1SCIENTIFIC REPORTS | (2018) 8:3107 | DOI:10.1038/s41598-018-21221-3

www.nature.com/scientificreports

Spectral features of the tunneling-induced transparency and the Autler-Townes doublet and triplet in a triple quantum dotXiao-Qing Luo1, Zeng-Zhao Li1, Jun Jing2, Wei Xiong1, Tie-Fu Li1,3 & Ting Yu1,4

We theoretically investigate the spectral features of tunneling-induced transparency (TIT) and Autler-Townes (AT) doublet and triplet in a triple-quantum-dot system. By analyzing the eigenenergy spectrum of the system Hamiltonian, we can discriminate TIT and double TIT from AT doublet and triplet, respectively. For the resonant case, the presence of the TIT does not exhibit distinguishable anticrossing in the eigenenergy spectrum in the weak-tunneling regime, while the occurrence of double anticrossings in the strong-tunneling regime shows that the TIT evolves to the AT doublet. For the off-resonance case, the appearance of a new detuning-dependent dip in the absorption spectrum leads to double TIT behavior in the weak-tunneling regime due to no distinguished anticrossing occurring in the eigenenergy spectrum. However, in the strong-tunneling regime, a new detuning-dependent dip in the absorption spectrum results in AT triplet owing to the presence of triple anticrossings in the eigenenergy spectrum. Our results can be applied to quantum measurement and quantum-optics devices in solid systems.

Quantum coherence and interference effects can lead to considerably interesting phenomena of quantum optics such as lasing without inversion1,2, coherent population trapping3, correlated spontaneous emission4, and elec-tromagnetically induced transparency (EIT)5–9. As a phenomenon closely related to EIT, Autler-Townes (AT) splitting10,11 is indicated by a level anticrossing in the eigenenergy spectrum and a transparency window owing to the AT doublet rather than the quantum interference. This phenomenon has been utilized to measure the state of the electromagnetic field12–14, as well as the AT triplet and multiplet spectroscopy15. Both EIT and AT split-ting have been investigated theoretically and experimentally in different quantum systems, including atomic and molecular systems8,16,17, solid-state and metamaterials systems18,19, superconducting quantum circuits20–26 and whispering-gallery-mode optical resonators27–29. It is also interesting to investigate EIT and AT splitting in sem-iconductor nanostructures because the trapped carriers behave like atoms and can be conveniently manipulated via external fields.

In the semiconductor quantum dot (QD), excitons form bound states and play an important role in the optical properties of these systems10,30–35. Moreover, tunneling-induced transparency (TIT)36–38 can occur for the exci-tonic states, which is similar to EIT in a three-level atomic system, but no pump field is needed to apply to the excitonic system. As shown by Borges et al.37, there is an evidence regarding the coexistence of both TIT and AT doublet in the intermediate regime, when the tunneling coupling is slightly above a threshold in a double-QD sys-tem. However, a triple-QD system can offer new possibilities to study intriguing phenomena that are not observed in single and double-QD systems39–41. In the present paper, we show that when the electron is resonant-tunneling in such a triple-QD system, which is in contrast to the case considered in a double-QD system, the coexistence of both TIT and AT doublet regime does not occur in this system and the threshold of the tunneling coupling just corresponds to a transition point. More specifically, we find that, by analyzing the eigenenergy spectrum of the

1Beijing Computational Science Research Center, Beijing, 100193, China. 2Department of Physics, Zhejiang University, Hangzhou, 310027, Zhejiang, China. 3Institute of Microelectronics, Department of Microelectronics and Nanoelectronics and Tsinghua National Laboratory of Information Science and Technology, Tsinghua University, Beijing, 100084, China. 4Department of Physics and Engineering Physics, Center for Controlled Quantum Systems, Stevens Institute of Technology, Hoboken, New Jersey, 07030, USA. Correspondence and requests for materials should be addressed to Z.-Z.L. (email: zengzhaoli09@gmail.com) or T.-F.L. (email: litf@tsinghua.edu.cn)

Received: 14 September 2017Accepted: 31 January 2018Published: xx xx xxxx

OPEN

PHYSICAL REVIEW B 96, 155438 (2017)

Floquet engineering of long-range p-wave superconductivity: Beyond the high-frequency limit

Zeng-Zhao Li,1,2,3,* Chi-Hang Lam,4,† and J. Q. You3,‡1Division of Solid State Physics and NanoLund, Lund University, Box 118, S-22100 Lund, Sweden

2Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany3Quantum Physics and Quantum Information Division, Beijing Computational Science Research Center, Beijing 100094, China

4Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China(Received 8 April 2017; published 16 October 2017)

It has been shown that long-range p-wave superconductivity in a Kitaev chain can be engineered via an ac fieldwith a high frequency [M. Benito et al., Phys. Rev. B 90, 205127 (2014)]. For its experimental realization, however,theoretical understanding of Floquet engineering with a broader range of driving frequencies becomes important.In this paper, focusing on the ac-driven tunneling interactions of a Kitaev chain, we investigate effects from theleading correction to the high-frequency limit on the emergent p-wave superconductivity. Importantly, we find newengineered long-range p-wave pairing interactions that can significantly alter the ones in the high-frequency limitat long interaction ranges. We also find that the leading correction additionally generates nearest-neighbor p-wavepairing interactions with a renormalized pairing energy, long-range tunneling interactions, and, in particular,multiple pairs of Floquet Majorana edge states that are destroyed in the high-frequency limit.

DOI: 10.1103/PhysRevB.96.155438

I. INTRODUCTION

Floquet engineering, being a promising quantum engi-neering and quantum control technology, has attracted muchattention in recent years. Its main idea is to design appropriatetime-periodic driving protocols to engineer properties of atime-independent effective Hamiltonian that governs the timeevolution of periodically driven quantum systems. Besidesusual properties analogous to their static counterparts, Floquetquantum systems possess additional unique features thatresult from the periodicity of their quasienergy spectrum.Given the versatility of driving protocols, Floquet engineeringopens up new possibilities for exploring and observing exoticphenomena unreachable in static systems [1– 15]. For example,this technique has been employed to realize dynamicallocalization [16– 18], photon-assisted tunneling [19– 21], andnovel topological band structures [22– 29].

A promising quantum system for implementing Floquetengineering is a Kitaev chain. It is a one-dimensional p-wavesuperconducting wire supporting Majorana zero modes at itsends [30] and can potentially be realized in realistic quantumsystems such as a quantum nanowire [31– 37]. Similar tothe static case, a driven Kitaev chain hosts end states calledFloquet Majorana zero modes which may realize fault-tolerantquantum computation [38,39] and also provide a new way todetect these elusive Majorana states [5,40].

Interestingly, a very recent work has demonstrated thatlong-range p-wave superconductivity (or equivalently long-range many-body spin interactions) can be engineered byperiodically driving the tunneling interactions in a Kitaevchain [29]. It has been shown, for example, that the generatednext-neighbor interactions become comparable to the first-neighbor interactions as the driving amplitude increases [29].These interesting observations, however, are based on the high

*zeng-zhao.li@ftf.lth.se†C.H.Lam@polyu.edu.hk‡jqyou@csrc.ac.cn

driving frequency limit. A thorough understanding of Floquetengineering with a broader range of driving frequencies isimportant for the experimental realization of the effectivelong-range p-wave superconductivity.

