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Decoherence Effects Break Reciprocity in Matter Transport

Decoherence Effects Break Reciprocity in Matter Transport Decoherence E ects Break Reciprocity in Matter Transport P. Bredol, H. Boschker, D. Braak, and J. Mannhart Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany (Dated: September 24, 2021) The decoherence of quantum states de nes the transition between the quantum world and classical physics. Decoherence or, analogously, quantum mechanical collapse events pose fundamental ques- tions regarding the interpretation of quantum mechanics and are technologically relevant because they limit the coherent information processing performed by quantum computers. We have discov- ered that the transition regime enables a novel type of matter transport. Applying this discovery, we present nanoscale devices in which decoherence, modeled by random quantum jumps, produces fundamentally novel phenomena by interrupting the unitary dynamics of electron wave packets. Noncentrosymmetric conductors with mesoscopic length scales act as two-terminal recti ers with unique properties. In these devices, the inelastic interaction of itinerant electrons with impurities acting as electron trapping centers leads to a novel steady state characterized by partial charge separation between the two leads, or, in closed circuits to the generation of persistent currents. The interface between the quantum and the classical worlds therefore provides a novel transport regime of value for the realization of a new category of mesoscopic electronic devices. I. INTRODUCTION tum transport and describe the charge carriers as coher- ent wave packets of nite size (Fig. 1b). Existing for- malisms assessing this regime are usually based on the The measurement process is a mysterious phenomenon Kubo formalism, which ensemble-averages the e ects of at the heart of quantum physics. Starting with the collapses and dephasing processes [20]. Such e ects in- Copenhagen interpretation [1, 2], numerous approaches clude the broadening of energy levels [21] or \washing have attempted to describe this phenomenon. For an out of states" [22] and adding noise to wave functions' overview, see, e.g. [3]. The Copenhagen interpretation, amplitudes and phases [23{25]. These methods nd a for example, states that a measurement causes a sponta- smooth crossover from the quantum regime to the clas- neous collapse of the wave function [4], whereas the de- sical world [26, 27] because decoherence is modeled as a coherence theory attributes the apparent collapse to the decay process phenomenologically. entanglement of the system with its environment [5{7]. Hitherto unknown processes are also considered to cause In contrast, the quantum trajectory approach treats physically real quantum collapses, which for macroscopic an inelastic event (a \quantum jump") as an individual systems create an observer-independent reality [8]. In event that does or does not take place. It breaks time- this work, we assume that physically real decoherence or reversal symmetry by initiating a collapse of the wave collapse events are initiated by inelastic interactions and function and occurs in real space at a distinct location show that they impact mesoscopic transport without a and time [15]. The quantum jumps can equivalently be macroscopic measurement process. This concept under- described by an appropriate Lindblad master equation lies the quantum trajectories approach widely and suc- for the reduced density matrix of the observed system [28] cessfully used in quantum optics [9{14]. We shall show which in our case consists of the electrons in a nanoscopic that this method, applied to solid state systems on the device. The spontaneous breaking of time-reversal invari- nanoscale, predicts nonreciprocal transport properties of ance of the microscopic dynamics does not t the assump- electrons. tions underlying Onsager's reciprocity relations [29, 30]. We analyze the mesoscopic transport of electrons by It is therefore not guaranteed that the transport must be considering the event-type character of inelastic scatter- reciprocal if quantum collapse processes are relevant. ing, which initiates collapse processes [15]. The idea This irreversibility is caused on a microscopic level by behind this approach is presented in Fig. 1. Whereas inelastic scattering events. It is therefore impossible to electron transport in the quantum regime is described model it appropriately by a macroscopic procedure like by propagating extended waves and interference e ects tracing over unobserved degrees of freedom, as imple- (Fig. 1a), it is characterized in the classical world by mented by the methods mentioned above. Instead we scattering events of particles (Fig. 1c). These two trans- shall use a description taking into account those events port regimes are exempli ed by the Landauer-Buttik  er directly (see below). We exploit this intrinsic irreversibil- formalism [16, 17] and the Drude-Sommerfeld model, de- ity to design novel electronic devices. These nonunitary scribed by the Boltzmann equation [18, 19]. We focus quantum electronic devices work as follows. To start, here on the transition regime between classical and quan- the devices are nanostructured into noncentrosymmet- ric shapes. The spatial asymmetry induces a temporal asymmetry; the transit time  of a wave packet de- tr pends on whether the electron travels forward or back- corresponding author, o[email protected] ward through the device as shown in section II. This is arXiv:1912.11948v2 [cond-mat.mes-hall] 23 Sep 2021 2 point of the state evolution is unstable against the col- lapse dynamics or decoherence. Novel time-independent states are established which show charge separation be- tween the two leads, and, for closed systems, persistent currents. (a) (b) (c) Figure 1. Illustration of electron transport through a ring structure in (a) the quantum world, (c) the classical or semi- II. NONRECIPROCAL DYNAMICS OF WAVE classical world, and (b) in the transition regime between both PACKETS FOR i tr worlds. (a) In the quantum world, plane electron waves inter- fere without any events. (c) In the classical world, localized Figure 2 shows the structure of the devices consid- particles undergo numerous inelastic scattering events (red di- ered. The conductors connect two ports, L and R, amonds), which quickly erase the phase memory. (b) In the and are shaped asymmetrically perpendicular to or in transition regime, wave packets interfere and undergo possible the direction of the current ow (transversal and longi- inelastic scattering events (hallled diamond), which cause only a partial loss of coherence. tudinal asymmetry). We compare these conductors to Aharonov{Bohm rings [36, 37] with a transversal asym- metry and to symmetric devices. To nd the electron dynamics in the devices, we solve Schr odinger's equa- because the transit time depends on the interference of tion for the given device geometries by exact diagonal- the wave function due to the unitary dynamics. On the ization of the tight-binding Hamiltonian (see Appendix other hand, decoherence is caused by inelastic scattering C). Electron-electron interactions clearly modify the events as characterized by the average time  between electron dynamics but, following the arguments used in such events. Therefore, the number of quantum jumps per passage of the electron through the device depends Landauer's theory of ballistic transport [16, 17], can be neglected to rst approximation in these nanoscopic de- on  = and is thus nonreciprocal. In one direction the i tr electron transport is more coherent, in the other less so. vices. Likewise, we do not consider additional elastic impurity scattering because this mechanism only adds The intrinsic irreversibility of the quantum mechanical the time-reversal invariant weak localization correction decoherence process leads to nonreciprocity of electron to the ballistic dynamics but does not a ect the break- transport. ing of the time-reversal invariance which is the focus of Here, we demonstrate this device concept by modeling the investigation. the electron transport in a set of noncentrosymmetric The electron waves emitted by L or R into the one- devices [31{33]. These include devices with longitudi- dimensional conductor are described by Gaussian wave nal asymmetry, devices with transverse asymmetry sub- packets with momenta centered at k = =(3a), a being ject to a magnetic eld, and two-path interferometers. F the lattice constant. To consider transport across devices The devices are constructed from idealized model ma- with a similar size as the electron wave packets, small, terials that are described by the single particle picture dispersive wave packets were chosen. The description in and by perfect screening. The calculations were done by applying the Lindblad equation [28], or using a direct, terms of wave packets, which forms the basis of the well- established semiclassical description of electron dynamics stochastic implementation of the wave-function collapse, by computing the quantum trajectory of the particle [9{ [38], is central to our calculations, because Bloch-waves lack time dependence. Since we use the exact eigenfunc- 14]. Both kinds of calculations cover the entire range from coherent quantum transport to classical di usive tions of the single-particle Hamiltonian to compute the time evolution of the states, the results in this section are transport. As we work in the single particle picture, the obtained in the unitary quantum regime. Fermi statistics of the electron plays no role. For a re- cent proposal to implement Lindblad dynamics in the Figure 3 displays the probability that the electron has Meir-Wingreen transport formula, see [34, 35]. Nonre- reached one of the ports of the conductor with the trans- ciprocal transport of wave packets is achieved in all de- verse asymmetry as a function of time t after the electron vices in a well de ned window of scattering rates, yield- emission. As demanded by the unitarity of the scattering ing a peak of the nonreciprocity at a decoherence rate of matrix, these probabilities are reciprocal in the long-time = 2=  1= (see section III). Crucially, in the limits limit [21, 26]. In contrast, the time evolution does depend i tr of large and small scattering rates the transport is recip- on the travel direction [31{33]. The left-right symmetry rocal. Therefore, this device concept illustrates the emer- is broken because the phase shifts of the individual plane gence of new functionality in electron transport right in waves are nonreciprocal. As the phase shifts are also k- the transition regime between quantum physics and clas- dependent, they in uence the temporal behavior of the sical physics. Having established the nonreciprocity for wave packet, which comprises many plane waves with excited states (wave packets), we address in section IV di erent k-values. This leads to  6=  , where L!R R!L the question whether the e ect persists close to thermal  denotes the time spent in the device by a wave L!R equilibrium as described by a thermal density matrix. It packet coming from port L before it leaves through port turns out that the thermal equilibrium taken as a starting R (see also videos 8, 9). 3 localized nature of the interaction, we introduce no con- straint on the nature of the decoherence or collapse pro- cesses, which may be caused e.g. by local lattice distor- tions (local phonons) or the internal degrees of freedom of an impurity giving rise to a localized resonant level. We are interested in a description of the electron dynam- ics, i.e. the time evolution of the reduced density matrix (t) of the (single) electron. This reduced matrix is ob- Figure 2. Symmetric and asymmetric nanoscale conductors. tained from the full density matrix by tracing over the The gure shows a symmetric line, conductors with a trans- unobserved degrees of freedom. The general form of the verse asymmetry and a longitudinal asymmetry, and an ex- resulting time evolution is a time-local master equation ample of an asymmetric Aharonov-Bohm ring (from left to for  (t) that preserves complete positivity [28]: right). d i = [H ;  ] + D[P ]( ): (1) 0 e r r e dt ~ We nd that a nonreciprocal temporal dependence of particle transport is a generic property of many quantum Here, H is the single particle Hamiltonian of the sys- devices with appropriately broken symmetries. Fig. 3a tem without the impurities. The Lindblad dissipators shows a device that is symmetric in the transversal di- D[P ]() depend on the index r of the site at which an rection, but asymmetric in the longitudinal direction. impurity and the concomitant electronic level are located. In these devices, the re ected electrons follow a nonre- They are generally de ned as ciprocal temporal dependence even without an applied magnetic eld. The nonreciprocal temporal dependence y y y ^ ^ ^ ^ ^ ^ ^ has been predicted for asymmetric Rashba rings and D[A]( ) = A A (A A +  A A); (2) e e e e for Aharonov{Bohm rings biased with magnetic elds [31, 32] (Fig. 3c, videos 9a and 9b). Note that the re ec- where A denotes the so-called jump operators. For de- tion probabilities P , P are equal for the systems L!L R!R vices A, B and D we choose hermitean jump operators in Fig. 3b and Fig. 3c, while the transmission proba- P that project the electron wave function onto the site bilities P , P are equal in the system of Fig. 3a L!R R!L y 2 ^ ^ ^ ^ r, i.e. P = jrihrj, with P = P = P . This mechanism r r r r without magnetic eld. This feature re ects the sym- corresponds to a measurement of the occupation of site metries of the scattering matrix under spatial inversion r without read-out. For device C we chose a projection and time-reversal [33]. In contrast, conductors without onto the full subspace spanned by the sites of one inter- adequately broken symmetries (Fig. 3d) show reciprocal ferometer arm. In both cases the localization erases the dynamics. information about the momentum and phase of the elec- tron trapped at the site r. All jump operators are multi- plied with the r-independent rate = = 2= , where r i III. NONRECIPROCAL BEHAVIOR OF WAVE is the mean time between two phase-breaking scat- PACKETS FOR ANY tering events. After release from the trap, the electron state evolves again according to the Schr odinger equation In order to calculate the transport in the transition given by H . regime between unitary quantum physics and classical To quantify the e ects of the decoherence on the trans- physics, we now add decoherence to the dynamics of the port, we prepare the electron at time t = 0 as a Gaussian wave packets. This decoherence is commonly charac- wave packet which enters the system either at port L or terized by the phase-breaking time  . We expect the port R. We integrate the time-dependent local proba- physics of the transition regime to emerge when the de- bility current at site r in the left or right port until L;R fect density causes  to be of the order of the average time t to obtain transmission and re ection probabil- electron transmission time ( +  )=2. As shown, L!R R!L ities (see Appendix B). Reciprocal transport demands the electron transmission time is not necessarily equal P = P for suciently large t . Figure 4 shows L!R R!L n for both transmission directions. In those cases, elec- the nonreciprocity f = P P as function of s L!R R!L trons that pass the device in the slower direction su er computed with Eq. 1 and a set of localization centers stronger decoherence than those that pass in the faster at positions r ; : : : r within the devices (see Appendix 1 n one. The transmission probabilities for both directions, C). For