Driven quantum transport on the nanoscale

Sigmund Kohler Corresponding author. [email protected] Jörg Lehmann Peter Hänggi Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany

We explore the prospects to control by use of time-dependent fields quantum transport phenomena in nanoscale systems. In particular, we study for driven conductors the electron current and its noise properties. We review recent corresponding theoretical descriptions which are based on Floquet theory. Alternative approaches, as well as various limiting approximation schemes are investigated and compared. The general theory is subsequently applied to different representative nanoscale devices, like the non-adiabatic pumps, molecular gates, molecular quantum ratchets, and molecular transistors. Potential applications range from molecular wires under the influence of strong laser fields to microwave-irradiated quantum dots.

quantum transport, driven systems, noise
05.60.Gg, 85.65.+h, 05.40.-a, 72.40.+w

, , and thanks: Present address: Departement für Physik und Astronomie, Universität Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland



wire site index,
, lead index
wire site attached to lead : ,
Floquet state indices
side-band/Fourier index
complex quasienergy
driving (angular) frequency
, driving period
Boltzmann constant times temperature
spectral density of lead
self energy
wire site,
, Floquet state for finite self energy
th Fourier coefficient of Floquet state
Floquet state for self energy
, eigenenergy and eigenstate of a static Hamiltonian
, single particle density matrix in Floquet basis
, Fermi function

1 Introduction

As anticipated by Richard Feynman in his visionary lecture “There’s plenty of room at the bottom” [1], we witness an ongoing progress in the study of physical phenomena on ever smaller scales. Partly, this has been made possible by the continuous technical achievements in fabrication and miniaturization of electronic devices. However, it was the invention of scanning probe microscopes [2], which brought about the realization of Feynman’s dream, namely the selective manipulation of matter on the nanoscale. Since then, much progress has been made in nano sciences. In particular, the field of molecular electronics has emerged, which deals with the realization of electronic devices based on the properties of a single or a few molecules. The theoretical proposal of a molecular rectifier by Aviram and Ratner [3] has been trend-setting for investigating the distinct features of electrical transport on the nanoscale. On the experimental side, an ancestor of molecular electronics was the pioneering work by Mann and Kuhn [4] on transport through hybrid acid-salt surface adlayers. The ongoing advance in contacting single molecules by nano-electrodes allows one to perform transport measurements [5, 6, 7, 8, 9]. In these experiments, the quantum nature of the electrons and the quantum coherence across the wire, which is connected to adjacent macroscopic lead electrodes, influence various physical properties such as the conductance and the corresponding current noise statistics. The rapid evolution of molecular conduction is documented by recent monographs and article collections [10, 11, 12, 13].

For the corresponding theoretical investigations, two lines of research are presently pursued. A first one starts out from the ab-initio computation of the orbitals relevant for the motion of excess charges through the molecular wire [14, 15, 16, 17, 18] At present, however, the results of such computations generally differ by more than one order of magnitude from experimental data, possibly due to the equilibrium treatment of exchange correlations [19]. The second line employs corresponding phenomenological models in order to gain a qualitative understanding of the transport mechanisms involved [20, 21, 22, 23, 24, 25]. Two particular problems addressed within model calculations are the conduction mechanism in the presence of electron-phonon coupling [26, 27, 28, 29, 21, 30, 22, 23, 31, 32, 33] and the length dependence of the current-voltage characteristics [20, 24]. The present work also employs rather universal models: We describe the molecules by a linear arrangement of tight-binding levels with the terminating sites attached to leads. Still it is possible to suitably parametrize such tight-binding models in order to obtain qualitative results for real systems [34, 35, 36]. Furthermore, these models also capture the physics of the so-called artificial molecules, i.e. coupled quantum dots and quantum dot arrays [37, 38].

One particular question that arises in this context is the influence of excitations by electromagnetic fields and gate voltages on the electron transport. Such excitations bear intriguing phenomena like photon-assisted tunneling [39, 40, 38, 41] and the adiabatic [42, 43, 44] and non-adiabatic pumping [45, 46] of electrons. From a fundamental point of view, these effects are of interest because the external fields enable selective electron excitations and allow one to study their interplay with the underlying transport mechanism. In practical applications, time-dependent effects can be used to control and steer currents in coherent conductors. However, such control schemes can be valuable only if they operate at tolerable noise levels. Thus, the corresponding current noise is also of equal interest.