In this paper, we consider a driving protocol where thetunneling interactions of a Kitaev chain are modulated by an acfield. We focus on the leading correction to the high-frequencylimit. We find that the leading correction results in long-range p-wave pairing interactions with interaction strengthsdepending nontrivially on the driving amplitude and frequencyof the applied ac field. Compared with the ones correspondingto the high-frequency limit, these new interactions becomeimportant when the interaction range increases. We also findthat the long-range tunneling interactions, the nearest-neighborpairing interactions, and multiple Floquet Majorana edge statescan now be engineered with the leading correction. We believethat our results are important for the experimental realizationof Floquet-engineered long-range p-wave pairing interactionsand tunneling interactions. They also provide new insightsabout the quantum engineering of new exotic phases in realisticquantum systems.

This paper is organized as follows. In Sec. II, we introducea model of a time-dependent Kitaev chain. In Sec. III,we present our driving protocol by considering periodicallydriven tunneling interactions of a Kitaev chain and derivean effective time-independent Hamiltonian. In Sec. IV, weillustrate the important role that the leading correction plays inthe engineered long-range pairing interactions. In Sec. V, wedemonstrate that multiple Floquet Majorana edge states couldbe engineered when corrections to the high driving frequencylimit are included. Discussions and conclusions are finallygiven in Sec. VI.

II. THE MODEL

The model under our consideration is a Kitaev chain [30]with tunneling interactions which are periodically driven via,for example, gate voltages applied on a quantum wire. The

2469-9950/2017/96(15)/155438(12) 155438-1 ©2017 American Physical Society

PHYSICAL REVIEW A 95, 052509 (2017)

Probing weak dipole-dipole interaction using phase-modulated nonlinear spectroscopy

Zeng-Zhao Li,1 Lukas Bruder,2 Frank Stienkemeier,2 and Alexander Eisfeld1,*

1Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden, Germany2Physics Department, University of Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany

(Received 22 March 2017; published 26 May 2017)

Phase-modulated nonlinear spectroscopy with higher harmonic demodulation has recently been suggested toprovide information on many-body excitations. In the present work we theoretically investigate the applicationof this method to infer the interaction strength between two particles that interact via weak dipole-dipoleinteraction. To this end we use a full numerical solution of the Schrödinger equation with time-dependent pulses.For interpretation purposes we also derive analytical expressions in perturbation theory. We find one can detectdipole-dipole interaction via peak intensities (in contrast to line-shifts which typically are used in conventionalspectroscopy). We provide a detailed study on the dependence of these intensities on the parameters of the laserpulse and the dipole-dipole interaction strength. Interestingly, we find that there is a phase between the first andsecond harmonic demodulated signal whose value depends on the sign of the dipole-dipole interaction.

DOI: 10.1103/PhysRevA.95.052509

I. INTRODUCTION

The study of many-body effects in weakly interactingensembles of particles poses challenges to experiments. Inmany cases strong single-particle signals or broadening effectsmask the weak collective signals. Recently, coherent time-resolved nonlinear spectroscopy introduced a very sensitiveapproach to probe weak many-body effects with double-quantum two-dimensional (2D) spectroscopy [1– 5] wherethe two-body response is background-free detected, hencethe mere presence of a double-quantum signal indicates theexistence of interparticle interactions in the system. Doublequantum coherence measurements in 2D spectroscopy arevery useful to detect weak interactions. However, most 2Dschemes do not provide sufficient sensitivity to investigatedilute samples.

A multidimensional spectroscopy approach suitable forhighly dilute samples is based on phase modulation (PM) [6].This approach has been demonstrated in 2D electronic spec-troscopy [6], but also in electronic wave-packet interferome-try [7]. In the latter scheme, one records the signal from anobservable (such as fluorescence intensity, photocurrent, orphotoionization [7– 9]) emitted by the particles subject to twoshort laser pulses with a specific time-delay and a slowly modu-lated relative phase. Demodulation with respect to this relativephase and a subsequent Fourier transformation with respect tothe time-delay yield a complex-valued spectrum. By using avariant of this technique, in Ref. [10] the demodulation wasperformed by using higher harmonics of the reference signal.

In the present work we aim at understanding the effect oflong-range dipole-dipole interaction on the spectra resultingfrom higher-order demodulation. To this end we investigate aminimal model consisting of two particles which can interactvia transition dipole-dipole interaction. The model that weuse is, in particular, related to interacting dye moleculeswhich have a parallel arrangement. To infer the dipole-dipoleinteraction in such systems is of considerable interest (see,e.g., Refs. [11– 16]). The model of our measurement resultseffectively in a one-body measurement operator (number of

*eisfeld@pks.mpg.de

detected photons). Our work will thus provide reference forthe predictions from such a single-particle operator.

In our theoretical investigations we will, on the onehand, use a nonperturbative fully numerical approach, onthe other hand, we will derive analytical formulas validif a perturbation treatment with respect to the light-matterinteraction is applicable. In our numerical calculations we aimat following the experimental procedure as closely as possible.

The structure of this paper is as follows. In Sec. II, weintroduce a model of multiple particles that interact withlaser fields and also a theoretical background of the phase-demodulation technique used to reveal collective behaviors ofparticles. Taking two particles as a typical example, in Sec. IIIwe present the fully numerical simulation results for Gaussianpulses. In Sec. IV, based on perturbation theory, we developanalytical formulas of phase-(de)modulated fluorescence sig-nal responsible for that measured in the experiment and presentanalytical results of signals for rectangular pulses in theAppendix. Discussions and conclusions are finally given inSec. VI. In the Supplemental Material [17] we provide detailsabout our numerical and analytical procedures. Throughoutthe paper we set h = 1.

II. MODEL

A. System and interaction with laser pulses

We consider an ensemble of particles that do not move andthat are so far apart that only long-range interaction of thedipole-dipole type has to be taken into account. The Hamilto-nian for N particles coupled to a pulsed laser field reads

H =N!

n=1

Hn +!

nn′

Vnn′ +N!

n=1

H intn (t). (1)

The Hamiltonian Hn describes the individual particles, Vnn′

is the dipole-dipole interaction between particle n and n′, andH int

n is the interaction of particle n with the electromagneticfield. In the following subsections we discuss the various partsof the Hamiltonian above. Throughout the work we focuson the case of two particles (i.e., N = 2). Our basic setup issketched in Fig. 1.

2469-9926/2017/95(5)/052509(12) 052509-1 ©2017 American Physical Society

F U L L P A P E R

Hierarchy of equations to calculate the linear spectra ofmolecular aggregates: Time-dependent and frequencydomain formulation

Pan-Pan Zhang | Zeng-Zhao Li | Alexander Eisfeld

Max Planck Institute for the Physics of

Complex Systems, Division Finite Systems,

N€othnitzer Strasse 38, Dresden D-01187,

Germany

CorrespondenceZeng-Zhao Li, Max Planck Institute for the

Physics of Complex Systems, Division Finite

Systems, N€othnitzer Straße 38, Dresden

01187, Germany.

Email: zengzhao@pks.mpg.de

AbstractIn a recent publication (Ritschel et al., J. Chem. Phys. 2015, 142, 034115) we have derived a hier-

archy of coupled differential equations in time domain to calculate the linear optical properties of

molecular aggregates. Here, we provide details about issues concerning the numerical implementa-

tion. In addition we present the corresponding hierarchy in frequency domain.