the ballistic regime, i.e. for small , f indeed however, are guaranteed to be equal only if the degree vanishes. of phase-breaking is equal in both directions. Therefore, the transmission probabilities are expected to be possibly However, sorting occurs in the asymmetric devices A, nonreciprocal in devices with nonreciprocal decoherence. B and C in a speci c window of . In this window, elec- The interaction with the environment is mediated by trons that are emitted by L and R reach port R with a one or more sites of the tight-binding lattice that we use higher probability than port L. The window for nonre- to model the device, see Appendix C. Apart from the ciprocal transport matches the inverse device transit time 4 0.8 0.8 L→R L→L R→R R→L 0.4 0.4 L→L L→R R→L R→R 0.0 0.0 0 20 40 0 40 80 time t (fs) time t (fs) (a) (b) 0.8 0.8 L→R L→R R→L R→L 0.4 0.4 L→L L→L R→R R→R 0.0 0.0 0 100 200 0 40 80 time t (fs) time t (fs) (c) (d) Figure 3. Layouts and time resolved transmission and re ection probabilities of unitary conductors with (a) a longitudinal asymmetry, (b) a transversal asymmetry, (c) an asymmetric Aharonov{Bohm loop, and (d) a symmetric device. The devices (b) 2 2 and (c) with areas 28 a and 398 a are penetrated by magnetic uxes  = 1:4 h=e and 0:4 h=e, respectively, where a = 0:3 nm is the unit length of the tight-binding lattice and h=e is the magnetic ux quantum. The structures connect two ports L and R. In the plots, the transmission probabilities P for electrons emitted as Gaussian wave packets (k = =3a,  = 0:9 nm m!n 0 x for (a,b,d) and 24 nm for (c)) by port m to reach port n calculated as a function of time after emission are presented for the scattering-free, unitary transport. The devices' lengths are (a) 1:8 nm, (b) 3:0 nm, (c) 12 nm and (d) 2:4 nm. which is a function of the properties of the chosen wave electrons and the defect or phonon systems. Between the packet. Furthermore, we observe a restoration of reci- \collapse times" t and t the wave function evolves j j+1 procity at high inelastic scattering rates , corresponding unitarily with the Hamiltonian H . At t and t the 0 j j+1 to the crossover to classical di usive transport. Notably, wave function changes according to the corresponding the sorting is completely absent for the symmetric device jump operator P . Two outcomes can be distinguished: D. In the rst case, the wave function is projected onto an eigenstate of the operator P , it becomes jri with proba- The physics behind the nonreciprocal transport hap- bility p = jh jrij as given by the Born rule [4]. In the pening in a well-de ned window of may be intu- second case the \measurement result" is negative: The itively understood by unraveling the Lindblad equa- state is projected onto the orthogonal complement of jri tion Eq. 1 as a stochastic evolution of the wave func- and changes to jr i with probability p = 1 p . For tion. Following this concept, we consider events at t > t , the wave packet evolves again unitarily until it j+1 randomly chosen times t : : : t with a mean spacing 1 n undergoes a second random inelastic scattering event at ht t i =  = 2=. As before, the rate 1= of this j+1 j i i t or leaves the system via the two ports (see video 10). j+2 Poisson process is proportional to the coupling between probability P probability P probability P probability P 5 Due to the stochastic time-evolution, the transmission in the larger framework of transport theory which refers probability P is itself a random variable. To ob- to the states in the vicinity of thermal equilibrium, char- L!R tain its distribution function n = n(P ), we con- acterized by a well-de ned constant temperature on the L!R L!R sider an ensemble of many stochastic quantum trajecto- nanoscopic scale? In equilibrium, electron wave pack- ries. Figure 4a shows the average hP ihP i for ets may occur as uctuations, e.g. by the thermally ex- L!R R!L 6:410 such evolutions for device A. As shown in the cited release of electrons from trapping sites. Indeed, the gure, the results obtained by this stochastic implemen- success of the semi-classical theory of electron transport tation of the wave-function collapse are quantitatively shows unambiguously that physically relevant thermal consistent with the calculation based on the Lindblad- excitations from the Fermi sea are wave packets of nite formalism. size. According to Onsager's reciprocity relations, such uctuations always relax back to the equilibrium state, We use the stochastic approach to illustrate the tran- sition between the di erent regimes of electron trans- which is the starting point of both the semi-classical and quantum theories of linear transport [16{20, 22{27]. port. Figure 5a shows the distribution n(P ; ) of the tr transmission probability P of device A as a function tr For the devices presented here, however, these uctua- of = 2= averaged over the two travel directions. 12 1 tions do not necessarily relax back into the standard ther- For . 510 s , in the unitary quantum regime, the mal equilibrium state. Wave packets that occur as uc- probability for P is peaked at 0:3, indicating that indi- tr tuations around the thermal equilibrium may be sorted vidual electrons leave the system in a state that is a su- by the devices as discussed in Sections II and III and perposition of 70 % being located in the port of origin and 12 1 15 1 therefore lead to a charge accumulation at one side of 30 % in the other port. For 510 s . . 310 s the device. To demonstrate this numerically, we employ | in the transition regime | the distribution varies a minimized version of the models used in the previous rapidly with changing and superpositions of any com- 15 1 17 1 sections. This system, sketched in Fig. 6, is given by a position are found. For 310 s . . 310 s , chain of 9 sites with the site basis vectors j1i;j2i : : :j9i. in the classical regime, the distribution is peaked at A ramp-shaped electric potential is applied to sites j4i, P = 0 and P = 1. Each individual electron is ei- tr tr j5i and j6i to break the symmetry. The potential at the ther re ected or transmitted by the device. States with remaining sites is zero (see Appendix C 3). a coherent superposition are no longer possible. Finally, 17 1 for & 310 s , the probability has a single peak at Because the existence of wave packets is crucial, we P = 0. This is due to the quantum Zeno e ect [39]: tr modify the jump operators in Eq. 1 to now describe the the scattering rate is so high that electrons cannot pass capture of the electron at site r and the subsequent re- through the device. Figure 5b shows the di erence of the lease from this site into the state j (p)i which is a wave distributions belonging to the two directions. We draw packet. The wave packet is generated at r with momen- attention to three features of the graph. First, the largest tum p which is determined by the recoil of the trap which di erence is observed at P = 0:3, the most probable tr now carries the momentum p. For the momentum p transmission in the case without scattering. This im- the value =3a is used because it is in the range of mo- plies that electrons traveling in one of the directions are menta for which relevant interference e ects take place indeed more frequently scattered inelastically than elec- in the device. The following results do not depend on the trons traveling in the opposite direction. Second, pos- choice of p in a qualitative manner. The corresponding itive di erences are found at higher values of P than tr ^ ^ jump operators A are non-hermitean, A = j (p)ihrj r r r negative di erences. This implies nonreciprocity of the with the Lindblad equation average transmission. Finally, the nonreciprocity is eas- ily observed when approaching the classical regime: the d i = [H ;  ] + D[A ]( ): (3) peak at P = 0 is negative and the peak at P = 1 is tr tr 0 e r r e dt ~ positive. Therefore, more electrons are transmitted when r originating from L and more electrons are re ected when Fig. 7a shows the time-dependent probability for the elec- originating from R. The stochastic time evolution of sin- tron being in the left, right or central part of the chain, gle electrons provides an understanding of the dynamics as derived from the dynamics of Eq. 3 with wave packets in the transition regime and con rms the reasoning that being generated at site j5i with the rate = 1:510 Hz. the observed nonreciprocity is caused by direction depen- The initial state is given by the thermal density matrix dent decoherence. E =kT (0) = jE ihE je , where jE i are the eigen- e i i i states of H with energy E and kT = 1 eV. As shown 0 i by Fig. 7, this state is not stable, but evolves into a steady IV. THE CHARGE-SEPARATED STEADY state with a lower entropy and partial charge separation STATE between the two ports. The chain spontaneously devel- ops a charge imbalance with net current zero. The I -V In the previous two sections we have shown that nonre- characteristic contains therefore a new, constant term, ciprocal transport is present when electrons are described as wave packets. How are these results to be interpreted V (I ) = V + RI +O(I ): (4) 0 6 0.03 0.015 11 13 15 17 10 10 10 10 decoherence rate Γ = 2 /τ (1/s) (a) 0.06 0.03 − 0.03 11 13 15 17 10 10 10 10 decoherence rate Γ = 2 /τ (1/s) (b) 0.03 0.015 11 13 15 17 10 10 10 10 decoherence rate Γ = 2 /τ (1/s) (c) Figure 4. Nonreciprocity of the devices introduced in Fig. 3 calculated with the Lindblad formalism (continuous lines) and the Monte-Carlo wave function method (dotted). (a) Nonreciprocal transport through the device A (Fig. 3a) occurs for 12 1 15 1 510 s < < 310 s . As anticipated, both methods yield coincident results. The Monte-Carlo data was obtained from 6:410 realizations of the time evolution. (b) Nonreciprocal transport of device B (orange, Fig. 3b) in the transition regime and return to reciprocal behavior for large and small . The transport through the symmetric device D (gray, Fig. 3d) is fully reciprocal, however. (c) Nonreciprocal transmission of device C (Fig. 3c) as obtained with the Monte-Carlo wave function method. The central assumption underlying Onsager's reciprocity etry is responsible for the charge separation and the relations is therefore not satis ed in the case discussed. circulating current, the state evolution of a chain and Furthermore, this I -V characteristic suggests a net cur- a ring with symmetric potential barriers were calcu- rent to ow in case the chain is bent into a ring, with lated. As expected, these displayed no charge separa- the left and right chain ends being shorted. Figs. 7b and tion (Fig. 14). With hermitean jump operators (Eq. 1), 7c, showing the results of the corresponding calculation, also no charge separation is found in the long time limit, reveal that a circulating persistent current is indeed gen- even when starting from a non-equilibrium density ma- 14 16 erated for  510 Hz : : : 210 Hz, corresponding to trix (Fig. 15). This underlines the importance of the the transition regime. This persistent current induces a existence of wave packets with non-zero momentum for corresponding magnetic moment. Notably, this steady- the charge-separated steady state. To explore whether state current vanishes for very small and large rates . the unconventional steady state generated by Eq. 3 is This behavior is analogous to the behavior of the wave- destroyed by other inelastic processes that may exist and packet transmission probabilities, which are nonrecipro- drive the system towards standard equilibrium, we have cal only in a well-de ned window of the decoherence rate. also added such processes to the master equation and found that the novel steady state persists also in this To investigate whether the asymmetric device geom- non-reciprocity f non-reciprocity f non-reciprocity f s s s 7 Q Q Q L C R j1i j2i j3i j4i j5i j6i j7i j8i j9i Figure 6. Sketch of the tight-binding chain used for most of the calculations in Section IV. The blue circles represent the tight-binding sites. The ll color of the sites sketches the local value of the electrostatic potential. The darker the shade, the higher is the potential at that site. The sites j4i, j5i and (a) j6i implement an asymmetric potential barrier. The central site j5i highlighted by the red arrows is coupled to a trap, which may absorb the particle and release it again as a wave packet. The observables Q ; Q and Q are used to measure L R C the charge density in the left, right and central region of the open system shown, respectively. required in order to achieve recti cation as described by Eq. 4. The optical analog of such devices is presented in [40]. The e ects presented di er from the nonrecipro- cal behavior of standard diodes [41], quantum rings [42], quantum dots [43], chiral structures [44], Weyl semimet- (b) als [45], noncentrosymmetric superconductors [46], and multiferroics [47]. In those cases, the nonreciprocity is Figure 5. Colormaps showing the transmission probability achieved by nonlinear, higher-order processes; the volt- distribution as a function of P and decoherence rate for tr age for I ! 0 vanishes, V = 0. In those cases, also no the device A. (a) The direction-independent mean of the dis- unconventional steady state exists besides the standard tributions for L ! R and R ! L directions illustrate the thermal equilibrium. di erent transport regimes. (b) The di erence of the distri- In conclusion, we have presented a device concept in butions for the two directions shows the direction dependent e ect of decoherence, which leads to nonreciprocal dynamics which nonreciprocal matter transport emerges when the on long time scales in the transition regime. inverse decoherence rate is of the order of the character- istic time for unitary transport through the device. This situation exists exactly in the transition regime between case (see Appendix C 3). quantum physics and classical physics. The nonrecip- rocal matter transport is expected to occur not only in top-down patterned devices but also in molecules with V. DISCUSSION AND OUTLOOK appropriate asymmetric structures and in crystals with suitable lattices. The phenomena found are explorable by experiments on mesoscopic electronic or photonic de- The transition between the quantum and classical vices. The described mechanism underlying nonrecipro- worlds is of intense interest. It harbors fundamental ques- cal, directed dynamics could even be responsible for the tions concerning the appropriate description of decoher- proper operation of biomolecules and thus for living sys- ence and the measurement process. The devices we have tems. discussed operate precisely at the border between these two worlds, because they utilize a small number of ran- dom phase-breaking events that interrupt the otherwise unitary evolution of wave packets. Our work shows that VI. ACKNOWLEDGEMENT in the unitary regime, electrons ow through devices with nonreciprocal velocities if the devices are shaped with ap- propriate asymmetries. Our results have been obtained The authors gratefully acknowledge useful discussions by using several assumptions and are only valid in those with A. Alavi, A. Brataas, T. Kopp, P. Schneeweiss, and cases in which these assumptions apply. In particular, we support by L. Pavka, T. Whittles, and the computer ser- have considered idealized model systems following a strict vice group of MPI-FKF. J. Mannhart acknowledges sup- single particle picture with perfect screening and without port by the Center for Integrated Quantum Science and disorder. For those real materials that may be in uenced Technology (IQST). The numerical calculations were per- by the phenomena described, e.g. lightly doped semi- formed using the Kwant [48] and QuTIP [49, 50] Python conductors, nothing more than nanostructuring a lm is packages. 8 0.45 0.3 0.15 0.6 0.15 − 0 0 − 0.15 − 0.6 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) (a) (b) 14 15 16 10 10 10 rate γ (Hz) (c) Figure 7. (a) Plots showing the time evolution of charges and the total current of the open 9-site chain. The initial state is given by a thermal density matrix with kT = 1 eV. The thermal density matrix is not a steady state of the dynamics and the system transitions within  10 s to the new charge-separated steady state. The initial charge separation is caused by the asymmetric electrostatic potential. This charge di erence is reversed through the nonreciprocal collapse dynamics: the electrochemical potential is no longer the same in both ports. (b) The plot of the total current J as a function of time t and wave-packet generation rate for the closed (shorted) device shows that a nite steady-state current is achieved for 14 16 12 510 Hz : : : 210 Hz. The time scale of the transition to the new steady state is again  10 s. (c) Steady-state current J plotted as function of the scattering rate . The current develops a peak at  210 Hz. This peak is the steady counterpart of the peak shown by the nonreciprocity of the wave-packet transmission probability as function of (Fig. 4a). basd asd asd asd asdas dasd asd asda sdasd asd asd asd basd asd asd asd asdas dasd asd asda sdasd asd asd asd Q Q charge Q L R J (μA) steady current J (μA) 9 [1] N. Bohr, Can quantum-mechanical description of phys- [24] Y. Blanter and M. Buttik  er, Shot noise in mesoscopic ical reality be considered complete? Phys. Rev. 48, 696 conductors, Physics Reports 336, 1 (2000). (1935). [25] A. E. Hansen, A. Kristensen, S. Pedersen, C. B. Srensen, [2] W. Heisenberg, Physik und Philosophie. (Hirzel, and P. E. Lindelof, Mesoscopic decoherence in Aharonov- Stuttgart, 2000). Bohm rings, Phys. Rev. B 64, 045327 (2001). [3] J. Wheeler and W. Zurek, Quantum Theory and Mea- [26] C. Beenakker and H. van Houten, in Semiconductor Het- surement (Princeton University Press, Princeton, 1983). erostructures and Nanostructures , Solid State Physics, [4] J. Von Neumann, Mathematische Grundlagen der Quan- Vol. 44, edited by H. Ehrenreich and D. Turnbull (Aca- tenmechanik, Vol. 38 (Springer-Verlag, 2013). demic Press, 1991) pp. 1{228. [5] H. D. Zeh, On the interpretation of measurement in quan- [27] S. Washburn and R. A. Webb, Quantum transport in tum theory, Foundations of Physics 1, 69 (1970). small disordered samples from the di usive to the ballistic [6] W. H. Zurek, Decoherence, einselection, and the quantum regime, Reports on Progress in Physics 55, 1311 (1992). origins of the classical, Rev. Mod. Phys. 75, 715 (2003). [28] G. Lindblad, On the generators of quantum dynamical [7] H. Everett, Relative state formulation of quantum me- semigroups, Commun.Math. Phys. 48, 119 (1976). chanics, Rev. Mod. Phys. 29, 454 (1957). [29] L. Onsager, Reciprocal relations in irreversible processes. [8] G. C. Ghirardi, A. Rimini, and T. Weber, Uni ed dy- I. Phys. Rev. 37, 405 (1931). namics for microscopic and macroscopic systems, Phys. [30] Y. Tokura and N. Nagaosa, Nonreciprocal responses from Rev. D 34, 470 (1986). non-centrosymmetric quantum materials, Nature Com- [9] N. Gisin, Quantum measurements and stochastic pro- munications 9, 3740 (2018). cesses, Phys. Rev. Lett. 52, 1657 (1984). [31] J. Mannhart, Non-reciprocal interferometers for matter [10] L. Diosi, Quantum stochastic processes as models for waves, Journal of Superconductivity and Novel Mag- state vector reduction, Journal of Physics A: Mathemat- netism 31, 1649 (2018). ical and General 21, 2885 (1988). [32] J. Mannhart and D. Braak, Lossless currents at high tem- [11] J. Dalibard, Y. Castin, and K. Molmer, Wave-function peratures, Journal of Superconductivity and Novel Mag- approach to dissipative processes in quantum optics, netism 32, 17 (2019). Phys. Rev. Lett. 68, 580 (1992). [33] P. Bredol, Nonreciprocity of the wave-packet scattering [12] H. Carmichael, An open systems approach to quantum delay in ballistic two-terminal devices, Phys. Rev. B 103, optics (Springer Science, 1992). 035404 (2021). [13] A. J. Daley, Quantum trajectories and open many-body [34] T. Jin, M. Filippone, and T. Giamarchi, Generic trans- quantum systems, Advances in Physics 63, 77 (2014), port formula for a system driven by markovian reservoirs, https://doi.org/10.1080/00018732.2014.933502. Phys. Rev. B 102, 205131 (2020). [14] K. Molmer, Y. Castin, and J. Dalibard, Monte Carlo [35] T. Jin, J. S. Ferreira, M. Filippone, and T. Giamarchi, wave-function method in quantum optics, J. Opt. Soc. Exact description of quantum stochastic models as quan- Am. B 10, 524 (1993). tum resistors, arXiv 2106.14765 (2021). [15] R. Haag, On the sharpness of localization of individual [36] Y. Aharonov and D. Bohm, Signi cance of electromag- events in space and time, Foundations of Physics 43, 1295 netic potentials in the quantum theory, Phys. Rev. 115, (2013). 485 (1959). [16] R. Landauer, Spatial variation of currents and elds due [37] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. to localized scatterers in metallic conduction, IBM Jour- Laibowitz, Observation of h=e Aharonov-Bohm oscilla- nal of Research and Development 1, 223 (1957). tions in normal-metal rings, Phys. Rev. Lett. 54, 2696 [17] M. Buttik  er, Absence of backscattering in the quantum (1985). Hall e ect in multiprobe conductors, Phys. Rev. B 38, [38] N. W. Ashcroft and N. D. Mermin, Solid State Physics 9375 (1988). (Holt, Rinehart and Winston, New York, 1976). [18] P. Drude, Zur Elektronentheorie der Metalle, Annalen [39] B. Misra and E. C. G. Sudarshan, The Zeno's paradox der Physik 306, 566 (1900). in quantum theory, Journal of Mathematical Physics 18, [19] A. Sommerfeld, Zur Elektronentheorie der Metalle auf 756 (1977), publisher: American Institute of Physics. Grund der Fermischen Statistik, Zeitschrift fur  Physik [40] D. Braak and J. Mannhart, Fermi's golden rule and the 47, 1 (1928). second law of thermodynamics, Foundations of Physics [20] R. Kubo, M. Yokota, and S. Nakajima, Statistical- 50, 1509 (2020). mechanical theory of irreversible processes. ii. re- [41] L. Brillouin, Can the recti er become a thermodynamical sponse to thermal disturbance, Journal of the demon? Phys. Rev. 78, 627 (1950). Physical Society of Japan 12, 1203 (1957), [42] R. Leturcq, D. S anchez, G. G otz, T. Ihn, K. Ensslin, https://doi.org/10.1143/JPSJ.12.1203. D. C. Driscoll, and A. C. Gossard, Magnetic eld sym- [21] Y. Imry, Introduction to mesoscopic physics (Oxford Uni- metry and phase rigidity of the nonlinear conductance in versity Press on Demand, 2002). a ring, Phys. Rev. Lett. 96, 126801 (2006). [22] Y. Gefen, Y. Imry, and M. Y. Azbel, Quantum oscilla- [43] C. A. Marlow, R. P. Taylor, M. Fairbanks, I. Shorubalko, tions and the Aharonov-Bohm e ect for parallel resistors, and H. Linke, Experimental investigation of the break- Phys. Rev. Lett. 52, 129 (1984). down of the Onsager-Casimir relations, Phys. Rev. Lett. [23] A. Stern, Y. Aharonov, and Y. Imry, Phase uncertainty 96, 116801 (2006). and loss of interference: A general picture, Phys. Rev. A [44] G. L. J. A. Rikken, J. F olling, and P. Wyder, Electrical 41, 3436 (1990). magnetochiral anisotropy, Phys. Rev. Lett. 87, 236602 10 (2001). [45] T. Morimoto and N. Nagaosa, Chiral anomaly and giant magnetochiral anisotropy in noncentrosymmetric Weyl semimetals, Phys. Rev. Lett. 117, 146603 (2016). [46] Y. Saito, T. Nojima, and Y. Iwasa, Highly crystalline 2d superconductors, Nature Reviews Materials 2, 16094 (2016). [47] B. B. van Aken, J.-P. Rivera, H. Schmid, and M. Fiebig, Observation of ferrotoroidic domains, Nature 449, 702 (2007). [48] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: a software package for quantum transport, New Journal of Physics 16, 063065 (2014). [49] J. Johansson, P. Nation, and F. Nori, Qutip: An open- source python framework for the dynamics of open quan- tum systems, Computer Physics Communications 183, 1760 (2012). [50] J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Computer Physics Communications 184, 1234 (2013). [51] T. Sch apers, Phase-coherent states, in Nanotechnology, edited by R. Waser (Wiley-VCH, 2008) Chap. 1, pp. 1{ 35. 11 Appendix A: Videos Videos of selected time evolutions. Fig. 8 shows the starting con guration of the time evolution of a wave packet passing a device unitarily with a transverse asym- metry that is shown in the videos. Fig. 9 shows the cor- responding con guration for an asymmetric Aharonov- Bohm ring. Fig. 10 shows the starting image of the video display the wave packet propagating across an asym- metric Aharonov-Bohm ring interrupted by collapse pro- (a) cesses. The videos are available at https://fkf.mpg. de/mannhart. (b) Figure 9. (a) Unitary propagation of a wave packet (blue) across an asymmetric Aharonov-Bohm ring (gold) biased with a magnetic ux penetrating the hole of the ring. The wave packet, presented as j (r; t)j , arrives from the left and passes the ring in a straightforward manner. The data have been obtained by exact diagonalization as described in the main text. (b) The wave packet arriving from the right passes the ring only after having been re ected back and forth. (a) Figure 10. Propagation of a wave packet in a closed system (b) provided by two contacts and an asymmetric Aharonov-Bohm ring biased with a magnetic ux penetrating the hole of the Figure 8. (a) Unitary propagation of a wave packet (blue) ring. The wave packet is represented by j (r; t)j . The uni- across a conducting line (gold) with a transverse asymmetry. tary propagation is interrupted by three collapse processes A magnetic eld is applied perpendicular to the conducting highlighted in purple which correspond to negative and to plane. The wave packet, presented as j (r; t)j , arrives from positive result measurements. The data have been obtained the left and is partially re ected. The data have been obtained by the method described in the main text. by exact diagonalization as described in the main text. (b) The wave packet arriving from the right. 12 Appendix B: Time-dependent wave packets and In the steady state, the current j (x ) of state k;n 0 k;n currents injected from the left reservoir at some point x  R reads ev(k) In this part of the supplement we show that the time- j (x ) = ; (B7) k;n dependent currents of wave packets are not reciprocal in a two-terminal device. Nevertheless, the total transported with the electron velocity v(k) = ~k=m . The total cur- charge is reciprocal in the long time limit. Unitary evo- rent injected from the left and passing the point x is lution yields therefore reciprocal transport probabilities independent from the shape of the electronic wave func- tion in the steady state. l 2 1 I (x ) = e v(k)N = eN dE (E) v (k (E)) L : 0 y We consider a one-dimensional wire along the x-axis on k;n a two-dimensional substrate with a local potential V (^r) (B8) con ned to a nite region in the x/y-plane and a homo- geneous magnetic eld perpendicular to the plane. The Here, N is the number of transversal channels and wire is attached to charge reservoirs (ports) on the left (E) = 2L= (hv(E)) is the one-dimensional density of and the right side with chemical potentials  and  , states per channel. It follows that l r respectively. The single-particle Hamiltonian reads then 2e (with electron charge q = e and mass m ) e e I (x ) = N  : (B9) 0 y l p ^ A(^r) The total current at x contains a contribution from elec- H = + V (^r) + V (^r): (B1) 0 conf 2m trons which are re ected at the central region and those transmitted from the right reservoir: The potential V (^r) con nes the electrons to the wire, conf 0 1 whereas the potential V (x; y) = 0 for jxj > R. With l r Z Z T 2 2 2e the choice A(x; y) = (0; Bx; 0) , we can approximate the l r @ A I (x ) = N dE T dE T : 0 y k(E) k(E) eigenfunctions of Eq. B1 in the wire outside the interac- 1;2 1;2 0 0 tion region by plane waves (x; y) =  (y) (x) with (B10) 1;2 (x) / exp(ikx) The transversal quantum number is denoted by n. We assume for simplicity that the energy To obtain the linear response for stationary states, one 1;2 2 2 E = ~ k =(2m ) of does not depend on it. k e k;n l;r averages T over E. Using Eq. B6 we obtain [51], k(E) 1;2 The asymptotic form of (x) for jxj  R reads in general 2e I (x ) = N T (  ) (B11) 0 y l r ( k(E) ikx l ikx e + R e x  R 1 k 2 2e = (B2) k l l ikx = N T (V V ); (B12) N T e x  R y l r k(E) r ikx 1 T e x  R 2 k where V (V ) are the voltages of the left (right) reservoir. l r = (B3) ikx r ikx N e + R e x  R One sees that because of Eq. B6, the coherent linear re- sponse for stationary states is reciprocal and I (x ) = 0 if V = V . with the normalization factor N / L, where L  R is l r the length of the wire. A wave function with energy E(k) is in general a superposition of incoming waves from the Now we consider wave packets instead of the time- left with amplitudes A and from the right with ampli- independent stationary eigenstates of the system, e.g., of Gaussian form. At the initial time t = 0, the packet tudes A (k  0). Both waves are scattered in the region is localized in the left part of the wire around x  R of V (r) 6= 0, with direction-dependent transmission and in s s with momentum expectation value ~k > 0, moving to re ection amplitudes T , R , s = r; l. The S-matrix of 0 k k the right. the system reads l 2 (xx ) l r in R T +ik x k k 2 1=4 2 ^ 2 S = : (B4) x (x; 0) = ( ) e (B13) l r l T R k k 1 + 2 ^ = dk A (x) + A (x) : (B14) Unitarity of S requires: l;k k l;k k 2 2 s s T = 1 R s = r; l (B5) k k If the packet is suciently narrow, the coecients A 2 2 l r l;k T = T : (B6) k k may be computed using the asymptotic expressions given 13 in Eq. B2 and Eq. B3 to obtain at the point x = x . The result is l r 1 1 1=2 Z Z 2 l N x (kk ) ikx e~ 0 in 2 0 0  i(kk )x p r A = e (B15) j (x ;t) = < dk dk k A A e l;k r r 3=4 r;k r;k m N 0 0 + l r A = (B A R )=T (B16) k k 0   0 l;k l;k l;k i(k+k )x 0  i(k+k )x r r + C A e k A C e r;k r;k 1=2 r;k r;k l 2 l N x ik x (k+k ) +ikx 0 0 in in B = p e e : (B17) 0 0 l;k  i(k k)x i E(k)E(k ) t=~ r ( ) 3=4 + C C e e ; (B25) r;k r;k The state given in Eq. B14 is time-dependent. The asso- r l with C = T A + A R . By comparing Eq. B20 r;k ciated current at some point x  R on the right side of k r;k r;k k and Eq. B25, one sees that the currents are re ection- the interacting region reads at time t l r symmetric, i.e., j (x ; t) = j (x ; t), if T = T and r r l r k k l r R = R . This follows from the re ection symmetry of j (x ; t) = < (t) (x x )p ^ (t) ; (B18) k k l r l r x l the Hamiltonian where <z denotes the real part of z. Now H (p ^ ; p ^ ; x; ^ y ^;A(x; ^ y ^)) x y = R(H ) 1 + 2 iE(k)t=~ (x; t) = dk A (x) + A (x) e ; = H (p ^ ; p ^ ;x; ^ y ^;A(x; ^ y ^)); (B26) l x y k k l;k l;k because then (B19) itH=~ R( )(t) = e R( (0)) and to compute j (x ; t) we may use the asymptotics of l r 1;2 itH=~ (x). This yields = R e (0) = R (t) : (B27) 1 1 Z Z e~ 0 However, if Eq. B26 is not satis ed, we have in general 0 0  i(k k)x j (x ;t) = < dk dk k C C e l r l;k l;k m N l r i l r i# k k T = T e ; R = R e : (B28) 0 0 k k k k 0 0 + i(k+k )x 0  + i(k+k )x r r + A C e k C A e l;k l;k l;k l;k The left and right transmission and re ection coecients 0 0 di er by phase factors, which are allowed by the unitarity + + i(kk )x i E(k)E(k ) t=~ r ( ) + A A e e ; (B20) l;k l;k of the S-matrix S . If the  , # do not vanish, it follows k k k j (x ; t) 6= j (x ; t), i.e., the time-dependent currents r r l r l r with C = T A + A R . We consider now a second are not reciprocal. l;k k l;k l;k k Nevertheless, the total charge transported from the initial state (x; 0) obtained from (x; 0) by re ection r l of x at the origin: (x; 0) = (x; 0), centered around left to the right over a suciently long time equals the r l total charge owing from the right to the left, so that x  R. in the steady state has reciprocal transport characteristics, l 2 (x+x ) in in accordance with the result Eq. B12, which follows ik x 2 1=4 2 (x; 0) = ( ) e (B21) from Eq. B6. The charge of initial state (x; 0) ow- ing through the point x in the time interval [0; T ] to the 1 + 2 right is given by = dk A (x) + A (x) : (B22) r;k k r;k k The state (x; t) has average momentum ~k and Q (x ; T ) = dtj (x ; t); (B29) l!r r l r r 0 moves to the left. The coecients A read r;k whereas the charge of state (x; 0) owing to the left A = A ; (B23) r;k l;k through point x reads + + r l + A = (B A R )=T ; B = B : (B24) k k r;k r;k r;k r;k l;k For this state, we calculate the current at a time t and Q (x ; T ) = dtj (x ; t): (B30) r!l r r r The limit T ! 