## Abstract

Scattering theory is a standard tool for the description of transport phenomena in mesoscopic systems. Here, we provide a detailed derivation of this method for nano-scale conductors that are driven by oscillating electric or magnetic fields. Our approach is based on an extension of the conventional Lippmann–Schwinger formalism to systems with a periodically time-dependent Hamiltonian. As a key result, we obtain a systematic perturbation scheme for the Floquet scattering amplitudes that describes the transition of a transport carrier through a periodically driven sample. Within a general multi-terminal setup, we derive microscopic expressions for the mean values and time-integrated correlation functions, or zero-frequency noise, of matter and energy currents, thus recovering the results of earlier studies in a unifying framework. We show that this framework is inherently consistent with the first and the second law of thermodynamics and prove that the mean rate of entropy production vanishes only if all currents in the system are zero. As an application, we derive a generalized Green–Kubo relation, which makes it possible to express the response of any mean currents to small variations of temperature and chemical potential gradients in terms of time integrated correlation functions between properly chosen currents. Finally, we discuss potential topics for future studies and further reaching applications of the Floquet scattering approach to quantum transport in stochastic and quantum thermodynamics.

## 1 Introduction

At room temperature, transport in macroscopic systems is a stochastic process, where carriers undergo ceaseless collisions that randomly change their velocity and direction of motion. This irregular behavior is the microscopic origin of both the finite resistance of a normal conductor and the fluctuations of induced currents. The fundamental relationship between these two phenomena is described by the fluctuation–dissipation theorem, a cornerstone result of statistical mechanics, which goes back to the pioneering works of Einstein, Nyquist, and Onsager and was later derived in a unified manner by Callen and Welton. Green and Kubo further expanded this approach and showed that, close to equilibrium, linear transport coefficients, which describe the response of a system to a small external field or thermal perturbation, can be expressed in terms of time integrated correlation functions of the corresponding currents, i.e. the zero-frequency noise [1], [2], [3]. This universal structure can be recovered even for systems in non-equilibrium steady states by introducing more general correlation functions that involve a current and a suitably chosen conjugate variable [4].

Reducing the temperature of a conductor increases the average distance that carriers can travel between two consecutive collisions. Coherent transport sets in when this mean free path becomes comparable to the dimensions of the sample. In this regime, which is realized in mesoscopic systems at millikelvin temperatures, the transfer of carriers becomes a reversible process governed by Schrödinger’s equation. As a result, the properties of mesoscopic conductors are dominated by quantum effects such as conductance quantization or coherent resistance oscillations, which can no longer be understood in terms of classical stochastic trajectories [5], [6], [7].

Scattering theory provides a quantum mechanical description of open systems that are subject to a constant in- and outflow of particles. Therefore, it is a well-suited tool to explore the principles of coherent transport. This approach was first proposed by Landauer and has since then evolved into a powerful theoretical framework, which has been extensively tested in experiments and shaped our modern understanding of transport phenomena in small-scale conductors. At the core of this framework lies the Landauer–Büttiker formula. It connects the scattering amplitudes of a mesoscopic sample, which describe the elastic deflection of incoming carriers, with the matter and energy currents that emerge in the system under external biases. Hence, it provides a direct link between microscopic and macroscopic quantities [8], [9], [10], [5], [6], [7].

As a key application, the scattering approach to quantum transport enables systematic investigations of the elementary principles that govern the thermodynamics of mesoscopic conductors and the performance of autonomous nano-machines such as thermoelectric heat engines or refrigerators [11], [12], [13], [14]. Cyclic machines like charge pumps or quantum motors, however, require the input or extraction of mechanical work; therefore, they must be driven by time-dependent electric or magnetic fields, which alter the energy of carriers inside the sample. Floquet theory provides an elegant way to take this effect into account by introducing a new type of scattering amplitudes that describe inelastic transitions, where carriers exchange photons with the external fields. This Floquet scattering approach yields a generalized Landauer–Büttiker formula for periodically driven systems [15], [16], [17], [18], [19]. Among other applications, this result enables quantitative models for cyclic nano-machines, which can be used to develop practical devices or to explore fundamental performance limits, two central topics in the field of quantum thermodynamics [20], [21].

The Floquet scattering approach also leads to explicit microscopic expressions for the time integrated correlation functions of matter and energy currents in periodically driven quantum conductors [22], [23], [24]. It thus provides a powerful tool to investigate the complex interplay between dissipation, thermal, and quantum fluctuations in mesocopic systems. This topic includes the search for generalizations of the well-established Green–Kubo relations as well as the quest for quantum extensions of the recently discovered thermodynamic uncertainty relations [25].

## 2 Objective and Outline

Our aim is to provide a thorough and general derivation of the Floquet scattering approach to coherent transport in mesoscopic conductors. This article is supposed to serve as both a step-by-step introduction for new users of the formalism and a compact reference text for experts in the field. We do not attempt to give a complete overview of the existing literature. Instead, our objective is to complement earlier works by focusing on the development of an algebraic scattering theory for periodically driven mesoscopic conductors and applications in stochastic and quantum thermodynamics.

We proceed as follows. In Section 3, we set the stage for our analysis by introducing the multi-terminal model as a general basis for the discussion of coherent transport. This section is followed by a brief recap of the algebraic scattering theory for autonomous systems in Section 4, which is based on common textbooks [26], [6], [7], [27]. We then show how the Floquet theorem makes it possible to extend this framework to periodically driven systems in Section 5. Following the approach of earlier studies, we construct an extended Hilbert space, which was originally proposed for closed systems [28], to derive a generalized Lippmann–Schwinger equation for Floquet scattering states [29], [30], [31]. This result naturally leads to a systematic perturbation scheme for the crucial Floquet scattering amplitudes and to explicit expressions for the corresponding scattering wave functions, which enable a transparent physical interpretation of the formalism.

