal as the movement of the oscillators trajectories and therewith theeffective energy levels. Different effects show up and the identification of their originis not always straightforward. As well as the correlations between the individualfunctions.

The first row in Fig. 3.14 shows the behaviour of the oscillating effective levels. Inthese graphs the dotted lines represent the position of the lead’s chemical potentials.For this parameter regime, the left effective level is situated always outside of thetransport window. In contrast, the right level is situated within. Caused by thesymmetric choice for the energy levels, νR = −νL, the left effective level is maximalwhen the right effective level reaches its minima. Following from that, the distancebetween the effective levels itself is in that case minimal.

As mentioned above and depicted in the first graph of the second row, the currentfor the left lead is maximal when the effective level is minimal. The current decreaseswhen the left effective level increases its distance to the transport window. Here, wesee an effect of the nonadiabatic approach which does not appear for the adiabaticansatz: the current becomes negative. Following from that, the electrons are flowingagainst the bias. In this regime, the real part of BLL becomes negative as well,corresponding to transitions from the left energy level to the left lead.

The effect of current flowing against the bias appears also in the current measuredat the right lead depicted in graph B2, but for a later time, when the right effectivelevel reaches its minima. Note, that for the direction of the right current the sign

78


2

4

2

4

-4

-2

-4

-2

0

0.08

0

0.08

0

0.1

0

0.1

-0.03

0

0.03

-0.03

0

0.03

0

0.04

0

0.04

-0.02

-0.01

-0.02

-0.01

0

0.01

0

0.01

985 990 995 1000

0.01

0.02

0.01

0.02

985 990 995 1000

-0.02-0.02

-0.015

PSfrag

ν L−λxt

ν R+λxt

I L(t)

I R(t)

Dℜ RL(t)

Dℑ RL(t)

Bℜ RL(t)

Bℜ LR(t)

BℜLL Bℜ

RRnL nR

d dtnL(t)

d dtnR(t)

t [1/ω0]t [1/ω0]

LEFT RIGHTA1

A2

A3

A4

A5

B1

B2

B3

B4

B5

Figure 3.14.: Time-evolution for the current and the corresponding correlation func-tions for |Tc|2 = ω0 in the nonadiabatic limit cycle case. The verticaldotted lines correspond to the maxima/minima of the effective levelνα = να ∓ gxt/l0. The first row depicts the position for the effec-tive left (A1) and right (B1) level and the dotted line corresponds tothe chemical potentials µL = ω0 and µR = −5ω0. The results for theleft/right current are plotted in graph A2/B2 together with the occupa-tion for the left/right level and the real part of the correlation functionBLL/RR. The time derivative of the level occupation is depicted inrow three. Row four shows the result for the real (A4) and the imagi-nary (B4) part of the level-transition function DRL. The real parts ofBLR/RL are depicted in row five. The dimensionless coupling constantis chosen as g = 2.5 and the internal bias voltage as Vint = 5ω0, whereasνL = −νR = eVint/2.

79


5

6

7

8

5

6

7

8

988 988.5 989 989.5 990 990.5 991time t

-0.007

-0.006

-0.005

-0.004

-0.003

988 988.5 989 989.5 990 990.5 991

-0.007

-0.006

-0.005

-0.004

-0.003

Dℜ R

L(t

)ω

R

∆t = 0.902 sω = 6.966 /s

∆t = 0.897 sω = 7.005 /s

∆t = 0.795 sω = 7.903 /s

∆t = 0.827 sω = 7.598 /s

∆t = 0.769 sω = 8.171 /s

Figure 3.15.: Detail of the DReRL function

plot from Fig. 3.14.

is the other way around as for the leftcurrent. The current reaches its mini-mum although the position of the effec-tive level is still above the right Fermiedge. This appears due to the broaden-ing of the effective levels and therewiththe branch of it is already beyond theedge.

The additional current peak (arrowsin A2 of Fig. 3.14) near the maximum ofthe left effective level does not appearin an adiabatic approach where the cur-rent follows the position of the levels.This peak is related to internal coher-ent electronic oscillations between thetwo levels. These oscillations are vis-ible in the real part of the DRL func-tion depicted in graph A4 of Figure 3.14(denoted by dashed boxes), with fre-quencies that match the time-dependentRabi frequency [27, 136]

ωR(t) =√νL(t)− νR(t) + 4|Tc|2. (3.30)

The lower graph in Fig. 3.15 depicts a part of the regime in A4, where oscillationsbetween the two electronic levels are resolved. The frequencies of the oscillations areobtained from their extrema and the results are denoted in the graph. The resultsmatch quite good the Rabi frequency shown in the upper graph of Fig. 3.15.

3.3.3. A brief summary for the two-level system

For the two-level system there exist two different kinds of long-time behaviour. De-pending on the initial conditions and the chosen parameters, the oscillator getsdamped and reaches a fixed point or performs periodic oscillations. This behaviourmatches the results of other theoretical works, as discussed at the beginning of thischapter.

The adiabatic approach also includes these dynamical features. Furthermore, wefind a qualitative good agreement to the results obtained with the nonadiabaticansatz. But it is also clear, that the adiabatic approach does not catch the wholedynamics of the oscillator, especially for small times and when the system performsoscillations in the long-time limit. For instance, the regime where periodic oscillationsappear is much smaller than those for the nonadiabatic approach.

Finally, from the foregoing investigation of the current and the transition functionswe can state, that the nonadiabatic approach contains additional information aboutthe inner processes of the system.

80

4. Large external spin coupled to a

single electronic level

A quantum dot typically consists of 105 atoms, electrons tunneling through suchdevices experience hyperfine interaction [137]. Combinations of huge numbers ofspins can be described as a large external effective spin system which interacts withthe single electron spin. In experiments with quantum dots features like a largeOverhauser field and self-sustained current oscillation have been observed [138].

The combination of electron spin and large spin shows interesting nonlinear effectsand the system is known to exhibit chaotic behaviour [139, 140, 141, 142]. Theseeffects appear for the closed system with anisotropic coupling and an external mag-netic field. In [46] the authors coupled these kind of system to two external leads,which are assumed to be polarised. Within a rate equation approach they found,that the chaotic behaviour survives for small magnetic fields.

Using our nonadiabatic approach we can extend this approach to the finite biasregime, which is not accessible within the rate equation method. Furthermore, theirmethod is also restricted to first order transitions, which we also could extend withour nonadiabatic approach and thus probe the rate equation ansatz.

The transfer and modification of the nonadiabatic approach, obtained for the de-scription of nanoelectromechanical systems, is straightforward. Especially for thespin system, because it is similar to a parallel two-level system. We like to em-phasise that this approach is transferable to a large amount of models, where a(semiclassical) system is strongly coupled to a nonequilibrium environment.

4.1. Single-level with spin and linear potential

As mentioned in the introduction to this chapter, we like to apply our nonadiabaticapproach to another kind of transport system. The similarity to the nanoelectrome-chanical systems of the last two chapters lies in the combination of an externalsystem, whose quantum fluctuations are assumed to be small, coupled to a nonequi-librium environment. Instead of an oscillator we consider a large external spin, whichcan be interpreted as an effective spin of a collective spin system.

This section starts with a detailed description of the used model and the adaptationof the nonadiabatic approach. We can directly derive the rate equation approach fromour nonadiabatic method, which is presented in the third part of this section, togetherwith a brief summary of the dynamical features which appear in [46]. Afterwards,we compare the results of both methods in the infinite bias regime.

81

4. Large external spin coupled to a single electronic level

4.1.1. Model

We assume that a vertical magnetic field Bz is applied to the Fano Anderson modelfrom Sec. 2.1.1, where from now on we consider as well the spin of the electrons. Themagnetic field leads to a Zeemann splitting of the electronic level εd into two parallellevels [143], corresponding to εσ = εd ± 1

2Bz, with σ ∈↑, ↓ and the z-componentof the magnetic field Bz. Without further interactions this solely leads to two spin-dependent current channels. Only electrons with spin-up (-down) can tunnel throughthe upper (lower) energy level. These energy levels are broadened due to the couplingto the leads and for small magnetic fields an overlap of both channels exists, but thereis no communication established between the two energy levels and spin-flips cannotoccur. Here we enable transitions between the two energy levels with the help of alarge external spin, which interacts with the electronic spin. Fig. 4.1 depicts a sketchof the considered model. There, S denotes the electronic spin operator for the levels,which components are defined via

Sx =1

2

(d†↑d↓ + d†↓d↑

), Sy =

1

2i

(d†↑d↓ − d†↓d↑

), Sz =

1

2

(d†↑d↑ − d†↓d↓

). (4.1)

The vertical magnetic field Bz couples to the Sz operator, leading to the splitting ofthe initial electronic level. The Hamiltonian for the Fano-Anderson model reads

HFA =∑

σ

εd d†σdσ +∑

kασ

εkασ c†kασ ckασ +

∑

kασ

(Vkασ c

†kασdσ + V ∗

kασd†σ ckασ

)+BzSz.

(4.2)Note, that the left and right lead’s operators now became spin-dependent. Thisenables us to consider polarised leads, where the density of states for spin-up andspin-down electrons differ. For simplicity, we include the prefactor of the last term inEq. (4.2), containing the electronic g-factor, into the definition of the magnetic field.

As well as the electronic spin operators, the large external spins z-component Jzcouples to the magnetic field. The free motion for it is described by

HJ = BzJz, (4.3)

where we assume the same g-factor and therewith the same magnetic field as forthe electronic spin operators. This is a simplification in theoretical approaches, cf.[46, 104], but a generalisation is straightforward.

Here, a large external spin means an effective spin describing a big ensemble ofspins, for example, the collective spin of the nuclei in a quantum dot. Electronswhich tunnel through this device, experience an interaction with the effective spinof the whole system. This interaction is described by

V =∑

i

λiSi Ji, i = x, y, z; (4.4)

introducing the coupling constant λi. If these coupling constants are equal for allcomponents λi = λj , one speaks of an isotropic coupling. There, the two spins

82


B

S

L R

J

Vbias

ε↑

ε↓

Figure 4.1.: Sketch of a single-level system coupled to an large external spin.

decouple in the long time limit. In the anisotropic case, where at least for twocomponents λi 6= λj is valid, interesting dynamical behaviour was observed [142].

We treat the interaction of the two spins in a semiclassical manner. Therefore, wehave to assume that the quantum fluctuations in the system are small. This shouldbe valid as long the external spin is large and its fluctuation can be neglected. Asa consequence of this assumption, we have no spin decay due to dissipation and thelarge spin is conserved.

Using a mean-field approximation for Eq. (4.4) we obtain

VMF =∑

i

λi

(Si 〈Ji〉+ Ji 〈Si〉 − 〈Si〉〈Ji〉

), i = x, y, z. (4.5)

Thereby, the fluctuations δAi = Ai − 〈Ai〉, A ∈ S, J have been neglected. Now wecan build up a closed system of equations for the considered system.

For the large spin we use the commutation relations[Ji, Jj

]= i∑

k

ǫijkJk, (4.6)

with the Levi-Civita symbol εijk [144], which is 1 if (i, j, k) is an even permutation of(x, y, z), −1 if it is an odd permutation, and 0 if any index is repeated. Therewith,the equation of motion for the large spin operator are derived via the Heisenbergequation. We obtain

〈 ˙Jx〉 = −(λz〈Sz〉+Bz

)〈Jy〉+ λy〈Sy〉〈Jz〉,

〈 ˙Jy〉 =(λz〈Sz〉+Bz

)〈Jx〉 − λx〈Sx〉〈Jz〉,

〈 ˙Jz〉 = λx〈Sx〉〈Jy〉 − λy〈Sy〉〈Jx〉. (4.7)

This is a strongly nonlinear system, since the electronic spin components 〈Si〉 dependon the large spin components, even in an adiabatic approach, where the movement ofthe large spin is assumed to be slow compared to changes in the electronic subsystem.

83


In the next part of this section we also derive equations of motion for the electronicspin operators using our nonadiabatic approach.

4.1.2. Nonadiabatic approach

In the framework of the nonadiabatic approach, we calculated all system quantitiesof the nanoelectromechanical systems by considering the full time-dependence of theoscillator’s trajectories. Now we want to transfer this method to our spin model,where the large spin takes the place of the oscillator as the semiclassical system.

In the following we assume an anisotropic coupling λy = 0 and λx = λz = λ, andtogether with the mean-field approach we obtain an effective Hamiltonian for theelectronic subsystem

Heff ≡∑

σ

εσ d†σdσ +λ

2〈Jx〉

(d†↑d↓ + d†↓d↑

)+HT +Hres, (4.8)

with εσ ≡ εd ± Bz

2 ± λ2 〈Jz〉 (σ =↑ / ↓). The last two terms describe the coupling

to the leads and the reservoir electrons. Based on this effective Hamiltonian we candescribe the effects arising due to the coupling to a large external spin. The 〈Jz(t)〉 -component solely leads to an additional shift of the electronic levels, but the couplingto the 〈Jx(t)〉 - component enables transitions between both levels. This Hamiltoniancorresponds to a two-level or a parallel double dot system.

The derivation of the equation of motions for the spin operators is similar to thetwo-level system case, therefore we placed the details in the Appendix section, seeSec. B.3.1. The results for the spin operators yield (Γ = Γ↓ = Γ↑)

d

dt〈Sx(t)〉 =− Γ〈Sx(t)〉 −

(Bz + λ〈Jz(t)〉

)〈Sy(t)〉

+∑

α

∫dωRe

[Bα

↑↓(ω, t) +Bα↓↑(ω, t)

],

d

dt〈Sy(t)〉 =− Γ〈Sy(t)〉+

(Bz + λ〈Jz(t)〉

)〈Sx(t)〉 − λ〈Jx(t)〉〈Sz(t)〉

+∑

α

∫dωIm

[Bα

↑↓(ω, t)−Bα↓↑(ω, t)

],

d

dt〈Sz(t)〉 =− Γ〈Sz(t)〉+ λ〈Jx(t)〉〈Sy(t)〉

+∑

α

∫dωRe

[Bα

↑↑(ω, t)−Bα↓↓(ω, t)

], (4.9)

with the definition Bασσ′(ω, t) = i

∑k Vkασδ(ω − εkασ)e

iεkασt〈c†kασ(0)dσ′ (t)〉 for thelead-transition functions

d

dtBα

σσ(ω, t) =− i(εσ(t)− ω − i

2Γ)Bα

σσ(ω, t)− iλ

2〈Jx(t)〉Bα

σσ′ (ω, t) +Γασ

2πfα(ω),

d

dtBα

σσ′(ω, t) =− i(εσ′ (t)− ω − i

2Γ)Bα

σσ′(ω, t)− iλ

2〈Jx(t)〉Bα

σσ(ω, t). (4.10)

84


The time-dependent z-component of the large spin is included into the effective levelsεσ(t) = εd ± Bz

2 ± λ2 〈Jz(t)〉. Here, the Fermi function fα(ω) does not depend on the

spin due to the assumption, that the chemical potentials for spin-up and -downelectrons in each lead are equal.

For the numerical calculations we decompose the lead-transition functions intotheir real and imaginary part and discretise the integration over the lead energiesin N intervals. Therefore, we obtain [6 + 16(N + 1)] coupled equations, includingthe equations of motion for the large spin, cf. Eq. (4.7). To receive numerical stableresults the number of equations can reach high values, e.g. up to 105 − 106 for smallvalues of the tunnelling rate Γ together with a high external bias.

Using the definitions for the lead-transitions function Eq. (4.10), the electroniccurrent is obtained from

Iασ(t) = e∑

k

Vkασi〈c

†kασ dσ〉 − V ∗

kασi〈d†σ ckασ〉

= e∑

k

iVkασe

iεkασt〈c†kασ(0)dσ〉 − |Vkασ|2t∫

0

dt′eiεkασ(t−t′)〈d†σ dσ〉+ h.c.

= e

∫dω 2 Re [Bα

σσ(ω, t)]− Γασ〈nσ(t)〉, (4.11)

in the second step we applied the definition for the spin dependent tunnelling ratesΓασ = 2π

∑k |Vkασ|2 δ(ω − εkασ). The equation of motion for the occupation yields

d

dt〈nσ(t)〉 =− Γ〈nσ(t)〉 ± λ〈Jx(t)〉〈Sy(t)〉+

∑

α

∫dω 2 Re [Bα

σσ(ω, t)] . (4.12)

Here, the upper/lower lower sign refers to ↑ / ↓ - electrons. Note, that the totaloccupation number of the dot 〈N(t)〉 = 〈n↑(t)〉 + 〈n↓(t)〉 still couples to the spinoperators via the transition functions. In the next section we derive a rate equationapproach where this dependence is omitted.

4.1.3. From Berlin to Madrid: rate equation approach

The Fano-Anderson model coupled to a large external spin was already in the scopeof interest in a collaborative work of López-Monís from Madrid and co-workers fromBerlin [46]. There, the authors worked in the infinite bias regime and used a rateequation approach to describe the system’s dynamics. They observed nontrivialdynamical behaviour like self-sustained oscillations and chaos.