K E YWORD S

hierarchy of equations, linear spectra, molecular aggregates

1 | INTRODUCTION

Molecular aggregates are assemblies of molecules where interaction

between transition dipoles of different molecules (monomers) leads to

a delocalization of electronic excitation over several monomers.[1–3]

Linear optical properties (absorption, linear- and circular-dichroism) can

provide information about the degree of excitonic delocalization, the

arrangement of the monomers and the strength of the dipole–dipole

couplings. However, the interpretation of the spectra is complicated by

the coupling to vibrational modes.[4] The inclusion of vibrational modes

makes the calculation of aggregate spectra quite challenging and vari-

ous methods have been developed to handle vibrations (see e.g., Refs.

5–23). In particular, the numerical treatment of many weakly damped

molecular vibrations is demanding.[24] Additional complications arise,

when finite temperature effects have to be accounted for.

In Ref. 25 some of us have developed a hierarchy of differential

equations that allows one to efficiently handle this situation. The

method is based on an open quantum system description, where elec-

tronic (excitonic) degrees of freedom are included in the system and all

vibrational modes are part of the environment. From the solution of

the hierarchy one can then construct the time-dependent dipole-corre-

lation function, whose Laplace transformation is related to absorption

and dispersion spectra.

While in Ref. 25 the focus was on the derivation of the hierarchy

at finite temperature, in the present work we discuss aspects on the

numerical implementation of this hierarchy. In addition we show that

the time-dependent hierarchy of Ref. 25 can be directly transformed to

frequency domain to obtain a hierarchy of coupled linear equations,

instead of coupled differential equations. This hierarchy provides a dif-

ferent viewpoint and offers an alternative way of numerically calculat-

ing the spectrum. It could also help to develop new analytical

approximations.

The article is organized as follows: first, we briefly provide the

relevant formulas used to model the aggregate. In section 3, we pres-

ent the time-dependent hierarchy of Ref. 25 and its relation to the

absorption spectrum in a concise manner. In this section, we also

derive the hierarchy in frequency domain and discuss some aspects

of the numerical implementation of both hierarchies. Finally, we con-

clude by relating to the hierarchy used in Ref. 21 for the calculation

of absorption spectra.

2 | MODEL

We briefly repeat our model of the aggregate. Details can be found

in Ref. 25. For each monomer we take two electronic states into

account, the ground state j/gni and the excited state j/e

ni; here n

labels the monomer. The electronic ground state of the aggregate is

simply a product state where all monomers are in their electronic

ground state. Since we are interested in linear optical properties, we

will restrict the excited state basis to states jpni5j/eniQN

m6¼n j/gmi;

where one monomer is electronically excited (e) and all other mono-

mers are in the ground electronic state (g). The excited state Hamilto-

nian is then given by

Int J Quantum Chem. 2017;117:e25386.https://doi.org/10.1002/qua.25386

http://q-chem.org VC 2017Wiley Periodicals, Inc. | 1 of 4

Received: 7 February 2017 | Revised: 27 March 2017 | Accepted: 28 March 2017

DOI: 10.1002/qua.25386

1Scientific RepoRts | 5:11416 | DOi: 10.1038/srep11416

www.nature.com/scientificreports

Probing Majorana bound states via counting statistics of a single electron transistorZeng-Zhao Li1,2, Chi-Hang Lam2 & J. Q. You1

We propose an approach for probing Majorana bound states (MBSs) in a nanowire via counting statistics of a nearby charge detector in the form of a single-electron transistor (SET). We consider the impacts on the counting statistics by both the local coupling between the detector and an adjacent MBS at one end of a nanowire and the nonlocal coupling to the MBS at the other end. We show that the Fano factor and the skewness of the SET current are minimized for a symmetric SET configuration in the absence of the MBSs or when coupled to a fermionic state. However, the minimum points of operation are shifted appreciably in the presence of the MBSs to asymmetric SET configurations with a higher tunnel rate at the drain than at the source. This feature persists even when varying the nonlocal coupling and the pairing energy between the two MBSs. We expect that these MBS-induced shifts can be measured experimentally with available technologies and can serve as important signatures of the MBSs.

Majorana fermions are particles that are their own antiparticles. In high-energy physics, neutrino being an elementary particle was suggested as a Majorana fermion1. Experiments aiming to prove this pro-posal are still on going. Besides the high-energy context where they arose, it is believed that Majorana fermions can also emerge as quasiparticles in condensed-matter systems2,3. The search for Majorana bound states (MBSs) in these systems has attracted much interest not only due to their exotic proper-ties (e.g., non-Abelian statistics) but also because they are promising candidates for topological quan-tum computation4,5. Several physical systems have been suggested to support MBSs, including fractional quantum Hall states6–8, chiral p-wave superconductors/superfluids8,9 surfaces of three-dimensional (3D) topological insulators in proximity to an s-wave superconductor10, superfluids in the 3He-B phase11,12, and helical edge modes of 2D topological insulators in proximity to both a ferromagnet and a super-conductor13. More recently, it has been shown that a spin-orbit coupled semiconducting 2D thin film14 or a 1D nanowire15–19 with Zeeman spin splitting, which is in proximity to an s-wave superconductor, can also host MBSs.

Providing experimental evidences for the realization of MBSs is of great importance. Techniques proposed to detect MBSs include the analysis of the tunneling spectroscopy20–23, the verification of the nature of nonlocality13,24 or the observation of the periodic Majorana-Josephson current25. In particular, the very recent observation of a zero-bias peak in the differential conductance through a semiconduc-tor nanowire in contact with a superconducting electrode indicated the possible existence of a midgap Majorana state26. Such a zero-bias peak was also observed in subsequent experiments27–29. However, this zero-bias peak could be due to the Kondo resonance30 and also occur in the presence of either disorders31 or a singlet-doublet quantum phase transition32, corresponding to ordinary Andreev bound states rather than MBSs. Moreover, a study of a more realistic model of a nanowire with MBSs further indicates a different origin for this observed zero-bias peak33. There are several recent works34–39 developed, for example, to distinguish between the Majorana and Kondo origins of the zero-bias conductance peak, but

1Laboratory for Quantum Optics and Quantum Information, Beijing Computational Science Research Center, Beijing 100094, China. 2Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China. Correspondence and requests for materials should be addressed to C.H.L. (email: C.H.Lam@polyu.edu.hk) or J.Q.Y. (email: jqyou@csrc.ac.cn)

Received: 08 January 2015

Accepted: 22 May 2015

Published: 22 June 2015

OPEN

PHYSICAL REVIEW B 89, 134505 (2014)

Probing zero modes of a defect in a Kitaev quantum wire

Sheng-Wen Li (!!!),1,2,3 Zeng-Zhao Li (!!"),1 C. Y. Cai (!!!),3,4 and C. P. Sun (!!!)1,3

1Beijing Computational Science Research Center, Beijing 100084, China2School of Engineering and Information Technology, University of New South Wales at the Australian Defense Force Academy,

Canberra, ACT 2600, Australia3Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China,

Hefei, Anhui 230026, China4State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, University of Chinese Academy of Sciences,

Beijing 100190, China(Received 10 December 2013; revised manuscript received 12 February 2014; published 9 April 2014)

The Kitaev quantum wire (KQW) model with open boundary possesses two Majorana edge modes. When thelocal chemical potential on a defect site is much higher than that on other sites and than the hopping energy, theelectron hopping is blocked at this site. We show that the existence of such a defect on a closed KQW also givesrise to two low-energy modes, which can simulate the edge modes. The energies of the defect modes vanish tozero as the local chemical potential of the defect increase to infinity. We develop a quantum Langevin equation tostudy the transport of KQW for both open and closed cases. We find that when the lead is contacted with the sitebeside the defect, we can observe two splitted peaks around the zero-bias voltage in the differential conductancespectrum, whereas if the lead is contacted with the bulk of the quantum wire far from the the defect or the openedges, we cannot observe any zero-bias peak.