1 exists (0  Q(x; T )  e) because we consider an open system without periodic boundary con- ditions. Thus the electron may pass the point x and never come back - otherwise Q(x;1) would be either zero or 14 in nity. Reciprocity of the steady state corresponds to Q (x ;1) = Q (x ;1). Because ˆ l!r r r!l r 1...9 itE=~ lim dte = (E=~) + iP ; (B31) T!1 where P denotes the principal part, we obtain 1 Figure 11. Sketch of the tight-binding implementation of the h i 2 2 device A. The circles represent the system sites, the red sites Q (x ;1) = dk C A ; (B32) l!r r l;k l;k correspond to inelastic scattering centers. These centers me- diate the coupling to the environment. and Z moving packet, j i located in the right port R.  (t) h i k 2 2 is then numerically evolved with the Lindblad equation Q (x ;1) = dk C A : (B33) r!l r r;k r;k Eq. 1 to compute the quantities P and P as func- L!R R!L tion of . For the devices A, B, and D we have used nine r l l r di erent scattering centers located at the sites indicated Using the identity T R + T R = 0, which follows k k k k in Fig. 11 with the same coupling = . For the de- from the unitarity of S , one may show that the inte- vice C we regard the entire lower interferometer arm as grands of Eq. B32 and Eq. B33 are the same. Therefore a scattering center. In this case, there is only one projec- tion operator onto the subspace spanned by all sites in Q (x ;1) = Q (x;1); (B34) l!r r r!l the lower interferometer arm. The layout of the devices is further illustrated in Fig. 12. as anticipated. 2. Monte-Carlo wave functions Appendix C: Numerical implementation As in Eq. B18, we calculate the time-dependent prob- 1. Lindblad formalism ability currents at sites r = (25; 0) and r = (25; 0) L R to obtain the total transmitted and re ected charges as in Eq. B29 and Eq. B30. The time is discretized in steps The numerical implementation starts with the tight- of width t = minf2:610 s;  =20g to obtain an ad- binding Hamiltonian: equate temporal resolution in the time integrals of the 0 1 currents and the dynamical collapse events. X X y y @ A H = 4t c c t c c + h.c. ; (C1) The collapse events are computed in the following way: 0 i j i i i2 A collapse occurs with probability p per unit time, the hi;ji2 rate constant of the corresponding Poisson process is then where t denotes the hopping energy, hi; ji a pair of neigh- 1= = p , where  is the average time between collapses c c c boring sites and c , c the creation and annihilation op- occurring at times t and t . The rate constant 1= c;j c;j+1 c is proportional to the coupling between the electron and erators on site i. In the following, we con ne ourselves to the single-particle subspace. For t we take 1 eV, cor- the localized degree of freedom with which it interacts inelastically. The inelastic event itself is treated like a responding to typical dwelling times of several fs in the measurement process: The electron with wave function device. The lattice  de nes the system geometry and (t ) becomes localized (positive measurement result) size. The following paragraphs refer to the systems used c;j and acquires one of the wave functions at time t + in Sections II and III. The implementation details of c;j loc i 2 the minimized system shown in Section IV can be found with probability p = jh (t )j ij . The index i runs i c;j loc over the inelastic scattering centers as depicted in Fig. 11. in Section C 3 of this appendix. The leads are imple- mented as long lines. The systems contain up to 5065 In case of a negative measurement result, the state sites to ensure that during the observation time t the at t +  is the projection of (t ) onto the orthog- n c;j c;j ? i wave packets do not return back into to device. Gaus- onal complement of the sum of all . This loc loc happens with probability 1 p . From the time sian wave packets with  = 3a and 8a and k = =(3a), x 0 i where a = 0:3 nm is the lattice constant of , serve as t + onwards, the state develops according to the time- c;j dependent Schr odinger equation until the next collapse initial wave functions for the devices A, B, D and C, re- spectively. We start from the density matrix of a pure event at time t , where the wave function changes c;j+1 again discontinuously. state  (0) = jihj where ji is either a right-moving wave packet placed in the left port L, j i, or a left- For Figs. 5, 13 we use the following collapse sce- 0 15 nario: = jri, r being one of the nine lattice points trajectories j (t)i obtained in this way yields the same loc f(0; 0); (1; 0); (0;1); (1;1); (1;1)g. In case of a result for the reduced density matrix  (t) as computed negative measurement, the wave function is projected via the Lindblad equation Eq. 1. First, we consider the onto the orthogonal complement of the span of all nine case of a single scattering center at site r. Let's assume jri. that for the l-th trajectory the wave function at time t is We demonstrate now that averaging over the stochastic j (t)i. At time t + t, the wave function reads then < 1 i H j (t)i with probability 1 tp ; 0 l c 00 2 j (t + t)i = jri tp jh (t)jrij ; (C2) l c l ? 00 2 jr i tp (1jh (t)jrij ): c l This entails the following mixed density matrix  at time The accuracy of the calculations of the unitary dynam- t + t belonging to the l-th trajectory up to time t (ne- ics in our numerical implementation is limited by the - glecting terms of order (t) ), nite size of the leads, which sets an upper bound to the observation time t due to recurrence of the waves after (t + t) =(1 tp )j (t)ih (t)j l c l l they have been re ected by the lead ends. In our case, t = 1:58 ps. t [H ;j (t)ih (t)j] 0 l l The statistical accuracy of the collapse dynamics obvi- ously increases with the number of sampled trajectories. ^ ^ + tp P j (t)ih (t)jP c r l l r In the calculations, a minute fraction of trajectories had ^ ^ + tp (1 P )j (t)ih (t)j(1 P ); (C3) c r l l r to be discarded because accumulation of numerical dis- cretization errors led to division-by-zero errors or minute which yields the following equation for the derivative of negative probabilities. For Fig. 5, 6:410 trajectories (t) at time t, were used for each transmission map, 448 trajectories were discarded. The same discretization errors cause the d (t ) i calculated total probability of all trajectories to deviate = [H ;  (t)] 0 l dt ~ from 1. On average, the probability conservation viola- tion equals 510 . For 99 % of the trajectories it is bet- ^ ^ ^ ^ + p 2P  (t)P P  (t)  (t)P : c r l r r l l r ter than 110 , the largest violation being 0:1. A third (C4) systematic error concerns the fact that at t the wave function is not completely zero inside the device, such The average over all trajectories (t) = N  (t) l that Q (t ) +Q (t ) 6= 1, where Q , Q are the charges A 0 B 0 A B therefore satis es the same di erential equation for all in the contacts. Less than 110 of the trajectories times, from which we obtain the Lindblad equation Eq. 1 leave a residual charge Q (t )+Q (t )1 > 110 (all A 0 B 0 for a single scattering center with the identi cation p = numbers referring to the calculations shown in Fig. 5). 1= = =2. The generalization to several scattering c r Figure 13 shows, analogous to Fig. 5, the transmission centers is straightforward. probability histograms of the symmetric device shown in Figs. 3d, 12d. No sorting is observed. The presented stochastic unraveling of the Lindblad equation is not unique. Another stochastic process, equivalent to Eq. 1, considers only positive measure- 3. Minimized model used to calculate steady states ments, the \quantum jump" projects the wave function with probability p always onto one of the , but never loc ? The calculations of steady states in Section IV were onto . To account for null measurements, the deter- loc done using the smaller tight-binding system sketched in ministic evolution of j (t)i between collapse events pro- Fig. 6. The Hamiltonian of this system is given by ceeds not with the hermitean Hamiltonian H but with ^ 0 1 the non-hermitean operator H = H ip P . For this, an 0 c r X X additional normalization of j (t)i during the evolution y y @ A H = (2t + V (i)) c c t c c + h.c. ; 0 i j i i is required because H does not conserve the norm of the i2 hi;ji2 wave function [12]. The corresponding master equation (C6) for (t) reads then where the tight-binding lattice  is given by d i ^ ^ = (H H ) + 2p P P ; (C5) c r r = fj1i;j2i : : :j9ig, hi; ji are again the pairs of dt ~ neighboring sites. Sites that are connected by solid which is again Eq. 1 [14]. lines in Fig. 6 are of course neighbors. In the ring 16 con guration used to demonstrate the existence of such that the chain contains one carrier n = 1=(9a) = 6 1 persistent currents, the sites at the left and right ends 3:710 cm with the lattice constant a = 0:3 nm. The thermal relaxation processes mentioned in the last of the chain are nearest neighbors as well. The c and c are the creation and annihilation operators on site i and paragraph of Section IV are implemented by additional V (i) is the electric potential at site i. The asymmetric jump operators B = jE ihE j, where jE i is the eigen- ij j i i state of the Hamiltonian H with energy E . The corre- potential barrier used for the calculations shown in the 0 i main text is given by sponding rates are given by = exp(E =kT ). The ij th j resulting Lindblad equation reads 3t if i = 4 X X d i < ^ ^ = [H ;  ] + D[A ]( ) + D[B ]( ): 2t if i = 5 0 e r r e ij ij e dt ~ V (i) = : (C7) asym i;j t if i = 6 (C12) 0 else Fig. 16 shows the temporal behavior of the charge im- balance for the chain with the thermalizing jump opera- The symmetric potential barrier used in Fig. 14 is given tors. The initial state  (0) is the unique steady state of by Eq. C12 without the impurity dissipators D[A ](). It is clear that the novel steady state persists even with these 2t if 4  i  6 relaxation processes. V (i) = : (C8) sym 0 else Appendix D: Further gures The two wave packet states used for the jump operators Figures 12, 13 and 14 further illustrate the time evo- are given by lution of the electron waves in several devices. Figure 14 illustrates that the time evolution of the open 1   T i i 3 3 j (p)i = 0; 0; 0; e ; 2; e ; 0; 0; 0 (C9) r and closed chains (Fig. 6) for various degrees of asymme- try. As shown by the gure, the device asymmetry is mandatory for charge separation or persistent currents The electrical current J associated with a density opera- to occur. Figure 15 illustrates for the case of a device tor  is calculated using the velocity operator v ^, with longitudinal asymmetry that a a hermitean jump- operator does not generate a charge-separated steady- J = en tr(v ^); (C10) state. v ^ = [x; ^ H ] (C11) Figure 16 shows that a charge-separated steady state is generated by the open 9-site chain even if additional jump with the elementary charge e and the 1D carrier density operators that drive the system to thermal equilibrium n and the position operator x ^. The density is chosen are present in the Lindblad master equation. 17 L R (a) L R (b) L R (c) L R (d) Figure 12. Layout of the devices sketched in Fig. 3 as modeled numerically. The dots present the sites used in the tight-binding model. 18 (a) (b) Figure 13. Transmission probability histograms calculated as a function of the decoherence rate for the symmetric device as shown in Figs. 3d, 12d and the inset. Nine localization centers in the center of the device mediate the coupling to the environment. The central device has a length of 2:4 nm. The data of the two colormaps have been obtained from 2:910 trajectories. (a) The average over both directions shows one single probability for small , a broad range of probabilities in the transition regime and only P = 0 or P = 1 in the classical limit of large , comparable to Fig. 5a. (b) However, the tr tr di erence plot shows the statistical noise only, lacking any direction dependent features. 19 symmetric − 3 asymmetry 10 0.04 − 2 asymmetry 10 0.02 0.45 0.3 0.00 0.15 − 0.02 0.02 − 0.04 0 0 − 0.02 − 10 − 0.04 − 0.02 0.00 0.02 0.04 − 15 − 14 − 13 − 12 − 11 10 10 10 10 10 time t (s) (a) 0 Q − 40 Q − 80 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) (b) Figure 14. Plots of the time evolution of charges in the open system as shown in Fig. 6, but with fully symmetric (solid lines) and slightly asymmetric (dashed and dotted lines) potential barriers. The dashed line corresponds to a barrier similar to Eq. C7 but with the potentials 1:999t, 2t, 2:001t at sites j4i, j5i, j6i. The dotted line corresponds to potentials 1:99t, 2t, 2:01t at sites j4i, j5i, j6i. (a) While no charge separation occurs in the fully symmetric open system, charge separation starts to occur with increasing barrier asymmetry. (b) No persistent current is owing in the fully symmetric shorted system. Again, with increasing barrier asymmetry, persistent currents occur. The calculational method used is completely identical with the one used in Fig. 7, the only di erence being the symmetry of the barrier. Some of the blue Q curves are not visible because they fully overlap with the corresponding green Q curves. Q Q charge Q L R current J (nA) current J (nA) 20 0.8 C 0.4 0.025 − 0.025 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) Figure 15. Plots showing the time evolution of charges in the device with longitudinal asymmetry (Fig. 3a) as calculated using Eq. 3. The upper panel shows the total charge Q in the left lead, in the right lead Q , and within the central device Q . L R C The bottom panel shows the charge di erence between the left and right leads, which is zero for large and small times. For intermediate times, however, the di erence is nite, revealing a transient charge separation. 0.45 0.3 γ > 0 th 0.15 γ = 0 th 0.04 0.15 0.5 0.02 − 0 0 0.00 − 0.5 − 0.15 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) − 0.02 Figure 16. Plots showing the time evolution of charges and currents in the open 9-site chain with additional jump operators that are speci cally chosen to drive the system towards the state given by the thermal density matrix (see C 3). The rate of the wave-packet generating jump operators and the rate of the additional thermalizing jump operators are given by − 0.04 th 15 14 = 1:510 Hz and = 310 Hz, respectively. The steady-state charge imbalance indicates that the novel steady state th persists even in the presence of thermalizing processes. The dashed lines show the case without the additional jump-operators, which is the same data as shown in Fig. 7a. − 0.04 − 0.02 0.00 0.02 0.04 Q Q charge Q L R Q Q charge Q L R current J (μA) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Condensed Matter arXiv (Cornell University)

Decoherence Effects Break Reciprocity in Matter Transport

Bredol, P.; Boschker, H.; Braak, D.; Mannhart, J.