In Section 6 we switch from the single-particle picture that had been used in the foregoing sections to a many-particle description. To this end we first show how the operators *t*, can be connected to the previously discussed Floquet scattering states. We then derive microscopic expressions for the mean currents and the time integrated current correlation functions, or noise power, which are given by

where angular brackets denote the average over all possible quantum sates of the system. We thereby recover the results of earlier studies [22], [15], [24].

Moving on, in Section 7 we show how the Floquet scattering approach can be furnished with a thermodynamic structure. To this end, we formulate the first and the second law and show that the scattering formalism is inherently consistent with these constraints. As an application of this theory, we derive a generalization of the Green–Kubo relations for periodically driven systems far from equilibrium. Finally, we discuss open problems and potential starting points for future studies in Section (8).

## 3 The Multi-Terminal Model

The multi-terminal model provides a universal platform for the description of coherent transport in mesoscopic systems. The key idea is thereby to divide the conductor into a scattering region, where carriers are affected by the potential landscape of the sample and periodic driving fields, and a set of *N* ideal leads, which can be traversed freely (Fig. 1). For the sake of simplicity, we assume throughout this article that the leads are effectively one-dimensional.^{[1]}

Each lead is connected to a thermochemical reservoir with a fully transparent interface, which injects a continuous beam of thermalized, non-interacting carriers into the system. Inside the conductor, these carriers follow a deterministic time evolution governed by Schrödinger’s equation until they are absorbed again into one of the reservoirs. Hence, all irreversible processes are relegated to the reservoirs, while the transfer of carriers between them is coherent. Once the system has reached a steady state, each lead *α* is traversed by a periodically modulated beam of incoming and outgoing carries, which gives rise to a matter and an energy current. The corresponding mean values and fluctuations are given by the formulas (1). As we will see in the following sections, these quantities are completely determined by the scattering amplitudes of the driven sample and the energy distribution of the carriers injected by the reservoirs.

## 4 Standard Scattering Theory

### 4.1 Scattering States

Without external driving, the carrier dynamics in a multi-terminal system is governed by the Hamiltonian

Here *P* and *M* are the carrier momentum and mass and *U* accounts for the potential landscape of the scattering region as well as the coupling to external magnetic fields. The scattering of individual carriers with fixed energy *E* > 0 is described by solutions of the time-dependent Schrödinger equation that have the form

The outgoing and incoming states, *α*, respectively. These scattering states satisfy the stationary Schrödinger equation

and the boundary conditions

Here, the plane waves

describe the free propagation of carriers inside the leads and the scattering amplitudes, *β* and *α*^{[2]}. The coordinate *β* in radial direction and the factor

has been introduced for normalization [32].

The outgoing and incoming states as defined by the conditions (4) and (5) obey the orthogonality relations

and form two complete bases of the single-particle Hilbert space *ℋ*; for simplicity, we assume throughout this article that no bound states exist inside the scattering region^{[3]}.

The scattering states are not normalizable and carry a finite probability current. Therefore, they cannot be interpreted in the same manner as bound states, whose wave function corresponds to the probability amplitude for finding a particle at a given position. Instead, we may regard the scattering states as a quantum mechanical description of a homogeneous sequence of carriers that emerge from a distant source and travel through the system one by one before being absorbed by a distant sink [26]. This interpretation does not imply that the states *E* that is injected into the terminal *α* to leave the system through the terminal *β*. Analogously, the square modulus of *E* that escapes through the terminal *α* to originate form the terminal *β*.

### 4.2 Scattering Amplitudes

To ensure the conservation of probability currents, the scattering amplitudes have to obey the unitarity condition [32]

Furthermore, they provide a link between outgoing and incoming states by means of the relation

which can be easily verified in position representation using the boundary conditions (5) and (9).

Upon applying the orthogonality relation (8), (10) implies an algebraic expression for the scattering amplitudes in terms of the scattering states given by

This result makes it possible to establish a universal symmetry, which follows from the observation that outgoing and incoming states are connected by time reversal, i.e.

where Θ denotes the anti-unitary time-reversal operator [35] and tildes indicate the reversal of external magnetic fields, see Figure 2. Consequently, we have

and therefore, given the (11),

Hence, for systems without magnetic fields, the scattering amplitudes for forward and backward transitions between any two terminals *α* and *β* are identical.

### 4.3 Lippmann–Schwinger Theory I: Autonomous Systems

The scattering states and amplitudes can, in principle, be determined by rewriting the stationary Schrödiner equation (4) in position representation, calculating the wave function inside the scattering region and matching it with the boundary conditions (5). This procedure, however, becomes impractical when the scattering wave functions cannot be found exactly and perturbation methods must be applied. It is then more convenient to follow an algebraic approach, which we develop next.

We first divide the Hamiltonian (2) into a free part *H*_{0} and a perturbation *V* acting only on the scattering region,

where we assume that the scattering states for *H*_{0} can be determined exactly. Next, we combine the stationary Schrödinger equations for the free and the perturbed scattering states,

into a single inhomogeneous linear equation,

This equation can be formally solved for the vector

where *ε* > 0 and the limit *ε* → 0 must be taken after physical observables have been calculated. Note that the sign of the complex shift is important to ensure the correct correspondence between free and perturbed outgoing and incoming states, for details see [26].

By construction, the solutions

which ensures that the perturbed states *H*, while the solutions of the stationary Schrödinger equation are unique only up to linear combinations of scattering states with the same energy [26].

The Lippmann–Schwinger equation (18) can be formally solved by iteration. This procedure yields

where the last line follows by noting that

The expression (20) provides a systematic expansion of the scattering states *V*. Moreover, it implies that the solutions of the Lippmann–Schwinger equation obey the same orthogonality relation as the free states, as

Here, we have first inserted (20) for

where we have used the relation [36]

which must be understood in the sense of distributions, and *H*_{0}. Comparing (23) with (11) yields the formula

which makes it possible to calculate the full scattering amplitudes order by order in *V* by using the expansion (20) of the scattering states

## 5 Floquet Scattering Theory

### 5.1 Floquet Theory

The carrier dynamics in a driven multi-terminal system is governed by a Hamiltonian with the general form

where the dynamical potential *V _{t}* accounts for time-dependent external fields acting on the scattering region. If the driving is periodic with frequency

where *E* here plays the role of a continuous quantum number and *α* stands for any discrete quantum number [28], [37], [38]. The Floquet states

and form an orthogonal basis of the single-particle Hilbert space *ℋ* at every fixed time *t*.