An intuitive description of the coherent electron transport through time-independent mesoscopic systems is provided by the Landauer scattering formula [47] and its various generalizations. Both the average current [48, 49, 50, 51] and the transport noise characteristics [52] can be expressed in terms of the quantum transmission coefficients for the respective scattering channels. By contrast, the theory for driven quantum transport is less developed. Scattering of a single particle by arbitrary time-dependent potentials has been considered [53, 54, 55] without relating the resulting transmission probabilities to a current between electron reservoirs. Such a relation is indeed non-trivial since the driving opens inelastic transport channels and, therefore, in contrast to the static case, an ad hoc inclusion of the Pauli principle is no longer unique. This gave rise to a discussion about “Pauli blocking factors” [56, 57]. In order to resolve such conflicts, one should start out from a many-particle description. In this spirit, within a Green function approach, a formal solution for the current through a time-dependent conductor has been presented [58, 59] without taking advantage of the full Floquet theory for the wire and without obtaining a “scattering form” for the current in the general driven case. The spectral density of the current fluctuations has been derived for the low-frequency ac conductance [60, 61] and the scattering by a slowly time-dependent potential [62]. For arbitrary driving frequencies, the noise can be characterized by its zero-frequency component. A remarkable feature of the current noise in the presence of time-dependent fields is its dependence on the phase of the transmission amplitudes [62, 63, 64]. By clear contrast, both the noise in the static case [52] and the current in the driven case [63] depend solely on transmission probabilities.

In Section 3, we derive within a Floquet approach explicit expressions for both the current and the noise properties of the electron transport through a driven nanoscale conductor under the influence of time-dependent forces [63, 64]. This approach is applicable to arbitrary periodically driven tight-binding systems and, in particular, is valid for arbitrary driving strength and extends beyond the adiabatic regime. The dynamics of the electrons is solved by integrating the Heisenberg equations of motion for the electron creation and annihilation operators in terms of the single-particle propagator. For this propagator, in turn, we provide a solution within a generalized Floquet approach. Such a treatment is valid only for effectively non-interacting electrons, i.e., in the absence of strong correlations. Moreover, this Floquet scattering approach cannot be generalized straightforwardly to the case with additional electron-vibrational coupling. Better suited for this situation is a quantum kinetic equation formalism which, however, is perturbative in both the wire-lead coupling and the electron-vibrational coupling [65, 66].

An experimental starting point for the investigation of the influence of electromagnetic fields on molecular conduction is the excitation of electrons to higher orbitals of the contacted molecule. In molecular physics, specific excitations are usually performed with laser fields. The resulting changes of the current through a contacted molecule due to the influence of a laser field are studied in Section 5. In particular, we focus on the modification of the length dependence of the conductivity [67, 68].

An intriguing phenomenon in strongly driven systems is the so-termed ratchet or Brownian motor effect [69, 70, 71, 72, 73, 74], originally discovered for overdamped classical Brownian motion in asymmetric non-equilibrium systems. Counter-intuitively to the second law of thermodynamics, one then observes a directed transport although none of the acting forces possesses any net bias. This effect has been established also within the regime of dissipative, incoherent quantum Brownian motion [75, 74, 76]. A mesoscopic device related to ratchets is an electron pump [42, 77, 43, 44, 45, 46, 78] which indeed might be regarded as a localized ratchet. Such systems have already been realized in the quantum domain, but almost exclusively operating in the regime of incoherent tunneling [79, 80, 81, 82, 83]. In Section 6, we study the possibilities for molecular wires to act as coherent quantum ratchets and explore the crossover from electron pumps to quantum ratchets. This requires to investigate thoroughly such quantum ratchet systems in the coherent tunneling regime [84, 65].

The tunneling dynamics of a particle in a bistable potential can be altered significantly by ac fields. In particular, it is possible to bring tunneling to a standstill by the purely coherent influence of a time-periodic driving [85, 86]. This so-called coherent destruction of tunneling has also been found in other systems [87, 88, 89]. In Section 7, we address the question whether a related effect exists also for the electron transport through a driven conductor between two leads. Moreover, we study the noise properties of the resulting transport process [90, 63, 91, 64].