Condensed Matter , Volume 2021 (1912) – Dec 27, 2019
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10.1103/PhysRevB.104.115413
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Abstract

Decoherence E ects Break Reciprocity in Matter Transport P. Bredol, H. Boschker, D. Braak, and J. Mannhart Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany (Dated: September 24, 2021) The decoherence of quantum states de nes the transition between the quantum world and classical physics. Decoherence or, analogously, quantum mechanical collapse events pose fundamental ques- tions regarding the interpretation of quantum mechanics and are technologically relevant because they limit the coherent information processing performed by quantum computers. We have discov- ered that the transition regime enables a novel type of matter transport. Applying this discovery, we present nanoscale devices in which decoherence, modeled by random quantum jumps, produces fundamentally novel phenomena by interrupting the unitary dynamics of electron wave packets. Noncentrosymmetric conductors with mesoscopic length scales act as two-terminal recti ers with unique properties. In these devices, the inelastic interaction of itinerant electrons with impurities acting as electron trapping centers leads to a novel steady state characterized by partial charge separation between the two leads, or, in closed circuits to the generation of persistent currents. The interface between the quantum and the classical worlds therefore provides a novel transport regime of value for the realization of a new category of mesoscopic electronic devices. I. INTRODUCTION tum transport and describe the charge carriers as coher- ent wave packets of nite size (Fig. 1b). Existing for- malisms assessing this regime are usually based on the The measurement process is a mysterious phenomenon Kubo formalism, which ensemble-averages the e ects of at the heart of quantum physics. Starting with the collapses and dephasing processes [20]. Such e ects in- Copenhagen interpretation [1, 2], numerous approaches clude the broadening of energy levels [21] or \washing have attempted to describe this phenomenon. For an out of states" [22] and adding noise to wave functions' overview, see, e.g. [3]. The Copenhagen interpretation, amplitudes and phases [23{25]. These methods nd a for example, states that a measurement causes a sponta- smooth crossover from the quantum regime to the clas- neous collapse of the wave function [4], whereas the de- sical world [26, 27] because decoherence is modeled as a coherence theory attributes the apparent collapse to the decay process phenomenologically. entanglement of the system with its environment [5{7]. Hitherto unknown processes are also considered to cause In contrast, the quantum trajectory approach treats physically real quantum collapses, which for macroscopic an inelastic event (a \quantum jump") as an individual systems create an observer-independent reality [8]. In event that does or does not take place. It breaks time- this work, we assume that physically real decoherence or reversal symmetry by initiating a collapse of the wave collapse events are initiated by inelastic interactions and function and occurs in real space at a distinct location show that they impact mesoscopic transport without a and time [15]. The quantum jumps can equivalently be macroscopic measurement process. This concept under- described by an appropriate Lindblad master equation lies the quantum trajectories approach widely and suc- for the reduced density matrix of the observed system [28] cessfully used in quantum optics [9{14]. We shall show which in our case consists of the electrons in a nanoscopic that this method, applied to solid state systems on the device. The spontaneous breaking of time-reversal invari- nanoscale, predicts nonreciprocal transport properties of ance of the microscopic dynamics does not t the assump- electrons. tions underlying Onsager's reciprocity relations [29, 30]. We analyze the mesoscopic transport of electrons by It is therefore not guaranteed that the transport must be considering the event-type character of inelastic scatter- reciprocal if quantum collapse processes are relevant. ing, which initiates collapse processes [15]. The idea This irreversibility is caused on a microscopic level by behind this approach is presented in Fig. 1. Whereas inelastic scattering events. It is therefore impossible to electron transport in the quantum regime is described model it appropriately by a macroscopic procedure like by propagating extended waves and interference e ects tracing over unobserved degrees of freedom, as imple- (Fig. 1a), it is characterized in the classical world by mented by the methods mentioned above. Instead we scattering events of particles (Fig. 1c). These two trans- shall use a description taking into account those events port regimes are exempli ed by the Landauer-Buttik  er directly (see below). We exploit this intrinsic irreversibil- formalism [16, 17] and the Drude-Sommerfeld model, de- ity to design novel electronic devices. These nonunitary scribed by the Boltzmann equation [18, 19]. We focus quantum electronic devices work as follows. To start, here on the transition regime between classical and quan- the devices are nanostructured into noncentrosymmet- ric shapes. The spatial asymmetry induces a temporal asymmetry; the transit time  of a wave packet de- tr pends on whether the electron travels forward or back- corresponding author, o[email protected] ward through the device as shown in section II. This is arXiv:1912.11948v2 [cond-mat.mes-hall] 23 Sep 2021 2 point of the state evolution is unstable against the col- lapse dynamics or decoherence. Novel time-independent states are established which show charge separation be- tween the two leads, and, for closed systems, persistent currents. (a) (b) (c) Figure 1. Illustration of electron transport through a ring structure in (a) the quantum world, (c) the classical or semi- II. NONRECIPROCAL DYNAMICS OF WAVE classical world, and (b) in the transition regime between both PACKETS FOR i tr worlds. (a) In the quantum world, plane electron waves inter- fere without any events. (c) In the classical world, localized Figure 2 shows the structure of the devices consid- particles undergo numerous inelastic scattering events (red di- ered. The conductors connect two ports, L and R, amonds), which quickly erase the phase memory. (b) In the and are shaped asymmetrically perpendicular to or in transition regime, wave packets interfere and undergo possible the direction of the current ow (transversal and longi- inelastic scattering events (hallled diamond), which cause only a partial loss of coherence. tudinal asymmetry). We compare these conductors to Aharonov{Bohm rings [36, 37] with a transversal asym- metry and to symmetric devices. To nd the electron dynamics in the devices, we solve Schr odinger's equa- because the transit time depends on the interference of tion for the given device geometries by exact diagonal- the wave function due to the unitary dynamics. On the ization of the tight-binding Hamiltonian (see Appendix other hand, decoherence is caused by inelastic scattering C). Electron-electron interactions clearly modify the events as characterized by the average time  between electron dynamics but, following the arguments used in such events. Therefore, the number of quantum jumps per passage of the electron through the device depends Landauer's theory of ballistic transport [16, 17], can be neglected to rst approximation in these nanoscopic de- on  = and is thus nonreciprocal. In one direction the i tr electron transport is more coherent, in the other less so. vices. Likewise, we do not consider additional elastic impurity scattering because this mechanism only adds The intrinsic irreversibility of the quantum mechanical the time-reversal invariant weak localization correction decoherence process leads to nonreciprocity of electron to the ballistic dynamics but does not a ect the break- transport. ing of the time-reversal invariance which is the focus of Here, we demonstrate this device concept by modeling the investigation. the electron transport in a set of noncentrosymmetric The electron waves emitted by L or R into the one- devices [31{33]. These include devices with longitudi- dimensional conductor are described by Gaussian wave nal asymmetry, devices with transverse asymmetry sub- packets with momenta centered at k = =(3a), a being ject to a magnetic eld, and two-path interferometers. F the lattice constant. To consider transport across devices The devices are constructed from idealized model ma- with a similar size as the electron wave packets, small, terials that are described by the single particle picture dispersive wave packets were chosen. The description in and by perfect screening. The calculations were done by applying the Lindblad equation [28], or using a direct, terms of wave packets, which forms the basis of the well- established semiclassical description of electron dynamics stochastic implementation of the wave-function collapse, by computing the quantum trajectory of the particle [9{ [38], is central to our calculations, because Bloch-waves lack time dependence. Since we use the exact eigenfunc- 14]. Both kinds of calculations cover the entire range from coherent quantum transport to classical di usive tions of the single-particle Hamiltonian to compute the time evolution of the states, the results in this section are transport. As we work in the single particle picture, the obtained in the unitary quantum regime. Fermi statistics of the electron plays no role. For a re- cent proposal to implement Lindblad dynamics in the Figure 3 displays the probability that the electron has Meir-Wingreen transport formula, see [34, 35]. Nonre- reached one of the ports of the conductor with the trans- ciprocal transport of wave packets is achieved in all de- verse asymmetry as a function of time t after the electron vices in a well de ned window of scattering rates, yield- emission. As demanded by the unitarity of the scattering ing a peak of the nonreciprocity at a decoherence rate of matrix, these probabilities are reciprocal in the long-time = 2=  1= (see section III). Crucially, in the limits limit [21, 26]. In contrast, the time evolution does depend i tr of large and small scattering rates the transport is recip- on the travel direction [31{33]. The left-right symmetry rocal. Therefore, this device concept illustrates the emer- is broken because the phase shifts of the individual plane gence of new functionality in electron transport right in waves are nonreciprocal. As the phase shifts are also k- the transition regime between quantum physics and clas- dependent, they in uence the temporal behavior of the sical physics. Having established the nonreciprocity for wave packet, which comprises many plane waves with excited states (wave packets), we address in section IV di erent k-values. This leads to  6=  , where L!R R!L the question whether the e ect persists close to thermal  denotes the time spent in the device by a wave L!R equilibrium as described by a thermal density matrix. It packet coming from port L before it leaves through port turns out that the thermal equilibrium taken as a starting R (see also videos 8, 9). 3 localized nature of the interaction, we introduce no con- straint on the nature of the decoherence or collapse pro- cesses, which may be caused e.g. by local lattice distor- tions (local phonons) or the internal degrees of freedom of an impurity giving rise to a localized resonant level. We are interested in a description of the electron dynam- ics, i.e. the time evolution of the reduced density matrix (t) of the (single) electron. This reduced matrix is ob- Figure 2. Symmetric and asymmetric nanoscale conductors. tained from the full density matrix by tracing over the The gure shows a symmetric line, conductors with a trans- unobserved degrees of freedom. The general form of the verse asymmetry and a longitudinal asymmetry, and an ex- resulting time evolution is a time-local master equation ample of an asymmetric Aharonov-Bohm ring (from left to for  (t) that preserves complete positivity [28]: right). d i = [H ;  ] + D[P ]( ): (1) 0 e r r e dt ~ We nd that a nonreciprocal temporal dependence of particle transport is a generic property of many quantum Here, H is the single particle Hamiltonian of the sys- devices with appropriately broken symmetries. Fig. 3a tem without the impurities. The Lindblad dissipators shows a device that is symmetric in the transversal di- D[P ]() depend on the index r of the site at which an rection, but asymmetric in the longitudinal direction. impurity and the concomitant electronic level are located. In these devices, the re ected electrons follow a nonre- They are generally de ned as ciprocal temporal dependence even without an applied magnetic eld. The nonreciprocal temporal dependence y y y ^ ^ ^ ^ ^ ^ ^ has been predicted for asymmetric Rashba rings and D[A]( ) = A A (A A +  A A); (2) e e e e for Aharonov{Bohm rings biased with magnetic elds [31, 32] (Fig. 3c, videos 9a and 9b). Note that the re ec- where A denotes the so-called jump operators. For de- tion probabilities P , P are equal for the systems L!L R!R vices A, B and D we choose hermitean jump operators in Fig. 3b and Fig. 3c, while the transmission proba- P that project the electron wave function onto the site bilities P , P are equal in the system of Fig. 3a L!R R!L y 2 ^ ^ ^ ^ r, i.e. P = jrihrj, with P = P = P . This mechanism r r r r without magnetic eld. This feature re ects the sym- corresponds to a measurement of the occupation of site metries of the scattering matrix under spatial inversion r without read-out. For device C we chose a projection and time-reversal [33]. In contrast, conductors without onto the full subspace spanned by the sites of one inter- adequately broken symmetries (Fig. 3d) show reciprocal ferometer arm. In both cases the localization erases the dynamics. information about the momentum and phase of the elec- tron trapped at the site r. All jump operators are multi- plied with the r-independent rate = = 2= , where r i III. NONRECIPROCAL BEHAVIOR OF WAVE is the mean time between two phase-breaking scat- PACKETS FOR ANY tering events. After release from the trap, the electron state evolves again according to the Schr odinger equation In order to calculate the transport in the transition given by H . regime between unitary quantum physics and classical To quantify the e ects of the decoherence on the trans- physics, we now add decoherence to the dynamics of the port, we prepare the electron at time t = 0 as a Gaussian wave packets. This decoherence is commonly charac- wave packet which enters the system either at port L or terized by the phase-breaking time  . We expect the port R. We integrate the time-dependent local proba- physics of the transition regime to emerge when the de- bility current at site r in the left or right port until L;R fect density causes  to be of the order of the average time t to obtain transmission and re ection probabil- electron transmission time ( +  )=2. As shown, L!R R!L ities (see Appendix B). Reciprocal transport demands the electron transmission time is not necessarily equal P = P for suciently large t . Figure 4 shows L!R R!L n for both transmission directions. In those cases, elec- the nonreciprocity f = P P as function of s L!R R!L trons that pass the device in the slower direction su er computed with Eq. 1 and a set of localization centers stronger decoherence than those that pass in the faster at positions r ; : : : r within the devices (see Appendix 1 n one. The transmission