In order to formulate a systematic scattering theory for periodically driven systems, it is convenient to introduce the extended Hilbert space [28]

where *τ*-periodic functions. In time representation, the elements *τ*-periodic single-particle state vectors, i.e.

The scalar product in

This framework makes it possible to cast the Floquet–Schrödinger equation (28) into the form of a stationary Schrödinger equation given by

where *m* runs over all integers. The Floquet vectors

and the effective Hamiltonian *Ĥ*, which is defined as

is a self-adjoint operator on *m*, was introduced to ensure that the solutions of (32) are complete in

### 5.2 Lippmann–Schwinger Theory II: Driven Systems

Replacing the stationary Schrödinger equation (4) with (32), we can now extend the Lippmann–Schwinger theory of autonomous systems to systems with periodic driving. The dynamical potential *V _{t}* thereby plays the role of the perturbation and the free states are replaced by the Floquet vectors

where *H* of the Hamiltonian (26). The free Floquet scattering vectors

where the free effective Hamiltonian is defined as

Furthermore, using (8) and (31), it is straightforward to verify the orthogonality relation

Note that the quantum numbers *E* and *α* have now been identified with the energy and the terminal of either an incident (+) or an escaping (−) carrier.

The full Floquet scattering vectors

that reduce to the corresponding free vectors

which can be derived along the same lines as (18); the perturbation operator on the extended Hilbert space is thereby defined as

Using the (40) and (41), we can now establish the orthogonality relation for the Floquet scattering vectors,

and the connecting relations between outgoing and incoming vectors,

Here, we followed the same steps as in the derivations of the (22) and (23). In the (43), *H*.

### 5.3 Floquet Scattering Amplitudes I: General Properties

The Floquet scattering amplitudes are defined as

where

and the symmetry relation

where the double tilde indicates the reversal of both external magnetic fields and driving protocols. In the following, we will show how these results can be derived within the framework of Floquet scattering theory. Note that, throughout this article, we understand that sums over the mode index run over all integers and that the Floquet scattering amplitudes are zero if their energy argument is not positive.

The unitarity conditions (45) follow from the completeness relation for the Floquet scattering vectors,

where 1 stands for the identity operator on the extended Hilbert space and the symbolic notation

has been introduced for convenience. We thus have^{[4]}

and shifting the summation index *n* yields the result (45).

To derive the symmetry relation (46), we first observe that the free outgoing and incoming Floquet scattering vectors are connected by time reversal, i.e.

as can be easily verified with the help of (35) and the definition of the time-reversal operator on the extended Hilbert space, ^{[5]}

where we have used the identity

and thus, by comparison with the definition (44), the symmetries (46).

### 5.4 Floquet Scattering Amplitudes II: Perturbation Theory

The framework of our Floquet–Lippmann–Schwinger theory makes it possible to derive a systematic expansion of the Floquet scattering amplitudes in powers of the dynamical potential. To this end, we first compare the definitions (44) with the relations (43) to obtain the explicit expressions

Inserting the series representation (41) of the Floquet scattering vector

This result is analogous to the Born series in standard scattering theory [27]. Taking into account only first-order corrections gives the Floquet–Born approximation

which is justified if the amplitude of the external potential variations are small compared to the carrier energy.

### 5.5 Scattering Wave Functions

The physical content of the Floquet scattering states can be understood from their asymptotic wave functions. To derive their structure, we first use the Floquet–Lippmann–Schwinger (40) and the completeness relation for the free Floquet scattering vectors,

to connect the lead wave functions of the Floquet scattering states with the lead wave functions (5) of the stationary scattering states,

This expression shows that the wave functions

Here, we have used *Lemma 1c* of App. 9 and the symbol ≍ indicates asymptotic equality in the limit

This result shows that the outgoing and incoming Floquet scattering states, *λ*_{E} in the lead *α*. Hence, they represent a carrier with energy *E* that either enters or leaves the system through the terminal *α*. The Floquet scattering amplitude *α* to the terminal *β* under the absorption *m* units of energy *E* in the terminal *α* was injected into the terminal *β* with an energy surplus *m* quanta

We stress that the lead wave functions (59) have not been used to define the Floquet scattering states in our approach; in fact, their structure results from the continuity condition

Note that the lead wave functions (59) can be used as boundary conditions to determine the incoming and outgoing solutions of the Floquet–Schrödinger equation (28) in position representation. For sufficiently simple dynamical potentials, the Floquet scattering amplitudes can thus be found by calculating the Floquet wave functions inside the scattering region and solving a spatio-temporal boundary value problem [39], [40], [41].

## 6 Matter and Energy Currents

### 6.1 Current Operators

On the single-particle level, the matter and energy currents that flow at the position *r _{α}* of the lead

*α*into a multi-terminal conductor are represented by the operators [42], [43]

Here, *R* and *P* are the position and momentum operators, *M* denotes the carrier mass and curly brackets indicate the usual anti-commutator. Note that, for convenience, we notationally suppress the dependence of the current operators on the coordinate *r _{α}* throughout.

As the transport carriers are indistinguishable, the many-body quantum state of a mesoscopic conductor must be either symmetric or antisymmetric under the exchange of two arbitrary carriers. An elegant method to take this constraint into account is provided by the language of second quantization, which can be adopted to our present setup as follows. We first introduce the scattering field operators *t*, these operators obey the commutation relations

where we focus on Fermions for the sake of concreteness; the theory for Bosonic carriers can be developed analogously. The many-particle current operators can now be expresses as

where

with *τ*-periodic functions of *t* and can thus be expanded in a Fourier series,

where the coefficients

### 6.2 Mean Currents

We are now ready to calculate the average steady-state currents of matter and energy in a periodically driven multi-terminal conductor. To this end, we recall the general formula (1a) for the mean currents,

The Heisenberg-picture operator *t* and at a given position *r _{α}* in the lead

*α*; angular brackets denote the ensemble average over all possible quantum states of the system.