DOI: 10.1103/PhysRevB.89.134505 PACS number(s): 74.50.+r, 73.23.−b, 74.78.Na

I. INTRODUCTION

The emergent Majorana fermion in condensed mattersystem has attracted much attention for its novel non-Abelianstatistical property and potential application in topologicalquantum computation [1– 3]. The Kitaev quantum wire (KQW)model with open boundary possesses two localized Majoranaedge modes at the two ends [4]. The realization of thisquantum wire model was also reported with the help of strongspin-orbit coupling and Zeeman field in proximity to an s-wavesuperconductor [5– 9].

To detect the existence of the Majorana fermion in the quan-tum wire, people can measure the differential conductance intransport experiments [10– 13]. It was predicted that there is azero-bias peak (ZBP) in the dI/dV profile in the topologicalphase when the quantum wire is contacted with a normallead, and the height of the peak is 2e2/h at zero temperature.Moreover, if there exists finite coupling between the Majoranafermions at the two ends, this ZBP would split into two peaks[12]. However, it was recognized that such feature of a singleZBP is not an unambiguous evidence, because similar ZBPmay be also induced by different mechanisms, such as theKondo effect [14– 18].

When the local chemical potential µp on a defect site ismuch higher than that on other sites and the hopping energy,the electron hopping is blocked at this site. Thus, the existenceof such a defect on a closed wire is similar to cutting off thewire at this position and generating new boundaries. Sucha “cutoff” for a closed KQW also gives rise to a pair oflow-energy modes (we call them the defect modes) [19– 21].These defect modes have many similar properties to theMajorana edge modes as follows:

(1) when the defect becomes “strong,” i.e., the chemicalpotential µp of the defect becomes quite large, the energies ofthe defect modes approach zero;

(2) the energies of the defect modes are gapped from thebulk band of the quantum wire;

(3) the defect modes are superpositions of both electronand hole modes with equal weight, localized around the defectsite.

If µp approaches infinity, electron hopping is fully blockedand the quantum wire can be regarded as completely cut off,thus the defect modes become Majorana edge modes. In thissense, a close quantum wire with a defect is equivalent to ahomogenous open wire.

However, in practice, the strength of the defect is finite,thus there remains a small energy splitting between the twodefect modes. In contrast, the energy splitting of Majoranaedge modes in an open wire is practically too small to beobserved even for a short chain [22]. Throughout this paper,we call both the edge and defect modes the zero modes.

With the above understanding, in this paper, we study thetransport measurement in KQW for two kinds of configura-tions, i.e., a homogenous open wire and a closed wire with adefect. The detection of such zero modes induced by defectalso helps test the existence of Majorana fermions [23,24].We derive a quantum Langevin equation to describe theelectrical current transport of the quantum wire contacted withtwo normal leads, which could give exact numerical results[22,25– 28]. We obtain the differential conductance when oneof the leads is contacted with different sites of the quantum wire[29]. We find that if the lead is contacted beside the defect, wecan observe two splitted ZBPs in the dI/dV profile. Moreover,if the lead is contacted with other sites in the bulk of the chainfar from the defect or the open edges, we cannot observe anyZBP, because the zero modes, both the edge and defect modes,are localized.

We arrange our paper as follows. In Sec. II we give abrief review of KQW. We give a demonstration of the energyspectrum and spatial distribution of the edge and defect modes.In Sec. III we derive a quantum Langevin equation for thetwo contacts measurement setup and obtain the steady currentformula. In Sec. IV, we show the dI/dV profiles for different

1098-0121/2014/89(13)/134505(8) 134505-1 ©2014 American Physical Society

PHYSICAL REVIEW A 90, 022122 (2014)

Approach to solving spin-boson dynamics via non-Markovian quantum trajectories

Zeng-Zhao Li,1,2 Cho-Tung Yip,2 Hai-Yao Deng,2,3 Mi Chen,1,4 Ting Yu,1,5 J. Q. You,1,6,* and Chi-Hang Lam2,*

1Beijing Computational Science Research Center, Beijing 100084, China2Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

3International Center for Materials Nanoarchitechtonics, National Institute for Materials Science, Namiki 1-1, Tsukuba 305-0044, Japan4Department of Physics, Fudan University, Shanghai 200433, China

5Center for Controlled Quantum Systems and Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken,New Jersey 07030, USA

6Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei,Anhui 230026, China

(Received 12 June 2014; published 26 August 2014)

We develop a systematic and efficient approach for numerically solving the non-Markovian quantum statediffusion equation for an open quantum system that can be strongly coupled to an environment. As an importantapplication, we consider a real-time simulation of a spin-boson model in a strong-coupling regime that is difficultto deal with using conventional methods. We show that the non-Markovian stochastic Schrodinger equation canbe efficiently implemented as a real-time simulation for this model, so as to give an accurate description ofspin-boson dynamics beyond the rotating-wave approximation.

DOI: 10.1103/PhysRevA.90.022122 PACS number(s): 03.65.Yz, 42.50.Lc

I. INTRODUCTION

The dynamics of open quantum systems has been exten-sively studied in the last decades due to its pivotal importancein the areas of quantum optics, quantum dissipative dynamics,and quantum information [1– 3]. The Lindblad master equa-tions under the Born-Markov approximations are the majortheoretical tools in depicting quantum evolution under theinfluence of external noises, but they are doomed to fail whenthe system-environment coupling becomes strong or when theenvironment is a structured medium [4]. Moreover, the widelyused rotating-wave approximation (RWA) ceases to be validat a strong-coupling regime [1,5– 7]. It becomes clear that tocorrectly explain the novel quantum-mechanical phenomenaarising from the strong-coupling physics, the counter-rotatingterms neglected in the RWA must be taken into accountproperly. In addition, the counter-rotating terms are knownto be important in understanding quantum Zeno and anti-Zenoeffects [8– 10]. All the current researches going beyond theRWA and Markov approximation have shown the necessity ofdeveloping a powerful approach to dealing with new physicsarising from the strong coupling between the open quantumsystem of interest and its environment [9– 13].

A stochastic Schrodinger equation, named the non-Markovian quantum state diffusion (QSD) equation, derivedfrom a microscopic model has several advantages over theexact master equations. While the exact master equationsexist only for a few solvable models (see, e.g., Ref. [14]),the exact QSD equation has been established for a genericclass of quantum open systems [15]. However, the applicationsof the exact QSD equation are severely limited unless thistime-nonlocal integro-differential equation can be cast into anumerically implementable time-local form [15– 18].