1.1 Experimental motivation

1.1.1 Coupled quantum dots

The experimental achievement of the coherent coupling of quantum dots [37] enabled the measurement of intriguing phenomena in mesoscopic transport [38]. A remarkable feature of coupled quantum dots—the so-called artificial molecules with the single dots representing the atoms—is that the energy levels of each “atom” can be controlled by an appropriate gate voltage. In particular, the highest occupied levels of neighboring dots can be tuned into resonance. At such resonances, the conductance as a function of the gate voltage exhibits a peak. This behavior is modified by the influence of microwave radiation: With increasing microwave intensity, the resonance peaks become smaller and side-peaks emerge. The distance between the central peak and the side-peaks is determined by the frequency of the radiation field which provides evidence for photon-assisted tunneling [39, 40, 38, 41]. Photon-assisted tunneling through quantum dots is, in comparison to its counterpart in superconductor-insulator-superconductor junctions [92], a potentially richer phenomenon. The reason for this is that quantum dots form a multi-barrier structure which permits real occupation and resonant tunneling. Therefore, a theoretical description requires to also take into account the influence of the field on the dynamics of the electrons localized in the central region between the barriers. The quantum dot setup used for the observation of photon-assisted tunneling can also be employed as an implementation [93] of the theoretically suggested non-adiabatic pump [59, 94, 95].

Related experiments have been performed also with single quantum dots exposed to laser pulses which resonantly couple the highest occupied orbital and the lowest unoccupied orbital of the quantum dot [96]. Such a pulse can create an electron-hole pair which in turn is transformed by a transport voltage into a current pulse. Depending on their duration, pulses may not only excite an electron but also coherently de-excite the electron and thereby reduce the resulting current [97]. In the ideal case, the electron-hole pair is excited with probability unity and finally yields a dc current consisting of exactly one electron per pulse. This effect might be employed for the realization of a current standard. At present, however, the deviations from the ideal value of the current are still of the order of a few percent.

1.1.2 Molecular wires

During the last years, it became possible to chemisorb organic molecules via thiol groups to a metallic gold surface. Thereby a stable contact between the molecule and the gold is established. This enables reproducible measurements of the current not only through artificial but also through real molecules. Single molecule conductance can be achieved in essentially two ways: One possible setup is an open break junction bridged by a molecule [5, 98, 7]. This setup can be kept stable for several hours. Moreover, it provides evidence for single molecule conductance because asymmetries in the current-voltage characteristics reflect asymmetries of the molecule [7, 99]. Alternatively, one can use a gold substrate as a contact and grow a self-assembled monolayer of molecules on it. The other contact is provided by a gold cluster on top of a scanning tunneling microscope tip which contacts one or a few molecules on the substrate [100, 6]. Yet another interesting device is based on the setup of a single-molecule chemical field effect transistor in which the current through a hybrid-molecular diode is controlled by nanometer-sized charge transfer complex which is covalently linked to a molecule in a scanning tunneling microscope junction [101]. Therein, the effect is due to an interface dipole which shifts the substrate work function. Naturally, the experimental effort with such molecular wires is accompanied by vivid theoretical interest [24, 10, 8].

Typical energy scales of molecules lie in the infrared regime where most of today’s lasers work. Hence, lasers represent a natural possibility to excite the electrons of the molecular wire and, thus, to study the corresponding changes of the conduction properties. At present, such experiments are attempted, but still no clear-cut effect has been reported. The molecule-lead contacts seem stable even against relatively intense laser fields, but a main problem is the exclusion of side effects like, e.g. heating of the break junction which might distort the molecule-tip setup and, thus, be responsible for the observed enhancement of the conductance [102].

In a recent experiment, Yasutomi et al. measured the photocurrent induced in a self-assembled monolayer of asymmetric molecules [83]. They have found that even the current direction depends on the wavelength of the irradiating light. Albeit not a single-molecules experiment, this measurement represents a first experimental demonstration of a ratchet-like effect in molecular wires.

2 Basic concepts

Before going in medias res and addressing specific quantum transport situations, we introduce the reader to our archetypal working model and the main theoretical methods and tools.

2.1 Model for driven conductor coupled to leads

The entire setup of our nanoscale system is described by the time-dependent Hamiltonian


where the different terms correspond to the wire, the leads, and the wire-lead couplings, respectively. We focus on the regime of coherent quantum transport where the main physics at work occurs on the wire itself. In doing so, we neglect other possible influences originating from driving-induced hot electrons in the leads, dissipation on the wire and, as well, electron-electron interaction effects. Then, the wire Hamiltonian reads in a tight-binding approximation with orbitals


For a molecular wire, this constitutes the so-called Hückel description where each site corresponds to one atom. The fermion operators , annihilate and create, respectively, an electron in the orbital . Note that in the absence of driving a diagonalization of the system Hamiltonian would yield the stationary eigenvalues of the wire levels. The influence of an externally applied ac field with frequency results in a periodic time-dependence of the wire Hamiltonian: . In an experiment, the driving is switched on at a specific time and, thus, the Hamiltonian is, strictly speaking, not time-periodic. This can be modeled by a slowly time-dependent driving amplitude that assumes its ultimate value after a transient stage in the “infinite past”. Within this work, however, we focus on the transport properties at asymptotically long times where the amplitude has already settled [86, 103] and, thus, the driving can be assumed periodic. This provides the basis for the applicability of a Floquet transport theory.

The leads are modeled by ideal electron gases,


where () creates an electron in the state () in the left (right) lead. The tunneling Hamiltonian


establishes the contact between the sites , and the respective lead, as depicted with Fig. 1. This tunneling coupling is described by the spectral density


of lead which becomes a smooth function if the lead modes are dense. If the leads are modeled by a tight-binding lattice, the assume a semi-elliptic shape, the so-called Newns-Anderson density of states [104], which is sometimes employed in the context of molecular conduction [20, 105, 106]. Within the present context, however, we are mainly interested in the influence of the driving field on the conductor and not in the details of the coupling to the leads. Therefore, we later on often choose for a rather generic form by assuming that in the relevant regime, it is practically energy-independent,


 Level structure of a nano-conductor with

Figure 1: Level structure of a nano-conductor with orbitals. The end sites are coupled to two leads with chemical potentials and .

To fully specify the dynamics, we choose as an initial condition for the left (right) lead a grand-canonical electron ensemble at temperature and electro-chemical potential . Thus, the initial density matrix reads


where is the number of electrons in lead and denotes the Boltzmann constant multiplied by the temperature. An applied voltage maps to a chemical potential difference with being the electron charge. Then, at initial time , the only nontrivial expectation values of the lead operators read where denotes the Fermi function.

Below, we specify the wire Hamiltonian as a tight-binding model composed of sites as sketched in Fig. 1. Each orbital is coupled to its nearest neighbor by a hopping matrix element , thus, the single-particle wire Hamiltonian reads


where stands for the on-site energies of the tight-binding levels. Although the theoretical approach derived below is valid for an arbitrary periodically driven wire Hamiltonian, we always assume that the time dependence results from the coupling to an oscillating dipole field that causes the time-dependent level shifts , where denotes the scaled position of site . The energy is determined by the electrical field strength multiplied by the electron charge and the distance between two neighboring sites.

An applied transport voltage is mapped to a symmetric shift of the leads’ chemical potentials, . Moreover, for the evaluation of the dc current and the zero-frequency noise, we restrict ourselves to zero temperature. The zero-temperature limit is physically well justified for molecular wires at room temperature and for quantum dots at helium temperature since in both cases, thermal electron excitations do not play a significant role.

In a realistic wire molecule, the hopping matrix element is of the order . Thus, a typical wire-lead hopping rate yields a current  Ampère and corresponds to a laser frequency in the near infrared, i.e., to wavelengths of the order . For a typical distance of Å between two neighboring sites, a driving amplitude is equivalent to an electrical field strength of . It has to be emphasized that the amplitude  is determined by the local electrical field between the contacts. The difference to the incident field can be huge: Model calculations demonstrated that the presence of metallic tips enhances the local field by several orders of magnitude [107, 108]. This explains the observation that the Raman scattering intensity increases drastically once the molecules are adsorbed to a metallic surface [109, 110]. Coupled quantum dots typically [37, 40, 38] have a distance of less than while the coupling matrix element is of the order of which corresponds to a wavelength of roughly . The dipole approximation inherent to the time-dependent part of the Hamiltonian (8) neglects the propagation of the electromagnetic field and, thus, is valid only for wavelengths that are much larger than the size of the sample [111]. This condition is indeed fulfilled for both applications we have in mind.

2.2 AC transport voltage

Within this work, we focus on models presented in the previous subsection, i.e., models where the driving enters solely by means of time-dependent matrix elements of the wire Hamiltonian while the leads and the wire-lead couplings remain time-independent. However, it is worthwhile to demonstrate that a setup with an oscillating external voltage can be mapped by a gauge transformation to the model introduced above. Consequently, it is possible to apply the formalism derived below also to situations with an oscillating voltage.

We restrict the discussion to a situation where the electron energies of the left lead are modified by an external -periodic voltage with zero time-average, thus


The generalization to a situation where also the levels in the right lead are -periodically time-dependent, is straightforward. Since an externally applied voltage causes a potential drop along the wire [112, 113, 114], we have to assume for consistency that for an ac voltage, the wire Hamiltonian also obeys a time-dependence. Ignoring such a time-dependent potential profile enables a treatment of the transport problem within the approach of Refs. [115, 116]. In the general case, however, we have to resort to the approach put forward with this work.

We start out by a gauge transformation of the Hamiltonian (1) with the unitary operator




describes the phase accumulated from the oscillating voltage. The transformation (10) has been constructed such that the new Hamiltonian possesses a time-independent tunnel coupling. Since, the operator transforms as , the matrix elements of the wire Hamiltonian acquire an additional time-dependence,


The second term in the Hamiltonian (12) stems from . Owing to the zero time-average of the voltage , the phase is -periodic. Therefore, the transformed wire Hamiltonian is also -periodic while the contact and the lead contributions are time-independent, thus, is of the same form as the original Hamiltonian (1).

2.3 Tien-Gordon theory

In order to explain the steps in the current-voltage characteristics of microwave-irradiated superconductor-insulator-superconductor junctions [92], Tien and Gordon [117] proposed a heuristical theoretical treatment which is of appealing simplicity but nevertheless captures some essential features of driven transport. The central idea of this approach is to model the influence of the driving fields by a periodic shift of the energies in the, e.g. left lead according , cf. Eq. (9). Then the corresponding lead eigenstates evolve as


where denotes the th order Bessel function of the first kind. The interpretation of the Fourier decomposition (14) is that each state consists of sidebands whose energies are shifted by multiples of . For the evaluation of the dc current, this is equivalent to replacing the Fermi function of the left lead by


and formally treating the system as time-independent [117]. While this effective static treatment indeed captures the photon-assisted dc current, it naturally fails to describe any time-dependent response.

For time-dependent wire-lead models where the driving shifts all wire levels simultaneously, it is possible to map the driving field by a gauge transformation to oscillating chemical potentials. Then, the average current can be evaluated from an effective electron distribution like the one in Eq. (15) [118, 119, 120]. However, generally the time-dependent field also influences the dynamics of the electrons on the wire. In particular, this is the case for the dipole driving (8). Then, a treatment beyond Tien-Gordon theory becomes necessary. Deriving an approach which is valid in the general case is the objective of Section 3.

2.4 Scattering approach for static conductors

In the absence of a driving field, the computation of the coherent transport through mesoscopic structures has become a standard procedure [48, 49, 50, 51]. The crucial idea goes back to Landauer who postulated already in 1957 [47] that in the absence of both inelastic effects and electron-electron interaction, conduction can be described as a coherent scattering process of independent electrons. Then, an infinitesimal voltage causes the current with the (linear) conductance


of a one-dimensional conductor, where is the transmission probability of an electron at the Fermi surface. Since conductors may have non-vanishing reflection probability , the transmission probability does not necessarily assume an integer value. The prefactor is the so-called conductance quantum.

Originally [47], the conductance (16) has been proposed with replaced by . In the beginning of the 1980’s, there has been a theoretical debate [121, 122, 123] whether or not, the reflection coefficient has to be included. The controversy was resolved by considering four-terminal devices where two terminals act as voltage probes and are considered as a part of the mesoscopic conductor [124, 125]. Then, represents the probed voltage and the factor indeed is justified. In a two-terminal device, however, denotes the externally applied voltage and the conductance includes a contact resistance and is given by Eq. (16).

With the same ideas, Landauer theory can be generalized to the case of a finite voltage for which the current reads


with being the electron transmission probability at energy . The electron distribution in the left (right) lead is given by the Fermi function with the chemical potential whose difference is determined by the applied voltage. The linearization for small voltages yields the conductance (16). The current formula (17) and the conductance (16) have been derived from Kubo formula [124, 122, 126, 123, 127] and by means of non-equilibrium Green function methods [128, 126, 129, 130] for various microscopic models. In doing so, one usually starts by defining a current operator, e.g. as the change of the electron charge in the left lead, i.e. . Finally, one obtains the expected expression for the current together with the relation


between the transmission probability and the Green function of the electrons. The trace sums over all single-particle states of the wire and denotes the imaginary part of the self-energy of the terminating wire sites which results from the coupling to the respective leads.

In order to obtain an expression for the related current noise, one considers the symmetrized correlation function


of the current fluctuation operator , where the anticommutator ensures hermiticity. For a stationary process, the correlation function is a function of only the time difference. Then, the noise strength can be characterized by the zero-frequency component


which obeys according to the Wiener-Khinchine theorem. In terms of the transmission function , the noise strength reads [52]


A dimensionless measure for the relative noise strength, is the so-called Fano factor [131]


Note that in a two-terminal device, both the absolute value of the average current and the noise strength are independent of the contact . Historically, the zero-frequency noise (20) contains a factor , i.e., one considers , resulting from a different definition of the Fourier transform. Then, the Fano factor is defined as . The definition (22) is such that a Poisson process corresponds to .

The generalization of the noise expression (21) to driven systems must also account for absorption and emission. Owing to this energy non-conserving processes, the zero-frequency noise is no longer given solely in terms of transmission probabilities but also depends on the phases of the transmission amplitudes [62, 63, 64]; cf. Eq. (3.2.2), below.

2.5 Master equation

A different strategy for the computation of stationary currents relies on the derivation of a master equation for the dynamics of the wire electrons. There, the central idea is to consider the contact Hamiltonian (4) as a perturbation, while the dynamics of the leads and the wire, including the external driving, is treated exactly. From the Liouville-von Neumann equation for the total density operator one obtains by standard techniques [132, 133] the approximate equation of motion


The tilde denotes operators in the interaction picture with respect to the molecule and the lead Hamiltonian without the molecule-lead coupling, , where is the propagator without the coupling. For the evaluation of Eq. (23) it is essential to use an exact expression for the zeroth-order time evolution operator . The use of any approximation bears the danger of generating artifacts, which, for instance, may lead to a violation of fundamental equilibrium properties [134, 135].

In order to make practical use of equation (23), one has to trace over the lead degrees of freedom and thereby obtains a master equation for the reduced density operator of the wire electrons. Subsequently, the reduced density operator is decomposed into the eigenstates of the wire Hamiltonian —or the corresponding Floquet states if the system is driven. As a further simplification, one might neglect off-diagonal matrix elements and, thus, obtain a master equation of the Pauli type, i.e., a closed equation for the occupation probabilities of the eigenstates [136, 137, 95]. For driven systems close to degeneracies of the quasienergies, however, such a Pauli master equation is not reliable as has been exemplified in Ref. [66].

3 Floquet approach to the driven transport problem

In the following, we present the Floquet approach for our working model of Section 2.1. This derivation is rigorous and exact: It is equivalent to an exact treatment in terms of a Keldysh Green function calculation [58]. However, the chosen Floquet derivation is here more direct and technically less cumbersome.

We start out from the Heisenberg equations of motion for the annihilation operators in lead , i.e.,


where denotes the conductor site attached to lead , i.e., and . These equations are straightforwardly integrated to read


Inserting (25) into the Heisenberg equations for the wire operators yields in the asymptotic limit


where the lead response function results from the Fourier transformation of the spectral density (5),


In the wide-band limit (6), one obtains and, thus, the equations of motion for the wire operators are memory-free. The influence of the operator-valued Gaussian noise


is fully specified by the expectation values


which for the uncorrelated initial state (7) follow from the definition (29). It is convenient to define the Fourier representation of the noise operator, whose correlation function


is obtained directly from Eq. (31).

3.1 Retarded Green function

The equations of motion (26) and (27) represent a set of linear inhomogeneous equations and, thus, can be solved with the help of the retarded Green function which obeys


where . At this stage, it is important to note that in the asymptotic limit , the l.h.s. of this equation is periodic in . As demonstrated in the Appendix, this has the consequence that for the propagator of the homogeneous equations obeys and, accordingly, the retarded Green function


is also -periodic in the time argument. Thus, we can employ the Fourier decomposition , with the coefficients


Physically, describes the propagation of an electron with initial energy under the absorption (emission) of photons for (). In the limiting case of a time-independent situation, becomes independent of and, consequently, identical to while all sideband contributions with vanish.

From the definition of the Green function, it can be shown that the solution of the Heisenberg equation (26), (27) reads


Inserting for the Fourier representation (34), one obtains the form


which proves more convenient.

Below, we need for the elimination of back-scattering terms the relation


A proof of this relation starts from the definition of the Green function, Eq. (33). By Fourier transformation with respect to , we obtain


which we multiply by from the left. The difference between the resulting expression and its hermitian adjoint with and interchanged is relation (38).

3.2 Current through the driven nano-system

The (net) current flowing across the contact of lead into the conductor is determined by the negative change of the electron number in lead multiplied by the electron charge . Thus, the current operator reads , where denotes the corresponding electron number. By using Eqs. (25) and (29), we obtain


This operator-valued expression for the time-dependent current is a convenient starting point for the evaluation of expectation values like dc current, ac current, and current noise.

3.2.1 Average current

In order to evaluate the current , we insert the solution (37) of the Heisenberg equation into the current operator (40) and use the expectation values (32). The resulting expression


still contains back-scattering terms and, thus, is not of a “scattering form”. Indeed, bringing (41) into a form that resembles the static current formula (17) requires some tedious algebra. Such a derivation has been presented for the linear conductance of time-independent systems [126], for finite voltage in the static case for tunneling barriers [128] and mesoscopic conductors [129], a wire consisting of levels that couple equally to both leads [58], and for weak wire-lead coupling [56]. For the general time-dependent case in the absence of electron-electron interactions, such an expression has been derived only recently [63, 64].

Inserting the matrix element of equation (38), eliminates the back-scattering terms and we obtain for the time-dependent current the expression




denotes the charge oscillating between the left lead and the wire. Obviously, since is time-periodic and bounded, its time derivative cannot contribute to the average current. The corresponding charge arising from the right lead, , is a priori unrelated to ; the actual charge on the wire reads . The time-dependent current is determined by the time-dependent transmission probability


The corresponding expression for follows from the replacement . We emphasize that (42) obeys the form of the current formula obtained for a static conductor within a scattering formalism. In particular, consistent with Refs. [56, 49], no “Pauli blocking factors” appear in our derivation. In contrast to a static situation, this is in the present context relevant since for a driven system generally


such that a contribution proportional to would not cancel [56, 57].

In order to obtain an expression for the dc current, we insert for the Green function the Fourier representation (35) followed by performing the average over time . Then, the average current becomes




denote the transmission probabilities for electrons from the right lead, respectively from the left lead, with initial energy and final energy , i.e., the probability for an scattering event under the absorption (emission) of photons if ().

For a static situation, the transmission probabilities and are identical and contributions with vanish. Thus, it is possible to write the current (46) in the form (17) as a product of a single transmission probability , which is independent of the direction, and the difference of the Fermi functions, . We emphasize that in the driven case this is no longer true.

3.2.2 Noise power

Like in the static case, we characterize the noise power by the zero-frequency component of the current-current correlation function (19). However, in the driven case, is still time-dependent. Since it shares the time-periodicity of the driving, it is possible to characterize the noise level by the zero-frequency component of averaged over the driving period,


It can be shown [64] that for driven two-terminal devices, is independent of the contact , i.e., .

We start by writing with the current operator (40) and insert the solution (37) of the Heisenberg equations. We again employ relation (38) and finally obtain

We have defined and . It can be shown (cf. Section 3.5.1) that in the undriven limit, the noise power (3.2.2) depends solely on the transmission probabilities and is given by Eq. (21). In the time-dependent case, however, the noise expression (3.2.2) cannot be brought into such a convenient form and, thus, generally depends on the phase of the transmission amplitude.

3.2.3 Floquet decomposition

For energy-independent wire-lead coupling, i.e. in the so-called wide-band limit the lead response function (28) reads . Consequently, the integro-differential equation (33) for the Green function becomes a pure differential equation. Then, determining the Green function is equivalent to computing a complete set of solutions for the equation


where the self-energy


results from the coupling to the leads. Equation (50) is linear and possesses time-dependent, -periodic coefficients. Thus, following the reasoning of Appendix A, it is possible to construct a complete solution with the Floquet ansatz


The so-called Floquet states obey the time-periodicity of and have been decomposed into a Fourier series. In a Hilbert space that is extended by a periodic time coordinate, the so-called Sambe space [138], they obey the Floquet eigenvalue equation [139, 140]


Due to the Brillouin zone structure of the Floquet spectrum [141, 138, 139], it is sufficient to compute all eigenvalues of the first Brillouin zone, . Since the operator on the l.h.s. of Eq. (54) is non-Hermitian, the eigenvalues are generally complex valued and the (right) eigenvectors are not mutually orthogonal. Thus, to determine the propagator, we need to solve also the adjoint Floquet equation yielding again the same eigenvalues but providing the adjoint eigenvectors . It can be shown that the Floquet states together with the adjoint states form at equal times a complete bi-orthogonal basis: and . A proof requires to account for the time-periodicity of the Floquet states since the eigenvalue equation (54) holds in a Hilbert space extended by a periodic time coordinate [142, 139]. For details, see Appendix A.

For the special case [59] of a wire with sites which couple equally strong to both leads, i.e., , the self-energy is proportional to the unity matrix. Consequently, the Floquet states become independent of the self-energy.

Using the Floquet equation (54), it is straightforward to show that with the help of the Floquet states the propagator can be written as


where the sum runs over all Floquet states within one Brillouin zone. Consequently, the Fourier coefficients of the Green function read


For the exact computation of current and noise, we solve numerically the Floquet equation (54). With the resulting Floquet states and quasienergies, we obtain the Green function (35). In the zero temperature limit, the Fermi functions in the expressions for the average current (46) and the zero-frequency noise (3.2.2) become step functions. Then, the remaining energy integrals can be performed analytically since the integrands are rational functions.

3.3 Symmetries

A system obeys a discrete symmetry if its Hamiltonian is invariant under a symmetry operation , i.e, if . Then the corresponding transition amplitude in position representation fulfills the relation


such that the corresponding transmission probabilities are identical. The complex conjugation in Eq. (58) holds if includes time inversion [143]; then the r.h.s. becomes . If , relation (58) means that two different scattering processes occur with the same probability. Correspondingly, in a time-dependent transport problem as defined by the Hamiltonian (1), the presence of a symmetry implies that two different transport channels have equal transmission probability.

Here, we identify the channel which is related to given a certain symmetry is present. In particular, we consider systems that are invariant under the transformations studied in the Appendix A.3 which are combinations of the transformations


For the tight-binding model sketched in Fig. 1, the parity operation (59) maps the lead states and the wire sites according to


where labels the wire sites and () the states in the left (right) lead. Both the parity and the time inversion can be generalized by an additional shift of position and time, respectively. Alternatively, one can place the origin of the corresponding axis properly. For convenience, we choose the latter option.

It should be mentioned that for the periodic driving considered in this work, the system contains a further symmetry, namely the time-translation by a full driving period. This has already been taken into account when deriving a Floquet transport theory and cannot be exploited further.

Transmission of an electron with energy

Figure 2: Transmission of an electron with energy under the absorption of photons (solid line) and its symmetry related process (dashed) for (a) time-reversal symmetry, (b) time-reversal parity, and (c) generalized parity. The sketched processes occur with equal probability.

3.3.1 Time-reversal symmetry

If the Hamiltonian obeys time-reversal symmetry , i.e., if , Eq. (58) yields . Inserting into the definition of the Green function , Eqs. (34) and (35), results in the relation , where we have shifted the limits of the -integration using the relation . Thus, the transmission probabilities obey


i.e., the scattering processes sketched in Fig. 2a occur with equal probability.

A time-independent system in the absence of magnetic fields presents a particular case of time-reversal symmetry since all transmissions probabilities with vanish and, thus, .

3.3.2 Time-reversal parity

Systems driven by a dipole force with purely harmonic time-dependence obey the so-called time-reversal parity , i.e., a combination of time-reversal symmetry and parity. This of course implies that the static part of the Hamiltonian has to obey spatial parity which requires identical wire-lead couplings, . The consequences for the Floquet states are discussed in the Appendix A.3 while here, we derive the consequences for the transmission probabilities.

By the same reasoning as in the case of time-reversal symmetry discussed above, but with additionally interchanging left and right, we find which yields equal transmission probabilities for the scattering events sketched in Fig. 2b, i.e.


Interestingly, time-reversal parity relates two scattering events that both go into the same direction. Therefore, relation (64) has no obvious consequence for any dc current. Still time-reversal parity entails an intriguing and more hidden consequence for non-adiabatic electron pumping by harmonic mixing as a function of the wire-lead coupling [65]. We discuss this effect in the context of non-adiabatic electron pumping in Section 6.3.

3.3.3 Generalized parity

A further spatio-temporal symmetry that has an impact on the transmission properties is the so-called generalized parity , i.e., a parity operation combined with a time shift by half a driving period. This symmetry also explains qualitatively the quasienergy spectra found in the context of driven quantum tunneling [85, 86, 144, 145].

If the wire-lead Hamiltonian is invariant under , the time evolution operator obeys