The formula (66) can be evaluated in two steps. First, transforming the current operators (62) into the Heisenberg picture yields

where the unitary operator U_{t} generates the evolution of the many-particle system from the time 0 to the time *t*. The second line in (67) follows from the time evolution laws for the field operators,

which, in turn, are a consequence of the fact that the outgoing scattering states ^{[6]}

Here, *U _{t}* is the single-particle time evolution operator. Note that the time argument 0 is omitted throughout for simplicity.

Second, to evaluate the ensemble average in (66), we recall that the outgoing Floquet scattering states *T _{α}* and chemical potential

*μ*. Hence, provided that all reservoirs are mutually independent, the quantum-statistical average of an ordered pair of one creation and one anihilation operator is given by the grand canonical rule

_{α}denotes the Fermi function of the reservoir *α* and Boltzmann’s constant is set to 1 throughout; averages of products that contain different numbers of creation and annihilation are zero [8], [9], [24].

Inserting (67) into the formula (66) and using (70) yields

where we have used the Fourier expansion (64) for the second identity. Upon recalling the matrix elements (65), the mean currents can now be expressed in terms of the Floquet scattering amplitudes of the conductor and the Fermi functions of the attached reservoirs,

This formula, which holds arbitrary far from equilibrium, shows that the conductance properties of a coherent multi-terminal system are fully determined by its Floquet scattering amplitudes. In the limit

The physical consistency of the current formula (72) derives from the sum rules

which follow directly from the unitarity conditions for the Floquet scattering amplitudes, (45). By using the first of these relations, (72) can be rewritten in the form

This result shows that the mean currents indeed vanish in equilibrium, i.e. if all reservoirs are at the same temperature and chemical potential and the external driving fields are turned off. Furthermore, by summing both sides of (72) over the terminal index and using the second sum rule in (73), we recover the fundamental conservation laws for matter and energy,

The average power that is injected into the system through the external driving,

### 6.3 Zero-Frequency Noise

The zero-frequency noise, or noise power, of the matter and energy currents in a multi-terminal conductor is given by the general formula

for

has been introduced for the correlation function of the observables *A* and *B*. The quantity

with dots being inserted to improve readability. The correlation function of the scattering field operators in (79) can be evaluated using the finite-temperature version of Wick’s theorem [44], which implies

Here, we have used the commutation rules (61) and the grand canonical averaging rule (70) for the last identity. After inserting (80) and the Fourier expansion of the current matrix elements (64) into (79), we can carry out the time integrals. This step yields

Upon taking the limit *t* → ∞ with the help of *Lemma 2* of App. 9, this expression simplifies to the compact result

where we have applied the relation

The zero-frequency noise can now be expressed in terms of the Floquet scattering amplitudes of the driven conductor and the Fermi functions of the reservoirs. To this end, we insert the matrix elements (65) into (82). After some algebra, we thus obtain the explicit formula

where we have introduced the abbreviations

for convenience.^{[7]}

In order to analyze the physical content of the key result (83), it is instructive to divide the noise power into two contributions, ^{[8]}

Here, the thermal noise, or Nyquist–Johnson noise, ^{[9]}. By contrast, the non-equilibrium noise

As a final remark for this section, we note that, although we have focused here on matter and energy currents, our analysis applies to any set of generalized currents that can be represented by operators of the form

with real coefficients

where

for

## 7 Thermodynamics

### 7.1 The First Law

The first law for periodically driven multi-terminal conductors follows directly from the conservation laws (75) and can be formulated as

where *μ* denotes the base level of the chemical potential. It governs the balance between the thermal energy that is injected into the system by the reservoirs through the heat currents

### 7.2 The Second Law

The second law requires that the average rate of entropy production that is caused by the transport process is non-negative, that is [45]

A simple demonstration that the Floquet scattering approach is consistent with this constraint uses only the sum rules (73) and the fact that the Fermi distribution is the derivative of a convex function, for details see [46], [47]. In the following, we provide an alternative proof, which also shows that the dissipation rate *σ* can only become zero if all currents in the system vanish.

Our proof is inspired by methods that are usually employed to derive bounds on quantum entropy functions, for details see [48]. The key idea is to express the rate of entropy production in terms of the binary entropy function

and its first derivative, where *σ* can then be obtained from a simple argument involving Taylor’s theorem. We proceed as follows. First, we use the formula (72) for the mean currents and the sum rules (73) to rewrite *σ* as

with *g* between

Since the Fermi function takes only values between 0 and 1, the number *g* must also lie in this interval. Hence, we have

upon combining the (92) and (93).

The bound (94) shows that, first, the rate of entropy production can indeed not become negative within the Floquet scattering approach and, second, that *σ* is zero if and only if the integrand in (94) vanishes for all energies *E* and all combinations of the indices

Finally, we note that the rate or entropy production (90) is in fact the mean value of a generalized current that is represented by the operator

Therefore, the formalism developed in Section 6.3 can be applied to investigate whether not only the average but also the fluctuations, or even higher-oder cumulants, of the entropy production are subject to universal bounds. This problem has recently been studied for stationary mesoscopic conductors [49], [50]. We leave it to future research to extend this approach to periodically driven systems.

### 7.3 Green–Kubo Relations

The Green–Kubo relations are a cornerstone result of non-equilibrium statistical mechanics. They make it possible to express the linear response coefficients that quantify the variations of mean currents due to a small changes of the thermodynamic forces that drive the system away from equilibrium in terms of integrated equilibrium correlation functions of the involved currents [45], [2]. As our final topic in this article, we will now show how this fundamental relationship arises naturally within the framework of Floquet scattering theory.