*Corresponding authors: jqyou@csrc.ac.cn; C.H.Lam@polyu.edu.hk

In real-world problems, solving the exact dynamical equa-tions in a strong-coupling regime is very difficult. Therefore itis imperative to develop an efficient perturbative method thatcan be implemented to solve open system dynamics dictatedby the strong coupling and structured medium.

In this paper, we develop a systematic and efficientapproach to solving the non-Markovian QSD equations foropen quantum systems up to arbitrary orders of noises.The major breakthrough is to convert the non-MarkovianQSD equation into a set of coupled stochastic ordinarydifferential equations (SODEs) which efficiently evaluates aseries expansion of the previously unsolvable O operator up toarbitrarily high orders. The method can be generally applied toan arbitrary finite-state open system coupled to a bosonic bathwith a Lorentzian noise spectrum at zero temperature. As animportant example, our method is used to solve a spin-bosonmodel with a Lorentzian environment at zero temperaturein the strong-coupling regime that is previously intractablewhen real-time quantum dynamics is needed.

II. EXACT QSD EQUATION

To put our discussion into perspective, we first consider ageneric open quantum system with the following Hamiltonian(setting ! = 1)[15]:

Htot = Hsys +!

k

(gkLb†k + g∗

kL†bk) +

!

k

ωkb†kbk, (1)

where Hsys is the Hamiltonian of the system under con-sideration, L is the Lindblad operator, and bk denotes theannihilation operator of the kth mode of the bosonic bath.The state of the bath may be specified by a set of complexnumbers {zk} labeling the (Bargmann) coherent states of allmodes. The function zt that characterizes time-dependentstates of the bath may be defined by the Fourier expansionz∗t ≡ − i

"k g∗

k z∗ke

iωk t . When zk is interpreted as a Gaussianrandom variable, then zt becomes a Gaussian process with

1050-2947/2014/90(2)/022122(8) 022122-1 ©2014 American Physical Society

Detector-induced backaction on thecounting statistics of a double quantumdotZeng-Zhao Li1, Chi-Hang Lam2, Ting Yu3 & J. Q. You1

1Beijing Computational Science Research Center, Beijing 100084, China, 2Department of Applied Physics, Hong Kong PolytechnicUniversity, Hung Hom, Hong Kong, China, 3Center for Controlled Quantum Systems and Department of Physics and EngineeringPhysics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA.

Full counting statistics of electron transport is of fundamental importance for a deeper understanding of theunderlying physical processes in quantum transport in nanoscale devices. The backaction effect from adetector on the nanoscale devices is also essential due to its inevitable presence in experiments. Here weinvestigate the backaction of a charge detector in the form of a quantum point contact (QPC) on thecounting statistics of a biased double quantum dot (DQD). We show that this inevitable QPC-inducedbackaction can have profound effects on the counting statistics under certain conditions, e.g., changing theshot noise from being sub-Poissonian to super-Poissonian, and changing the skewness from being positiveto negative. Also, we show that both Fano factor and skewness can be either enhanced or suppressed byincreasing the energy difference between two single-dot levels of the DQD under the detector-inducedbackaction.

Current fluctuations in nanoscale systems provide key insights into the nature of charge transfer beyondwhat is obtainable from a conductance measurement alone (see, e.g., Refs. 1 and 2 for recent reviews). Anin-depth understanding, however, may require us to go beyond the first-order and even the second-order

current correlation functions (corresponding to the average current and the shot noise respectively) to study thefull counting statistics3,4 which yields all zero-frequency correlation functions at once. Real-time detection of thetunneling of individual electrons, an important step towards experimental measurement of the full countingstatistics, has recently been achieved in various QD systems5–7. In particular, since its measurement in a single QDfor the first time8, counting statistics has become an important experimental tool to examine interaction andcoherence effects in nanoscale systems under out-of-equilibrium conditions9–13. More recently, counting statisticswas applied to characterize correlations in both classical and quantum systems14.

However, a pronounced effect known as backaction on the counting statistics of electron transport2,15 isinevitably introduced during measurements made by even most noninvasive detectors such as a quantum pointcontact (QPC)16. Very recently, such backaction has been investigated experimentally in a single QD16,17. Incontrast to a single QD, a double quantum dot (DQD)18 involves coherent coupling between two different dots,and therefore can be used to demonstrate prominent coherent effects19. The counting statistics for DQDs has beenstudied theoretically20 and experimentally only for noise properties21,22. In a DQD measured by a QPC, both thecurrent and the shot noise of the QPC have been previously investigated23–26. In addition, for a zero-bias DQD, theeffect of charge-detector-induced backaction was studied theoretically27 to explain experimental observations ofinelastic electron tunneling28. However, to the best of our knowledge, the impacts of charge-detector-inducedbackaction on the full counting statistics in these QD systems have not yet been studied.

Here we investigate the counting statistics of electron transport through a biased DQD under measurement bya charge detector. We demonstrate that this inevitable backaction can indeed have profound effects on thecounting statistics under certain conditions for the DQD. In particular, it can change the nature of the shot noisefrom being sub-Poissonian to super-Poissonian and also change the skewness from being positive to negative.Moreover, we show that when the energy difference between two single-dot levels of the DQD increases, bothFano factor and skewness can be either enhanced or suppressed under the detector-induced backaction. TheseQPC-backaction-induced effects are expected to be experimentally observable with currently existing technolo-gies. Apart from a deeper understanding of experimental observations, this study may also shed light on how tocontrol these QD systems using the backaction of a charge detector.

OPEN

SUBJECT AREAS:QUANTUM

INFORMATION

QUANTUM DOTS

Received

16 July 2013

Accepted

4 October 2013

Published

23 October 2013

Correspondence andrequests for materials

should be addressed toJ.Q.Y. (jqyou@csrc.ac.

cn)

SCIENTIFIC REPORTS | 3 : 3026 | DOI: 10.1038/srep03026 1

PHYSICAL REVIEW B 85, 235420 (2012)

Probing the quantum behavior of a nanomechanical resonator coupled to a double quantum dot

Zeng-Zhao Li,1,2 Shi-Hua Ouyang,1,2 Chi-Hang Lam,2 and J. Q. You1,3,*

1Department of Physics, State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China2Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

3Beijing Computational Science Research Center, Beijing 100084, China(Received 7 January 2012; revised manuscript received 1 April 2012; published 7 June 2012)

We propose a current correlation spectrum approach to probe the quantum behaviors of a nanomechanicalresonator (NAMR). The NAMR is coupled to a double quantum dot (DQD), which acts as a quantum transducerand is further coupled to a quantum-point contact (QPC). By measuring the current correlation spectrum of theQPC, shifts in the DQD energy levels, which depend on the phonon occupation in the NAMR, are determined.Quantum behaviors of the NAMR could, thus, be observed. In particular, the cooling of the NAMR into thequantum regime could be examined. In addition, the effects of the coupling strength between the DQD and theNAMR on these energy shifts are studied. We also investigate the impacts on the current correlation spectrum ofthe QPC due to the backaction from the charge detector on the DQD.

DOI: 10.1103/PhysRevB.85.235420 PACS number(s): 85.85.+j, 03.67.Mn, 42.50.Lc

I. INTRODUCTION

The observation of quantum-mechanical behaviors in na-noelectromechanical systems, in particular, nanomechanicalresonators (NAMRs) for testing the basic principles ofquantum mechanics1–3 has become a topic of considerableinterest and activity. Besides their wide range of poten-tial applications,1,2 e.g., serving as ultrasensitive sensorsin high-precision displacement measurements, and detectionof gravitational waves, quantized NAMRs are potentiallyuseful for quantum-information processing. For example,NAMRs may serve as a unique intermediary for transferringquantum information between microwave and optical domainsbecause they can be coupled to electromagnetic waves of anyfrequency.4

At very low temperatures (in the milli-Kelvin range),NAMRs of high-vibration frequencies (gigahertz range) haverecently been experimentally verified to approach the quantumlimit.5–11 However, low-frequency (!100-MHz) mechanicaloscillators have the distinct advantages of high-quality factors,long phonon lifetimes, and large motional state displacements,which are important for future testing of quantum theory12

and other applications. A formidable challenge (see, e.g.,Refs. 5–10) in this field is to detect the quantum quivering(zero-point motion) of an NAMR so as to quantitatively verifywhether it has been cooled into the quantum-mechanicalregime or not. To directly detect the extremely small dis-placements of an NAMR vibrating at gigahertz frequenciesby using available displacement-detection techniques is verydifficult.8–11 The usual position-measurement method is alsoseverely limited by the “zero-point displacement” fluctuationsin the quantum regime,13 although near-Heisenberg-limitedmeasurements have been performed in recent experiments.14

In this paper, we propose a current spectroscopic approachto study the behaviors of an NAMR. It is based on the detectionof the current correlation spectrum in a charge detector, e.g.,a quantum-point contact (QPC), which is indirectly coupledto an NAMR via a double quantum dot (DQD) acting asa quantum-electro-mechanical transducer.15 Based on thisproposal, we show that one can observe the quantum behaviorsof the NAMR and can further verify whether it has been cooled

into the quantum regime. In contrast to a previous approachbased on the superconducting qubit coupled to a cavity, whichinvolves Rabi splitting,16 our proposed setup is expected toprovide better tunability via the gate voltages. Moreover, wealso study the effects of the backaction from the charge detectoron the DQD.

We consider a coupled NAMR-DQD system in the strongdispersive regime where the coupling strength g is muchsmaller than the frequency detuning δ between the DQD andthe NAMR. In this regime, energy quanta in the NAMR areonly virtually exchanged between the DQD and the NAMR.Thus, the coupling of the DQD to the NAMR does not changethe occupation state of the electron in the DQD, but only resultsin phonon-number-dependent frequency shifts in the DQDenergy levels. These shifts are analogous to Stark shifts andcan be further detected by measuring the current correlationspectrum of the QPC.

This paper is organized as follows. In Sec. II, the coupledsystem is explained. The effective dispersive Hamiltonian isderived in Sec. III. The quantum dynamics of the coupledNAMR-DQD system in the presence of the QPC are derivedin Sec. IV. In Sec. V, results related to the observation ofquantum behaviors, the verification of the ground-state coolingof an NAMR, as well as the backaction from the QPC on theDQD are analyzed. Conclusions are given in Sec. VI.

II. THE COUPLED NAMR-DQD-QPC SYSTEM

The device layout of an NAMR capacitively coupled to alateral DQD, which is further measured by a QPC, is presentedin Fig. 1(a). Here, we consider a Coulomb-blockade regimewith strong intradot and interdot Coulomb interactions so thatonly one electron is allowed in the DQD. The states of theDQD are denoted by occupation states |1⟩ and |2⟩, representingone electron in the left and the right dots, respectively. Thestereographical diagram of this device is shown in Fig. 1(b).The lateral DQD is formed by properly tuning the voltagesapplied to the gates. Also, an electron can be injected fromthe left reservoir to the DQD by changing the gate voltages.A metallic NAMR is fabricated above the DQD, and thedisplacement of the NAMR from its equilibrium position

235420-11098-0121/2012/85(23)/235420(6) ©2012 American Physical Society

August 2011

EPL, 95 (2011) 40003 www.epljournal.orgdoi: 10.1209/0295-5075/95/40003

Cooling a nanomechanical resonator by a triple quantum dot

Zeng-Zhao Li1,2, Shi-Hua Ouyang1,2, Chi-Hang Lam2 and J. Q. You1(a)

1Department of Physics and State Key Laboratory of Surface Physics, Fudan University - Shanghai 200433, China2Department of Applied Physics, Hong Kong Polytechnic University - Hung Hom, Hong Kong, China

received 13 May 2011; accepted in final form 7 July 2011published online 3 August 2011

PACS 03.65.Ta – Foundations of quantum mechanics; measurement theoryPACS 42.50.Gy – Effects of atomic coherence on propagation, absorption, and amplification of

light; electromagnetically induced transparency and absorptionPACS 73.21.La – Quantum dots

Abstract – We propose an approach for achieving ground-state cooling of a nanomechanicalresonator (NAMR) capacitively coupled to a triple quantum dot (TQD). This TQD is an electronicanalog of a three-level atom in Λ configuration which allows an electron to enter it via lower-energystates and to exit only from a higher-energy state. By tuning the degeneracy of the two lower-energy states in the TQD, an electron can be trapped in a dark state caused by destructivequantum interference between the two tunneling pathways to the higher-energy state. Therefore,ground-state cooling of an NAMR can be achieved when electrons absorb readily and repeatedlyenergy quanta from the NAMR for excitations.

Copyright c⃝ EPLA, 2011

Introduction. – Nanomechanical resonators(NAMRs) with high resonance frequencies and largequality factors are currently attracting considerableattention owing to their wide range of potential applica-tions (see, e.g., [1,2]). Moreover, quantized NAMRs arepotentially useful for quantum information processing.For example, quantized motion of buckling nanoscalebars has been proposed for qubit implementation [3]and also for creation of quantum entanglement [4–6].However, the dynamics of the NAMRs must approachthe quantum regime, which is usually difficult to arriveat due to interactions with other components as well asthe environments.One way to achieve the quantum regime for an NAMR

is to increase the resonance frequency so that an energyquantum of the NAMR is larger than the thermal energy.Recently, NAMRs based on metallic beams [7] and carbonnanotubes [8] with resonance frequencies about severalhundred megahertz have been developed. However, atemperature lower than 10mK (below the typical dilutionrefrigerator temperature) is required for an NAMR with afrequency of 200MHz to operate in the quantum regime.Therefore, one still needs to cool the NAMR further,for instance via coupling to an optical or an on-chipelectronic system. Various experiments on the coolingof a single NAMR via radiation pressure or dynamical

(a)E-mail: jqyou@fudan.edu.cn

backaction have been reported (see, e.g., [9–15]). Othercooling mechanisms based on a Cooper pair box [16] or athree-level flux qubit [17] with periodic resonant couplinghave also been theoretically proposed. In these approaches,a strong resonant coupling between the NAMR and thequbit is required for the NAMR to achieve its ground-state cooling.In the weak coupling regime, a conventional method for

cooling an NAMR is the sideband cooling approach (see,e.g., refs. [18–23] ), where an NAMR is usually coupledto a two-level system (TLS) with the two states beingelectronic states in quantum dots [18–20], photonic statesin a cavity [21–23], or charge states in superconductingcircuits [24]. In order to achieve ground-state cooling,the resolved-sideband cooling condition ωm≫ Γ (with ωmdenoting the oscillating frequency of the NAMR and Γ thedecay rate of the TLS) must be followed to selectively drivethe lowest sideband of the TLS. However, the frequency ofa typical NAMR is about 100MHz [7,8] which is in generalof the same order of the decay rate of the two-level system,indicating that the resolved-sideband cooling condition isnot easy to be fully fulfilled.In this letter, we propose a different approach to cool

an NAMR, in the non–resolved-sideband cooling regime,via quantum interference in a capacitively coupled triplequantum dot (TQD) (see fig. 1(a)). Here we focus on thestrong Coulomb-blockade regime with at most one electronbeing allowed in the TQD at one time. The TQD acts

40003-p1

DOI: 10.1142/S021798491002269X

March 19, 2010 9:1 WSPC/147-MPLB S021798491002269X

Modern Physics Letters B, Vol. 24, No. 7 (2010) 619–626c⃝ World Scientific Publishing Company

DECOHERENCE OF A DRIVEN QUBIT APPROACHED

VIA THE MASTER EQUATION OF REDFIELD FORM

ZENG-ZHAO LI∗, CHUN-MEI YAO† and XIAN-TING LIANG∗,‡

∗Department of Physics and Institute of Modern Physics,

Ningbo University, Ningbo 315211, China†Department of Physics and Electronic Sciences,

Hunan University of Arts and Science, Changde 415000, China‡xtliang@ustc.edu

Received 22 May 2009Revised 21 September 2009

By solving the master equation of Redfield form, we investigate the decoherence of areduced open-driven qubit. It is shown that when the driving field has not too largefrequency to compare to the Rabi frequency of the qubit system, the decoherence of thequbit is affected by the amplitude but not by the frequency of the driving field. We canprolong the decoherence time of the qubit as we choose an appropriate driving field tooperate the qubit.

Keywords: Driven qubit; decoherence; Redfield equation.

PACS Number(s): 73.63.Kv, 03.65.Yz, 03.67.Lx

1. Introduction

The dynamics of a quantum system interacting with its environment has been in-vestigated for decades. Many methods are used and many results are obtained inthe field. It is an elegant choice, in my opinion, to investigate the reduced quantumsystem by using the phenomenological environment as proposed by Caldeira andLeggett.1 In this approach, the environment of the interested quantum system ismodeled by a bath which is supposed being made up with a set of harmonic oscilla-tors. The dynamics of a driven quantum system interacting with its environment2–5

is a developed model based on the simple open-quantum system. The model of theopen-driven quantum system has many physical correspondences, for example, itcan be used to model electron and proton transfer in external field. In particular, asinvestigated recently, most of the theoretical models of solid qubits, which are cellsfor making quantum computer in future, are supposed being driven by externalfields.

For a general open-driven quantum system, Grifoni et al. gave a deeper investi-gations in Ref. 6. Aiming at quantum manipulation by the driven fields, Thorwartet al. investigated the dynamics of a two-level atom7 and a quantum XOR8 in

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Physics Letters A 373 (2009) 4028–4032

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

The entanglement dynamics of two coupled qubits in different environment

Zeng-Zhao Li, Xian-Ting Liang ∗, Xiao-Yin PanDepartment of Physics and Institute of Modern Physics, Ningbo University, Ningbo 315211, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 July 2009Received in revised form 27 August 2009Accepted 7 September 2009Available online 9 September 2009Communicated by R. Wu

PACS:03.65.Yz03.67.Lx03.67.Mn

Keywords:EntanglementQubitsMaster equationRedfield formEntanglement sudden deathEntanglement sudden birth

In this Letter, the entanglement dynamics of two interacting qubits in a common bath and in twoindependent baths, at finite and zero temperature are investigated. Entanglement sudden death (ESD) andentanglement sudden birth (ESB) are observed when the two qubits are embedded in two independentbaths at finite temperature. At zero temperature, the entanglement of the two qubits may evolveto a steady state with non-zero value when the two qubits are embedded in a common bath, theentanglement sudden birth does not occur when the qubits are embedded in two independent baths.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Entanglement is one of the most puzzling features of quantummechanics [1,2]. Dynamical behavior of entanglement for openquantum systems has attracted a great deal of attention, becauseit is not only an essential distinction from classical physics [3] butalso a central issue in quantum information [4–6]. Usually, theentanglement can be produced by the interactions [7,8]. In thisLetter, we investigate the entanglement dynamics between twoqubits in various environment. Since the entanglement is fragile,it is important to study the problem. During the past decade, thistopic has been investigated extensively [9–17]. Entanglement sud-den death (ESD) and sudden birth (ESB) have been observed whenthe two interacting qubits are embedded in some environments.However, to our knowledge it is not clear what physical mecha-nism results in the phenomena of ESD and ESB.

In this Letter, we first study the entanglement evolutions[18–24] of two interacting qubits based on the spin–boson model.The environment is modelled by either the Ohmic bath or sub-Ohmic bath. We show that when the qubits are embedded in two

* Corresponding author. Tel.: +86 574 87600783; fax: +86 574 87600744.E-mail addresses: xtliang@ustc.edu (X.-T. Liang), panxiaoyin@nbu.edu.cn

(X.-Y. Pan).

independent baths, then ESD and ESB are observed. However, novisible ESD and ESB are observed when the coupled qubits areembedded in a common bath.

At zero temperature, the thermal fluctuation is frozen. To inves-tigate the influence of the environment on entanglement, we nextinvestigate the entanglement dynamics at zero temperature [25].We find that ESD occur only when the two coupled qubits are em-bedded in two independent baths, while no ESB has been observedin these models.

In our investigations, we focus on the regime where the bathcorrelation time is short compared to the relaxation time of thecoupled qubit systems, and the non-Markovian effects are negli-gible. This then allows us to study the entanglement dynamics ofthe coupled qubits by using the master equation of Redfield form,which is a perturbation method based on the Markovian approxi-mation [12–14]. For convenience, we denote the model when thetwo qubits are embedded in (i) a common bath as model A, (ii) intwo independent but different baths as model B1, and (iii) in twoindependent but same baths as model B2.

This Letter is organized as follows. In Section 2 we introducethe qubit models by referring to Ref. [25]. In Section 3 we de-rive the master equation of Redfield form, then we present theentanglement dynamics in Section 4. Finally, a short conclusionsare given in Section 5.

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2009.09.015

Commun. Theor. Phys. (Beijing, China) 51 (2009) pp. 345–348c⃝ Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 2, February 15, 2009

Decoherence of a Double Quantum Dot Charge Qubit Approached with Redfield

Equation∗

LI Zeng-Zhao† and LI Zhi-Qiang

Department of Physics and Institute of Modern Physics, Ningbo University, Ningbo 315211, China

(Received April 14, 2008; Revised July 2, 2008)

Abstract In this paper, we investigate the decoherence time of a double quantum dot (DQD) charge qubit in threekinds of baths through solving dynamics of the qubit. The dynamics of the qubit is investigated with Redfield masterequation. It is shown that the decoherence time of the qubit in Ohmic bath has the same order of magnitude as theexperiments reported. When the environment is modeled with the supra-Ohmic bath the decoherence time of the qubitis shorter than the experimental result. And when modeled with the sub-Ohmic bath the decoherence time of the qubitis longer than the experimental result.

PACS numbers: 73.63.Kv, 03.65.Yz, 03.67.LxKey words: quantum dot, decoherence, redfield equation

1 Introduction

There has been wide interest in the study of quan-tum information science and in building a practical quan-tum computer in particular,[1,2] since Shor showed thata purely quantum mechanical computer can be used toachieve exponential speedup (compared to classical com-puters) in solving the prime factoring problem. Some ofthe most prominent proposals are based on solid statestructures. Among these solid state qubits, the doublequantum dot (DQD)[3−7] charge qubit is a promising one.Two low-energy states are used as the local states |0⟩and |1⟩ in the qubit which can be controlled via exter-nal voltage sources. There are some effective schemes toprepare the initial states and readout the final states ofthe qubit.[8] However, this kind of qubit has a shorter de-coherence time because of a large number of couplings toits environment. Therefore, it is considered that the deco-herence may be the major impediment for the qubit to betaken as the cell of quantum computer. Finding out theprimary origin or dominating mechanism of decoherencefor the qubit is a basal task for overcoming the impedi-ment. In order to find out the mechanism, Fedichkin et

al.[9] and Vorojtsov et al.[10] investigated electron-phonondecoherence with the Born–Markov approximation at alonger time. However, it is shown that the Born–Markovapproximation may be inappropriate at large tunnelingamplitude and at low temperature. In order to avoidthe usage of the approximation, Wu et al.[11] investigatedthe dynamics and decoherence of the qubit in terms of aperturbative treatment based on a special unitary trans-formation. Thorwart et al.[12] and Liang[13] investigatedthe same problem in longer times with an exact numeri-cal path integral, quasi-adiabatic propagator path integral

(QUAPI) method.[14] It is shown that the phonon cou-plings to the qubit play a subordinate role, resulting in thedecoherence of the qubit. So it is still a problem what isthe dominating mechanism of decoherence for the qubit.In this paper, we propose a mechanism that the Ohmic(or supra-Ohmic and sub-Ohmic) noise may be the ma-jor reason to result in decoherence of the qubit. Here, themaster equation with the Redfield form is used to performour proposal.

2 The Model

In this section, we shall introduce the model Hamil-tonian for the double quantum dot and then develop theRedfield equation theory leading to the explicit expressionfor the density matrix element, the evolution of which wefocus on. The DQD charge qubit consists of left and rightdots connected through an interdot tunneling barrier. Dueto the Coulomb blockade, at most one excess electron isallowed to occupy the left and right dot, which defines twobasis vectors |0⟩ and |1⟩. The energy difference betweenthese two states can be controlled by the source-drain volt-age. Neglecting the higher-order tunneling between leadsand the dots, the effective Hamiltonian in the manipula-tion process reads[11,12]

H = Hs + Hb + Hr , (1)

where H is the Hamiltonian of the system (qubit and theenvironment), which consists of three terms, the unper-turbated Hamiltonian Hs is the Hamiltonian of the QDQcharge qubit with eigenvalues ϵn and corresponding eigen-functions |n⟩, Hb is the Hamiltonian for the phonon bath,and Hr describes the electron-phonon interaction that isresponsible for decoherence. More explicitly, we have

Hs = hTcσx , (2)

∗The project supported by National Natural Science Foundation of China under Grant Nos. 10675066 and K.C. Wong Magna Foundation

in Ningbo University†Corresponding author, E-mail: focus-mit@hotmail.com

Study of the decoherence of a double quantum dot charge qubit via theRedfield equation

Zeng-Zhao Li, Xiao-Yin Pan, Xian-Ting Liang !

Department of Physics and Institute of Modern Physics, Ningbo University, Ningbo 315211, China

a r t i c l e i n f o

Article history:Received 14 May 2008Received in revised form3 July 2008Accepted 15 July 2008Available online 23 July 2008

PACS:73.63.Kv03.65.Yz03.67.Lx

Keywords:Quantum dotDecoherenceRedfield equation

a b s t r a c t

By using the Redfield form of the master equation, we investigate the decoherence times of a doublequantum dot charge qubit (DQDCQ) in three different cases, namely when it is coupled to (I) thepiezoelectric coupling phonon bath (PCPB), (II) the deformation coupling phonon bath (DCPB), and (III)the Ohmic bath. It is found that our results for cases (I) and (II) are in the same magnitude with thoseobtained via the exact path integral methods, while for case (III), the decoherence time is in wellagreement with the experimental value.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The double quantum dot (DQD) charge qubit [1–5] is one of thequbits that are considered to be promising candidates for therealization of the building blocks of quantum informationprocessing. Its two low-energy states are denoted as the localstates j0i and j1i, which could be controlled via external voltagesources. There exist some effective schemes to prepare the initialstates and readout the final states of the qubit [6]. It is also knownthat this kind of qubit is coupled to its environment inevitably.Therefore, it is believed that the decoherence might be the centralimpediment for the qubit to be taken as the cell of quantumcomputer. Hence, finding out the primary origin or dominatingmechanism of decoherence for the qubit is a basic task toovercome the difficulty.

The recently experimental realization of the coherent manip-ulation of electronic states in a double-dot system [7,8], whichwas implemented in a GaAs/AlGaAs heterostructure containing atwo-dimensional electron gas, stimulated a lot of theoreticalinterests. Various methods have been tried to study the system.For instance, through density matrix simulation, Fujisawa et al. [9]tried to explain its transportation property. The Born–Markov-type electron–phonon decoherence at large times due to sponta-neous phonon emission of the quantum dot charge qubits was

investigated by Fedichkin et al. [10]. In 2005, Vorojtsov et al. [11]studied the decoherence of the DQD charge qubit by employingthe Born–Markov approximation. Based on a unitary transforma-tion, Wu et al. [12] investigated the decoherence in terms of aperturbative treatment. Thorwart et al. [13] investigated thedecoherence of the DQD charge qubit in a longer time with anumerical exact iterative quasiadiabatic propagator path integral(QUAPI) method [14], while Liang [15] used an iterative tensormultiplication (ITM) method [14] derived from the QUAPI to studydecoherence of a DQD charge qubit in both the piezoelectriccoupling phonon bath (PCPB) and the deformation couplingphonon bath (DCPB). They found that the decoherence times ofthe qubit are shorter than the reported experimental ones whenthe qubit in PCPB and DCPB.

In this paper, we investigate the decoherence times of the(DQD) charge qubit coupled to PCPB, DCPB and the Ohmic bathwith another method, through solving the master equations. Themodel Hamiltonian and the spectral functions for the differentbaths are introduced in Section 2. We then develop the Redfieldform of the master equation and obtain the decoherence times inSection 3. Conclusions are given in the last section.

2. QDQ charge qubit model

The DQD charge qubit consists of left and right dots connectedthrough an interdot tunneling barrier. Due to the Coulomb

ARTICLE IN PRESS

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journal homepage: www.elsevier.com/locate/physe

Physica E

1386-9477/$ - see front matter & 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physe.2008.07.005

! Corresponding author. Tel.: +86 574 87600783; fax: +86 574 87600744.E-mail addresses: liangxianting@nbu.edu.cn, xtliang@ustc.edu (X.-T. Liang).

Physica E 41 (2008) 220–223