Analysis of lense-governed Wignersigned particle quantum dynamics

Paul Ellinghaus1, Josef Weinbub2, Mihail Nedjalkov*,1, and Siegfried Selberherr1

1 Institute for Microelectronics, TU Wien, Austria2 Christian Doppler Laboratory for High Performance TCAD, Institute for Microelectronics, TU Wien, Austria

Received 5 April 2017, revised 3 May 2017, accepted 15 May 2017Published online 30 May 2017

Keywords coherence, electronic transport, particles, quantum dynamics, Wigner formalism

*Corresponding author: e-mail nedjalkov@iue.tuwien.ac.at, Phone: þ43 1 58801-36044, Fax: þ43 1 58801-36099

We present a Wigner signed particles analysis of the lense-governed electron state dynamics based on the quantitativetheory of coherence reformulated in phase space terms.Electrostatic lenses are used for manipulating electron evolutionand are therefore attractive for applications in novel engineeringdisciplines like entangletronics. The signed particle model of

Wigner evolution enables physically intuitive insights into theprocesses maintaining coherence. Both, coherent processes andscattering-caused transitions to classical dynamics are unified bya scattering-aware particle model of the lense-controlled stateevolution. Our approach bridges the fairly new theory ofcoherence with the Wigner signed particle method.

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1 Introduction The refraction of an electron state onthe surface between two domains with different potentials V1

and V2 is described by a notion equivalent to Snell’s law:k1sinu1 ¼ k2sinu2 or sinu2sinu1 ¼ ek1=ek2, where x is alongand y normal to the interface, kj j is the wave vector length, andu is the angle between the wave vector and the y-direction.These simple relations give rise to an approach for electronstate control, where specially shaped potentials (acting aslenses) are used to focus, guide, reshape, or split an electronstate into components. In particular, the electron state can bedecomposed by the lense into well established density peakswhich may be directed to propagate in desired directions. Ofimportance are scenarios where the evolution process retainsinitial coherencewhich, however, is destroyed by decoherenceprocesses like scattering with lattice vibrations, that isphonons. The electron dynamics, being a complicatedinterplay between such processes, can be convenientlyanalyzed in phase space using the Wigner theory [1].

Quantum coherence is the underlying concept ofquantum information disciplines and for emerging quantumengineering disciplines such as entangletronics. It is thussurprising that the rigorous theory for quantification ofcoherence has been suggested only recently [2]. Thedeveloped theory follows the ideas of the correspondingquantification theory of entanglement. Moreover, it has beendemonstrated that the two concepts are quantitatively

equivalent [3], that is, any nonzero amount of coherence in asystem can be converted into an equal amount ofentanglement between that system and another initiallyincoherent one [4]. This means that the two concepts whichdescribe very different physical notions have a commonmathematical foundation. The mathematical approach hasbeen developed in the framework of operator mechanics, interms of Hilbert spaces, eigenbasis sets, and tensor products.The formal quantification of coherence in terms of theinformation theory is based on identifying a set of states Iwith the label “incoherent” and a class of incoherentoperations which map I onto I .

The first goal of our work is to show that the Wignerformalism provides a legitimate theoretical frameworkwhich presents the basic notions of the quantification theoryof coherence in phase space terms. Furthermore, theresulting criteria for coherence in phase space in conjunctionwith a Wigner signed particle method [5] are applied toanalyze a splitting mechanism by the lense electrondynamics. The main results reveal (i) how the splittedparts interact with each-other to maintain the coherence; (ii)how phonons break this interaction, striving to imposeclassical evolution; and (iii) how incoherent states� definedby the novel theory of coherence � are incorporated in thispicture. The analysis is supported by scattering-awareWigner simulations of the splitting process.

Phys. Status Solidi RRL 11, No. 7, 1700102 (2017) / DOI 10.1002/pssr.201700102

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In the next section, we show that the basic notions fromthe quantitative theory of coherence can be presented inphase space by the Weyl–Wigner map.