The thermodynamic forces, or affinities, for a transport process are defined as gradients in the thermodynamic variables that form entropy-conjugate pairs with the conserved quantities of the system. For a multi-terminal conductor, these objects can be identified with the thermochemical biases between the external reservoirs,

where, *μ* and *T* denote the base chemical potential and temperature. Using these definitions, the rate of entropy production (90) can be divided into a mechanical part,

Several proposals were made to extend this structure to the total rate of entropy production by associating the mechanical perturbation with an effective current and a generalized affinity, which, depending on the scheme, corresponds to the mean applied work [51] or either the amplitude [52], [53], [54] or the frequency [55], [47] of the periodic driving fields. For the purpose of our analysis, however, it is sufficient to focus on the conventional thermal currents and affinities appearing in (97).

To establish the Green–Kubo relations for multi-terminal systems we first calculate the response coefficients

where we have used the current formula (72) and the symbol

Hence, the symmetric part of the response coefficients (98) is identical to the thermal noise, even if the transport process takes place far from equilibrium. In equilibrium, i.e. for

In order to extend the result (100) to non-equilibrium situations and systems with broken time-reversal symmetry, we have to express the coefficient

A minimal choice for such a variable is given by

where the energy dependent weights,

are found by replacing the scattering wave functions *k _{E}*. This operator can be easily shown to satisfy the condition (101) by following the lines of Section 6.3. It describes the gross influx of matter

*α*and thus provides a physically transparent non-equilibrium generalization of the Green–Kubo relation (100), which covers even systems with broken time reversal symmetry. From a practical perspective, the result (101) makes it possible to infer the time-integrated correlation function between net currents and gross influx, which are otherwise hard to access, by measuring the variations of mean currents in response to small changes of the thermochemical biases (96).

We conclude this section by pointing out that the bilinear decomposition (97) of *σ*_{th} into affinities and currents is not unique. In fact, for any set of generalized currents and affinities,

the thermal rate of entropy production assumes the standard form

as can be easily verified by inspection.

## 8 Perspectives and Challenges

### 8.1 Adiabatic Perturbation Theory

In Section 5.4, we have shown how the Floquet scattering amplitudes can be calculated order by order in the dynamical potential. This approach is well justified if the periodic variations of the scattering potential are small compared to the typical carrier energies. For practical purposes, however, an adiabatic perturbation scheme, where the frequency rather than the amplitude of the driving fields plays the role of the expansion parameter, is often more suitable.

Such a theory can be developed as follows. Consider an approaching or escaping carrier with energy *E* in the terminal *α*. If the dynamical potential is practically constant during the dwell time of this carrier inside the sample, its transition through the system at the time *t* is described by the frozen scattering states

and satisfy the boundary conditions

with the frozen scattering amplitudes given by

The corresponding quasi-static Floquet scattering amplitudes are the Fourier components of these objects, i.e.

This result follows by comparing (107) with (59) and assuming that the carrier energy is practically constant during the transition through the sample.

The expression (109) can be interpreted as the zeroth order of an expansion of the Floquet scattering amplitudes in the photon energy

Here, the first term accounts for small changes in the carrier energy during the transition and the correction

This scheme proved quite effective for various practical applications [15], [24]. How it can be derived from a systematic perturbation theory, which would make it possible to also calculate higher-order terms, however, is not immediately clear. As a first attempt, we might try to adapt the Lippmann–Schwinger formalism of Section 5.2 by mimicking the adiabatic perturbation theory for systems with discrete spectrum [58], [59], [34]. To this end, the free scattering vectors (35) have to be replaced with their frozen counterparts,

The roles of the free effective Hamiltonian and the perturbation are then assumed by the operators

Hence, we indeed recover the zeroth- and fist-order terms (109) and (110). However, this result must be taken with a grain of salt, as the correction term in (112), which involves the time derivative of the frozen scattering state

The singular behavior of the last term in (112) arises because the time derivative

### 8.2 Thermal Machines

The Floquet scattering formalism provides a general platform to explore the performance of thermal nano-devices. As a concrete example, we might consider a quantum heat engine that consists of a driven sample and two reservoirs with equal chemical potential *μ* and different temperatures

which follows from the second law, ^{[10]}.

From a practical perspective, it is therefore important to determine the maximum efficiency, at which a nano-engine can deliver a given power output. For autonomous, i.e. thermoelectric, heat engines such bounds have been found by seeking constraints on the total rate of entropy production that go beyond the second law [66], [67], [68], [69], [70], or by explicitly optimizing the scattering amplitudes of the sample [71], [72], [73], [74], [75]. The first strategy has also been applied in studies of piston-type heat engines, which use a closed working system, and lead to the general trade-off relation

between efficiency *η* and power output −Π; here, Θ > 0 is a system-specific constant [52], [53], [68], [69]. First steps towards an extension of this bound to paddle-wheel type quantum engines, which are driven by a continuous flow of carriers, have been made under the assumptions of slowly varying driving fields and small thermochemical biases [55], [47]. A universal and physically transparent performance bound that covers also devices operating far from equilibrium is, however, still lacking.

### 8.3 Thermodynamic Uncertainty Relations

Thermodynamic uncertainty relations describe a trade-off between dissipation and precision in non-equilibrium processes. Specifically, for a time-homogeneous Markov process that obeys detailed balance, the inequality

holds for arbitrary currents with mean value *J* and fluctuations, or noise power, *P*, where *σ* denotes the total rate of entropy production and *ε* the relative uncertainty of the current *J* [76], [77]. This bound, which was first discovered for biomolecular processes, does, however, not apply to periodically driven systems, systems with broken time-reversal symmetry or in the quantum regime [78], [79], [80], [81], [82], [83], [84]. In order to close these gaps, a whole variety of generalized thermodynamic uncertainty relations have been proposed over the last years, see for instance [85], [86], [87], [88], [25], [89], [90], [91], [92], [93].