probabilities for both directions, C). For the ballistic regime, i.e. for small , f indeed however, are guaranteed to be equal only if the degree vanishes. of phase-breaking is equal in both directions. Therefore, the transmission probabilities are expected to be possibly However, sorting occurs in the asymmetric devices A, nonreciprocal in devices with nonreciprocal decoherence. B and C in a speci c window of . In this window, elec- The interaction with the environment is mediated by trons that are emitted by L and R reach port R with a one or more sites of the tight-binding lattice that we use higher probability than port L. The window for nonre- to model the device, see Appendix C. Apart from the ciprocal transport matches the inverse device transit time 4 0.8 0.8 L→R L→L R→R R→L 0.4 0.4 L→L L→R R→L R→R 0.0 0.0 0 20 40 0 40 80 time t (fs) time t (fs) (a) (b) 0.8 0.8 L→R L→R R→L R→L 0.4 0.4 L→L L→L R→R R→R 0.0 0.0 0 100 200 0 40 80 time t (fs) time t (fs) (c) (d) Figure 3. Layouts and time resolved transmission and re ection probabilities of unitary conductors with (a) a longitudinal asymmetry, (b) a transversal asymmetry, (c) an asymmetric Aharonov{Bohm loop, and (d) a symmetric device. The devices (b) 2 2 and (c) with areas 28 a and 398 a are penetrated by magnetic uxes  = 1:4 h=e and 0:4 h=e, respectively, where a = 0:3 nm is the unit length of the tight-binding lattice and h=e is the magnetic ux quantum. The structures connect two ports L and R. In the plots, the transmission probabilities P for electrons emitted as Gaussian wave packets (k = =3a,  = 0:9 nm m!n 0 x for (a,b,d) and 24 nm for (c)) by port m to reach port n calculated as a function of time after emission are presented for the scattering-free, unitary transport. The devices' lengths are (a) 1:8 nm, (b) 3:0 nm, (c) 12 nm and (d) 2:4 nm. which is a function of the properties of the chosen wave electrons and the defect or phonon systems. Between the packet. Furthermore, we observe a restoration of reci- \collapse times" t and t the wave function evolves j j+1 procity at high inelastic scattering rates , corresponding unitarily with the Hamiltonian H . At t and t the 0 j j+1 to the crossover to classical di usive transport. Notably, wave function changes according to the corresponding the sorting is completely absent for the symmetric device jump operator P . Two outcomes can be distinguished: D. In the rst case, the wave function is projected onto an eigenstate of the operator P , it becomes jri with proba- The physics behind the nonreciprocal transport hap- bility p = jh jrij as given by the Born rule [4]. In the pening in a well-de ned window of may be intu- second case the \measurement result" is negative: The itively understood by unraveling the Lindblad equa- state is projected onto the orthogonal complement of jri tion Eq. 1 as a stochastic evolution of the wave func- and changes to jr i with probability p = 1 p . For tion. Following this concept, we consider events at t > t , the wave packet evolves again unitarily until it j+1 randomly chosen times t : : : t with a mean spacing 1 n undergoes a second random inelastic scattering event at ht t i =  = 2=. As before, the rate 1= of this j+1 j i i t or leaves the system via the two ports (see video 10). j+2 Poisson process is proportional to the coupling between probability P probability P probability P probability P 5 Due to the stochastic time-evolution, the transmission in the larger framework of transport theory which refers probability P is itself a random variable. To ob- to the states in the vicinity of thermal equilibrium, char- L!R tain its distribution function n = n(P ), we con- acterized by a well-de ned constant temperature on the L!R L!R sider an ensemble of many stochastic quantum trajecto- nanoscopic scale? In equilibrium, electron wave pack- ries. Figure 4a shows the average hP ihP i for ets may occur as uctuations, e.g. by the thermally ex- L!R R!L 6:410 such evolutions for device A. As shown in the cited release of electrons from trapping sites. Indeed, the gure, the results obtained by this stochastic implemen- success of the semi-classical theory of electron transport tation of the wave-function collapse are quantitatively shows unambiguously that physically relevant thermal consistent with the calculation based on the Lindblad- excitations from the Fermi sea are wave packets of nite formalism. size. According to Onsager's reciprocity relations, such uctuations always relax back to the equilibrium state, We use the stochastic approach to illustrate the tran- sition between the di erent regimes of electron trans- which is the starting point of both the semi-classical and quantum theories of linear transport [16{20, 22{27]. port. Figure 5a shows the distribution n(P ; ) of the tr transmission probability P of device A as a function tr For the devices presented here, however, these uctua- of = 2= averaged over the two travel directions. 12 1 tions do not necessarily relax back into the standard ther- For . 510 s , in the unitary quantum regime, the mal equilibrium state. Wave packets that occur as uc- probability for P is peaked at 0:3, indicating that indi- tr tuations around the thermal equilibrium may be sorted vidual electrons leave the system in a state that is a su- by the devices as discussed in Sections II and III and perposition of 70 % being located in the port of origin and 12 1 15 1 therefore lead to a charge accumulation at one side of 30 % in the other port. For 510 s . . 310 s the device. To demonstrate this numerically, we employ | in the transition regime | the distribution varies a minimized version of the models used in the previous rapidly with changing and superpositions of any com- 15 1 17 1 sections. This system, sketched in Fig. 6, is given by a position are found. For 310 s . . 310 s , chain of 9 sites with the site basis vectors j1i;j2i : : :j9i. in the classical regime, the distribution is peaked at A ramp-shaped electric potential is applied to sites j4i, P = 0 and P = 1. Each individual electron is ei- tr tr j5i and j6i to break the symmetry. The potential at the ther re ected or transmitted by the device. States with remaining sites is zero (see Appendix C 3). a coherent superposition are no longer possible. Finally, 17 1 for & 310 s , the probability has a single peak at Because the existence of wave packets is crucial, we P = 0. This is due to the quantum Zeno e ect [39]: tr modify the jump operators in Eq. 1 to now describe the the scattering rate is so high that electrons cannot pass capture of the electron at site r and the subsequent re- through the device. Figure 5b shows the di erence of the lease from this site into the state j (p)i which is a wave distributions belonging to the two directions. We draw packet. The wave packet is generated at r with momen- attention to three features of the graph. First, the largest tum p which is determined by the recoil of the trap which di erence is observed at P = 0:3, the most probable tr now carries the momentum p. For the momentum p transmission in the case without scattering. This im- the value =3a is used because it is in the range of mo- plies that electrons traveling in one of the directions are menta for which relevant interference e ects take place indeed more frequently scattered inelastically than elec- in the device. The following results do not depend on the trons traveling in the opposite direction. Second, pos- choice of p in a qualitative manner. The corresponding itive di erences are found at higher values of P than tr ^ ^ jump operators A are non-hermitean, A = j (p)ihrj r r r negative di erences. This implies nonreciprocity of the with the Lindblad equation average transmission. Finally, the nonreciprocity is eas- ily observed when approaching the classical regime: the d i = [H ;  ] + D[A ]( ): (3) peak at P = 0 is negative and the peak at P = 1 is tr tr 0 e r r e dt ~ positive. Therefore, more electrons are transmitted when r originating from L and more electrons are re ected when Fig. 7a shows the time-dependent probability for the elec- originating from R. The stochastic time evolution of sin- tron being in the left, right or central part of the chain, gle electrons provides an understanding of the dynamics as derived from the dynamics of Eq. 3 with wave packets in the transition regime and con rms the reasoning that being generated at site j5i with the rate = 1:510 Hz. the observed nonreciprocity is caused by direction depen- The initial state is given by the thermal density matrix dent decoherence. E =kT (0) = jE ihE je , where jE i are the eigen- e i i i states of H with energy E and kT = 1 eV. As shown 0 i by Fig. 7, this state is not stable, but evolves into a steady IV. THE CHARGE-SEPARATED STEADY state with a lower entropy and partial charge separation STATE between the two ports. The chain spontaneously devel- ops a charge imbalance with net current zero. The I -V In the previous two sections we have shown that nonre- characteristic contains therefore a new, constant term, ciprocal transport is present when electrons are described as wave packets. How are these results to be interpreted V (I ) = V + RI +O(I ): (4) 0 6 0.03 0.015 11 13 15 17 10 10 10 10 decoherence rate Γ = 2 /τ (1/s) (a) 0.06 0.03 − 0.03 11 13 15 17 10 10 10 10 decoherence rate Γ = 2 /τ (1/s) (b) 0.03 0.015 11 13 15 17 10 10 10 10 decoherence rate Γ = 2 /τ (1/s) (c) Figure 4. Nonreciprocity of the devices introduced in Fig. 3 calculated with the Lindblad formalism (continuous lines) and the Monte-Carlo wave function method (dotted). (a) Nonreciprocal transport through the device A (Fig. 3a) occurs for 12 1 15 1 510 s < < 310 s . As anticipated, both methods yield coincident results. The Monte-Carlo data was obtained from 6:410 realizations of the time evolution. (b) Nonreciprocal transport of device B (orange, Fig. 3b) in the transition regime and return to reciprocal behavior for large and small . The transport through the symmetric device D (gray, Fig. 3d) is fully reciprocal, however. (c) Nonreciprocal transmission of device C (Fig. 3c) as obtained with the Monte-Carlo wave function method. The central assumption underlying Onsager's reciprocity etry is responsible for the charge separation and the relations is therefore not satis ed in the case discussed. circulating current, the state evolution of a chain and Furthermore, this I -V characteristic suggests a net cur- a ring with symmetric potential barriers were calcu- rent to ow in case the chain is bent into a ring, with lated. As expected, these displayed no charge separa- the left and right chain ends being shorted. Figs. 7b and tion (Fig. 14). With hermitean jump operators (Eq. 1), 7c, showing the results of the corresponding calculation, also no charge separation is found in the long time limit, reveal that a circulating persistent current is indeed gen- even when starting from a non-equilibrium density ma- 14 16 erated for  510 Hz : : : 210 Hz, corresponding to trix (Fig. 15). This underlines the importance of the the transition regime. This persistent current induces a existence of wave packets with non-zero momentum for corresponding magnetic moment. Notably, this steady- the charge-separated steady state. To explore whether state current vanishes for very small and large rates . the unconventional steady state generated by Eq. 3 is This behavior is analogous to the behavior of the wave- destroyed by other inelastic processes that may exist and packet transmission probabilities, which are nonrecipro- drive the system towards standard equilibrium, we have cal only in a well-de ned window of the decoherence rate. also added such processes to the master equation and found that the novel steady state persists also in this To investigate whether the asymmetric device geom- non-reciprocity f non-reciprocity f non-reciprocity f s s s 7 Q Q Q L C R j1i j2i j3i j4i j5i j6i j7i j8i j9i Figure 6. Sketch of the tight-binding chain used for most of the calculations in Section IV. The blue circles represent the tight-binding sites. The ll color of the sites sketches the local value of the electrostatic potential. The darker the shade, the higher is the potential at that site. The sites j4i, j5i and (a) j6i implement an asymmetric potential barrier. The central site j5i highlighted by the red arrows is coupled to a trap, which may absorb the particle and release it again as a wave packet. The observables Q ; Q and Q are used to measure L R C the charge density in the left, right and central region of the open system shown, respectively. required in order to achieve recti cation as described by Eq. 4. The optical analog of such devices is presented in [40]. The e ects presented di er from the nonrecipro- cal behavior of standard diodes [41], quantum rings [42], quantum dots [43], chiral structures [44], Weyl semimet- (b) als [45], noncentrosymmetric superconductors [46], and multiferroics [47]. In those cases, the nonreciprocity is Figure 5. Colormaps showing the transmission probability achieved by nonlinear, higher-order processes; the volt- distribution as a function of P and decoherence rate for tr age for I ! 0 vanishes, V = 0. In those cases, also no the device A. (a) The direction-independent mean of the dis- unconventional steady state exists besides the standard tributions for L ! R and R ! L directions illustrate the thermal equilibrium. di erent transport regimes. (b) The di erence of the distri- In conclusion, we have presented a device concept in butions for the two directions shows the direction dependent e ect of decoherence, which leads to nonreciprocal dynamics which nonreciprocal matter transport emerges when the on long time scales in the transition regime. inverse decoherence rate is of the order of the character- istic time for unitary transport through the device. This situation exists exactly in the transition regime between case (see Appendix C 3). quantum physics and classical physics. The nonrecip- rocal matter transport is expected to occur not only in top-down patterned devices but also in molecules with V. DISCUSSION AND OUTLOOK appropriate asymmetric structures and in crystals with suitable lattices. The phenomena found are explorable by experiments on mesoscopic electronic or photonic de- The transition between the quantum and classical vices. The described mechanism underlying nonrecipro- worlds is of intense interest. It harbors fundamental ques- cal, directed dynamics could even be responsible for the tions concerning the appropriate description of decoher- proper operation of biomolecules and thus for living sys- ence and the measurement process. The devices we have tems. discussed operate precisely at the border between these two worlds, because they utilize a small number of ran- dom phase-breaking events that interrupt the otherwise unitary evolution of wave packets. Our work shows that VI. ACKNOWLEDGEMENT in the unitary regime, electrons ow through devices with nonreciprocal velocities if the devices are shaped with ap- propriate asymmetries. Our results have been obtained The authors gratefully acknowledge useful discussions by using several assumptions and are only valid in those with A. Alavi, A. Brataas, T. Kopp, P. Schneeweiss, and cases in which these assumptions apply. In particular, we support by L. Pavka, T. Whittles, and the computer ser- have considered idealized model systems following a strict vice group of MPI-FKF. J. Mannhart acknowledges sup- single particle picture with perfect screening and without port by the Center for Integrated Quantum Science and disorder. For those real materials that may be in uenced Technology (IQST). The numerical calculations were per- by the phenomena described, e.g. lightly doped semi- formed using the Kwant [48] and QuTIP [49, 50] Python conductors, nothing more than nanostructuring a lm is packages. 8 0.45 0.3 0.15 0.6 0.15 − 0 0 − 0.15 − 0.6 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) (a) (b) 14 15 16 10 10 10 rate γ (Hz) (c) Figure 7. (a) Plots showing the time evolution of charges and the total current of the open 9-site chain. The initial state is given by a thermal density matrix with kT = 1 eV. The thermal density matrix is not a steady state of the dynamics and the system transitions within  10 s to the new charge-separated steady state. The initial charge separation is caused by the asymmetric electrostatic potential. This charge di erence is reversed through the nonreciprocal collapse dynamics: the electrochemical potential is no longer the same in both ports. (b) The plot of the total current J as a function of time t and wave-packet generation rate for the closed (shorted) device shows that a nite steady-state current is achieved for 14 16 12 510 Hz : : : 210 Hz. The time scale of the transition to the new steady state is again  10 s. (c) Steady-state current J plotted as function of the scattering rate . The current develops a peak at  210 Hz. This peak is the steady counterpart of the peak shown by the nonreciprocity of the wave-packet transmission probability as function of (Fig. 4a). basd asd asd asd asdas dasd asd asda sdasd asd asd asd basd asd asd asd asdas dasd asd asda sdasd asd asd asd Q Q charge Q L R J (μA) steady current J (μA) 9 [1] N. Bohr, Can quantum-mechanical description of phys- [24] Y. Blanter and M. Buttik  er, Shot noise in mesoscopic ical reality be considered complete? Phys. Rev. 48, 696 conductors, Physics Reports 336, 1 (2000). (1935). [25] A. E. Hansen, A. Kristensen, S. Pedersen, C. B. Srensen, [2] W. Heisenberg, Physik und Philosophie. (Hirzel, and P. E. Lindelof, Mesoscopic decoherence in Aharonov- Stuttgart, 2000). Bohm rings, Phys. Rev. B 64, 045327 (2001). [3] J. Wheeler and W. Zurek, Quantum Theory and Mea- [26] C. Beenakker and H. van Houten, in Semiconductor Het- surement (Princeton University Press, Princeton, 1983). erostructures and Nanostructures , Solid State Physics, [4] J. Von Neumann, Mathematische Grundlagen der Quan- Vol. 44, edited by H. Ehrenreich and D. Turnbull (Aca- tenmechanik, Vol. 38 (Springer-Verlag, 2013). demic Press, 1991) pp. 1{228. [5] H. D. Zeh, On the interpretation of measurement in quan- [27] S. Washburn and R. A. Webb, Quantum transport in tum theory, Foundations of Physics 1, 69 (1970). small disordered samples from the di usive to the ballistic [6] W. H. Zurek, Decoherence, einselection, and the quantum regime, Reports on Progress in Physics 55, 1311 (1992). origins of the classical, Rev. Mod. Phys. 75, 715 (2003). [28] G. Lindblad, On the generators of quantum dynamical [7] H. Everett, Relative state formulation of quantum me- semigroups, Commun.Math. Phys. 48, 119 (1976). chanics, Rev. Mod. Phys. 29, 454 (1957). [29] L. Onsager, Reciprocal relations in irreversible processes. [8] G. C. Ghirardi, A. Rimini, and T. Weber, Uni ed dy- I. Phys. Rev. 37, 405 (1931). namics for microscopic and macroscopic systems, Phys. [30] Y. Tokura and N. Nagaosa, Nonreciprocal responses from Rev. D 34, 470 (1986). non-centrosymmetric quantum materials, Nature Com- [9] N. Gisin, Quantum measurements and stochastic pro- munications 9, 3740 (2018). cesses, Phys. Rev. Lett. 52, 1657 (1984). [31] J. Mannhart, Non-reciprocal interferometers for matter [10] L. Diosi, Quantum stochastic processes as models for waves, Journal of Superconductivity and Novel Mag- state vector reduction, Journal of Physics A: Mathemat- netism 31, 1649 (2018). ical and General 21, 2885 (1988). [32] J. Mannhart and D. Braak, Lossless currents at high tem- [11] J. Dalibard, Y. Castin, and K. Molmer, Wave-function peratures, Journal of Superconductivity and Novel Mag- approach to dissipative processes in quantum optics, netism 32, 17 (2019). Phys. Rev. Lett. 68, 580 (1992). [33] P. Bredol, Nonreciprocity of the wave-packet scattering [12] H. Carmichael, An open systems approach to quantum delay in ballistic two-terminal devices, Phys. Rev. B 103, optics (Springer Science, 1992). 035404 (2021). [13] A. J. Daley, Quantum trajectories and open many-body [34] T. Jin, M. Filippone, and T. Giamarchi, Generic trans- quantum systems, Advances in Physics 63, 77 (2014), port formula for a system driven by markovian reservoirs, https://doi.org/10.1080/00018732.2014.933502. Phys. Rev. B 102, 205131 (2020). [14] K. Molmer, Y. Castin, and J. Dalibard, Monte Carlo [35] T. Jin, J. S. Ferreira, M. Filippone, and T. Giamarchi, wave-function method in quantum optics, J. Opt. Soc. Exact description of quantum stochastic models as quan- Am. B 10, 524 (1993). tum resistors, arXiv 2106.14765 (2021). [15] R. Haag, On the sharpness of localization of individual [36] Y. Aharonov and D. Bohm, Signi cance of electromag- events in space and time, Foundations of Physics 43, 1295 netic potentials in the quantum theory, Phys. Rev. 115, (2013). 485 (1959). [16] R. Landauer, Spatial variation of currents and elds due [37] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. to localized scatterers in metallic conduction, IBM Jour- Laibowitz, Observation of h=e Aharonov-Bohm oscilla- nal of Research and Development 1, 223 (1957). tions in normal-metal rings, Phys. Rev. Lett. 54, 2696 [17] M. Buttik  er, Absence of backscattering in the quantum (1985). Hall e ect in multiprobe conductors, Phys. Rev. B 38, [38] N. W. Ashcroft and N. D. Mermin, Solid State Physics 9375 (1988). (Holt, Rinehart and Winston, New York, 1976). [18] P. Drude, Zur Elektronentheorie der Metalle, Annalen [39] B. Misra and E. C. G. Sudarshan, The Zeno's paradox der Physik 306, 566 (1900). in quantum theory, Journal of Mathematical Physics 18, [19] A. Sommerfeld, Zur Elektronentheorie der Metalle auf 756 (1977), publisher: American Institute of Physics. Grund der Fermischen Statistik, Zeitschrift fur  Physik [40] D. Braak and J. Mannhart, Fermi's golden rule and the 47, 1 (1928). second law of thermodynamics, Foundations of Physics [20] R. Kubo, M. Yokota, and S. Nakajima, Statistical- 50, 1509 (2020). mechanical theory of irreversible processes. ii. re- [41] L. Brillouin, Can the recti er become a thermodynamical sponse to thermal disturbance, Journal of the demon? Phys. Rev. 78, 627 (1950). Physical Society of Japan 12, 1203 (1957), [42] R. Leturcq, D. S anchez, G. G otz, T. Ihn, K. Ensslin, https://doi.org/10.1143/JPSJ.12.1203. D. C. Driscoll, and A. C. Gossard, Magnetic eld sym- [21] Y. Imry, Introduction to mesoscopic physics (Oxford Uni- metry and phase rigidity of the nonlinear conductance in versity Press on Demand, 2002). a ring, Phys. Rev. Lett. 96, 126801 (2006). [22] Y. Gefen, Y. Imry, and M. Y. Azbel, Quantum oscilla- [43] C. A. Marlow, R. P. Taylor, M. Fairbanks, I. Shorubalko, tions and the Aharonov-Bohm e ect for parallel resistors, and H. Linke, Experimental investigation of the break- Phys. Rev. Lett. 52, 129 (1984). down of the Onsager-Casimir relations, Phys. Rev. Lett. [23] A. Stern, Y. Aharonov, and Y. Imry, Phase uncertainty 96, 116801 (2006). and loss of interference: A general picture, Phys. Rev. A [44] G. L. J. A. Rikken, J. F olling, and P. Wyder, Electrical 41, 3436 (1990). magnetochiral anisotropy, Phys. Rev. Lett. 87, 236602 10 (2001). [45] T. Morimoto and N. Nagaosa, Chiral anomaly and giant magnetochiral anisotropy in noncentrosymmetric Weyl semimetals, Phys. Rev. Lett. 117, 146603 (2016). [46] Y. Saito, T. Nojima, and Y. Iwasa, Highly crystalline 2d superconductors, Nature Reviews Materials 2, 16094 (2016). [47] B. B. van Aken, J.-P. Rivera, H. Schmid, and M. Fiebig, Observation of ferrotoroidic domains, Nature 449, 702 (2007). [48] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: a software package for quantum transport, New Journal of Physics 16, 063065 (2014). [49] J. Johansson, P. Nation, and F. Nori, Qutip: An open- source python framework for the dynamics of open quan- tum systems, Computer Physics Communications 183, 1760 (2012). [50] J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Computer Physics Communications 184, 1234 (2013). [51] T. Sch apers, Phase-coherent states, in Nanotechnology, edited by R. Waser (Wiley-VCH, 2008) Chap. 1, pp. 1{ 35. 11 Appendix A: Videos Videos of selected time evolutions. Fig. 8 shows the starting con guration of the time evolution of a wave packet passing a device unitarily with a transverse asym- metry that is shown in the videos. Fig. 9 shows the cor- responding con guration for an asymmetric Aharonov- Bohm ring. Fig. 10 shows the starting image of the video display the wave packet propagating across an asym- metric Aharonov-Bohm ring interrupted by collapse pro- (a) cesses. The videos are available at https://fkf.mpg. de/mannhart. (b) Figure 9. (a) Unitary propagation of a wave packet (blue) across an asymmetric Aharonov-Bohm ring (gold) biased with a magnetic ux penetrating the hole of the ring. The wave packet, presented as j (r; t)j , arrives from the left and passes the ring in a straightforward manner. The data have been obtained by exact diagonalization as described in the main text. (b) The wave packet arriving from the right passes the ring only after having been re ected back and forth. (a) Figure 10. Propagation of a wave packet in a closed system (b) provided by two contacts and an asymmetric Aharonov-Bohm ring biased with a magnetic ux penetrating the hole of the Figure 8. (a) Unitary propagation of a wave packet (blue) ring. The wave packet is represented by j (r; t)j . The uni- across a conducting line (gold) with a transverse asymmetry. tary propagation is interrupted by three collapse processes A magnetic eld is applied perpendicular to the conducting highlighted in purple which correspond to negative and to plane. The wave packet, presented as j (r; t)j , arrives from positive result measurements. The data have been obtained the left and is partially re ected. The data have been obtained by the method described in the main text. by exact diagonalization as described in the main text. (b) The wave packet arriving from the right. 12 Appendix B: Time-dependent wave packets and In the steady state, the current j (x ) of state k;n 0 k;n currents injected from the left reservoir at some point x  R reads ev(k) In this part of the supplement we show that the time- j (x ) = ; (B7) k;n dependent currents of wave packets are not reciprocal in a two-terminal device. Nevertheless, the total transported with the electron velocity v(k) = ~k=m . The total cur- charge is reciprocal in the long time limit. Unitary evo- rent injected from the left and passing the point x is lution yields therefore reciprocal transport probabilities independent from the shape of the electronic wave func- tion in the steady state. l 2 1 I (x ) = e v(k)N = eN dE (E) v (k (E)) L : 0 y We consider a one-dimensional wire along the x-axis on k;n a two-dimensional substrate with a local potential V (^r) (B8) con ned to a nite region in the x/y-plane and a homo- geneous magnetic eld perpendicular to the plane. The Here, N is the number of transversal channels and wire is attached to charge reservoirs (ports) on the left (E) = 2L= (hv(E)) is the one-dimensional density of and the right side with chemical potentials  and  , states per channel. It follows that l r respectively. The single-particle Hamiltonian reads then 2e (with electron charge q = e and mass m ) e e I (x ) = N  : (B9) 0 y l p ^ A(^r) The total current at x contains a contribution from elec- H = + V (^r) + V (^r): (B1) 0 conf 2m trons which are re ected at the central region and those transmitted from the right reservoir: The potential V (^r) con nes the electrons to the wire, conf 0 1 whereas the potential V (x; y) = 0 for jxj > R. With l r Z Z T 2 2 2e the choice A(x; y) = (0; Bx; 0) , we can approximate the l r @ A I (x ) = N dE T dE T : 0 y k(E) k(E) eigenfunctions of Eq. B1 in the wire outside the interac- 1;2 1;2 0 0 tion region by plane waves (x; y) =  (y) (x) with (B10) 1;2 (x) / exp(ikx) The transversal quantum number is denoted by n. We assume for simplicity that the energy To obtain the linear response for stationary states, one 1;2 2 2 E = ~ k =(2m ) of does not depend on it. k e k;n l;r averages T over E. Using Eq. B6 we obtain [51], k(E) 1;2 The asymptotic form of (x) for jxj  R reads in general 2e I (x ) = N T (  ) (B11) 0 y l r ( k(E) ikx l ikx e + R e x  R 1 k 2 2e = (B2) k l l ikx = N T (V V ); (B12) N T e x  R y l r k(E) r ikx 1 T e x  R 2 k where V (V ) are the voltages of the left (right) reservoir. l r = (B3) ikx r ikx N e + R e x  R One sees that because of Eq. B6, the coherent linear re- sponse for stationary states is reciprocal and I (x ) = 0 if V = V . with the normalization factor N / L, where L  R is l r the length of the wire. A wave function with energy E(k) is in general a superposition of incoming waves from the Now we consider wave packets instead of the time- left with amplitudes A and from the right with ampli- independent stationary eigenstates of the system, e.g., of Gaussian form. At the initial time t = 0, the packet tudes A (k  0). Both waves are scattered in the region is localized in the left part of the wire around x  R of V (r) 6= 0, with direction-dependent transmission and in s s with momentum expectation value ~k > 0, moving to re ection amplitudes T , R , s = r; l. The S-matrix of 0 k k the right. the system reads l 2 (xx ) l r in R T +ik x k k 2 1=4 2 ^ 2 S = : (B4) x (x; 0) = ( ) e (B13) l r l T R k k 1 + 2 ^ = dk A (x) + A (x) : (B14) Unitarity of S requires: l;k k l;k k 2 2 s s T = 1 R s = r; l (B5) k k If the packet is suciently narrow, the coecients A 2 2 l r l;k T = T : (B6) k k may be computed using the asymptotic expressions given 13 in Eq. B2 and Eq. B3 to obtain at the point x = x . The result is l r 1 1 1=2 Z Z 2 l N x (kk ) ikx e~ 0 in 2 0 0  i(kk )x p r A = e (B15) j (x ;t) = < dk dk k A A e l;k r r 3=4 r;k r;k m N 0 0 + l r A = (B A R )=T (B16) k k 0   0 l;k l;k l;k i(k+k )x 0  i(k+k )x r r + C A e k A C e r;k r;k 1=2 r;k r;k l 2 l N x ik x (k+k ) +ikx 0 0 in in B = p e e : (B17) 0 0 l;k  i(k k)x i E(k)E(k ) t=~ r ( ) 3=4 + C C e e ; (B25) r;k r;k The state given in Eq. B14 is time-dependent. The asso- r l with C = T A + A R . By comparing Eq. B20 r;k ciated current at some point x  R on the right side of k r;k r;k k and Eq. B25, one sees that the currents are re ection- the interacting region reads at time t l r symmetric, i.e., j (x ; t) = j (x ; t), if T = T and r r l r k k l r R = R . This follows from the re ection symmetry of j (x ; t) = < (t) (x x )p ^ (t) ; (B18) k k l r l r x l the Hamiltonian where <z denotes the real part of z. Now H (p ^ ; p ^ ; x; ^ y ^;A(x; ^ y ^)) x y = R(H ) 1 + 2 iE(k)t=~ (x; t) = dk A (x) + A (x) e ; = H (p ^ ; p ^ ;x; ^ y ^;A(x; ^ y ^)); (B26) l x y k k l;k l;k because then (B19) itH=~ R( )(t) = e R( (0)) and to compute j (x ; t) we may use the asymptotics of l r 1;2 itH=~ (x). This yields = R e (0) = R (t) : (B27) 1 1 Z Z e~ 0 However, if Eq. B26 is not satis ed, we have in general 0 0  i(k k)x j (x ;t) = < dk dk k C C e l r l;k l;k m N l r i l r i# k k T = T e ; R = R e : (B28) 0 0 k k k k 0 0 + i(k+k )x 0  + i(k+k )x r r + A C e k C A e l;k l;k l;k l;k The left and right transmission and re ection coecients 0 0 di er by phase factors, which are allowed by the unitarity + + i(kk )x i E(k)E(k ) t=~ r ( ) + A A e e ; (B20) l;k l;k of the S-matrix S . If the  , # do not vanish, it follows k k k j (x ; t) 6= j (x ; t), i.e., the time-dependent currents r r l r l r with C = T A + A R . We consider now a second are not reciprocal. l;k k l;k l;k k Nevertheless, the total charge transported from the initial state (x; 0) obtained from (x; 0) by re ection r l of x at the origin: (x; 0) = (x; 0), centered around left to the right over a suciently long time equals the r l total charge owing from the right to the left, so that x  R. in the steady state has reciprocal transport characteristics, l 2 (x+x ) in in accordance with the result Eq. B12, which follows ik x 2 1=4 2 (x; 0) = ( ) e (B21) from Eq. B6. The charge of initial state (x; 0) ow- ing through the point x in the time interval [0; T ] to the 1 + 2 right is given by = dk A (x) + A (x) : (B22) r;k k r;k k The state (x; t) has average momentum ~k and Q (x ; T ) = dtj (x ; t); (B29) l!r r l r r 0 moves to the left. The coecients A read r;k whereas the charge of state (x; 0) owing to the left A = A ; (B23) r;k l;k through point x reads + + r l + A = (B A R )=T ; B = B : (B24) k k r;k r;k r;k r;k l;k For this state, we calculate the current at a time t and Q (x ; T ) = dtj (x ; t): (B30) r!l r r r The limit T ! 1 exists (0  Q(x; T )  e) because we consider an open system without periodic boundary con- ditions. Thus the electron may pass the point x and never come back - otherwise Q(x;1) would be either zero or 14 in nity. Reciprocity of the steady state corresponds to Q (x ;1) = Q (x ;1). Because ˆ l!r r r!l r 1...9 itE=~ lim dte = (E=~) + iP ; (B31) T!1 where P denotes the principal part, we obtain 1 Figure 11. Sketch of the tight-binding implementation of the h i 2 2 device A. The circles represent the system sites, the red sites Q (x ;1) = dk C A ; (B32) l!r r l;k l;k correspond to inelastic scattering centers. These centers me- diate the coupling to the environment. and Z moving packet, j i located in the right port R.  (t) h i k 2 2 is then numerically evolved with the Lindblad equation Q (x ;1) = dk C A : (B33) r!l r r;k r;k Eq. 1 to compute the quantities P and P as func- L!R R!L tion of . For the devices A, B, and D we have used nine r l l r di erent scattering centers located at the sites indicated Using the identity T R + T R = 0, which follows k k k k in Fig. 11 with the same coupling = . For the de- from the unitarity of S , one may show that the inte- vice C we regard the entire lower interferometer arm as grands of Eq. B32 and Eq. B33 are the same. Therefore a scattering center. In this case, there is only one projec- tion operator onto the subspace spanned by all sites in Q (x ;1) = Q (x;1); (B34) l!r r r!l the lower interferometer arm. The layout of the devices is further illustrated in Fig. 12. as anticipated. 2. Monte-Carlo wave functions Appendix C: Numerical implementation As in Eq. B18, we calculate the time-dependent prob- 1. Lindblad formalism ability currents at sites r = (25; 0) and r = (25; 0) L R to obtain the total transmitted and re ected charges as in Eq. B29 and Eq. B30. The time is discretized in steps The numerical implementation starts with the tight- of width t = minf2:610 s;  =20g to obtain an ad- binding Hamiltonian: equate temporal resolution in the time integrals of the 0 1 currents and the dynamical collapse events. X X y y @ A H = 4t c c t c c + h.c. ; (C1) The collapse events are computed in the following way: 0 i j i i i2 A collapse occurs with probability p per unit time, the hi;ji2 rate constant of the corresponding Poisson process is then where t denotes the hopping energy, hi; ji a pair of neigh- 1= = p , where  is the average time between collapses c c c boring sites and c , c the creation and annihilation op- occurring at times t and t . The rate constant 1= c;j c;j+1 c is proportional to the coupling between the electron and erators on site i. In the following, we con ne ourselves to the single-particle subspace. For t we take 1 eV, cor- the localized degree of freedom with which it interacts inelastically. The inelastic event itself is treated like a responding to typical dwelling times of several fs in the measurement process: The electron with wave function device. The lattice  de nes the system geometry and (t ) becomes localized (positive measurement result) size. The following paragraphs refer to the systems used c;j and acquires one of the wave functions at time t + in Sections II and III. The implementation details of c;j loc i 2 the minimized system shown in Section IV can be found with probability p = jh (t )j ij . The index i runs i c;j loc over the inelastic scattering centers as depicted in Fig. 11. in Section C 3 of this appendix. The leads are imple- mented as long lines. The systems contain up to 5065 In case of a negative measurement result, the state sites to ensure that during the observation time t the at t +  is the projection of (t ) onto the orthog- n c;j c;j ? i wave packets do not return back into to device. Gaus- onal complement of the sum of all . This loc loc happens with probability 1 p . From the time sian wave packets with  = 3a and 8a and k = =(3a), x 0 i where a = 0:3 nm is the lattice constant of , serve as t + onwards, the state develops according to the time- c;j dependent Schr odinger equation until the next collapse initial wave functions for the devices A, B, D and C, re- spectively. We start from the density matrix of a pure event at time t , where the wave function changes c;j+1 again discontinuously. state  (0) = jihj where ji is either a right-moving wave packet placed in the left port L, j i, or a left- For Figs. 5, 13 we use the following collapse sce- 0 15 nario: = jri, r being one of the nine lattice points trajectories j (t)i obtained in this way yields the same loc f(0; 0); (1; 0); (0;1); (1;1); (1;1)g. In case of a result for the reduced density matrix  (t) as computed negative measurement, the wave function is projected via the Lindblad equation Eq. 1. First, we consider the onto the orthogonal complement of the span of all nine case of a single scattering center at site r. Let's assume jri. that for the l-th trajectory the wave function at time t is We demonstrate now that averaging over the stochastic j (t)i. At time t + t, the wave function reads then < 1 i H j (t)i with probability 1 tp ; 0 l c 00 2 j (t + t)i = jri tp jh (t)jrij ; (C2) l c l ? 00 2 jr i tp (1jh (t)jrij ): c l This entails the following mixed density matrix  at time The accuracy of the calculations of the unitary dynam- t + t belonging to the l-th trajectory up to time t (ne- ics in our numerical implementation is limited by the - glecting terms of order (t) ), nite size of the leads, which sets an upper bound to the observation time t due to recurrence of the waves after (t + t) =(1 tp )j (t)ih (t)j l c l l they have been re ected by the lead ends. In our case, t = 1:58 ps. t [H ;j (t)ih (t)j] 0 l l The statistical accuracy of the collapse dynamics obvi- ously increases with the number of sampled trajectories. ^ ^ + tp P j (t)ih (t)jP c r l l r In the calculations, a minute fraction of trajectories had ^ ^ + tp (1 P )j (t)ih (t)j(1 P ); (C3) c r l l r to be discarded because accumulation of numerical dis- cretization errors led to division-by-zero errors or minute which yields the following equation for the derivative of negative probabilities. For Fig. 5, 6:410 trajectories (t) at time t, were used for each transmission map, 448 trajectories were discarded. The same discretization errors cause the d (t ) i calculated total probability of all trajectories to deviate = [H ;  (t)] 0 l dt ~ from 1. On average, the probability conservation viola- tion equals 510 . For 99 % of the trajectories it is bet- ^ ^ ^ ^ + p 2P  (t)P P  (t)  (t)P : c r l r r l l r ter than 110 , the largest violation being 0:1. A third (C4) systematic error concerns the fact that at t the wave function is not completely zero inside the device, such The average over all trajectories (t) = N  (t) l that Q (t ) +Q (t ) 6= 1, where Q , Q are the charges A 0 B 0 A B therefore satis es the same di erential equation for all in the contacts. Less than 110 of the trajectories times, from which we obtain the Lindblad equation Eq. 1 leave a residual charge Q (t )+Q (t )1 > 110 (all A 0 B 0 for a single scattering center with the identi cation p = numbers referring to the calculations shown in Fig. 5). 1= = =2. The generalization to several scattering c r Figure 13 shows, analogous to Fig. 5, the transmission centers is straightforward. probability histograms of the symmetric device shown in Figs. 3d, 12d. No sorting is observed. The presented stochastic unraveling of the Lindblad equation is not unique. Another stochastic process, equivalent to Eq. 1, considers only positive measure- 3. Minimized model used to calculate steady states ments, the \quantum jump" projects the wave function with probability p always onto one of the , but never loc ? The calculations of steady states in Section IV were onto . To account for null measurements, the deter- loc done using the smaller tight-binding system sketched in ministic evolution of j (t)i between collapse events pro- Fig. 6. The Hamiltonian of this system is given by ceeds not with the hermitean Hamiltonian H but with ^ 0 1 the non-hermitean operator H = H ip P . For this, an 0 c r X X additional normalization of j (t)i during the evolution y y @ A H = (2t + V (i)) c c t c c + h.c. ; 0 i j i i is required because H does not conserve the norm of the i2 hi;ji2 wave function [12]. The corresponding master equation (C6) for (t) reads then where the tight-binding lattice  is given by d i ^ ^ = (H H ) + 2p P P ; (C5) c r r = fj1i;j2i : : :j9ig, hi; ji are again the pairs of dt ~ neighboring sites. Sites that are connected by solid which is again Eq. 1 [14]. lines in Fig. 6 are of course neighbors. In the ring 16 con guration used to demonstrate the existence of such that the chain contains one carrier n = 1=(9a) = 6 1 persistent currents, the sites at the left and right ends 3:710 cm with the lattice constant a = 0:3 nm. The thermal relaxation processes mentioned in the last of the chain are nearest neighbors as well. The c and c are the creation and annihilation operators on site i and paragraph of Section IV are implemented by additional V (i) is the electric potential at site i. The asymmetric jump operators B = jE ihE j, where jE i is the eigen- ij j i i state of the Hamiltonian H with energy E . The corre- potential barrier used for the calculations shown in the 0 i main text is given by sponding rates are given by = exp(E =kT ). The ij th j resulting Lindblad equation reads 3t if i = 4 X X d i < ^ ^ = [H ;  ] + D[A ]( ) + D[B ]( ): 2t if i = 5 0 e r r e ij ij e dt ~ V (i) = : (C7) asym i;j t if i = 6 (C12) 0 else Fig. 16 shows the temporal behavior of the charge im- balance for the chain with the thermalizing jump opera- The symmetric potential barrier used in Fig. 14 is given tors. The initial state  (0) is the unique steady state of by Eq. C12 without the impurity dissipators D[A ](). It is clear that the novel steady state persists even with these 2t if 4  i  6 relaxation processes. V (i) = : (C8) sym 0 else Appendix D: Further gures The two wave packet states used for the jump operators Figures 12, 13 and 14 further illustrate the time evo- are given by lution of the electron waves in several devices. Figure 14 illustrates that the time evolution of the open 1   T i i 3 3 j (p)i = 0; 0; 0; e ; 2; e ; 0; 0; 0 (C9) r and closed chains (Fig. 6) for various degrees of asymme- try. As shown by the gure, the device asymmetry is mandatory for charge separation or persistent currents The electrical current J associated with a density opera- to occur. Figure 15 illustrates for the case of a device tor  is calculated using the velocity operator v ^, with longitudinal asymmetry that a a hermitean jump- operator does not generate a charge-separated steady- J = en tr(v ^); (C10) state. v ^ = [x; ^ H ] (C11) Figure 16 shows that a charge-separated steady state is generated by the open 9-site chain even if additional jump with the elementary charge e and the 1D carrier density operators that drive the system to thermal equilibrium n and the position operator x ^. The density is chosen are present in the Lindblad master equation. 17 L R (a) L R (b) L R (c) L R (d) Figure 12. Layout of the devices sketched in Fig. 3 as modeled numerically. The dots present the sites used in the tight-binding model. 18 (a) (b) Figure 13. Transmission probability histograms calculated as a function of the decoherence rate for the symmetric device as shown in Figs. 3d, 12d and the inset. Nine localization centers in the center of the device mediate the coupling to the environment. The central device has a length of 2:4 nm. The data of the two colormaps have been obtained from 2:910 trajectories. (a) The average over both directions shows one single probability for small , a broad range of probabilities in the transition regime and only P = 0 or P = 1 in the classical limit of large , comparable to Fig. 5a. (b) However, the tr tr di erence plot shows the statistical noise only, lacking any direction dependent features. 19 symmetric − 3 asymmetry 10 0.04 − 2 asymmetry 10 0.02 0.45 0.3 0.00 0.15 − 0.02 0.02 − 0.04 0 0 − 0.02 − 10 − 0.04 − 0.02 0.00 0.02 0.04 − 15 − 14 − 13 − 12 − 11 10 10 10 10 10 time t (s) (a) 0 Q − 40 Q − 80 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) (b) Figure 14. Plots of the time evolution of charges in the open system as shown in Fig. 6, but with fully symmetric (solid lines) and slightly asymmetric (dashed and dotted lines) potential barriers. The dashed line corresponds to a barrier similar to Eq. C7 but with the potentials 1:999t, 2t, 2:001t at sites j4i, j5i, j6i. The dotted line corresponds to potentials 1:99t, 2t, 2:01t at sites j4i, j5i, j6i. (a) While no charge separation occurs in the fully symmetric open system, charge separation starts to occur with increasing barrier asymmetry. (b) No persistent current is owing in the fully symmetric shorted system. Again, with increasing barrier asymmetry, persistent currents occur. The calculational method used is completely identical with the one used in Fig. 7, the only di erence being the symmetry of the barrier. Some of the blue Q curves are not visible because they fully overlap with the corresponding green Q curves. Q Q charge Q L R current J (nA) current J (nA) 20 0.8 C 0.4 0.025 − 0.025 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) Figure 15. Plots showing the time evolution of charges in the device with longitudinal asymmetry (Fig. 3a) as calculated using Eq. 3. The upper panel shows the total charge Q in the left lead, in the right lead Q , and within the central device Q . L R C The bottom panel shows the charge di erence between the left and right leads, which is zero for large and small times. For intermediate times, however, the di erence is nite, revealing a transient charge separation. 0.45 0.3 γ > 0 th 0.15 γ = 0 th 0.04 0.15 0.5 0.02 − 0 0 0.00 − 0.5 − 0.15 − 15 − 14 − 13 − 12 − 11 − 10 10 10 10 10 10 10 time t (s) − 0.02 Figure 16. Plots showing the time evolution of charges and currents in the open 9-site chain with additional jump operators that are speci cally chosen to drive the system towards the state given by the thermal density matrix (see C 3). The rate of the wave-packet generating jump operators and the rate of the additional thermalizing jump operators are given by − 0.04 th 15 14 = 1:510 Hz and = 310 Hz, respectively. The steady-state charge imbalance indicates that the novel steady state th persists even in the presence of thermalizing processes. The dashed lines show the case without the additional jump-operators, which is the same data as shown in Fig. 7a. − 0.04 − 0.02 0.00 0.02 0.04 Q Q charge Q L R Q Q charge Q L R current J (μA)

Journal

Condensed Matter – arXiv (Cornell University)

Published: Dec 27, 2019

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