2 Coherence in phase space terms In the follow-ing, operators are denoted by the “hat” symbol ðÞ, while theindices i; j label the eigenstates of the chosen basis. Forsimplicity, in this section we consider a one-dimensionalphase space x; p.

The Weyl map A x; pð Þ ¼ W Að Þ:

W Að Þ ¼Z

dsdqh

Tr Aei�hðsxþqpÞ

� �e�

i�h sxþqpð Þ ð1Þ

defines an isomorphism from the algebra of operatorsA x; pð Þwith a product and a commutator [,] to the algebra ofphase space functions A x; pð Þ with a non-commutative star(�)-product and a Moyal bracket [,]M given by:

A�B ¼ W A � B� �; i�h A;B½ �M ¼ W A; B

� �� �: ð2Þ

We introduce the nondiagonal eigenvector Wignerfunction f ij x; pð Þ for the eigenvectors aiij of the operator Aof a given physical observable. The following correspond-ences characterize the map:

aiihaj�� �� $ f ij x; pð Þ ¼ 1

hWðjaiihajjÞ ð3aÞ

A aii ¼ aij jaii $ A�f iið Þ x; pð Þ ¼ aif ii x; pð Þ ð3bÞ

r ¼Xb

Pb cbihcb

�� �� $ f w x; pð Þ ¼ 1hW rð Þ ð3cÞ

drdt

¼ 1i�h

H ; r� � $ df w

dt¼ 1

i�hH; f w½ �M ð3dÞ

hAi ¼ Tr rA� � $ hAi ¼

Zf wAð Þ x; pð Þdxdp: ð3eÞ

In the case of A ¼ H x; pð Þ, Eqs. (3a) and (3b) involvestates and energies of the Hamiltonian function H.

b 2 B (cf. Eq. (3c)) labels the pure states rb ¼ cbihcb

�� ��with probabilities Pb, so that the Wigner function f w can beexpressed as a linear combination of pure state Wignerfunctions f bw x; pð Þ ¼ 1

hWðjcbihcbjÞ or via the nondiagonalfunctions f ij. The second equation shown in (3d), is theWigner equation. The relation in Eq. (3e) allows to use theclassical expressions A x; pð Þ for the physical observables.

The equation system (3a)–(3e) demonstrates the fullalgebraic equivalence of the operator and Wignerformalisms. Any operator mechanics results obtainedby algebraic considerations can be expressed in Wignerterms and vice-versa.

The Weyl map allows to transfer the concepts of theinformation theory of coherence in the Wigner phase spacepicture. In particular, the definition of coherence becomes:For a fixed basis aiij one has to use Eq. (3a) for the set ofeigenvector Wigner functions f ij to define the incoherentstates as

s ¼Xi

Pi aiihaij j $ f w;s ¼Xi

Pif ii ð4Þ

where Pi are probabilities. The set of such states is denotedby I . Any other state which cannot be written in this way isdefined coherent [4]. Incoherent states involve only diagonalelements f ii. From Eq. (3a) we conclude that coherence isdirectly related to the existence of nondiagonal elements f ijin the state representation of f w.

Finally the measure for coherence in phase spaceC f wð Þ ¼ minD f w; f w;s

� �where f w;s 2 I is introduced from

the measure for distance D. The latter can be based on thevon Neumann entropy or trace distance [2]. We note that thetime evolution of the linearized von Neumann entropy, orpurity, has been already used for the analysis of thereduction of coherence and time reversibility due to randominterfaces [6].

In this way, the rules and notions of the resource theoryof coherence have been reformulated in terms of phase spacefunctions, establishing theWigner formalism as a legitimateapproach for analysis of coherent processes.