A particularly transparent result was recently obtained in [89], where the frequency dependent bound

was derived for periodically driven Markov jump processes. Whether or not this result can be extended to coherent mesoscopic conductors, or whether the relation (115) can be generalized for such systems by other means are compelling questions, which can be systematically investigated within the theoretical framework presented in this article. Further research in this direction promises valuable insights on how quantum effects can be exploited to control the thermodynamic cost of precision in transport processes. However, this endeavor can be expected to be challenging, as general properties of the Floquet scattering amplitudes that go beyond the ones discussed in Section 5 are hard to establish and specific models for which they can be determined exactly are scarce.

## Acknowledgement

The author acknowledges insightful discussions with E. Potanina, M. Moskalets, K. Saito, and U. Seifert. The research leading to the results presented in this article has received funding from the Academy of Finland (Contract No. 296073), the Japan Society for the Promotion of Science through a Postdoctoral Fellowship for Research in Japan (Fellowship ID: P19026), the University of Nottingham, Nottingham Research Fellowship and from UK Research and Innovation through a Future Leaders Fellowship (Grant Reference: MR/S034714/1).

## Appendix A Appendix: Some Helpful Lemmas

*Let *

*in the limit *

We proceed in two steps. First, we close the integration path in the complex plane as shown in Figure 3 and observe that

for *x* → ∞, since the integrand on the left-hand side is exponentially suppressed in *x* on either the upper (+) or the lower (−) half plane. Second, using Cauchy’s theorem to evaluate the contour integral yields

*For *

*in the limit *

Set *w* < 0 and *w* > 0 and repeat the steps of the proof of Lemma 1. ⊡

*For *

*in the limit *

Change the integration variable to

*Let F _{u} be a test function on the real axis and define *

We first rewrite the left-hand side of (122) as

where

where *ε* > 0. Consequently, we have [36]

⊡

where we used that *m* ≠ 0.

## References

[1] R. Kubo, Rep. Prog. Phys. **29**, 255 (1966).10.1088/0034-4885/29/1/306Search in Google Scholar

[2] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II – Nonequilibrium Statistical Mechanics, 2nd ed., Springer, Tokyo 1998.Search in Google Scholar

[3] U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Phys. Rep. **461**, 111 (2008).10.1016/j.physrep.2008.02.002Search in Google Scholar

[4] U. Seifert, Phys. Rev. Lett. **104**, 138101 (2010).10.1103/PhysRevLett.104.138101Search in Google Scholar

[5] G. B. Lesovik and I. A. Sadovskyy, Phys.-Usp. **54**, 1007 (2014).10.3367/UFNe.0181.201110b.1041Search in Google Scholar

[6] P. A. Mello and N. Kumar, Quantum Transport in Mesoscopic Systems, 1st ed., Oxford University Press, Oxford 2004.10.1093/acprof:oso/9780198525820.001.0001Search in Google Scholar

[7] Y. V. Nazarov and Y. M. Blanter, Quantum Transport – Introduction to Nanoscience, 1st ed., Cambridge University Press, Cambridge 2009.10.1017/CBO9780511626906Search in Google Scholar

[8] Y. M. Blanter and M. Büttiker, Phys. Rep. **336**, 1 (2000).10.1016/S0370-1573(99)00123-4Search in Google Scholar

[9] M. Büttiker, Phys. Rev. B **46**, 12485 (1992).10.1103/PhysRevB.46.12485Search in Google Scholar
PubMed

[10] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B **31**, 6207 (1985).10.1103/PhysRevB.31.6207Search in Google Scholar

[11] G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Phys. Rep. **694**, 1 (2017).10.1016/j.physrep.2017.05.008Search in Google Scholar

[12] P. Gaspard, New J. Phys. **15**, 115014 (2013).10.1088/1367-2630/15/11/115014Search in Google Scholar

[13] P. Gaspard, New J. Phys. **17**, 045001 (2015).10.1088/1367-2630/17/4/045001Search in Google Scholar

[14] P. Gaspard, Ann. Phys. (Berlin) **527**, 663 (2015).10.1002/andp.201500121Search in Google Scholar

[15] M. Moskalets and M. Büttiker, Phys. Rev. B **66**, 205320 (2002).10.1103/PhysRevB.66.205320Search in Google Scholar

[16] M. V. Moskalets, Scattering Matrix Approach to Quantum Transport, 1st ed., Imperial College Press, London 2012.10.1142/p822Search in Google Scholar

[17] M. H. Pedersen and M. Buttiker, Phys. Rev. B **58**, 12993 (1998).10.1103/PhysRevB.58.12993Search in Google Scholar

[18] M. Wagner, Phys. Rev. Lett. **85**, 174 (2000).10.1103/PhysRevLett.85.174Search in Google Scholar
PubMed

[19] M. Wagner and F. Sols, Phys. Rev. Lett. **83**, 4377 (1999).10.1103/PhysRevLett.83.4377Search in Google Scholar

[20] M. F. Ludovico, L. Arrachea, M. Moskalets, and D. Sánchez, Entropy **18**, 419 (2016).10.3390/e18110419Search in Google Scholar

[21] S. Vinjanampathy and J. Anders, Contemp. Phys. **57**, 545 (2016).10.1080/00107514.2016.1201896Search in Google Scholar

[22] M. Moskalets, Phys. Rev. Lett. **112**, 206801 (2014).10.1103/PhysRevLett.112.206801Search in Google Scholar

[23] M. Moskalets and M. Büttiker, Phys. Rev. B **66**, 035306 (2002).10.1103/PhysRevB.66.035306Search in Google Scholar

[24] M. Moskalets and M. Büttiker, Phys. Rev. B **70**, 245305 (2004).10.1103/PhysRevB.70.245305Search in Google Scholar

[25] J. M. Horowitz and T. R. Gingrich, Nat. Phys. **16**, 15 (2020).10.1038/s41567-019-0702-6Search in Google Scholar

[26] L. E. Ballentine, Quantum Mechanics: A Modern Development, 1st ed., World Scientific, Singapore 1998.10.1142/3142Search in Google Scholar