We recover their ansatz by applying an adiabatic approximation to our approach.There, the large spin’s movement is considered to be slow compared to the elec-trons which are jumping through the system. As a consequence, we can neglect thetime-dependence of the large spin in the equations for the lead-transition functionsEq. (4.10), leading to εσ(t) ≡ εσ and a decoupling from the equations for the largespin, 〈Jx(t)〉 ≡ 〈Jx〉. The remaining two coupled equation can be solved with help

85


of the Laplace transformation. Note, that a Laplace transformation without theadiabatic approximation is not effective, due to the product of two time-dependentfunctions in time-space leads to a complex convolution in Laplace space.

Starting from Eq. (4.10) we obtain for t → ∞

Bασσ(ω) =

Γασ

2πfα(ω)

−i(εσ′ − ω − i2Γ)

(εσ′ − ω − i2Γ)(εσ − ω − i

2Γ)− λ2

4 〈Jx〉2,

Bασσ′(ω) =

Γασ

2πfα(ω)

iλ2 〈Jx〉(εσ′ − ω − i

2Γ)(εσ − ω − i2Γ)− λ2

4 〈Jx〉2. (4.13)

Before inserting these results into the equations for the spin operators Eq. (4.9), weseparate them into real and imaginary part and perform the integrations over ω.Assuming infinite bias, these integrals yield

∞∫

−∞

dω ReBασσ′(ω) =

Γασ

2δαLδσσ′ and

∞∫

−∞

dω ImBασσ′(ω) = 0 for σ 6= σ′. (4.14)

Therewith, the spin operator equations become

d

dt〈Sx(t)〉 =− Γ〈Sx(t)〉 −

(Bz + λ〈Jz(t)〉

)〈Sy(t)〉,

d

dt〈Sy(t)〉 =− Γ〈Sy(t)〉+

(Bz + λ〈Jz(t)〉

)〈Sx(t)〉 − λ〈Jx(t)〉〈Sz(t)〉,

d

dt〈Sz(t)〉 =− Γ〈Sz(t)〉+ λ〈Jx(t)〉〈Sy(t)〉+

1

2(ΓL↑ − ΓL↓) . (4.15)

This result coincides with [46]: the whole system has now been reduced to six coupledequations and the current simplifies to

Iασ(t) = eΓασ δαL − 〈nσ(t)〉 , (4.16)

with the occupation

d

dt〈nσ(t)〉 =− Γ〈nσ(t)〉 ± λ〈Jx(t)〉〈Sy(t)〉+ ΓLσ. (4.17)

As mentioned before, the total occupation number of the dot 〈N (t)〉 decouples fromthe remaining equations and becomes constant in the long-time limit

〈N (t → ∞)〉 = limt→∞

〈N(0)〉e−Γt + (ΓL↑ + ΓL↓)

∞∫

0

dt′eΓ(t′−t) =

(ΓL↑ + ΓL↓)

Γ. (4.18)

The equations for the spin operators Eq. (4.15) and the large external spin Eq. (4.7)provide further possibilities for an analytic investigation. The fixed points of thesystem can easily be calculated, see [46]. For a dynamical system with higher di-mensions the classification of fixed points becomes more complicated than in the

86


two-dimensional system. The Jacobian is still the starting point, but the classifi-cation and stability of the fixed points are not simply determined by its trace andits determinant. In Sec. 4.2 we discuss the system for finite bias and present resultsfrom a dynamical analysis. Here, in the infinite bias case, we use the classificationfrom [46] and compare their results to our nonadiabatic approach.

damped

damped

oscillations

oscillations

oscillations

& self-sustained

self-sustained

I

IIIII

Bz/λΓ/λ

j=

101 2

j=

10

j=10

3

2

00

10

20

30

40

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 4.2.: Regions of different dynam-ical behaviour concerning theparameters Γ and Bz.

By varying the magnetic field Bz andthe tunnelling rate Γ three regions ofdifferent dynamical behaviour were ob-tained from the rate equation approach.In Fig. 4.2 these regions are depicted.The solid black lines correspond to pa-rameters used in [46]. There, the lengthof the large spin equals j = |J | = 10and the tunnelling rates match 2ΓL↑ =ΓL↓ = Γ. This choice of tunnellingrates implies ΓR↓ = 0, because they aimfor the case where the current flows ex-clusively through the upper electroniclevel. Following from that, spin-downelectrons get trapped into the lower leveland it is possible to draw conclusions about spin-flips in the dot system.

As a complement, the grey lines in Fig. 4.2 illustrate the change of the first tworegions if the length of the large spin is varied, while the ratio of the tunnelling ratesis fixed. For a smaller spin (j = 10

1

2 ), region II decreases to the benefit of region I,which increases. Equally, the reverse situation can be seen, if the length of the spinis increased (j = 10

3

2 ). Modifying the ratio of the tunnelling rates shifts the verticalline separating region II from region III. In the subsequent analysis for the infinitebias regime we remain in the case were j = 10.

The several regions in Fig. 4.2 involve a variable number of fixed points. Forinstance, in region I two real, and therewith physically reasonable, fixed points exist.For the corresponding parameter range, the large spin component, as well as theelectronic spin operators, carry out damped oscillations and in the long-time limitthe trajectories run into

〈J∗1,2〉 =(0, 0,±j) , 〈S∗

1,2〉 =(0, 0,

ΓL↑ − ΓL↓2Γ

). (4.19)

These points exist in the whole parameter regime and are independent of the mag-netic field. But their classification depends on the chosen parameters. Considerregion III, there the oscillations are not damped down and self-sustained oscillationsrun about the upper fixed points. For the eigenvalues of the Jacobian the lattermeans, that in region III a pair of complex conjugate eigenvalues with negative realparts determines the behaviour of the spin components. This stable spiral attitudechanges in region II, there the fixed points are unstable and thus the dominanteigenvalues have a positive real part.

87


rate eq. nonadiabatic

λtλt

2〈Sx 〉

2〈Sy 〉

2〈Sz 〉

〈Jx〉/j

〈Jy〉/j

〈Jz〉/j

1

00

00

00

0.9

0.8

0.2

−0.2

0.6

−0.6

0.1

0.1

−0.1

−0.1

−0.4

−0.5

40004000 20002000

Figure 4.3.: Comparison of the nonadiabatic approach and the rate equation ap-proach results for regime I. The magnetic field equals Bz/λ = 0.1and the tunnelling rate is chosen as Γ/λ = 9. Initial conditions are

〈Jx(0)〉 = 〈Jy(0)〉 = 5(√5 − 1)/(2

√2), 〈Jz(0)〉 = 5/

√2√

5 +√5 and

〈Sy(0)〉 = 〈Sz(0)〉 = 0, 〈Sx(0)〉 = 0.5.

Four additional fixed points appear in region II, only there their imaginary part iszero and therewith they have physical meaning. Hence, the boundaries of region IIare determined by the existence of a finite imaginary part of these fixed points

〈J∗3−6〉 =

(Γ

BzB1,2,±

√j2 − Γ2

B2z

B21,2 −

B2z

λ2,−Bz

λ

), 〈S∗

3−6〉 =(0,B1,2,−

Bz

λ

),

(4.20)

where 〈(S, J)∗3,4〉/〈(S, J)∗5,6〉 correspond to B1/B2, which yield

B1,2 =±√

Bz

λ

[ΓL↓ − ΓL↑

2Γ− Bz

λ

]. (4.21)

This equation defines the borderline between region II and III in Fig. 4.2. There theratio of the tunnelling rates equals 2ΓL↑ = ΓL↓ ≡ Γ leading to

ΓL↓−ΓL↑

2Γ = 14 and

thus B1,2 ∈ C if Bzλ > 1

4 . In the same manner the border between region I and II isdefined, there concerning the imaginary part of 〈J∗

y,3−6〉.

88


As denoted in Fig. 4.2, the system performs damped and self-sustained oscillationsin region II. But one quite interesting thing about this region is the appearance ofchaotic behaviour for small values of the tunnelling rate Γ.

Therewith, we gave a brief review about the dynamics occurring in the infinite biascase by using the rate equation approach. But what happens if we apply our nona-diabatic method developed in Sec. 4.1.2? There, we include higher order transitionsbetween the dot and the leads, mediated through the lead-transition functions.

In Fig. 4.3 the results for regime I are depicted. The rate equation results describedamped oscillations as expected. For large times the z-component of the large spinbecomes polarised parallel to the magnetic field and one spin-down electron getstrapped in the lower energy level.

In the nonadiabatic case the dynamical behaviour is quite different. For small timesthe spin trajectories follow the damped results from the rate equation approach. Butthe damping decreases strongly after the time step tλ ≈ 1000. The same amount oftime steps later, the amplitude drops down for one oscillation period, followed by areturn to the initial oscillation. This behaviour is similar for all spin components. Ifwe consider the z-component for the large spin, the turning point appears when itapproaches its fixed point value 〈J∗

z,1〉 = j.

〈Jx〉/j 〈Jy〉/j

〈Jz〉/j 〈J∗〉

1

0

0

0

0.8

0.6

0.4

0.4

0.4

0.2

0.2

0.2

−0.2−0.2

−0.4−0.4

Figure 4.4.: 〈Ji〉 for Bz/λ = 0.01.

By varying the parameters we foundno damped oscillations in region I at all.For a certain range for the magnetic fieldand the tunnelling rate the dynamicsappear as in Fig. 4.3. There, the sys-tem seems to own two limit cycles, be-tween whom the spin components moveback and forth. This is clearly visible inFig. 4.4, where the x- and y- spin com-ponents are depicted for a low value ofthe magnetic field.

The reason for this behaviour couldbe related to the appearing of a torusbifurcation [145, 135], where a torus is spawned to which the trajectories in thesystem are attracted or repelled. These bifurcation appears if a stable limit cyclereverses its stability and becomes unstable.

In Fig. 4.4 the z- component oscillates close to its fixed point value 〈J∗z,1〉 = j,

similar to Fig. 4.3, but the turning points appear every tλ ≈ 300. If we vary Γslightly we observe a similar behaviour. But for Γ/λ ≈ 8 the oscillations disappearand we enter region II. Approximately, in the range of Γ/λ = 0.1 the system startsto exhibit chaotic behaviour.

In Fig. 4.5 the results for regime II are depicted. The rate equation approachpredicts three kinds of dynamical behaviour. For Bz/λ = 0.2 and small tunnellingrate Γ/λ = 0.7 the system performs fast damped oscillations. This result coincidesperfectly with the nonadiabatic result, see graphs in the left column of Fig. 4.5. Thespin components run into the fixed point 〈J∗

3 〉, cf. Eq. (4.20). The large spin becomes

89


rate eq. nonad.

I ασ/eΓ

α↑

λt λt λt

λt λt λt

x y z

2〈S

i〉

〈Ji〉/j

1

0

0

0

00

00

0

0

0

−1

0.5

−0.5

0.4

0.8

0.1

0.2 0.04

−0.04

20

20

40

40

60

60

100

100

200200

200200

400

400

600

600

Figure 4.5.: Behaviour of the spin components in regime II. The left column de-picts results for Bz/λ = 0.2 and Γ/λ = 0.7, where fast damping of thetrajectories appear in both approaches. The middle graphs show re-sults for Bz/λ = 0.1 and Γ/λ = 0.16 where the rate equations predictself-sustained oscillations, but for the nonadiabatic results already chaosappear. Just as in the right graphs, were the tunnelling rate is furtherincreased (Γ/λ = 0.015). For clarity, we omitted the rate equation re-sults for the chaotic regions. The lowest row depicts the correspondingcurrent results, for the chaotic region the green line denotes −IR↑, thelight green line IL↑ and the dark blue line IL↓. Initial conditions arechosen as for Fig. 4.3.

almost completely polarised perpendicular to the magnetic field.

In the first instance, by further increasing the tunnelling rate, the rate equationsforecast self-sustained oscillations for the systems trajectories, followed by a chaoticoscillating behaviour. Using the nonadiabatic approach, things change again. There,the chaotic behaviour appears already in the parameter region, where self-sustained

90


nonadiab.nonadiab.rate eq.rate eq.

λt

Γ/λ = 1Γ/λ = 10

x y z

2〈S

i〉

〈Ji〉/j

2〈Sy〉

2〈Sz 〉

〈Jx〉/j

〈Jy 〉/

j

2〈Sy〉

2〈S

z〉

〈Jx〉/j

〈Jy〉/j

0.5

0

0

0

0

0

0

0

−0.5

1

1

1

−1−1

−1

0.2

0.2

−0.2

−0.2

20 40

Figure 4.6.: Results for regime III, where self-sustained oscillations appear. In the leftgraphs (Γ/λ = 10) the oscillations are smoothed sinusoidal. For lowertunnelling rate, the 〈Si〉 trajectories perform nonsinusoidal, but periodicoscillations, depicted in the upper graph on the right side. Parametersare Bz/λ = 1.0, ǫd/λ = 0 and with the initial conditions 〈Jx(0)〉 =〈Jy(0)〉 = 3

√5.5, 〈Jz(0)〉 = −1 and 〈Sx(0)〉 = 〈Sz(0)〉 = 0, 〈Sy(0)〉 =

0.5.

oscillations were predicted by the rate equation approach.For higher values of the external magnetic field the system performs self-sustained

oscillations. The results are depicted in Fig. 4.6. There, the oscillation frequencyof the large spin is close to the Larmor frequency ωL = Bz/λ. In the case withoutcoupling between the two spin systems (λ = 0), the large spin oscillates exactly withthe frequency ωL in the xy−plane. The z-component is fixed due 〈Jz〉 = 0 andpossesses no coupling to the magnetic field.

The left graphs in Fig. 4.6, depict results for Γ/λ = 10. Here, the frequency ofthe electronic spin y-component matches the frequency of the large spin x- and y-component ωSy = ωJx = ωJy ≈ 0.8/λ. But the z-component of the electronic spinis almost twice the frequency of the other components ωSz ≈ 1.6λ. This behavioursurvives for a smaller tunnelling rate, right graphs in Fig. 4.6, but there the frequency

91


is even closer to the Larmor frequency, ωJx,Jy,Sy ≈ 0.96λ and ωSz ≈ 1.91λ.

The appearance of the double frequency for 〈Sz(t)〉 can easily be explained. Con-sider the equation of motion for the z-component of the electronic spin in Eq. (4.15).It does not directly couple to the magnetic field, but it couples to the product of thex-component for the large spin and the y-component of the electronic spin. The lat-ter develop sinusoidal in time with the same frequency ωs and without a phase shift,cf. Fig. 4.6. If we approximate their evolution in time with 〈Jeff

x (t)〉 = aj sinωst

and 〈Seffy (t)〉 = ae sinωst, we can estimate an effective solution for the z-component

of the electronic spin (As = ajae)

〈Seffz (t)〉 =〈Seff

z (0)〉e−Γt +

t∫

0

dt′[As sin

2(ωst′) +

1

2(ΓL↑ − ΓL↓)

]eΓ(t

′−t),

⇒ 〈Seffz (t → ∞)〉 =(ΓL↑ − ΓL↓ + 2As)

2Γ− ΓAs

8

[cos(2ωst) + 2ωs/Γ sin(2ωst)

ω2s − Γ2

4

].

(4.22)

Hence, the oscillation goes with twice the frequency of the other spin components.Note, that for a more general ansatz, e.g. 〈Jeff

x (t)〉 = aj sinωst + bj cosωst, theresult is similar and differs solely in the prefactor.

2I L

(t)/eΓ

λt

Γ

λ= 1 Γ

λ= 10

T = 3.8

T = 3.2

40 45 50 55

0.8

0.6

Figure 4.7.: Nonadiabatic current resultscorresponding to Fig. 4.6.

The electronic current in regime IIIperforms periodic oscillations. The re-sults for the rate equation approach co-incides with the nonadiabatic results,which are depicted in Fig. 4.7 for thesame parameters as in Fig. 4.6. Thedashed-dotted line represents the cur-rent for a small tunnelling rate, wheresmoothed oscillations appear. Forhigher Γ, solid grey line, the oscilla-tion is still periodic but not longer si-nusoidal. The frequency of these oscil-lations ω = 2π/T matches ωSz . Notsurprising, because this spin componentcorresponds to the occupation difference of the dot system 2〈Sz〉 = n↑−n↓. Insertingthe latter into the current equation Eq. (4.16) and using the solution for the totaloccupation number in the long-time limit, cf. Eq. (4.18), we obtain for the currentthrough lead α in the rate equation frame

Iα(t) = e

[δαL − ΓL↑ + ΓL↓

2Γ

(Γα↑ + Γα↓)− 〈Sz(t)〉(Γα↑ − Γα↓)

]. (4.23)

And with Γ = ΓLσ + ΓRσ it is obvious that IL(t) = −IR(t) for all times t andhence current conservation is ensured. The latter is also valid for the nonadiabaticapproach.

92


The investigation of the nonadiabatic parameter space costs a lot of numericaleffort. Depending on the obtained dynamical behaviour the discretisation schemehas to be aligned. In the chaotic case, the numerical stability is the most difficult.Thanks to the chaotic nature, every small deviation grows exponentially in time.Following from that, an enlargement of the integration interval or the modificationof the discretisation step-size implies a change of the trajectories for large times. Wechecked, that the chaotic behaviour stays alive by a modification of the discretisationscheme, but the actual evolution of the trajectories may vary from that in Fig. 4.5.