3 Wigner signed particles In the following, wesummarize the concepts of signed particles, needed for ouranalysis. First derived from the Wigner formalism [5],these concepts can be postulated to derive back theWignertheory [7]. Thus the signed particle approach is fullyequivalent to the Wigner formalism1. Furthermore, itenables considerable physical insights into variousquantum mechanical processes. The signed particleapproach: (i) point-like particles with classical features,such as drift over Newtonian (field-less) trajectories andof Boltzmann type scattering carry the quantum informa-tion by their positive or negative sign [8]; (ii) physicalaverages hAi, Eq. (3e), in a phase space region are given bythe sum

Pnsign nð ÞAn for all particles n in this region; (iii)

couples of one positive and one negative particles aregenerated by rules dictated from (3d) and propagate inspace by distinct but fixed momentum which may bechanged only by a scattering event; (iv) particles withopposite sign which meet in the phase space annihilateeach-other. The annihilation property is crucial forunderstanding how scattering strives to impose classical,incoherent behavior.

1 This is in full analogy with the Boltzmann particle model which can beused to derive the Boltzmann equation, but actually contains morephysical information, since the Boltzmann particle model describes alsoprocesses of noise and correlation.

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In the next section, the developed phase space notionsand the signed particle model are applied to analyze lense-governed scattering-aware quantum dynamics.

4 Lense-governed quantum dynamics The con-cept of electrostatic lenses was experimentally firstdemonstrated in 1990 accompanied by computer simula-tions [9, 10]. Coherent effects of focusing and splitting of asingle electron state have been investigated by Wignersimulations [11, 12]. We present scattering-aware simu-lations of lense-controlled electron dynamics. They are usedto support the analysis of the roots of coherence in terms ofsigned particles and validate the intuitive model of theprocess of decoherence. The simulations have beenconducted with VIENNAWD [13] in a two-dimensionalphase space, denoted by x ¼ x; yð Þ, and p ¼ ðpx; pyÞ.

Figure 1 shows a typical electron splitting experiment,where an initial Wigner pure state (corresponding to aminimum uncertainty wave packet Cmin) moves along they-direction towards the lense. The initial Gaussian-shaped envelope of plane waves (the chosen basis eipjx=�h)

identifies the nondiagonal Wigner basis as f ij x;pð Þ ¼eix pi�pjð Þd ðpi þ pjÞ=2� p

� �which has the ability to form

interference patterns. In contrast, incoherent states (cf. Eq.(4)) are represented by d pi � pð Þ which characterizes aclassical particle with fixed momentum and an evendistribution in the phase space. Thus classical andincoherent perceptions become synonyms so that a lossof coherence is equivalent to a transition to classicalbehavior. A major part of the state is already located in theregion of the lense potential after 75 fs of evolution. Thegeneration of particle couples takes place according to (iii)and thus negative particles begin to appear. In the process ofevolution the newborn particles move according to (i) in alldirections of the space, in turn generating couples of newparticles. After 150 fs the electron is outside the region of thelense and is split into two symmetric and well separateddensity peaks (cf. Fig. 2). It is important to note that the two

peaks cannot be considered as two separated electronswhich advance in the phase space in an eventually entangledway. Indeed, the pure state evolution is time reversible sothat if one of the peaks is evolved backwards to the timeorigin, it will become a subset cin 2 Cmin. However, cin isnot a physically admissible state, since the uncertaintyrelations are violated by subsets of the spatial (momentum)variables defining the minimum uncertainty state Cmin.

2

Thus, the task corresponds to a coherent evolution of asingle electron. By virtue of (ii) with A ¼ 1, the density in adomain Dx around x at time t is given by the summation ofthe signs of all particles there. Contributions give the locallygenerated particles which are still in Dx, together withparticles initialized in the past, t0 < t, at different phasespace points x0;p0, whose trajectories, determined byx0; p0; t0, cross Dx at the time of interest t. We note thatdomains with negligible physical density but finitedensity of signed particles also contribute to the dynamics.Furthermore, any changes of the density in the given domaincauses corresponding changes in the rest of the space asimposed by the normalization condition.