[27] R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer Science+Business Media, New York 1982.10.1007/978-3-642-88128-2Search in Google Scholar

[28] H. Sambe, Phys. Rev. A **7**, 2203 (1973).10.1103/PhysRevA.7.2203Search in Google Scholar

[29] J. S. Howland, Indiana U. Math. J. **28**, 471 (1979).10.1512/iumj.1979.28.28033Search in Google Scholar

[30] N. Moiseyev and R. Lefebvre, Phys. Rev. A **58**, 4218 (1998).10.1103/PhysRevA.58.4218Search in Google Scholar

[31] U. Peskin and N. Moiseyev, Phys. Rev. A **49**, 3712 (1994).10.1103/PhysRevA.49.3712Search in Google Scholar
PubMed

[32] H. U. Baranger and A. D. Stone, Phys. Rev. B **40**, 8169 (1989).10.1103/PhysRevB.40.8169Search in Google Scholar
PubMed

[33] M. Mintchev, L. Santoni, and P. Sorba, Ann. Phys. (Berlin) **529**, 1600274 (2017).10.1002/andp.201600274Search in Google Scholar

[34] L. I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill Book Company, New York 1968.Search in Google Scholar

[35] G. F. Mazenko, Nonequilibrium Statistical Mechanics, 1st ed., Wiley-VCH Verlag GmbH & Co KGaA, Weinheim, Weinheim 2006.10.1002/9783527618958Search in Google Scholar

[36] W. Appel, Mathematics for Physics and Physicists, 1st ed., Princeton University Press, Princeton, NJ 2007.Search in Google Scholar

[37] J. H. Shirley, Phys. Rev. **138**, B979 (1965).10.1103/PhysRev.138.B979Search in Google Scholar

[38] Y. B. Zel’dovich, Sov. Phys. JETP **24**, 1006 (1967).Search in Google Scholar

[39] W. Li and L. E. Reichel, Phys. Rev. B **60**, 15732 (1999).10.1103/PhysRevB.60.15732Search in Google Scholar

[40] D. F. Martinez and L. E. Reichl, Phys. Rev. B **64**, 245315 (2001).10.1103/PhysRevB.64.245315Search in Google Scholar

[41] M. Wagner, Phys. Rev. B **49**, 16544 (1994).10.1103/PhysRevB.49.16544Search in Google Scholar

[42] R. J. Hardy, Phys. Rev. **132**, 168 (1963).10.1103/PhysRev.132.168Search in Google Scholar

[43] A. Kugler, Z. Phys. **198**, 236 (1967).10.1007/BF01331237Search in Google Scholar

[44] G. Giulianni and G. Vignale, Quantum Theory of the Electron Liquid, Cambridge University Press, Cambridge 2005.10.1017/CBO9780511619915Search in Google Scholar

[45] H. B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd ed., John Wiley & Sons, New York 1985.Search in Google Scholar

[46] G. Nenciu, J. Math. Phys. **48**, 033302 (2007).10.1063/1.2712418Search in Google Scholar

[47] E. Potanina, M. Moskalets, C. Flindt, and K. Brandner, preprint arXiv:1906.04297 (2019).Search in Google Scholar

[48] M. Ohya and D. Petz, Quantum Entropy and Its Use, 1st ed., Springer-Verlang, Berlin, Heidelberg 1993.10.1007/978-3-642-57997-4Search in Google Scholar

[49] M. Mintchev, L. Santoni, and P. Sorba, Phys. Rev. E **96**, 052124 (2017).10.1103/PhysRevE.96.052124Search in Google Scholar
PubMed

[50] M. Mintchev, L. Santoni, and P. Sorba, Ann. Phys. (Berlin) **530**, 1800170 (2018).10.1002/andp.201800170Search in Google Scholar

[51] Y. Izumida and K. Okuda, Eur. Phys. J. B **77**, 499 (2010).10.1140/epjb/e2010-00285-0Search in Google Scholar

[52] K. Brandner, K. Saito, and U. Seifert, Phys. Rev. X **5**, 031019 (2015).10.1103/PhysRevX.5.031019Search in Google Scholar

[53] K. Brandner and U. Seifert, Phys. Rev. E **93**, 062134 (2016).10.1103/PhysRevE.93.062134Search in Google Scholar
PubMed

[54] K. Proesmans and C. van den Broeck, Phys. Rev. Lett. **115**, 090601 (2015).10.1103/PhysRevLett.115.090601Search in Google Scholar
PubMed

[55] M. F. Ludovico, F. Battista, F. von Oppen, and L. Arrachea, Phys. Rev. B **93**, 075136 (2016).10.1103/PhysRevB.93.075136Search in Google Scholar

[56] V. Gasparian, T. Christen, and M. Büttiker, Phys. Rev. A **54**, 4022 (1996).10.1103/PhysRevA.54.4022Search in Google Scholar

[57] C. Texier, Physica E **82**, 16 (2016).10.1016/j.physe.2015.09.041Search in Google Scholar

[58] V. Cavina, A. Mari, and V. Giovannetti, Phys. Rev. Lett. **119**, 050601 (2017).10.1103/PhysRevLett.119.050601Search in Google Scholar
PubMed

[59] M. S. Sarandy and D. A. Lidar, Phys. Rev. A **71**, 012331 (2005).10.1103/PhysRevA.71.012331Search in Google Scholar

[60] M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Phys. Rep. **697**, 1 (2017).10.1016/j.physrep.2017.07.001Search in Google Scholar

[61] P. Weinberg, M. Bukov, L. D’Alessio, A. Polkovnikov, S. Vajna, and M. Kolodrubetz, Phys. Rep. **688**, 1 (2017).10.1016/j.physrep.2017.05.003Search in Google Scholar

[62] M. Thomas, T. Karzig, S. V. Kusminskiy, G. Zaránd, and F. von Oppen, Phys. Rev. B **86**, 195419 (2012).10.1103/PhysRevB.86.195419Search in Google Scholar