The general numerical error for all results is influenced by different factors. Thepronounced one is the applied trapezoidal rule, which comes with a remainder

RTR ≤ h2D12

[ωmax − ωmin] maxωmin≤ω<ωmax

∣∣∣∣df2(ω)

dω2

∣∣∣∣ , (4.24)

as the upper bound for the discretisation error [144], where hD equals the step sizehD = [ωmax−ωmin]/ND and f(ω) represents all transition functions. A second factoris the solution of the differential equations, where we get a possible error ǫDQ = 10−10

for every time-step [132]. For our calculation the latter error is not that relevant,because the discretisation error is much larger. The upper bound for the completenumerical error, assuming that round-off errors are small, can be approximate withǫtotal ≤ t(RTR + ǫDQ) and therewith grows linearly in time for a fixed hD.

With this discussion we like to emphasise, that the numerical error tends to growfor large time steps and to decrease for smaller step size. The shape of the integrandis very important as well, which is denoted by the second deviation in Eq. (4.24).The limit for a small step size lays in the computation time. For example, thenonadiabatic results in Fig. 4.3, where we chose a small step size and a large interval,took about four months of computing. The similar behaviour is also obtained fora more coarse discretisation scheme, taking only two or three weeks to calculate,but than we see small deviations for large time-steps. But as in the chaotic case,the nature of the dynamical behaviour stays alive for an appropriate discretisationscheme.

We like to mention, that the analysis of the chaotic behaviour in regime II is notdone in the frame of this work. In [46] the authors analyse the frequency spectrumof the numerical data and they obtain an uniform frequency distribution, which theyinterpret as a signature of chaos. For a general analysis of chaotic behaviour oneshould rather calculate the Lyapunov-exponent. This quantity surely determineschaotic behaviour, it describes if a small disturbance grows exponentially or not[103]. This Lyapunov-exponent is not straightforwardly calculable, especially for thenonadiabatic approach. Furthermore, our focus lays in the comparison of the rateequation and the nonadiabatic approach. From this comparison we can state, thathigher order transitions, included in the nonadiabatic approach, do matter in theregime of low magnetic field.

93


4.2. Abundant dynamics in the finite bias regime

0 500

0.2

0.4

0.6

0 500 0 5000 500 0 5000 500 0 5000 500 0 5000 500

λtλt λtλtλt

Vbias = 2 Vbias = 14 Vbias = 16 Vbias = 130 Vbias = 400

2I/

eΓ

infinite

bias

increasing bias Vbias

Figure 4.8.: Current in region I as a function of Vbias in units of λ. The results wereobtained within the adiabatic approach.

In the last section our investigation was focused on the infinite bias regime. There,the interaction between the large spin and the electronic system leads to interestingnonlinear effects. The rate equation approach developed by López-Monís and co-workers [46] is restricted to this regime of high external bias. Within our nonadiabaticapproach we learned, that differences appear if one includes higher order transitionterms. In this section, we take the next step by studying the finite bias regime. Forthe two-level system coupled to an oscillator, we observed that this regime featureseven richer dynamics as for the infinite bias regime.

Due to the lack of an easy access to further analytic studies for the nonadiabaticapproach, we start with the introduction of the adiabatic approach based on KeldyshGreen’s functions. This enables us to clarify the effects of a finite transport window.

A foretaste of the differences which appear are shown in Fig. 4.8. There the currentfor regime I is depicted, whereby the external bias is increased from left to right.These results were obtained from an adiabatic approach and two important featuresare visible. The key point is the change of the dynamics for an intermediate biasregime, cf. third graph in Fig. 4.8, where oscillations, appearing in the regime oflow and high bias, disappear. The second point is, that we observe no dampedoscillations in region I as in the rate equation approach and no two limit cycle-likeoscillating behaviour as in the nonadiabatic approach. We discuss these deviationsin more detail at the end of this section.

4.2.1. Finite bias: adiabatic approach

Using an adiabatic approach for our system implies the assumption that the largespin’s movement is slower compared to the electrons jumping through the system.This is an additional assumption on top of the mean-field approach, where quantumfluctuations are neglected already. But even for the derivation of the rate equationsone needs an adiabatic approximation for the electrons tunnelling into the system.

94


We performed the latter in the last section to decouple the lead-transition functions,cf. Eq. (4.13). A full adiabatic approach also assumes that the electronic spin changeson a time-scale which is much smaller than the one of the large spin. This approxi-mation for the interaction between the electronic spin and the large spin reduces thenumber of dynamical equations to three, because solely the ones for the large spinremain. The equations of motion for the electronic spin operators are solved withthe help of Green’s functions.

In concrete terms, the expectation values of the spin operators in frequency spaceare obtained from

〈Si(ω)〉 = − i

2tr[G<(ω)σi

], i = x, y, z, (4.25)

containing the Pauli spin matrices

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

).

If we reconsider the effective Hamiltonian Eq. (4.8) for this system, we remember thesimilarity to a parallel two-level system and as in that case, the Green’s functionshave matrix character

G(ω) =

(G↑↑(ω) G↑↓(ω)G↓↑(ω) G↓↓(ω)

).

The off-diagonal functions refer to the coupling to the x-component of the large spin.Without this coupling we end up with two independent levels. The retarded and theadvanced Green’s function are obtained from their equation of motion, see Sec.A.2.3.For the derivation of the electronic spin we need the lesser Green’s function whosecalculation takes usage of the Keldysh equation

G<σσ′ =

∑

σ′′

GRσσ′′(ω) Σ<

σ′′(ω) GAσ′′σ′(ω), (4.26)

with the lesser self energy

Σ<σσ′(ω) = δσ,σ′

∑

α

Σ<ασ = δσ,σ′

∑

α

iΓασ fασ(ω). (4.27)

Again, we choose the tunnelling rates as Γασ = 2π∑

k |Vkασ|2 δ(ω − εkασ). Forpolarised leads the density of states for spin-up and spin-down electrons differ, hence(Γα↑ 6= Γα↓). Finally, the lesser Green’s functions read

G<σσ(ω) =i

∑

α

[(ω − εσ′)2 + (

Γσ′

2 )2]Γασfασ(ω) +

λ2

4 〈Jx〉2Γασ′fασ′(ω)∣∣∣[ω − εσ + i

Γσ

2

] [ω − εσ′ + i

Γσ′

2

]− λ2

4 〈Jx〉2∣∣∣2 ,

G<σσ′(ω) =i

λ

2〈Jx〉

∑

α

[(ω − εσ′ + i

Γσ′

2 )Γασfασ(ω) + (ω − εσ − iΓσ

2 )Γασ′fασ′(ω)]

∣∣∣[ω − εσ + i

Γσ

2

] [ω − εσ′ + i

Γσ′

2

]− λ2

4 〈Jx〉2∣∣∣2 .

(4.28)

95


Assuming Γ↑ = Γ↓ ≡ Γ and the same chemical potential for spin-up and spin-downelectrons in each lead the spin operators yield

〈Sx(t)〉 = λ〈Jx〉∫

dω

4π

∑

α

fα(ω)[(ω − ε↓)Γα↑ + (ω − ε↑)Γα↓]∣∣∣

[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2 ,

〈Sy(t)〉 =λ

2〈Jx〉

∫dω

4π

∑

α

fα(ω)Γ [Γα↓ − Γα↑]∣∣∣

[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2 ,

〈Sz(t)〉 =∫

dω

4π

∑

α

fα(ω)

[(ω − ε↓)2 + (Γ2 )

2 − λ2

4 〈Jx〉2]Γα↑

∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2

−

[(ω − ε↑)2 + (Γ2 )

2 − λ2

4 〈Jx〉2]Γα↓

∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2

. (4.29)

We derive the adiabatic current starting from the nonadiabatic result Eq. (4.11) bykeeping εσ and 〈Ji〉 fixed in time. Hence, the lead-transition function Bα

σσ can becalculated with the help of a Laplace transformation, resulting in Eq. (4.13) for thelong-time limit. We can identify the retarded Green’s function

Bασσ(ω) = i

Γασ

2πfασ(ω)G

Rσσ(ω). (4.30)

The second term in Eq. (4.11) contains the occupation, which corresponds to thelesser Green’s function 〈nσ(t)〉 = −iG<

σσ(t, t), and the rate Γασ = 2Im[ΣAασ

]. Addi-

tionally, we use the relations

2Re[iGR

σσ

]= −2Im

[GR

σσ

]= i(GR

σσ −GAσσ

),

and2Re

[ΣAασG

<σσ

]= iG<

σσ2Im[ΣAασ

]= G<

σσ

(ΣAασ − ΣR

ασ

).

Finally, the adiabatic result reads

Iασ(t) = e

∫dω

2π

iΓασfα(ω)

[GR

σσ(ω)−GAσσ(ω)

]+G<

σσ(ω)[ΣAασ(ω)− ΣR

ασ(ω)]

.

(4.31)

The expression including the explicit Green’s functions yields

Iασ(t) = e

∫dω

2π

Γασfα(ω)

Γ[(ω − εσ′)2 + 1

4(Γ2 + λ2〈Jx〉2)

]

∣∣∣[ω − εσ + iΓ2

] [ω − εσ′ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2

−Γασ

∑

α

fα(ω)

[(ω − εσ′)2 + (Γ2 )

2]Γασ + λ2

4 〈Jx〉2Γασ′

∣∣∣[ω − εσ + iΓ2

] [ω − εσ′ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2

.

(4.32)

96


4.2.2. Dynamical analysis

In Fig. 4.8 we see, that the adiabatic results for the current in the infinite bias regimedo not coincide with the ones obtained from the rate equation or the nonadiabaticapproach. This is still valid in the regime of finite bias. The reason therefore lays inthe lacking of higher order correction terms.

Remember the nanoelectromechanical systems, there we included a first adiabaticcorrection term, leading to an intrinsic friction, which is responsible for all the in-teresting dynamical features. Neglecting this friction, the multistability stays alive,but only centers, and therewith smooth periodic oscillations, appear in phase space.One important thing is, that the effective force term, which determines the numberof fixed points of the dynamical system, does not depend on the friction. Followingfrom that, the fixed points for the adiabatic approach with and without the fric-tion are equal. Also we observed good accordance to the nonadiabatic results in thelong-time limit, if the trajectories run into one of the fixed points.

Therewith we can estimate, that the fixed points of our adiabatic approach alsoappear in the nonadiabatic regime. For the nanoelectromechanical system, a com-plete stability analysis of these points is not possible without the adiabatic correctionterm, cf. Sec. 3.2.3. But the classification as saddle points or stable centers is possi-ble, because the determinant of the Jacobian does not vanish. This is similar for thesingle-level system coupled to a large external spin. Here we can use the adiabaticapproach to search for fixed points and also perform a rough characterisation of them.We see, that the predictions for the classification of the fixed points coincide muchbetter with the actually obtained nonadiabatic results as for the nanoelectromechan-ical system.

From now on, we neglect the language of a two-dimensional dynamical system,speaking of determinant and trace, instead we talk about the eigenvalues of theJacobian. Due to the low dimension of the equation of motion for the large spinEq. (4.7), a further analysis of the dynamical system is straightforward. The Jacobianof the dynamical system Eq. (4.7) yield

−λ∂Sz

∂JxJy −(λSz +Bz) −λ∂Sz

∂JzJy

(λSz +Bz) + λ∂Sz

∂JxJx − λ∂Sx

∂JxJz 0 −λSx + λ∂Sz

∂JzJx − λ∂Sx

∂JzJz

λ∂Sx

∂JxJy λSx λ∂Sx

∂JzJy

,

where we neglect the brackets for the expectation values 〈.〉 and the denotation asfixed points (.∗), for better overview. If we evaluate the Jacobian at the obtainedfixed points, we can make predictions for the spin dynamics.

Fixed point analysis of P±0 : center or saddle point

The first fixed points P±0 , where the large spin is completely polarised parallel to

the magnetic field, cf. Eq. (4.19), appear also for the adiabatic system. But thecorresponding values of the z-components for the electronic spin do now depend on

97


ℑ [E±] ℜ [E±]〈S∗z (±j)〉

Vbias/2λ Vbias/2λ

Vbias/2λ Vbias/2λ

j = 10

j = 10

j = −10

j = −10reg

imeI

regim

eII

0

0

0

0

0

0

0

0 4040 8080

1010 2020 -10-10 -20-20

0.2

0.2

0.20.2

-0.2

-0.2

-0.2-0.2

0.4

-0.4

Figure 4.9.: Results of the dynamical analysis for region I and II. Here, parametersare chosen as in Sec. 4.1.3. The solid black line denotes the z-componentof the electronic spin 〈Sz〉 and the external bias is chosen in a symmetricmanner µL = −µR = Vbias/2. Zero imaginary part appears in region I foralignment of 〈Jz〉 in the direction of Bz in the range of Vbias/2λ ∈ [−5; 60]and for alignment in the opposite direction the range yield Vbias/2λ ∈[10; 60]. For region II the imaginary part disappears in the range ofVbias/2λ ∈ [−5;∞] for 〈J∗

z 〉 = j and Vbias/2λ ∈ [6.4;∞] for 〈J∗z 〉 = −j.

several system parameters; for zero temperature they yield

〈S∗z (±j)〉 =

∑

α

Γα↑2πΓ

arctan

[µα − ε↑(±j)

Γ/2

]− Γα↓

2πΓarctan

[µα − ε↓(±j)

Γ/2

],

(4.33)

with ε↑,↓(j) = εd ± 0.5(Bz + λj). The x- and y-components are zero as for therate equation approach. For infinite bias the z-component solely depends on thetunnelling rates 〈S∗IB

z (±j)〉 = (ΓL↑ − ΓL↓)/2Γ, which coincides with Eq. (4.19). For

98


centercentercenter

saddle saddle saddleVbias = ∞

Bz/λ

Γ/λ

Vbias = 2 Vbias = 20 Vbias = 140

202020

404040

606060

0.10.10.1 0.20.20.2 0.30.30.3 0.40.40.4

Figure 4.10.: Classification of the fixed point P+0 . The external bias is increased from

left to right, the explicit values are denoted in the graphs (in units of[λ]). Additionally, the infinite bias result is plotted in the last graphon the right side.

the fixed points P±0 , one eigenvalue of the Jacobian is zero and the two other read

E±(±j) = ±i

√√√√√√[λ〈S∗

z (±j)〉+Bz

][λ〈S∗

z (±j)〉 +Bz

]∓ λj

∂〈Sx〉∂〈Jx〉

∣∣∣∣∣ 〈J∗x 〉=0,

〈J∗z 〉=±j

.

(4.34)

So two possible realisations can appear. If the root is real the eigenvalues are complexconjugate and the fixed points are therewith stable and can be classified as stablecenters. This means, that periodic oscillations for the spin dynamics are expected.The real part is equal to zero and therefore no damped oscillations should appear.This coincides with the adiabatic current result depicted in Fig. 4.8. There, weobserved smooth periodic oscillations followed by a regime where these oscillationsdisappear, which corresponds to the case, when the root in Eq. (4.34) is complex andthe eigenvalues are real with opposite sign and the trajectories end up in differentfixed points.

In Fig. 4.9 〈S∗z 〉 as a function of the applied bias is depicted, together with the

real (grey dashed-dotted lines) and the imaginary (cyan solid line) part of the corre-sponding eigenvalue. The two upper graphs show the behaviour in regime I for 〈J∗

z 〉polarised in the direction of the magnetic field (left, j = 10) as well as against it(right, j = −10). We observe regions where the imaginary part of the eigenvalues isequal to zero and therewith the oscillations disappear.

The ranges of finite real part are slightly different for a P+0 and P−

0 . These differ-ences appear for small bias, the intervals of finite real part are denoted in the captionof Fig. 4.9. For 〈J∗

z 〉 = −10 the intervals with finite real part are smaller than for〈J∗

z 〉 = 10. Depending on the existence of other fixed points, the regions where onlyone fixed point has eigenvalues with finite imaginary parts, are promising for theoccurrence of spin-flips.

Based on the chosen polarisation of the leads and therewith the different tunnellingrates for the spin-down electrons we obtain a system which is not symmetric. Hence,

99


the dynamical behaviour for negative detuning is different than for positive detuningVbias > 0. This is clearly visible in Fig. 4.9, where for large negative bias the imagi-nary part in region I is always unequal to zero. The infinite bias result for region Iequals 〈S∗

z 〉IB = ±0.25 as expected, whereby the sign depends on the detuning. Forregion I the system performs periodic oscillations for a bias Vbias/2λ > 60.

In contrast, in region II (2nd row in Fig. 4.9) no finite imaginary part for the highbias regime exists. This coincides with the infinite bias result for the nonadiabaticapproach, where the concerned fixed points do not appear as stable centers or spirals.But here, for a small bias range and also for negative detuning, we find, that theeigenvalues can become complex and therewith the fixed points 〈J∗

1,2〉 can exist as

centers. For positive detuning the fixed point corresponding to 〈J∗z 〉 = 10 has already

turned into a saddle point, but the 〈J∗z 〉 = −10 state is alive until Vbias/2λ = 6.4.

For region III, which means for Bz/λ > 0.25, the behaviour of the fixed points issimilar to region I, if the magnetic field and the tunnelling rate are small. But forhigher values of these parameters the real part of the eigenvalues is always zero. Thechange between the stable center situation and the saddle points can be monitored byvarying the magnetic field and the tunnelling rate. The density plots in Fig. 4.10 showthe regions where we obtain periodic oscillations, corresponding to stable centers andfinite imaginary part of the eigenvalues. The dark grey area denotes the saddle pointsolution.

Additionally, the right graph in Fig. 4.10 shows the saddle point area for the infinitebias case. The latter can be calculated analytically with the result for the deviationincluded in Eq. (4.34), which for the infinite bias case yields (εd = 0)


∣∣∣∣∣ 〈Jx〉=0,

〈Jz〉=±j

= λ(ΓL↑ − ΓL↓)

2Γ

(Bz ± λj)

[(Bz ± λj)2 + Γ2]. (4.35)

The border between the saddle point and stable spiral area is than defined via

Γ =

√[Bz ± λj]

∓λj

(ΓL↓ − ΓL↑)

[2ΓBz − (ΓL↓ − ΓL↑)]− [Bz ± λj]

. (4.36)

This result does not exactly coincide with the one directly obtained from setting they-component of the large spin in Eq. (4.20) to zero as done in [46], but the evolutionis quite similar. The existence of a real solution for the border is mainly determinedby the denominator of the first term. For the parameters used in Fig. 4.10 we obtainΓ ∼ [1/4 −Bz]

−1/2, which coincides with the denominator in [46].

Two conditions for the second fixed points P±SN

As we had seen, the investigation concerning the first two fixed points is straightfor-ward and we obtained nice analytic expressions for them and their eigenvalues. Thischanges, if we consider other fixed points of the system. They are obtained from

100


〈J∗〉〈S∗〉 E

〈Sx〉 = 0

(J∗x ,±

√j2 − J∗2

x − J∗2z , J∗

z ) (0, S∗y ,−Bz

λ ) ESN2 , ESN

3 P±SN

(0,±j, 0) (0, 0,−Bz

λ ) ±ES0 P±S0

(0,±√

j2 − J∗2z , J∗

z ) (0, 0,−Bz

λ ) ±ES1 P±S1

(J∗x ,±

√j2 − J∗2

x , 0) (0, S∗y ,−Bz

λ ) ES22 , ES2

3 P±S2

(J∗x , 0,±

√j2 − J∗2

x ) (0, S∗y ,−Bz

λ ) ES32,3 = 0 P±

S3

〈Sx〉 6= 0 (±j, 0, 0) (S∗x, S

∗y ,−Bz

λ ) EX2,3 = 0 P±

SX

Table 4.1.: Fixed points of the spin components under the condition 〈Sz〉+Bz/λ = 0.So far, these points can be complex or include complex ranges, where theyhave no physical meaning. Note, that the fixed points P±

S0−S3 are special

cases of P±SN . One eigenvalue E1 = 0 is always zero, therefore we denote

solely the two other eigenvalues E± in the table.

Eq. (4.7), if we set all derivations to zero

0 = −(λ〈Sz〉+Bz

)〈Jy〉,

0 =(λ〈Sz〉+Bz

)〈Jx〉 − λ〈Sx〉〈Jz〉,

0 = λ〈Sx〉〈Jy〉. (4.37)

This is a strong nonlinear system, because the electronic spin operators depend onthe large spin components 〈Si[〈Jx〉, 〈Jz〉]〉, cf. Eq. (4.29). These functions includefrequency integrals, leading to expressions with trigonometrical terms. From thatequations an analytic expression for the fixed points or the eigenvalues is not possible,instead we obtain transcendental equations which require numerical calculations.Even these numerical calculations can become complicated, if we have more than onetranscendental equation or complex arguments for the trigonometrical functions.

The third equation in Eq. (4.37), gives a good hint where we have to start in oursearch for fixed points. It includes the requirement that 〈Sx〉 = 0 and/or 〈Jy〉 = 0,keeping this in mind, we evaluated the fixed points of Tab. 4.1, where the condition〈Sz〉 + Bz/λ = 0 has to be complied with as well. We want to highlight, thatthese points are mathematical possible solutions of the equation system Eq. (4.37),but without calculating the transcendental equations for the spin operators and thederivations of them, which are needed to derive the eigenvalues, an exact forecast oftheir existence is impossible.

In the case 〈Sx〉 = 0, the general fixed points equal P±SN , are not straightforward to

handle in a further analysis, because for their existence two transcendental equationshave to be fulfilled. But we denote some special cases of P±

SN in Tab. 4.1. In theinfinite bias regime, P±

SN coincide with the four fixed points in region II Eq. (4.20),obtained from the rate equations.

101


ℑ [E±] ℜ [E±]〈J∗z 〉〈J∗

y 〉

Vbias/2λVbias/2λ

regim

eI reg

imeII0

0

0

0

1010

10

-5

-5

-5

-5

5

5

5

5

Figure 4.11.: Results of the dynamical analysis of P+S1 for region I and II. Outside

of the depicted range, the fixed point is complex. The real range islimited by 〈J∗

y 〉, which is only real inside of the dashed line. Parameteras in Sec. 4.1.3.

For all fixed points one eigenvalue is zero, due to the fact, that we actually dealwith a two-dimensional system. The eigenvalues from P±

SN yield

ESN2,3 =

λ

2〈Jy〉

(∂〈Sx〉∂〈Jz〉

− ∂〈Sz〉∂〈Jx〉

)±

√√√√(∂〈Sx〉∂〈Jz〉

+∂〈Sz〉∂〈Jx〉

)2

− 4∂〈Sx〉∂〈Jx〉

∂〈Sz〉∂〈Jz〉

∣∣∣∣∣∣∣P±SN

.

(4.38)

The eigenvalues denoted in Tab. 4.1 for the 〈Sx〉 = 0 case, can be calculated withthis equation, evaluated at the respective fixed points. We can also make a forecastfor the classification of these fixed points. If the first term is equal to zero the twoeigenvalues only differ in their sign and we can classify them as centers if they arecomplex or as saddle points if they are real. In the case of two different eigenvalues,we can expect stable/unstable spirals or nodes. If 〈J∗

y 〉 = 0 all eigenvalues are zero,

as for the point P±S3 and therewith it is not an isolated fixed point [103], instead it

corresponds to a plane of fixed points. This is similar for P±SX , where the electronic

spins x-component is unequal to zero. Only one possible solution P±SX for the large

spin exists there, corresponding to its parallel alignment to the x-axis.

As an example, we want to discuss one special case of P±SN . Because of its relative

simplicity, we choose P±S1, where 〈Jx〉 = 0 and the eigenvalues read

ES1± = ±iλ〈J∗

y 〉√



∣∣∣∣∣〈J∗

x〉=0,〈J∗z 〉, (4.39)

102


whereby the component 〈J∗z 〉 is obtained from the transcendental equation

−Bz

λ=∑

α

Γα↑2πΓ

arctan

[(µα − ε∗↑)

Γ/2

]− Γα↓

2πΓarctan

[(µα − ε∗↓)

Γ/2

], (4.40)

with ε∗↑,↓ = εd±0.5(Bz+λ〈J∗z 〉). This equation has no solution for 〈J∗

z 〉 in the infinite

bias case and therefore the fixed points P±S1 do not exist there. In the regime of a

low magnetic field, the right side of Eq. (4.40) has to be small. Following from that,the argument of the arctan function has to be small as well, leading to the estimatefor the evolution of the z-component as linear to the applied bias 〈J∗

z 〉 ∼ Vbias.

This linear behaviour is clearly visible in Fig. 4.11, where we plotted the behaviourof P+

S1 and its eigenvalues as a function of the applied bias. The arcs corresponds to

the 〈J∗y 〉 components, which determines whether the fixed point exists or not. Within

this arc the point is real and therewith physically reasonable. The radius of the arcis limited by the length of the large spin.

For both regions we observe small ranges with finite imaginary part, where thefixed point can be classified as a stable center. As well as ranges, where a saddlepoint occurs. In regime I, the point P+

S1 starts its existence for Vbias/2λ ≈ −5. Atthe bias value, P+

0 turns into a saddle point, cf. Fig. 4.9.

As mentioned above, the investigation of the fixed points P±SN is solely numerically

possible. In Fig. 4.12 we present results obtained for region I. The first column depicts〈Sx〉 as a function of the the large spins components for different bias values. Followedby the results for 〈Sz〉+Bz/λ in the middle column. The cycle shapes originate fromthe conservation of the large spin, hence the radius equals j. The yellow (blue) areain the first two columns corresponds to positive (negative) values. Along the bordersof these areas the functions are equal to zero. Outside of the cycle 〈Jy〉 becomescomplex and no real solution for P±

SN exists.

The condition 〈Sx〉 = 0 and 〈Sz〉 + Bz/λ = 0 must be fulfilled for the appear-ance of P±

SN . Because of this requirement, the last column in Fig. 4.12 shows thesuperposition of the first two columns. The points where both functions are zeroare denoted with red circles in the graphs. For a small bias, Vbias/2λ < 5, only onesolution exists, corresponding to the fixed points P±

S1 which we investigated above.

There, 〈Jx〉 is equal to zero and the fixed point is a stable center. The stability holdsuntil Vbias/2λ < 7, cf. Fig. 4.11.

By further increasing the bias another bifurcation appears, where P±S1 becomes

unstable and two other fixed points are created. These points are symmetric to the〈Jx〉 - axis and move with higher bias values further to 〈Jz〉 = ±j. When they reachthe border of the cycle, they disappear and 〈Sz〉+Bz/λ = 0 has no longer a realsolution, see middle graph in last row of Fig. 4.12.

This behaviour matches quite good those of P±0 . We can interpret P±

SN as thecomplementary points in the region, where P±

0 disappears. As we discussed before,for a certain bias region the stable solution P±

0 turns into a saddle point and we

103


Vbias=

16

Vbias=

10

Vbias=

20

Vbias=

30

Vbias=

140

> 0

> 0

> 0

> 0

> 0> 0

> 0

> 0

> 0

> 0

< 0

< 0

< 0

< 0

< 0< 0

< 0

< 0

< 0

< 0

C

C

C

C

CC

C

C

C

C

〈Jz〉

〈Jx〉

〈Sx〉〈Sz〉+ Bz

λP±SN

10

10

10

10

10

10

10

10

10

1010

1010

1010

1010

10

10

10

10

10

10

10

10

10

10

10

10

10

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

−10

−10

−10

−10−10

−10

−10

−10

−10 −10 −10

−10

−10

−10

−10

Figure 4.12.: Results for the search of P±SN in region I: numerically calculated den-

sity plots for 〈Sz〉 + Bz/λ and 〈Sx〉 as a function of the large spincomponents. The right column depicts the superposition of the firsttwo columns, there a contact of the blue and yellow area denotes a so-lution for P±

SN (red circles). The numerical results were sorted by theirsign, with the purpose of highlighting the positive and the negativeareas. The bifurcation appears approximately at Vbias/λ ≈ 14.

104


Vbias=

4Vbias=

10

Vbias=

20

Vbias=

−10

Vbias=

−4

> 0

> 0

> 0

> 0

> 0> 0

> 0

> 0

> 0

> 0

< 0

< 0

< 0

< 0

< 0< 0

< 0

< 0

< 0

< 0

C

C

C

C

CC

C

C

C

C

〈Jz〉

〈Jx〉

〈Sx〉〈Sz〉+ Bz

λP±SN

10

10

10

10

10

10

10

10

10

1010

1010

1010

1010

10

10

10

10

10

10

10

10

10

10

10

10

10

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

−10

−10

−10

−10−10

−10

−10

−10

−10 −10 −10

−10

−10

−10

−10

1

1

1

1

1

1

−1

−1

−1

3.6

4

−1

−1

−4

−8

Figure 4.13.: Numerical results for the transcendental equations in region II. Thefourth column depicts details of the superposition plots (third column).The contact points of the blue and yellow area represent a solution forPSN± , denoted with a red circle.The numerical results were sorted by

their sign, with the purpose of highlighting the positive and the negativeareas. The bifurcation appears shortly before Vbias/λ ≈ 4.

can estimate, that then the fixed points appearing in Fig. 4.12, correspond to stablesolutions of the dynamical system.

In Fig. 4.13 the results for regime II are depicted. Here, also the case of negativedetuning is considered. The upper row illustrates, that 〈Sz〉 + Bz/λ = 0 has nosolution for Vbias/2λ < −5. For higher values of the bias we obtain one solutioncorresponding to P±

S1. By further increasing the bias a bifurcation appears and twoadditional fixed points appear come up. There P±

S1 becomes unstable and subse-quently disappears if the bias is further increased. In contrast to region I, where thetwo fixed points, which are born after the bifurcation, disappear above a certain biasvalue, the two fixed points in region II stay alive in the high bias regime.

105


Are there more fixed points if 〈Sz〉+Bz 6= 0 ?

There is only one range of possible fixed points left, namely those where 〈Sz〉+Bz 6= 0.In this case, the second equation in Eq. (4.37) has to be fulfilled and the y-componentof the large spin must vanish, leading to the fixed point

P±0N : J(J∗

x , 0,±√

j2 − J∗2x ), S(S∗

x, S∗y , S

∗z ), (4.41)

where all components are non-zero, except 〈Jy〉. The corresponding eigenvalues yield

E0N± = ±i

λ〈Sx〉

[λ〈Sx〉 − λ〈Jx〉


+ λ〈Jz〉∂〈Sx〉∂〈Jz〉

]

+(λ〈Sz〉+Bz

)[(λ〈Sz〉+Bz

)+ λ〈Jx〉

∂〈Sz〉∂〈Jx〉

− λ〈Jx〉∂〈Sx〉∂〈Jx〉

] 1

2

∣∣∣∣∣∣P±0N

.

(4.42)

〈Jx〉

Vbias/2λ

T < 0

T > 054

60

66

1050-10 -5

Figure 4.14.: Condition for P±0N

Here, the case 〈Jx〉 = 0 correspondsto the first fixed points P±

0 discussed atthe beginning of this section. Similar toP±SN , these points depend on a transcen-

dental equation and have to be evalu-ated numerically. In concrete terms, wehave to calculate

T =(λ〈Sz〉+Bz

)− λ

〈Sx〉〈Jx〉

〈Jz〉,

(4.43)

for 〈Jx〉 6= 0. We used the relation

〈Jz〉 =√

j2 − 〈Jx〉2 and searched for re-gions were T equals zero as a functionof the applied bias and 〈Jx〉. In Fig. 4.14we plotted the results for region I. Thefunction T changes its sign for Vbias/2λ ≈ 60 over the whole range of 〈Jx〉. There-with, it does not really depend on 〈Jx〉. The bias voltage value coincides with thepoint, were P±

0 become a finite imaginary part, cf. Fig. 4.9, and turn from saddlepoints into stable centers.

Following from that analysis, we can state that P±0N appear almost independently

of the large spin and exists for the bias value Vbias/2λ ≈ 60. The eigenvalues fromP±0N are complete imaginary and therewith we can classify these points as stable

centers. Note, in region II these points do not exist.

106


P−0 P±

S1

P±0

Bz/λ

Γ/λ

A B

〈Jx〉

〈Jx〉

〈Jy〉

〈Jy〉

〈Jz〉

〈Jz〉

regio

nA

regio

nB

10

20

30

40

50

60

0.1 0.2 0.3 0.41

1

1

11

1

0.50.5

0.5

0.50.5

0.5

00

0

00

0

−0.5−0.5

−0.5

−0.5−0.5

−0.5

−1

−1

−1

−1

Figure 4.15.: Fixed points for unpolarised leads as a function of tunneling rate Γ andmagnetic field Bz. The symmetric bias is chosen as Vbias/2λ = 5. Theright graphs depict the adiabatic results for the large spin, in regionB the small cycles on the top/bottom correspond to P±

0 . In region A,instead of P+

0 the fixed point P+S1 appears (larger cycle). Magnetic field

for region A/B is Bz/λ = 0.1/0.3.

Fixed points for unpolarised leads

For completeness we briefly discuss our system also for unpolarised leads. There thetunneling rates become spin-independent Γ = ΓL + ΓR and only the fixed pointsP±0 and P±

S1 exist, as depicted in Fig. 4.15. In dependence of the tunneling rate andthe magnetic field, two main regions appear. The first one, region A, has the stablefixed points P−

0 and P±S1, these points can be characterized as centers. In this region

spin-flips of the large spin can be possible, because P+0 is no stable solution. But

the system is still multistable and therewith the trajectories can also end up in thefixed points P±

S1. Note, that for an total adiabatic ansatz, a complete spin-flip is notobservable, because the trajectories starting parallel to the magnetic field end up inP±S1, as shown in the right graphs of Fig. 4.15. If one reverses the direction of the

magnetic field, region A contains P+0 instead of P−

0 .

In region B solely P±0 are stable fixed points. By further increasing the bias, the

region where P±S1 exist gets smaller and in the infinite bias case only the second

region persists and all electronic spin components 〈Si〉 become zero. Than the twosystems decouple and the large spin oscillates with the Larmor frequency.

107


Vbias = ∞ P±SN

P−0 P±

S1

P+0 P±

S1

P±0

P−0 P±

SN

Bz/λ

Γ/λ

Vbias = 2 Vbias = 20 Vbias = 140

101010

202020

303030

404040

505050

606060

0.10.10.1 0.20.20.2 0.30.30.3 0.40.40.4

Figure 4.16.: Stable fixed points as a function of the tunneling rate and the magneticfield. The external bias is increased from left to right, the explicit valuesare denoted in the graphs (in units of [λ]). The infinite bias result isplotted in the last graph on the right side. The blue area denotes theregime where no oscillations appear and only the nodes P±

SN exist.

Expectations for the nonadiabatic regime

We received lot of information about the dynamical system, which hopefully help usto interpret the results we obtain with the nonadiabatic approach. So far, we alreadycan make some predictions, about possible features arising. For instance, differencesin the stability of P+

0 and P−0 , which lead to regions where the appearance of spin-

flips is possible. Additionally, we expect different dynamics for negative detuning.Due to the observation, that the fixed point P±

0 changes its stability, we expectoscillating and nonoscillating regions in dependence of the choice of the externalbias. Therewith the dynamical regions I-III, earlier defined for the rate equationapproach, change their shape in dependence of the applied bias.

In Fig. 4.10 it is clearly visible that the external bias is an important parameter forthis system. The graphs show the stable solutions of the system as a function of themagnetic field and the tunneling rate. Each coloured area corresponds to differentstable fixed points as denoted in the right legend. Here, all fixed points can beclassified as centers except of P±

SN , which are stable nodes. The unstable solutionsare not shown, e.g. the regime where P±

S1 becomes unstable and the fixed pointsP±SN are created. Following from that no area including P±

SN and P±S1 appears.

The left graph in Fig. 4.10 depicts the results for Vbias/λ = 2. There two main areasshow up, corresponding to fixed points which can be classified as centers, namely P±

0

and P±S1. From that we can estimate, that most of all oscillating solutions occur in

this regime of low bias. By increasing the bias the regime were no oscillations appeargrows (dark blue area: nodes P±

SN ). For high bias we recover the regions from therate equations. Not surprising, because the fixed points of both approaches coincidein this limit. However, the eigenvalues are still different for both approaches.

Finally, the regions where only one of the main fixed points P±0 exists are promising

for switching of the large spin as discussed above, but due to the fact that we dealwith a multistable system, there are always additional stationary solutions present.

108


4.2.3. Results for the nonadiabatic approach in the finite bias regime

What have we learnt from the foregoing analysis of the dynamical system? Whichkind of behaviour can we estimate for our nonadiabatic results?

We expect to obtain similar fixed points for the nonadiabatic approach as forthe adiabatic approach, and we can state, that two main series of points appear.The first series corresponds to P±

SN , where the z-component of the electronic spin

is proportional to the magnetic field 〈S∗z 〉 = −Bz/λ. Concerning only isolated fixed

points, the x-component of the electronic spin is always zero. These two componentshelp us to identify the appearing fixed points in our nonadiabatic results. For all othercomponents, we were not able to derive explicit expressions due to the transcendentalequations. They are only helpful for the detection, if they coincide with one of thespecial cases in Tab. 4.1.

The second series of fixed points P±0N appear for 〈S∗

z 〉 6= −Bz/λ. There, the y-component of the large spin is equal to zero, which could help us to identify thesefixed points. For the special case P±

0 , we want to show, that these fixed points alsoappear in the nonadiabatic regime. There, all lead-transition functions for differentspin Bα

σσ′ vanish and we obtain for the ones with equal spins the simple result

Bασσ(ω,±j) = i

Γασ

2π

fα(ω)

(ω − εσ(±j) + iΓ2 )= i

Γασ

2πGR

σ (ω,±j), (4.44)

containing the spin-dependent single-level Green’s function GRσ (ω,±j) without cou-

pling of the two electronic levels, due to 〈J∗x〉 = 0. Also the equation for the z-

component of the electronic spin is straightforwardly obtained from

〈S∗z (±j)〉 = 1

Γ

∑

α

∫dωRe

[Bα

↑↑(ω,±j) −Bα↓↓(ω,±j)

]

=1

2π

∑

σα

Γασ

Γ[2δσ↑ − 1] arctan

[(µα − εσ(±j))

Γ/2

]. (4.45)

This result coincides with Eq. (4.33) and the fixed points are identical to those ob-tained with the adiabatic approach. In the same manner, we can construct also theother fixed points. In the general stationary case, all lead-transition functions de-couple from the equation system and can be expressed by retarded Green’s function.The adiabatic expressions for electronic spin components can be reconstructed too.

This equivalence is not valid considering the adiabatic eigenvalues, they are notdirectly transferable to our nonadiabatic approach. As discussed in the beginningof the last section, the fixed points are long-time quantities and stationary solutionsof the dynamical system, in contrast the eigenvalues include higher order terms andtheir calculation requires the first derivations of the system’s variables.

But, as we see in the next pages, some predictions derived from the adiabaticeigenvalues also appear in the nonadiabatic approach. Therefore, we can estimatethat the behaviour for the full time-dependent solution is strongly influenced by theadiabatic eigenvalues.

109


Region I: disappearing of the oscillation for intermediate bias

We start our presentation and discussion of the nonadiabatic results with the onesobtained in region I, corresponding to a low value of the external magnetic fieldand a large tunnelling rate. In the case of infinite bias, we obtained an oscillatingbehaviour between two cycles. This is different to the rate equation results, wheredamped oscillations appear. From the adiabatic analysis of the last section, weobtained the prediction, that the fixed points of the system turn from centers intosaddle points for a certain bias range. Note, that these predictions originate fromthe adiabatic eigenvalues of P±

0 .

As depicted in Fig. 4.18, we obtain these features also in our nonadiabatic results.There, the spin components and the current are depicted for three different biasvalues. For Vbias/2λ = 2 we observe nonsinusoidal self-sustained oscillations. Theassignment of this behaviour to one of our fixed points is not obvious. 〈Sz〉 oscillatesaround Bz/λ, which could correspond to P±

S1 with complex conjugate eigenvalues.

If we increase the applied bias for the initial conditions corresponding to the resultsdepicted in Fig. 4.18, the oscillations disappear in the range of Vbias/2λ ≈ 8. Theyrun into one fixed point of the kind P±

SN , which is clearly visible in Fig. 4.18, where

〈Sx〉 = 0 and 〈Sz〉 = −Bz/λ.

By varying the initial conditions, we observe that not all oscillations have disap-peared for a bias in the range of Vbias/2λ ≈ 8. Choosing the initial conditions nearto the fixed point P−

0 , we observe oscillations, which are smooth sinusoidal and runaround P−

0 . For Vbias/2λ ≈ 10, the oscillations disappear and the trajectories runin the same fixed point as depicted in the middle graph of Fig. 4.18. This resultcoincides perfectly with the border predicted in the adiabatic analysis. With this wecan estimate, that the adiabatic eigenvalues are more useful than expected, becausethey defined this border.

IL IR

λt

0.2

0.3

2650 2700 2750

Figure 4.17.: Left and right cur-rent for Vbias/2λ ≈ 2in region I.

The bifurcation point, where a revival of theoscillations appear, is also correctly predicted bythe adiabatic analysis. For Vbias/2λ ≈ 60 weobserve again oscillations around P±

0 . These os-cillations slightly show the two limit cycle be-haviour as in the infinite bias case, but the dif-ferences between the radii of the cycles is notlarge. The latter is visible in the third columnof Fig. 4.18, where the results for Vbias/2λ = 100are depicted.

The lower row in Fig. 4.18, depicts the elec-tronic current Iασ(t), separated into its con-stituent parts. The right tunnelling amplitudeis equal to zero and following from that, as wellthe corresponding current channel. Therefore, the current IR↑ should be equal tothe total current through the system, hence current conservation remains valid. Aswe learnt from the nonadiabatic current of the two-level system, this is ensured for

110


I ασ/eΓ

α↑

L↑

R↑

L↓

R↓

λt

Vbias/2λ = 2 Vbias/2λ = 10 Vbias/2λ = 100

xy

z2〈S

i〉

〈Ji〉/j

λt λt

111

0 0

00

0

000

000

−1−1−1

0.4 0.40.4

0.40.40.4

−0.4 −0.4−0.4

−0.4−0.4−0.4

10050 40002700 2800 2000

Figure 4.18.: Nonadiabatic results obtained in region I for three different bias val-ues, which are chosen in a symmetric manner Vbias = µL/2 = −µR/2.The electronic current, depicted in the lowest row, is separated intoleft/right and spin-up/-down contributions. The magnetic field yieldBz/λ = 0.1 and Γ/λ = 9. Initial conditions as in Sec. 4.1.3 for regimeI.

the time-averaged current values, but can be different in the time-dependent case.There, we observed some accumulation of current in the central region.

This feature is also observed for the present system, we see deviations in the time-evolution of the current through the left and the right lead. Which is clearly visiblein Fig. 4.17, where the total current for each lead is depicted. Whenever the spintrajectories oscillate in time, these deviations appear, but the time-average coincideas expected.

When we consider the current channels for Vbias/2λ = 2 in Fig. 4.18, we notice,that all channels reach their minima if the 〈Jz〉 component is maximal and in thiscase close to j. For this low bias regime, the shift of the energy level, due to thecoupling to the large spin ±1/2(Bzλ+ 〈Jz〉), leads to a positioning of them slightlyoutside of the transport window. Therewith the current is minimal.

The current corresponding to the spin-up electrons flows in the direction of thebias. For the spin-down electrons the situation is more complicated, because theyare not allowed to leave the system through the right lead due to ΓR↓ = 0. The

111


Iασ/eΓα↑

L↑

R↑

L↓

R↓

x y z

2〈S

i〉

〈Ji〉/j

2〈S

i〉

〈Ji〉/j

λtλt

1

1

1

1

0

0

0

0

0

0

0

0

−1

−1−1

0.5

0.5

−0.5

0.02

−0.02

1002020 404040 606060 80

48404840 488048804880 4860

Figure 4.19.: Nonadiabatic results for region II. The upper row corresponds to asmall bias Vbias/2λ = 2, where the spin components oscillate aroundthe coordinates of P−

0 . Increasing the bias, leads to a disappearing ofthe oscillations and the fixed point P+

SN is present, see lower row forVbias/2λ = 5. The magnetic field yields Bz/λ = 0.2 and Γ/λ = 0.7.Note, that the time-intervals of the current and the spin operators differslightly. Initial conditions as in Sec. 4.1.3 for regime I.

electrons can either flip their spin, stay in the lower level or flow back into the leftlead. If the last process appears the electron moves against the bias and the currentbecomes negative. This feature is slightly visible in the lowest graph of the firstcolumn in Fig. 4.18, there the current IL↓ drops below zero for a quite small region.

Again, we can address this effect to the position of the large spin’s z-component.In the case IL↓ < 0, the shifting of the energy level is the other way around, because

〈Jz〉 is negative. Following from that, the spin-down level lays above the spin-uplevel and additionally in the neighbourhood of the left Fermi edge (µL = 2/λ) andelectrons can occupies empty states in the left lead. After 〈Jz〉 passed its minima,the spin-down level moves down and therewith its current channel drops as well.

Region II: Negative Detuning and spin-flip of the large spin

The feature of negative current is better visible in region II, as we see in the currentresults for Vbias/2λ = 2, which are depicted in the upper row of Fig. 4.19. There, thelarge spin’s z-component oscillates close to −j and therewith, both effective levelsare clearly situated outside the transport window and the spin-down level lays againabove the spin-up level. The oscillations of 〈Jz〉 are comparatively small and do

112


Vbias/2λ −10 −5 −4

Vbias/2

λ = −4

λt

〈Jz〉/j

〈Jx〉〈Jy〉

〈Jz〉

1

0

0

−1

100 200 300 4001

1

1

0.50.5

0.5

00

0

−0.5−0.5

−0.5

−1−1

−1

Figure 4.20.: The spin’s z-components for several bias values in regime II. Pa-rameter as for Fig. 4.19 and initial conditions are 〈Jx〉 = 〈Jy〉 =√

50− 9.92/2, 〈Jz〉 = 9.9 and 〈Sy〉 = 〈Sz〉 = 0, 〈Sx〉 = 0.5.

not influence the behaviour of the effective levels as much as for region I. They stayoutside of the transport window for all times.

We try to interpret the evolution in time for the current channels focusing ontransition between the levels. If IL↓ is negative, the right current reaches its minimumand the left current for spin-up electrons is maximal. Due to IR↑ < IL↑, we canassume that spin-flips from the lower (spin-up) to the upper level (spin-down) happenand a depletion of the upper level into the left lead appears, due to the negativity ofIL↓. This is in accordance with the evolution for 〈Sz〉, which decreases in this range.

The other way around, when 〈Sz〉 increases, the left current for spin-down electronsis maximal, as well as IR↑. But IL↑ is close to zero in this regime. Therefore, weinterpret this kind of current cycle in the following way. A spin-down electron entersthe upper level, flips into the lower level, due to the interaction with the large spin,and finally leaves the central system to the right lead. This would explain why thecurrent for spin-up electrons is maximal at the right lead and minimal at the leftlead.

If the bias is increased, the regions of negative spin-down current vanish. As wellas the oscillations of the spin components, as depicted in the lower row of Fig. 4.19.Remember, for the infinite bias regime, we found no periodic oscillations at all in thisregion. In accordance with the foregoing adiabatic analysis, the trajectories enterone fixed point of the P±

SN -series.

The adiabatic analysis, also predicts an earlier disappearing of the fixed point P+0

than the fixed point P−0 . From that we propose the possibility of spin-flips for the

large spin in the region, where only P−0 has finite imaginary part: Vbias/2λ ∈ [−5; 6.4].

To monitor how the first fixed point disappears we choose our initial conditionsclose to P+

0 and assume negative detuning. The results are depicted in Fig. 4.20.

113


I L↑/eΓ

L↑

I L↓/eΓ

L↓

I R↑/eΓ

R↑

Vbias/2λ = 2

λt λt

0.4

0.4

0.4

0.4

−0.4

−0.4

−0.4

−0.4

0

00

0

1500 2000 2500 3000 2470 2475 2480

Figure 4.21.: The graphs depict the different contributions to the current in regimeIII. The right graph presents a detail of the left graphs drawn togetherfor a short time range. Initial conditions and parameters as in Sec. 4.1.3for regime III.

For Vbias/2λ = −10 the trajectories perform smooth oscillations around P+0 . As

expected from Fig. 4.9, the z-component of the electronic spin 〈Sz〉 ≈ 0.2 and istherewith positive. As discussed in the last section, the observed system is notsymmetric and therewith the fixed point P+

0 stays alive by further decreasing thebias.

〈Jx〉/j 〈Jy〉/j

〈Jz〉/j

1

1

1

0

00

−1

−1 −1

0.5

0.50.5

−0.5

−0.5 −0.5

Figure 4.22.: Strange attractor.

Increasing the bias we see a different be-haviour, 〈Jz〉 drops down when the bias passesthe threshold Vbias/2λ = −5 as it is clearly visi-ble in Fig. 4.20. At first it seems that the trajec-tory runs into one of the fixed points P±

S1, lay-ing in the middle of both points P±

0 . But evenfor Vbias/2λ = −4, the spin components enterthe fixed point P−

0 . The spin-flip appears faster,if we increase the bias. For positive detuningVbias/2λ = 2 and closer to the disappearing ofthe oscillations, the trajectories move quite fastto the lower fixed point.

If we decrease the tunnelling rate, we observe a transition to chaotic oscillationsas in the infinite bias case, see Fig. 4.5. For small values of the bias, the chaoticbehaviour is a little suppressed and the trajectories oscillate comparatively smoothlywith a high frequency. For a range of small bias values, we observe first signs of chaos,

114


as depicted in Fig. 4.22. The shape of the large spin’s movement in space remains ofa strange attractor [103]. These strange attractors can occur in dynamical systemsbefore the crossover to chaos appears, for instance in the Rössler system [103].

Region III: oscillations and high frequency

In Sec. 4.1.3 we presented results for larger values of the magnetic field, where thetrajectories oscillate around P±

0 . This kind of behaviour is recovered for the finitebias regime. But we observe slight differences in the regime of small bias. In Fig. 4.21the current for regime III is depicted (Vbias/2λ = 2), the results for the left and theright current are separated into their contributions from spin-up and -down electrons.We observe that the current channel IL↓ oscillates around the zero axis and followingfrom that, the current is flowing in both direction. The frequency of the oscillationsis quite high and the spin components oscillate between two cycles, which signaturesare clearly visible in the left graphs in Fig. 4.21.

4.2.4. Which approach works best?

This question is not straightforward to answer. But we are able to discuss the ad-vantages and disadvantages of the used approaches. The rate equation approachfrom [46], is a quite practical method. The numerical effort is low and the dynamicalsystem can be investigated analytically. But this method neglects higher order tran-sitions and we learnt from the nonadiabatic results, that these are relevant in theregime of low magnetic field. In the latter region, the rate equations miss parts of thenonlinear dynamics. If the magnetic field increases, the results of both approachescoincide.

The limitation of the rate equation approach to high external bias, correspondsto another disadvantage. Because for finite bias, we observe even richer dynamicswithin the nonadiabatic approach. There, features like spin-flips of the large spinand suppression of the oscillations in dependence of the applied bias appear. Butdisadvantages belonging to the nonadiabatic approach, are the high numerical effortand its inaccessibility for further analytic investigations.

As for the nanoelectromechanical model we can use the adiabatic Green’s functionsto describe the system. In this work, we utilised this method to analyse our system.But we omitted the presentation of time-dependent results, because the completeadiabatic approach misses a lot of the dynamics even in the infinite bias regime.Chaotic, damped or two limit cycle-like oscillations are not observable within thisapproach. The dynamical system in the adiabatic approach reduces to three equa-tions of motion, which is not suitable to describe the system in the range of lowmagnetic field. Only if the system runs directly in a stable fixed point, the adiabaticapproach can catch the dynamics.

But the adiabatic Green’s functions are suitable for the analysis of the system.As we learnt in the last two sections for the finite bias regime, our initial adiabaticanalysis matched quite well the behaviour of the nonadiabatic approach.

115


adiabaticad. correction rate eq.

λt

〈Jx〉/j

〈Jy〉/j

〈Jz〉/j

0

0

0

500 1000 1500 2000 2500

0.4

−0.4

0.2

−0.2

1

0.8

Figure 4.23.: Regime I: adiabatic corrections vs. rate equations vs full adiabatic.

For the NEMS system we included adiabatic correction terms and therewith wefound a qualitatively good agreement with the nonadiabatic approach. These correc-tion terms originate from the expansion of the influence functional. The inclusion ofcorrection terms is also possible for the adiabatic Green’s function method. There-fore, we use a method from Bode and co-workers [104] who worked with a model foran anisotropic single-molecule magnet. Instead of anisotropic coupling, they assumean anisotropic potential for the molecular spin, or in our language, the large spin.But their method is straightforward transferable to our model.

The starting point for their calculations, is the formulation of the Dyson equationin terms of a so-called Wigner presentation to introduce their adiabatic approxima-tion. Because within this Wigner representation time-arguments can be separatedinto slow and fast time-scales. Assuming that the molecular spin changes on the slowtime-scale, they obtain for the lesser Green’s functions

G< =G<,0 +i

2

δωG<BGA −G<BδωGA + δωGRBG< −GRBδωG<

, (4.46)

with an effective magnetic field B containing the anisotropic potential. For clarityall arguments are suppressed, here the denoted functions correspond to matrices andare already in the frequency domain.

For our system the effective magnetic field yield B = σ·b with b = λ2

(〈Jx〉, 0, 〈Jz〉

).

The equation of motion for the large spin, including the adiabatic correction term,

116


becomes a Landau-Lifshitz-Gilbert equation

d

dtJ =− λ

[J× S0

]− [J× B]−

[J× A

d

dtJ

]. (4.47)

Here, A denotes a 3× 3 matrix, containing large expressions with Green’s functions.For the details of their derivation and their explicit form see Sec. B.3.3.

This approach should include the same fixed points as the adiabatic or nonadia-batic system, because all correction terms are correlated with the time-derivationsof the large spin. These disappear in the long-time limit. The obtained eigenvalueschange to those of the adiabatic system. The analytical calculation of the latteris possible, but is no fun due to the longish expressions of the correction term, seeSec. B.3.3.

We solved the equation of motion for the large spin Eq. (4.47) for infinite bias andthe results for region I are depicted in Fig. 4.23. Together with the results obtainedfrom the adiabatic and the rate equation approach. The appearing fixed point equalsP+0 . Remember, the nonadiabatic results showed here self-oscillations between two

cycles. The complete adiabatic approach misses the damping of the trajectories, theeigenvalues of P+

0 are complex conjugate and have no real part there.The spin results, including the adiabatic correction, coincide with the rate equation

solution. Both perform damped oscillations and end up in P+0 in the long-time

limit. Following from that, we can estimate that the inclusion of the correctionterms improves the adiabatic approach, but can not catch the whole (semiclassical)dynamics as the nonadiabatic approach.

The numerical solution of Eq. (4.47) is not as easy as it seems and depends stronglyon the parameter regime and the used integration routine. Therefore, we propose adifferent method, so to say an extension of the rate equation approach to the finitebias regime with less numerical effort as in the nonadiabatic case.

We had seen in Sec. 4.1.3, that the rate equations can be recovered from the nona-diabatic approach. In the performed adiabatic approach the lead-transition functionsdecouple from the remaining equation system and we were able to calculate theseequations for the infinite bias case. The way to finite bias is straightforward, insteadof the rates we include the complete transition functions. In other words, we startwith the nonadiabatic approach and perform an adiabatic approximation only forthe lead-transition functions. In the end, we obtain six coupled equations for thespin components containing the lead-transition function.

This would include the short-time dynamics for the electronic spin, which is miss-ing in the general adiabatic approach, and it includes the broadening due to thecoupling to the leads, which is missing in the rate equation approach. The analyticinvestigation is more complicated than in the adiabatic approach, but much simplerthan in the nonadiabatic regime.

In the scope of this work, we do not follow this idea. Finally we can state, thatthe nonadiabatic approach describes the system’s (semiclassical) dynamics in thecompletest manner. It includes the short-time dynamics of the spins and also highertransition terms.

117

5. Conclusion

We presented important improvements for the description of transport spectroscopywith semiclassical system, namely by going beyond the Master equation and beyondthe adiabatic approximation. With the developed nonadiabatic description of thesystem, we were able to probe the commonly used adiabatic approach.

For the NEMS models, we find, by comparing the outcomes of the adiabatic andthe nonadiabatic approach, a qualitative accordance. The same features arise inboth approaches and the stationary results predominantly coincide, but there is noquantitative accordance in the time-dependent regimes. The differences increasetogether with the complexity of the focused NEMS model.

By comparing the adiabatic and nonadiabatic results for the single-level systemwe obtained the best qualitative agreement. In principle, the same features arise,e.g. bistability and a hysteresis-like I − V characteristic. The largest deviations areobserved for small times, but in the long-time limit the results mostly coincide.

For the two-level case the differences are much larger. Qualitatively we observesimilar properties. The electron-oscillator interaction leads to multiple current chan-nels like in the single-level system. Additionally, we observe limit cycles of thedynamical system leading to periodic oscillations of the current. In this regime, thesystem acts as a DC-AC-transformer. But the quantitative predictions of the adi-abatic approach do not match the results for the nonadiabatic system, where theoscillator and the electrons act on the same timescale.

We showed, that the nonadiabatic approach is straightforwardly transferable toother coupled transport systems, which include a semiclassical treatment of the in-teraction arising due to the coupling. For the presented model of a large externalspin coupled to a single-level system we observed interesting nonlinear dynamicalfeatures, including some, which were not obtained within simpler approaches.

The disadvantage of the nonadiabatic approach is the necessary numerical effortand the limitation concerning further analytic studies. But it includes more informa-tion about the respective system, due to the Non-Markovian character of the methodand the inclusion of short-time dynamics.

In future works, it will be desirable to include the fluctuations of the system andto develop a more sophisticate treatment of the noise. In the last chapter, where westudied the spin system, we proposed a method how to improve the rate equationapproach on the basis of the nonadiabatic approach. The latter can also be suitablefor NEMS models, it would allow to include short time-dynamics and the broadeningdue to the coupling to the leads. In this framework, it should be possible to derivecorrection terms for the calculation of the electronic properties.

119

A. Theoretical concepts

A.1. Path Integral

A.1.1. Feynman-Vernon influence functional

The reduced density matrix ρosc(q, q′, t) can be written as a double path integral over

q (forward path) and q′ (backward path) weighted by the Feynman-Vernon influencefunctional

ρosc(q, q′, t) =

∫dq0

′dq′0 ρosc(q0′, q′0, t0

′)

q(t)∫

q0

Dq(τ)

q′(t)∫

q′0

D∗q′(τ) ei(Sq′−Sq′ )F [q; q′](τ),

(A.1)

introducing the influence functional in operator form

F [q; q′](t0, t) = trBU †[q′](t, t0)U [q](t, t0)ρB

, (A.2)

with

U [q′](t, t0) = T e−i

t∫0

dt′ [H0(t′)+V (t)]= T e

−it∫0

dt′Heff

B(t′)

,

containing the time ordering operator T , which arranges operators with later timesto the left. We introduce an effective interaction picture

U(t) = e−iH0t U(t) with iδtU(t) = V (t)U(t) → V (t) = eiH0t V (t) e−iH0t.(A.3)

Expanding the perturbation to second order leads to

U(t) = 1− i

t∫

0

dt′ V (t′)−t∫

0

dt′t′∫

0

V (t′)V (s) + ... . (A.4)

For the Feynman-Vernon influence functional the product of the time evolution op-erators is needed, which yields

U † [q′t′]U [qt′ ] = 1 + i

t∫

0

dt′V ′(t′)− V (t′)+t∫

0

dt′t′∫

0

ds[V ′(t′)− V (t′)V (s)

]

−t∫

0

dt′t′∫

0

ds[V ′(s)V ′(t′)− V (t′)

]+ ... . (A.5)

121


Depending on the chosen interaction picture with respect to HeffB (t) = H0(t) + V (t)

the further calculation is determined. For the adiabatic and the nonadiabatic ap-proach we transform to centre-of-mass and relative coordinates

xt =qt + q′t

2, yt = qt − q′t, (A.6)

to detach the quantum fluctuations yt from the classical path xt. For the adiabaticapproach we perform a Taylor expansion for the classical path xt ≈ x0 + tx0, incontrast the nonadiabatic approach contains the full time-dependence of xt. Thedecompositions of Heff

B (t) for both approaches yield

adiabatic: H0 = He − F x0, V (t) = −F (tx0 ±1

2yt),

nonadiabatic: H0(t) = He − F xt, V (t) = ∓1

2F yt. (A.7)

In the subsequent, we restrict our derivation to the nonadiabatic case. Hence, theinfluence functional reads

Fpert[qt; q

′t

]= 1 + i

t∫

0

dt′1

2yt′ + yt′〈F (t′)〉

−t∫

0

dt′t′∫

0

ds1

2yt′ + yt′

1

2ys

[〈F (t′) F (s)〉+ 〈F (s) F (t′)〉

]

= 1 + i

t∫

0

dt′ yt′〈F (t′)〉 −t∫

0

dt′t∫

0

ds yt′ ys 〈F (t′) F (s)〉. (A.8)

Reexponation and lowest order cluster expansion [98] yields

Fpert[qt; q

′t

]= e−Φ[xt;yt], (A.9)

with

Φ [xt; yt] = −i

t∫

0

dt′f(t′)yt′ +

t∫

0

dt′t∫

0

ds C(t′, s)yt′ys, (A.10)

andC(t′, s) = trB

(F (t′)− f(t′)

)(F (s)− f(s)

)≡ 〈δF (t′)δF (s)〉.

where f(t) = 〈F (t)〉 depends on the center of mass path xt. In the exponent thevariance δF (t′) appears due to the cumulant cluster expansion in lowest order. Therethe disconnected diagrams disappear while reexponation:

〈FtFt′〉Reexponation−→ 〈FtFt′〉Connect = 〈δFtδFt′ 〉.

122

A.1. Path Integral

A.1.2. Classical action and Langevin equation

With the oscillator potential V (qt) =12mω2

0q2t we obtain

S(qt) =

t∫

t0

dt′m2q2t′ − V (qt′)

. (A.11)

For the reduced density matrix we need the differences of the action in dependenceof the center of mass variables

Sq − Sq′ =

t∫

t0

dt′m

2

(xt′ +

1

2yt′)

2 − (xt′ −1

2yt′)

2 − V (xt′ +1

2yt′) + V (xt′ −

1

2yt′)

=

t∫

t0

dt′mxt′ yt′ − V ′(xt′)yt′

. (A.12)

Due to the quadratic potential this expression is exact, for arbitrary potential oneapproximates the difference with V ′(xt) +O(y3t ). With this, ρosc(x, y, t) yields

ρosc(x, y, t) = 〈xt +yt2|ρosc(t)|xt −

yt2〉 =

∫dx0

∫dy0 ρ0(x0, y0) J(xy, t;x0y0),

(A.13)with

J(xy, t;x0y0) =

x∫

x0

Dx

y∫

y0

Dy exp

i

t∫

t0

dt′ mxt′ yt′ − V ′(xt′)yt′

e−φ[xt′ ;yt′ ]

=

x∫

x0

Dx

y∫

y0

Dy exp

i

t∫

t0

dt′mxt′ yt′ −mω2

0xt′yt′

× exp

i

t∫

0

dt′ yt′ 〈F (t′)〉 −t∫

0

dt′t′∫

0

ds yt′ ys C(t′, s)

p.I.=

x∫

x0

Dx

y∫

y0

Dy eim(xtyt−x0y0) e−i

t∫0

dt′ K(t′) yt′e−

t∫0

dt′t′∫0

ds yt′ ys C(t′,s),

with K(t′) = (mxt′ +mω20 xt′ − 〈F (t′)〉). (A.14)

Finally, we obtained the Langevin equation for xt

mxt + V ′(xt)− f [xt′ , t] = ξt, (A.15)

with the electronic force f(t) = 〈F (t)〉. The adiabatic Langevin equation can bederived in a similar manner, see [42]. For the stochastic force ξt exists a correlationfunction 〈ξt′ξs〉 = C(t′, s) = 〈δF (t′)δF (s)〉.

123


A.2. Green’s functions in the adiabatic approach

The Green’s functions in the frequency domain are derived via the equations ofmotion method. There the Green’s function GR,A is defined as resolvent of theHamiltonian H0 via

(ω1−H0)GR,A(ω) = 1. (A.16)

Denoting the bath states with |φλ〉 = |λ〉 the previous equation yields

〈λ|(ω1−H0)GR,A(ω)|λ′〉 = δλ,λ′ . (A.17)

Taking the matrix elements and inserting the Hamiltonian leads to a set of equations,from whom the Green’s functions are derived. Here, the electron-phonon couplingis described adiabatically, thus the interaction part HSB came in by shifting therespective level energies εα → εα. Further details can be found in [92, 99].

A.2.1. Undisturbed Green’s functions of the leads

The time-dependent undisturbed Green’s function of the lead are defined via

g<kα(t, t′) ≡ i〈c†kα(t′)ckα(t)〉 = ifα(εkα) e

−it∫

t′

dt′′εkα(t′′)

,

gR,Akα (t, t′) ≡ ∓iΘ(±t∓ t′)〈

ckα(t

′), c†kα(t)〉 = ∓iΘ(±t∓ t′) e

−it∫

t′

dt′′εkα(t′′)

.(A.18)

Here, the curly brackets denote the anti-commutator. If the lead energies are time-independent, εkα(t

′′) = εkα, the undisturbed lesser Green’s function reads

g<kα(t− t′) = if(εkα) eiεkα(t

′−t). (A.19)

This expression depends only on a time-difference, hence we can perform a Fouriertransformation to obtain the lesser Green’s in the frequency domain

g<kα(ω) =

∞∫

−∞

dt g<kα(t, 0) eiωt = if(εkα)

∞∫

−∞

dt ei(ω−εkα)t = 2πi f(ω)δ(ω − εkα).

(A.20)

The advanced/retarded Green’s function read

gR,Akα (t− t′) = ∓iΘ(±t∓ t′) e−iεkα(t−t′), (A.21)

and in frequency space

gR,Akα (ω) =

1

ω − εkα ∓ iδ. (A.22)

124

A.2. Green’s functions in the adiabatic approach

A.2.2. Green’s functions of the two-level system

The Green’s functions for the two-level system yield

GR,AD (ω) =

(GR,A

LL (ω) GR,ALR (ω)

GR,ARL (ω) GR,A

RR (ω)

),

with the elements

GR,ALL (ω) =

ω − νR − ΣR,AR (ω)[

ω − νL − ΣR,AL (ω)

] [ω − νR − ΣR,A

R (ω)]− |Tc|2

,

GR,ARR (ω) =

ω − νL − ΣR,AL (ω)[


] [ω − νR − ΣR,A

R (ω)]− |Tc|2

,

GR,ALR (ω) =

Tc


GR,ARR (ω),

GR,ARL (ω) =

T ∗c

ω − νR − ΣR,AR (ω)

GR,ALL (ω). (A.23)

Here, the reduced/advanced self energy ΣR,Aα , (α ∈ R,L) is introduced corresponding

to the left or the right dot with

Σα(ω) =∑

k

|Vkα|2 gR,Akα,kα(ω). (A.24)

gR,Akα,kα is the undisturbed Green’s function for the leads. The derivation of the

lesser/greater Green’s function is taking usage of the Keldysh equation:

G≶αβ =

∑

γ

GRαγ(ω) Σ

≶γ (ω) G

Aγβ(ω), (A.25)

where we assume both levels initially unoccupied. The lesser/greater self energyfollows from

Σ≶γ (ω) =

∑

k

|Vkγ |2 g≶kγ,kγ(ω),

⇒ Σ<γ (ω) = iΓγ(ω) fγ(ω), Σ>

γ (ω) = −iΓγ(ω) [1− fγ(ω)] . (A.26)

A.2.3. Green’s functions of the spin system

The Green’s functions are obtained from the equation of motion

ωGR,Aλλ′ (ω)− δλ,λ′ =

(ελδλ∈σ + ελδλ∈kασ

)GR,A

λλ′ (ω)

+λ

2〈Jx〉δ↑,λGR,A

↓λ′ (ω) +λ

2〈Jx〉δ↓,λGR,A

↑λ′ (ω)

+∑

kα

V ∗kαδλ∈σG

R,Akαλλ′(ω) + Vλδλ∈kασG

R,Aσλ′ (ω). (A.27)

125


The Green’s functions read

GR,AD (ω) =

(GR,A

↑↑ (ω) GR,A↑↓ (ω)

GR,A↓↑ (ω) GR,A

↓↓ (ω)

),

with

GR,A↑↑ (ω) =

ω − ε↓ −ΣR,A↓↓ (ω)

[ω − ε↑ − ΣR,A

↑↑ (ω)] [

ω − ε↓ −ΣR,A↓↓ (ω)

]− λ2

4 〈Jx〉2,

GR,A↓↓ (ω) =

ω − ε↑ −ΣR,A↑↑ (ω)

[ω − ε↑ − ΣR,A

↑↑ (ω)] [

ω − ε↓ −ΣR,A↓↓ (ω)

]− λ2

4 〈Jx〉2,

GR,A↑↓ (ω) =

λ2 〈Jx〉

ω − ε↑ − ΣR,A↑↑ (ω)

GR,A↓↓ (ω),

GR,A↓↑ (ω) =

λ2 〈Jx〉

ω − ε↓ − ΣR,A↓↓ (ω)

GR,A↑↑ (ω). (A.28)

The retarded and the advanced Green’s function are calculated with the self energy

ΣR,Aσσ′ = δσ,σ′

∑

α

ΣR,Aασ = δσ,σ′

∑

kα

|Vkασ|2ω − εkασ ± i0

= δσ,σ′

∑

α

1

2[Λασ(ω)∓ iΓασ(ω)] .

(A.29)The lesser self energy is obtained from

Σ<σσ′(ω) = δσ,σ′

∑

α

Σ<ασ = δσ,σ′

∑

α

iΓασ(ω) fασ(ω). (A.30)

The Green’s functions for the spin system are similar to those of the two-level sys-tem which is concerned in this work, cf. Sec.A.2.2. One difference appears in therespective self energies, because for the spin systems the two-levels couple to the leftand the right lead. This corresponds to a parallel two-level system. In the case of theNEMS system, we investigated a two-level system in series. There, the self energiescontain no summation over the leads.

126

A.3. Basics for a fixed point analysis

A.3. Basics for a fixed point analysis

The fixed points of a non-linear two dimensional system can be investigated withstandard methods for linear dynamical system [103]. The general solution for a twodimensional linear system x = A x is

x(t) = c1eλ1tv1 + c2e

λ2tv2, (A.31)

and so determined by the eigenvalues λ1,2 of the matrix A. The constants c1,2 dependon the initial conditions and v1,2 are the corresponding eigenvectors. The eigenvaluescan be obtained from

λ1,2 =1

2(τ ±

√τ2 − 4∆), (A.32)

where τ corresponds to the trace and ∆ to the determinant of A. These two qualitiesdetermine the evolution of the trajectories in the phase plane. For a fixed point x∗

the condition x = 0 must be fulfilled. Various classes of fixed points exist, wherethe determinant ∆ decides which kind of point appears. In case of saddle points thedeterminant is negative and it is positive for spirals or nodes. The difference betweena spiral and a node is that for the second one the eigenvalues have no imaginary part.The trace τ defines the stability of nodes and spirals, this is caused by the fact thatτ determines the sign of the eigenvalue’s real part, for instance with negative realpart decaying oscillations occur and the fixed point is stable, see Eq. (A.31). Therealso exist some borderline cases, whereas the centers the most significant ones, theyoccur when the trace is equal to zero.Classification of fixed points:

• ∆ < 0 : eigenvalues are real → saddle point

• ∆ > 0 : eigenvalues are complex → spirals and centers or∆ > 0 : eigenvalues are real → nodes

Stability of fixed points:

• τ < 0 : λ1,2 have negative real part → stable

• τ > 0 : λ1,2 have positive real part → unstable

This analysis can be assigned to a two dimensional non-linear system x = f(x). Byassuming a small disturbance u = x − x∗ from a fixed point, we can investigate ifthis disturbance grows or decays by performing a Taylor expansion

u1 = f1(x∗1, x

∗2) + u1

∂f1∂x1

∣∣x∗1

+ u2∂f1∂x2

∣∣x∗2

+ h.t.,

u2 = f2(x∗1, x

∗2) + u1

∂f2∂x1

∣∣x∗1

+ u2∂f2∂x2

∣∣x∗2

+ h.t..

(A.33)

The first term is zero and higher terms (h.t.) can be neglected because the disturbanceis small. Hence we obtain a linearised system u = J∗u, containing the Jacobi matrixJ∗ evaluated at the fixed point coordinates.

127

B. Miscellaneous calculations

B.1. Single-level system

B.1.1. Once more: time-dependent single-level occupation

The occupation of the single-level was calculated with the Keldysh Green’s functions(Sec. 2.2.2) and with the equations of motion method (Sec. 2.2.2). The obtainedresults differ in their style, but we showed, that both results coincide. It is alsopossible to derive the expression for the occupation, obtained via Green’s functions,with the equations of motion method. Therefore we start from Eq. (2.52), whichyield

˙d(t) = −i

(ε(t)− i

Γ

2

)d(t)− i

∑

kα

V ∗kαe

−iεkαtckα(0),

˙d†(t) = i

(ε(t) + i

Γ

2

)d†(t) + i

∑

kα

Vkαeiεkαt c†kα(0). (B.1)

These inhomogeneous differential equations can easily be solved, the homogeneoussolution reads

dhom(t) = d(t = 0) e−i

t∫0

dt′ (ε(t′)−iΓ2), (B.2)

with this the inhomogeneous solution equals

dinhom(t) = e−i

t∫0

dt′ (ε(t′)−iΓ2)

t∫

0

dt′

−i

∑

kα

Vkαe−iεkαt

′

ckα(0) eit′∫0

dt′′ (ε(t′′)−iΓ2)

.

(B.3)

The total solution for the electronic level operator is obtained from

d(t) = dhom(t) + dinhom(t)

= d(0) e−i

t∫0

dt′ (ε(t′)−iΓ2)− i∑

kα

Vkα

t∫

0

dt′e−iεkαt′

ckα(0) eit′∫t

dt′′ (ε(t′′)−iΓ2),

(B.4)

129


and a similar expression can be calculated for d†(t):

d†(t) = d†(0) ei

t∫0

dt′ (ε(t′)+iΓ2)+ i∑

kα

V ∗kα

t∫

0

dt′eiεkαt′

c†kα(0) e−i

t′∫t

dt′′ (ε(t′′)+iΓ2).

(B.5)

With the assumption of an initial empty level N(0) = 〈d†(0)d(0)〉 = 0, the occupationfor the level can be calculated:

N(t) =∑

kα

|Vkα|2 〈c†kα(0)ckα(0)〉

t∫

0

dt′t∫

0

ds eiεkα(t′−s) e

−it′∫t

dt′′ (ε(t′′)+iΓ2)ei

s∫t

dt′′ (ε(t′′)−iΓ2)

=∑

kα

|Vkα|2∫

dω δ(ω − εkα) fα(ω)

t∫

0

dt′t∫

0

ds eiω(t′−s) e

−it′∫t

dt′′ (ε(t′′)+iΓ2)ei

s∫t

dt′′ (ε(t′′)−iΓ2)

=∑

α

Γα

∫dω

2πfα(ω)

t∫

0

dt′t∫

0

ds ei

t∫

t′

dt′′ (ε(t′′)−ω+iΓ2)e−i

t∫s

dt′′ (ε(t′′)−ω−iΓ2).

(B.6)

This coincides with the time-dependent Green’s function result (ε(t′′) = εd − λxt′′)

N(t) =∑

α

Γα

∫dω

2πfα(ω) |A(ω, t)|2, A(ω, t) = −i

t∫

0

dt′ e−i

t∫s

dt′′ (ε(t′′)−ω−iΓ2).

(B.7)

B.1.2. Check: Analytic result for the occupation

We start from the equation system Eq. (2.55) and neglect the time-dependence ofthe position ε(t) ≡ ε = εd − λx. This leads to

N(t) =∑

α

∫dω

t∫

0

dt′ 2 Re[Bα(ω, t

′)]eΓ(t

′−t),

Bα(ω, t′) =

Γα

2π

t′∫

0

ds fα(ω) ei

s∫

t′

dt′′(ε−ω−iΓ2)=

Γα

2π

t′∫

0

ds fα(ω) ei(ε−ω−iΓ

2)(s−t′).

(B.8)

130

B.1. Single-level system

Inserting the second equation into the result for the occupation leads to

N(t) =∑

α

Γα

π

∫dω fα(ω)

t∫

0

dt′t′∫

0

ds Re[ei(ε−ω−iΓ

2)(s−t′)

]eΓ(t

′−t)

=∑

α

Γα

π

∫dω fα(ω)

t∫

0

dt′ Re

[1− e−i(ε−ω−iΓ

2)t′

i(ε− ω − iΓ2

)]eΓ(t

′−t)

=∑

α

Γα

π

∫dω fα(ω)

t∫

0

dt′ Re

[eΓt

′ − e−i(ε−ω+iΓ2)t′

i(ε− ω − iΓ2

)]e−Γt

=∑

α

Γα

π

∫dω fα(ω) Re

[1− e−Γt

iΓ(ε− ω − iΓ2

) − e−i(ε−ω−iΓ2)t − e−Γt

(ε− ω)2 + Γ2

4

]

=∑

α

Γα

π

∫dω

fα(ω)

(ε− ω)2 + Γ2

4

× Re

[i

Γ

(e−Γt − 1

)(ε− ω + i

Γ

2

)− e−i(ε−ω−iΓ

2)t + e−Γt

]

=∑

α

Γα

π

∫dω

fα(ω)

(ε− ω)2 + Γ2

4

[1

2

(1− e−Γt

)+ e−Γt − Re

[e−i(ε−ω−iΓ

2)t]]

=∑

α

Γα

2π

∫dω

fα(ω)

(ε− ω)2 + Γ2

4

[(1 + e−Γt

)− 2 cos [(ε− ω) t] e−

Γ

2t]. (B.9)

For t → ∞ the occupation becomes

N(t) =∑

α

Γα

2π

∫dω

fα(ω)

(ε− ω)2 + Γ2

4

. (B.10)

This result coincides with the one obtained from the lesser Green’s function Eq. (2.16)

〈n(t)〉 = −iG<(t, t) =

∫dω

∑

α∈L,RΓα

fα(ω)

(ω − ε)2 + Γ2

4

T=0=

1

2− 1

π

∑

α

Γα

Γarctan

[ 2Γ(ε− µα)

]. (B.11)

131


B.2. Two-level system

B.2.1. Occupation difference

The occupation of the α-level can be calculated with help of the lesser Green’sfunction

〈nα〉 =−i

2π

∫dω G<

αα(ω)

=1

2π

∑

γ

∫dω fγ(ω) Γγ(ω) G

Rαγ(ω) G

Aγα(ω). (B.12)

For zero temperature the Fermi function equals a step function and by choosingconstant tunneling rates and ΓL = ΓR = Γ

2 , we obtain for the occupation difference

〈σz〉 =〈nL〉 − 〈nR〉

=Γ

4π

[ µL∫

−∞

dω(ω − νR)

2 +(Γ4

)2 − |TC |2N(ω)

−µR∫

−∞

dω(ω − νL)

2 +(Γ4

)2 − |TC |2N(ω)

],

(B.13)

with N(ω) = |D(ω)|2 and D(ω) = [ω − νL − iΓ/4] [ω − νR − iΓ/4] − |Tc|2. If weassume νR = −νL the results of the integrals yield

∫ µ

−∞dω

1

ω4 + 2Aω2 +B2

=1

2√C

[1√

A−√C

arctan( µ√

A−√C

)− 1√

A+√C

arctan( µ√

A+√C

)

+π

2

(√1

A−√C

−√

1

A+√C

)], (B.14)

∫ µ

−∞dω

ω

ω4 + 2Aω2 +B2=

1

2√C

[1

2ln(µ2 +A−

√C)− 1

2ln(µ2 +A+

√C)],

(B.15)

∫ µ

−∞dω

ω2

ω4 + 2Aω2 +B2

=1

2√C

[−√

A−√C arctan

( µ√A−

√C

)+

√A+

√C arctan

( µ√A+

√C

)

− π

2

(√A−

√C −

√A+

√C

)], (B.16)

with the abbreviations A = −(ν2L + |Tc|2 − Γ2/16), B = ν2L + |Tc|2 + Γ2/16 andC = A2 −B2.

132


B.2.2. Friction and occupation in the infinite bias case

In contrast to the single-level system the friction in the two-level system does notdisappear in the regime of infinite bias. We present the calculation for the sim-plified case of symmetric detuned energy levels νR = −νL, but a generalisation isstraightforward, leading to the result denoted in Eq. (3.16). In the infinite bias case,all contributing terms in Eq. (3.23) vanish except of the integral term, therefore weobtain

AIB(xt) = −2 |Tc|2λ2

πΓ2 νL

∞∫

−∞

dω1

(ω4 + 2Aω2 +B2)2

= − |Tc|2λ2

4πΓ2 νL

(A2 − 3B2 +A

√C)arctan x√

A−√C

B2C√C√

A−√C

−

(A2 − 3B2 −A

√C)arctan x√

A+√C

B2C√C√

A+√C

∣∣∣∣∣∣∣

∞

−∞

, (B.17)

with the assumptions A = (b − a), B = (a + b), C = A2 − B2 = −4ab, wherea = ν2L + |Tc|2 and b = Γ2/16. The appearing integral can also be solved with helpof the Residue theorem, but here we use the square root of a complex number

√z = ±

√|z|+ x

2+ i sgn(y)

√|z| − x

2

⇒√

A−√C = ±

(√b− i

√a). (B.18)

Hence, we can evaluate the appearing roots and obtain

AIB(xt) =− 1

4|Tc|2 λ2 Γ2 νL

1

B2C

[(A2 − 3B2 +A

√C)

√C√

A−√C

−

(A2 − 3B2 −A

√C)

√C√

A+√C

]

=− 1

4|Tc|2 λ2 Γ2 νL

2i

B2C3/2Im

[(A2 − 3B2 +A

√C)

√A−

√C

]

=− 1

4|Tc|2 λ2 Γ2 νL

2i

B3C3/2Im

[(A2 − 3B2 +A

√C)(√

A+√C

)]

=− 1

4|Tc|2 λ2 Γ2 νL

2i√a

B3C3/2

[2(b− a)b+ (b− a)2 − 3(a+ b)2

]

=− 1

8|Tc|2 λ2 Γ2 νL

(5b+ a)

(a+ b)3 b3/2. (B.19)

133


Inserting the expressions for a and b leads to

⇒ AIB(xt) = −8 |Tc|2λ2

ΓνL

ν2L + |Tc|2 + 5Γ2

16(ν2L + |Tc|2 + Γ2

16

)3 . (B.20)

For the occupation we can derive the infinite bias result in a similar manner

〈σz(xt)〉IB =Γ

4π

∞∫

−∞

dω(ω − νR)

2 +(Γ4

)2 − |TC |2N(ω)

=Γ

4π

∞∫

−∞

dω

[ω2

N(ω)+

(c+ b)

N(ω)

]

=Γ

4π

[π

2√C

[−√

A−√C +

√A+

√C

]

+π(c+ b)

2√C

[√1

A−√C

−√

1

A+√C

]]

=Γ

8√C

[2i Im

[√A+

√C

]+ (c+ b)

[√A+

√C −

√A−

√C√

A2 − C

]]

=Γi

4√C

[1 +

(c+ b)

B

]Im

[√A+

√C

]=

Γi

4√C

[1 +

(c+ b)

B

]√a

=Γ

8√b

(a+ c+ 2b)

(a+ b), (B.21)

where we abbreviated c = ν2L+ |Tc|2 and used that the integral∫ω/N(ω) disappears

in the infinite bias case. Finally the occupation difference reads

〈σz(xt)〉IB =ν2L + Γ2

16(ν2L + |Tc|2 + Γ2

16

) . (B.22)

B.2.3. Transition functions in the adiabatic limit

The adiabatic result for the lead-transition functions can directly be obtained fromEq. (3.9) by omitting the time-dependence of xt ≡ x. Hence, they decouple from theremaining equations and we can apply a Laplace transformation: (α 6= β)

zBαα(ω, z)−Bαα(ω, 0) = − i

(να − ω − i

Γ

4

)Bα,α(ω, z) −Bα,β(ω, z) +

Γ

4πfα(ω),

zBα,β(ω, z) −Bα,β(ω, z) = − i

(νβ − ω − i

Γ

4

)Bα,β(ω, z) + |Tc|2Bα,α(ω, z).

(B.23)

Assuming Bαα(0) = Bαβ(0) = 0 we can reformulate the second equation and insertthe result into the first equation. This leads to

Bαα(ω, z) =Γα

2πfα(ω)

z + i(νβ − ω − iΓ4 )

z[(z + i(να − ω − iΓ4 )(z + i(νβ − ω − iΓ4 ) + |Tc|2

] . (B.24)

134


The back transformation to the time domain is straightforward by performing anintegration over z

Bαα(ω, t) =Γα

4πifα(ω)

∞∫

−∞

dz eztz + i(νβ − ω − iΓ4 )

z[(z + i(να − ω − iΓ4 )(z + i(νβ − ω − iΓ4 ) + |Tc|2

] .

(B.25)

This can be solved with the help of the Residue theorem, the poles are

z1,2 = −Γ

4− i

[1

2(να + νβ)− ω ±

√1

4(να − νβ)2 + |Tc|2

]≡ −iP1,2 and z3 = 0.

(B.26)

Therewith, the residues can be calculated and finally, the level transition in the timedomain is obtained from

Bαα(ω, t) = iΓα

4πifα(ω)

[e−iP1t

[P1 − (νβ − ω − iΓ4 )

]

P1(P1 − P2)− e−iP2t

[P2 − (νβ − ω − iΓ4 )

]

P2(P1 − P2)

− (νβ − ω − iΓ4 )[(να − ω − iΓ4 )(νβ − ω − iΓ4 ) + |Tc|2

]]. (B.27)

The time-dependence goes with e−Γ/4t and the first two terms vanish for large times.Hence, we obtain in the long-time limit:

Bαα(ω, t → ∞) = − iΓα

4πifα(ω)

(νβ − ω − iΓ4 )[(να − ω − iΓ4 )(νβ − ω − iΓ4 ) + |Tc|2

]

=− iΓ

4πfα(ω)G

Rαα(ω). (B.28)

In the same fashion the lead-transition function Bαβ can be derived, the result reads

Bαβ(ω, t → ∞) =− Γ

4πfα(ω)

|Tc|2[(ω − νL + iΓ4 )(ω − νR + iΓ4 )− |Tc|2

]

=− Γ

4πfα(ω)

GRαβ(ω)

T(∗)c

. (B.29)

Here, T ∗c /Tc corresponds to GR

LR(ω)/GRRL(ω), see Eq. (A.23).

135


B.3. Spin system

B.3.1. Nonadiabatic equations for the coupling to a large spin

The Hamiltonian for the spin system can be formulated in an effective form

Heff =∑

σ

εσ d†σdσ +λ

2〈Jx〉

(d†↑d↓ + d†↓d↑

)+HT +Hres, (B.30)

with εσ = εd ± Bz

2 ± λ2 〈Jz〉. Therewith, the equation of motions for the operators of

the levels and the leads yield

˙dσ = i

[Heff , dσ

]= −iεσdσ − i

λ

2〈Jx〉dσ′ − i

∑

kα

V ∗kασ ckασ,

˙ckασ = i [Heff , ckασ ] = −iεkαckασ − i Vkασdσ. (B.31)

Solving the inhomogeneous differential equation for the lead operator leads to

ckασ(t) = e−iεkασtckασ(t = 0)− iVkασ

t∫

0

dt′e−iεkασ(t−t′) dσ(t′), (B.32)

which we can insert into the equation for the level operator:

˙dσ(t) = −i

(εσ(t)− i

Γσ

2

)dσ(t)− i

λ

2〈Jx(t)〉dσ′(t)− i

∑

kα

V ∗kασe

−iεkασtckασ(0).

(B.33)

With the definition Ckασ(t) = −iV ∗kασe

−iεkασtckασ(0) and its complex conjugateC∗kασ(t) the time evolution for different expectation values reads

d

dt〈d†σdσ〉 = −Γσ〈d†σ dσ〉 ± λ〈Jx〉〈Sy〉+ 2 Re

[∑

kα

〈C†kασdσ〉

],

d

dt〈d†σ dσ′〉 = i

(εσ − εσ′ +

i

2(Γσ + Γσ′)

)〈d†σ dσ′〉 ∓ iλ〈Jx〉〈Sz〉

+∑

kα

〈C†kασ dσ′〉+

∑

kα

〈d†σCkασ′〉,

d

dt〈C†

kασdσ〉 = −i

(εσ − εkασ − i

2Γσ

)〈C†

kασdσ〉 − iλ

2〈Jx〉〈C†

kασ dσ′〉

+∑

kα

|Vkασ|2 f(εkασ),

d

dt〈C†

kασdσ′〉 = −i

(εσ′ − εkασ − i

2Γσ′

)〈C†

kασdσ′〉 − iλ

2〈Jx〉〈C†

kασ dσ〉, (B.34)

where we suppressed the time-indices for better overview (upper/lower sign : ↑ / ↓).

136

B.3. Spin system

We end up with the equation system

d

dt〈Sx(t)〉 = −1

2(Γ↑ + Γ↓) 〈Sx(t)〉 − (ε↑(t)− ε↓(t))〈Sy(t)〉

+Re

[∑

kα

〈C†kα↑(t)d↓(t)〉+

∑

kα

〈C†kα↓(t)d↑(t)〉

],

d

dt〈Sy(t)〉 = −1

2(Γ↑ + Γ↓) 〈Sy(t)〉+ (ε↑(t)− ε↓(t))〈Sx(t)〉 − λ〈Jx(t)〉〈Sz(t)〉

+Im

[∑

kα

〈C†kα↑(t)d↓(t)〉 −

∑

kα

〈C†kα↓(t)d↑(t)〉

],

d

dt〈Sz(t)〉 = −1

2(Γ↑〈n↑(t)〉 − Γ↓〈n↓(t)〉) + λ〈Jx〉〈Sz(t)〉

+Re

[∑

kα

〈C†kα↑(t)d↑(t)〉 −

∑

kα

〈C†kα↓(t)d↓(t)〉

].

(B.35)

Multiplication with δ(ω − εkασ) and summing over all k states leads to an equationsystem for the spin operators (Γ = Γ↓ = Γ↑):

d

dt〈Sx(t)〉 =− Γ〈Sx(t)〉 − (ε↑(t)− ε↓(t))〈Sy(t)〉

+∑

α

∫dωRe

[Bα

↑↓(ω, t) +Bα↓↑(ω, t)

],

d

dt〈Sy(t)〉 =− Γ〈Sy(t)〉+ (ε↑(t)− ε↓(t))〈Sx(t)〉 − λ〈Jx(t)〉〈Sz(t)〉

+∑

α

∫dωIm

[Bα

↑↓(ω, t)−Bα↓↑(ω, t)

],

d

dt〈Sz(t)〉 =− Γ〈Sz(t)〉+ λ〈Jx(t)〉〈Sy(t)〉+

∑

α

∫dωRe

[Bα

↑↑(ω, t)−Bα↓↓(ω, t)

],

(B.36)

with the definition Bασσ′(ω, t) =

∑k δ(ω − εkασ)〈C†

kασ(t)dσ′(t)〉 and

d

dtBα

σσ(ω, t) = −i(εσ(t)− ω − i

2Γσ)B

ασσ(ω, t)− i

λ

2〈Jx(t)〉Bα

σσ′ (ω, t) +Γασ

2πfασ(ω),

d

dtBα

σσ′(ω, t) = −i(εσ′(t)− ω − i

2Γσ′)Bα

σσ′(ω, t)− iλ

2〈Jx(t)〉Bα

σσ(ω, t),

(B.37)

with εσ(t) = εd ± Bz

2 ± λ2 〈Jz(t)〉.

137


B.3.2. SDS: infinite bias results for the adiabatic approximation

Assuming infinite bias µL,R = ±∞ the spin operators in the adiabatic approach areobtained from

〈Sx〉 = λ〈Jx〉∞∫

−∞

dω

4π

[ω − ε↓] ΓL↑ + [ω − ε↑] ΓL↓∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2 ,

〈Sy〉 =λ

2〈Jx〉

∞∫

−∞

dω

4π

Γ [ΓL↓ − ΓL↑]∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2 ,

〈Sz〉 =∞∫

−∞

dω

4π

[ω − ε↓]2 ΓL↑ − [ω − ε↑]

2 ΓL↓ +14 (ΓL↑ − ΓL↓)

[Γ2 − λ2〈Jx〉2

]

∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣2 .

(B.38)

The decomposition of the denominator yield

N(ω) = |(ω −B1)(ω −B2)|2 , B1,2 =1

2

[(ε↓ + ε↑)±

√(ε↓ − ε↑)2 + λ2〈Jx〉2 + iΓ

].

(B.39)

Thus, we need the following integrals

I1 =

∞∫

−∞

dω1

N(ω)=

4π

Γ

1[(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)

] ,

I2 =

∞∫

−∞

dωω

N(ω)=

2π

Γ

(ε↓ + ε↑)[(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)

] ,

I3 =

∞∫

−∞

dωω2

N(ω)=

π

Γ

2(ε2↓ + ε2↑) + Γ2 + λ2〈Jx〉2[(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)

] . (B.40)

Hence, the spin operators are obtained from

〈Sx〉 = λ〈Jx〉1

2Γ

(ΓL↓ − ΓL↑)(ε↓ − ε↑)[(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)

] ,

〈Sy〉 = λ〈Jx〉1

2

(ΓL↓ − ΓL↑)[(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)

] ,

〈Sz〉 =(ΓL↑ − ΓL↓)

2Γ

(ε↓ − ε↑)2 + Γ2

[(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)

] , (B.41)

138

B.3. Spin system

and the occupation can be expressed in terms of the spin operator 〈Sz〉

〈nσ〉 = (δσ↑ − δσ↓) 〈Sz〉+(ΓLσ + ΓLσ′)

2Γ. (B.42)

The current in the infinite bias case reads

Iασ = eΓασ

δαL − 1

2Γ

2ΓLσ

((ε↓ − ε↑)2 + Γ2

)+ λ2〈Jx〉2(ΓLσ + ΓLσ′)[

(ε↓ − ε↑)2 + Γ2 + λ2〈Jx〉2)]

= eΓασ

[δαL − (δσ↑ − δσ↓) 〈Sz〉 −

(ΓLσ + ΓLσ′)

2Γ

]

= eΓασ [δαL − 〈nσ〉] . (B.43)

We can straightforwardly show that current conservation is ensured. The currentthrough the left lead yields

IL =∑

σ

ILσ = e ΓL↑ [1− 〈n↑〉] + ΓL↓ [1− 〈n↓〉] . (B.44)

And with 〈n↑〉 + 〈n↓〉 = (ΓL↑ + ΓL↓)/Γ and Γ = Γσ = ΓLσ + ΓRσ we obtain for theright current

IR =− e [Γ− ΓL↑] 〈n↑〉+ [Γ− ΓL↓] 〈n↓〉 = −IL. (B.45)

B.3.3. Adiabatic corrections

Expanding the Green lesser functions (2 × 2 matrix) according to Reference [104]yields

G< =G<,0 +i

2

δωG<BGA −G<BδωGA + δωGRBG< −GRBδωG<

, (B.46)

with B = σ · b and b = λ2

(〈Jx〉, 0, 〈Jz〉

).

The components of electronic spin are obtained via

Sj =− i

2tr[G<(t, t)σj

]= − i

2

∫dω

2πtr[G<,0σj

]+

∫dω

8πtr[G<,1σj

]. (B.47)

The adiabatic correction term G<,1 corresponds to the second term of Eq. (4.46),with an integration by parts this term can be simplified

∫dωδωG<BGA = G<BGA

∣∣∣+∞

−∞−∫

dωG<BδωGA,∫

dωGRBδωG< = GRBG<∣∣∣+∞

−∞−∫

dωδωGRBG<.

139


The first terms vanish and the adiabatic correction term Aj(t) yields

Aj(t) =

∫dω

4πtr[δωGRBG<σj −G<BδωGAσj

]

=

∫dω

4π

∑

k

λ

2〈Jk〉tr

[δωGRσkBG<σj −G<σkBδωGAσj

], (B.48)

whereby k ∈ x, z and j ∈ x, z due to the anisotropic coupling. The explicit expres-sions for the adiabatic correction terms are obtained from

Ax(t) =

∫dω

4πtr

[λ

2〈Jx〉G<

σxδωG

Rσx − σxδωGAσx

]

+ tr

[λ

2〈Jz〉G<

σxδωG

Rσz − σzδωGAσx

]

=

∫dω

4π

λ2〈Jx〉

[G<

↑↑δωG>↓↓ +G<

↓↑δωG>↓↑ +G<

↑↓δωG>↑↓ +G<

↓↓δωG>↑↑

]

+λ

2〈Jz〉

[G<

↑↑δωGR

↓↑ − δωGA↑↓−G<

↓↑δωGR

↓↓ + δωGA↑↑

+G<↑↓δωGR

↑↑ + δωGA↓↓−G<

↓↓δωGR

↑↓ + δωGA↓↑]

,

Az(t) =

∫dω

4πtr

[λ

2〈Jx〉G<

σzδωG

Rσx − σxδωGAσz

]

+ tr

[λ

2〈Jz〉G<

σzδωG

Rσz − σzδωGAσz

]

=

∫dω

4π

λ2〈Jx〉

[G<

↑↑δωGR

↑↓ − δωGA↓↑+G<

↓↑δωGR

↑↑ + δωGA↓↓

−G<↑↓δωGR

↓↓ + δωGA↑↑−G<

↓↓δωGR

↓↑ − δωGA↑↓]

+λ

2〈Jz〉

[−G<

↑↑δωG>↑↑ −G<

↓↑δωG>↑↓ −G<

↑↓δωG>↓↑ −G<

↓↓δωG>↓↓

].

(B.49)

Here, we used integration by parts, the relation GR −GA = G> −G< and the factthat the product of two Green functions evaluated at infinity is equal to zero. Aseparation Aj = λ/2 (〈Jx〉Ajx + 〈Jz〉Ajz) and assuming infinity bias leads to thefollowing equations (∆LR = ΓL↑ΓR↓ − ΓL↓ΓR↑ and ∆L = ΓL↓ − ΓL↑ /Γ2)

Axx =

∫dω

4π∆LR (ε↑ − ε↓)

[(ω − ε↑)(ω − ε↓)− Γ2

4 + λ2

4 〈Jx〉2]

∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣4

=∆LR

Γ3

[(ε↓ − ε↑)

2 + Γ2] (

2Γ2 − λ2〈Jx〉2)+ λ2〈Jx〉2

(2Γ2 + λ2〈Jx〉2

)

(ε↑ − ε↓)−1[(ε↓ − ε↑)

2 + λ2〈Jx〉2 + Γ2]3 , (B.50)

140

B.3. Spin system

Axz =

∫dω

4πλ〈Jx〉2Γ ΓL↓ − ΓL↑

[(ω − ε↑)(ω − ε↓)]∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣4

=λ〈Jx〉∆L

[(ε↓ − ε↑)

2[λ2〈Jx〉2 − 3Γ2

]+(Γ2 + λ2〈Jx〉2

)2]

[(ε↓ − ε↑)

2 + λ2〈Jx〉2 + Γ2]3 , (B.51)

Azx =

∫dω

4π

λ

2〈Jx〉Γ ΓL↑ − ΓL↓

[(ω − ε↑)2 + (ω − ε↓)2 +

Γ2

2 − λ2

2 〈Jx〉2]

∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣4

=λ〈Jx〉∆L

[(ε↓ − ε↑)

2(ε↓ − ε↑)

2 + λ2〈Jx〉2 + 4Γ2− Γ2

λ2〈Jx〉2 − 3Γ2

]

[(ε↓ − ε↑)

2 + λ2〈Jx〉2 + Γ2]3 ,

(B.52)

Azz =

∫dω

4π∆LR (ε↑ − ε↓)

λ2

4 〈Jx〉2∣∣∣[ω − ε↑ + iΓ2

] [ω − ε↓ + iΓ2

]− λ2

4 〈Jx〉2∣∣∣4

=λ2〈Jx〉2∆LR

Γ3

[(ε↓ − ε↑)

2 + λ2〈Jx〉2 + 5Γ2]

(ε↑ − ε↓)−1[(ε↓ − ε↑)

2 + λ2〈Jx〉2 + Γ2]3 . (B.53)

The equation of motion for the large spin, including the adiabatic correction term,equals a Landau-Lipshitz-Gilbert equation

d

dtJ =− λ

[J× S0

]− [J× B]−

[J× A

d

dtJ

], (B.54)

with

A =

Axx 0 Axz

0 0 0Azx 0 Azz

,

there the anisotropic coupling is already included. Finally, the equations of motionfor the large spin yield

〈 ˙Jx〉 = −(λ〈Sz〉+Bz

)〈Jy〉+ λy〈Sy〉〈Jz〉 −

(Azx〈 ˙Jx〉+Azz〈 ˙Jz〉

)〈Jy〉,

〈 ˙Jy〉 =(λ〈Sz〉+Bz

)〈Jx〉 − λ〈Sx〉〈Jz〉 −

(Axx〈 ˙Jx〉+Axz〈 ˙Jz〉

)〈Jz〉

+(Azx〈 ˙Jx〉+Azz〈 ˙Jz〉

)〈Jx〉,

〈 ˙Jz〉 = λ〈Sx〉〈Jy〉+(Axx〈 ˙Jx〉+Axz〈 ˙Jz〉

)〈Jy〉. (B.55)

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Coherent transport through nanoelectromechanical systems · Die Untersuchung von kohärentem Transport durch nanoelektromechanische Sys-teme (NEMS) steht im Fokus dieser Arbeit. NEMS