Any process which influences the free movement of thesigned particles not only modifies their distribution and thusthe expectation values of the physical quantities (cf. Eq.(3e)), it also reduces the interaction between the phase spaceregions and thus causes decoherence. The latter is oftenassociated and demonstrated by the emblematic model ofquantum Brownian motion, described by the Fokker–Planckequation. Since this equation is obtained as a limit of a moregeneral model, obtained by the inclusion of phononscattering in the Wigner electron evolution [14], weconsider the effect of the phonons on the splitting process.It is important to note that phonons strive to imposeequilibrium behavior and that the equilibrium distribution ofa quantum electron in a potential belongs to the set ofincoherent states (cf. Eq. (4)). Indeed, the Wignerrepresentation of the Gibbs operator is of the formP

bPbfbw, where Pb are given by the equilibrium distribution

Figure 1 Electron density after 75 fs evolution of the initialminimum uncertainty condition. The white shape shows theelectrostatic lense.

Figure 2 Electron density after 150 fs evolution. Two wellestablished density peaks leave the lense.

2 The case that cin is proportional to Cmin, is not possible, since theevolution would recover both peaks.

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function of the energy Eb. The latter, together with theeigenstate Wigner functions f bw, are given by Eq. (3b). Thus,phonons destroy the coherence, striving to transform thequantum state evolution towards an incoherent state.

The random change of the momentum not only preventsa particle to follow it’s coherent behavior impeding theinteraction between adjacent regions: Particles which wouldnever meet in the phase space under coherent conditions,now, in accordance with (iv) have a finite probability toannihilate. Annihilation effectively destroys the effect ofquantum generation. Furthermore, due to the random choiceof the after-scattering direction, generated signed particlesstay around the common place of birth, thus giving rise tolocalization. In this way scattering strives to turn thequantum evolution into a classical diffusive evolution.These processes are well demonstrated in Fig. 3, represent-ing the difference of the coherent and phonon-modifieddensities. The regions where the latter dominates (thedifference is negative) are marked in blue. The chosen

example illustrates well the transition from quantum toclassical behavior.

The quantum initial condition also may be interpretedas an initial classical distribution of electrons. Thegenerated couples of positive and negative particles areannihilated due to scattering, which effectively destroysthe quantum effect of the electric potential. Because ofthe lack of annihilated negative particles, the initialnumber of positive particles (but not the individual initialparticles) remains the same, so that the particledistribution can be described by a classical distributionfunction. This is pronounced even more after 150 fs ofevolution shown in Fig. 4. The spread of the phonon-aware density is restricted as compared to the quantumcounterpart, clearly demonstrating the scattering-inducedlocalization.

5 Summary It is shown that the Wigner formalismoffers not only a legitimate formulation of the recentlydeveloped theory of coherence. The signed particle model oftheWigner evolution, used for analysis of electrostatic lensecontrolled electron dynamics, provides an intuitive physicalpicture of the processes which maintain coherence. Inparticular coherence is associated with the existence ofnondiagonal Wigner basis states. In this picture, scatteringworks against the quantum action of the potential, pushingthe evolution towards classical behavior associated withincoherent states.

Acknowledgements The financial support by the AustrianScience Fund (FWF) project FWF-P29406-N30, the EC SUPER-AID7 project 688101, the Austrian Federal Ministry of Science,Research, and Economy, the National Foundation for Research,Technology and Development, and the Christian Doppler ResearchAssociation (www.cdg.ac.at) is gratefully acknowledged. Thecomputational results presented have been achieved using theVienna Scientific Cluster (VSC).

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Figure 3 Difference between the coherent densities and thoseaffected by phonons after 75 fs evolution. Phonon scatteringmechanisms for silicon are considered, however, the couplingconstants have been increased five times, resulting in a five timesincrease of the scattering frequency, which better illustrates theeffect of decoherence.

Figure 4 150 fs evolution of the difference between the coherentdensities and those affected by phonons. The latter resembles thepoint-like structure of spreading by scattering induced diffusion ofan initial distribution of classical particles.

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