[63] L. Arrachea and F. von Oppen, Physica E **82**, 247 (2016).10.1016/j.physe.2016.02.037Search in Google Scholar

[64] A. Bruch, S. V. Kusminskiy, G. Refael, and F. von Oppen, Phys. Rev. B **97**, 195411 (2018).10.1103/PhysRevB.97.195411Search in Google Scholar

[65] R. Bustos-Marún, G. Refael, and F. von Oppen, Phys. Rev. Lett. **111**, 060802 (2013).10.1103/PhysRevLett.111.060802Search in Google Scholar
PubMed

[66] K. Brandner and U. Seifert, Phys. Rev. E **91**, 012121 (2015).10.1103/PhysRevE.91.012121Search in Google Scholar
PubMed

[67] P. Pietzonka and U. Seifert, Phys. Rev. Lett. **120**, 190602 (2018).10.1103/PhysRevLett.120.190602Search in Google Scholar
PubMed

[68] N. Shiraishi and K. Saito, J. Stat. Phys. **174**, 433 (2019).10.1007/s10955-018-2180-0Search in Google Scholar

[69] N. Shiraishi, K. Saito, and H. Tasaki, Phys. Rev. Lett. **117**, 190601 (2016).10.1103/PhysRevLett.117.190601Search in Google Scholar
PubMed

[70] R. S. Whitney, Phys. Rev. B **87**, 115404 (2013).10.1103/PhysRevB.87.115404Search in Google Scholar

[71] P. P. Hofer and B. Sothmann, Phys. Rev. B **91**, 195406 (2015).10.1103/PhysRevB.91.195406Search in Google Scholar

[72] P. Samuelsson, S. Kheradsoud, and B. Sothmann, Phys. Rev. Lett. **118**, 256801 (2017).10.1103/PhysRevLett.118.256801Search in Google Scholar
PubMed

[73] R. Sánchez, B. Sothmann, and A. N. Jordan, Phys. Rev. Lett. **114**, 146801 (2015).10.1103/PhysRevLett.114.146801Search in Google Scholar
PubMed

[74] R. S. Whitney, Phys. Rev. Lett. **112**, 130601 (2014).10.1103/PhysRevLett.112.130601Search in Google Scholar
PubMed

[75] R. S. Whitney, Phys. Rev. B **91**, 115425 (2015).10.1103/PhysRevB.91.115425Search in Google Scholar

[76] A. C. Barato and U. Seifert, Phys. Rev. Lett. **114**, 158101 (2015).10.1103/PhysRevLett.114.158101Search in Google Scholar
PubMed

[77] T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Phys. Rev. Lett. **116**, 120601 (2016).10.1103/PhysRevLett.116.120601Search in Google Scholar
PubMed

[78] A. C. Barato and U. Seifert, Phys. Rev. X **6**, 041053 (2016).10.1103/PhysRevX.6.041053Search in Google Scholar

[79] K. Brandner, T. Hanazato, and K. Saito, Phys. Rev. Lett. **120**, 090601 (2018).10.1103/PhysRevLett.120.090601Search in Google Scholar
PubMed

[80] V. Holubec and A. Ryabov, Phys. Rev. Lett. **121**, 120601 (2018).10.1103/PhysRevLett.121.120601Search in Google Scholar
PubMed

[81] K. Ptaszyński, Phys. Rev. B **98**, 085425 (2018).10.1103/PhysRevB.98.085425Search in Google Scholar

[82] B. K. Agarwalla and D. Segal, Phys. Rev. B **98**, 155438 (2018).10.1103/PhysRevB.98.155438Search in Google Scholar

[83] S. Saryal, H. M. Friedman, D. Segal, and B. K. Agarwalla, Phys. Rev. E **100**, 042101 (2019).10.1103/PhysRevE.100.042101Search in Google Scholar
PubMed

[84] J. Liu and D. Segal, Phys. Rev. E **99**, 062141 (2019).10.1103/PhysRevE.99.062141Search in Google Scholar
PubMed

[85] A. C. Barato, R. Chetrite, A. Faggionato, and D. Gabrielli, New J. Phys. **20**, 103023 (2018).10.1088/1367-2630/aae512Search in Google Scholar

[86] F. Carollo, R. L. Jack, and J. P. Garrahan, Phys. Rev. Lett. **122**, 130605 (2019).10.1103/PhysRevLett.122.130605Search in Google Scholar
PubMed

[87] G. Guarnieri, G. T. Landi, S. R. Clark, and J. Goold, Phys. Rev. Research **1**, 033021 (2019).10.1103/PhysRevResearch.1.033021Search in Google Scholar

[88] Y. Hasegawa and T. Van Vu, Phys. Rev. Lett. **123**, 110602 (2019).10.1103/PhysRevLett.123.110602Search in Google Scholar
PubMed

[89] T. Koyuk and U. Seifert, Phys. Rev. Lett. **122**, 230601 (2019).10.1103/PhysRevLett.122.230601Search in Google Scholar
PubMed

[90] T. Koyuk, U. Seifert, and P. Pietzonka, J. Phys. A: Math. Theor. **52**, 02LT02 (2019).10.1088/1751-8121/aaeec4Search in Google Scholar

[91] K. Macieszczak, K. Brandner, and J. P. Garrahan, Phys. Rev. Lett. **121**, 130601 (2018).10.1103/PhysRevLett.121.130601Search in Google Scholar
PubMed

[92] K. Proesmans and C. Van den Broeck, Europhys. Lett. **119**, 20001 (2017).10.1209/0295-5075/119/20001Search in Google Scholar

[93] A. M. Timpanaro, G. Guarnieri, J. Goold, and G. T. Landi, Phys. Rev. Lett. **123**, 090604 (2019).10.1103/PhysRevLett.123.090604Search in Google Scholar
PubMed

**Received:**2020-02-22

**Accepted:**2020-03-08

**Published Online:**2020-04-27

**Published in Print:**2020-05-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston