{'fraction_non_alphanumeric': 0.040182755388940956, 'fraction_numerical': 0.009176819743830052, 'mean_word_length': 4.047102877414269, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Geometrical Pumping in Quantum Transport: Quantum Master Equation Approach

For an open quantum system, we investigate the pumped current induced by a slow modulation of control parameters on the basis of the quantum master equation and full counting statistics. We find that the average and the cumulant generating function of the pumped quantity are characterized by the geometrical Berry-phase-like quantities in the parameter space, which is associated with the generator of the master equation. From our formulation, we can discuss the geometrical pumping under the control of the chemical potentials and temperatures of reservoirs. We demonstrate the formulation by spinless electrons in coupled quantum dots. We show that the geometrical pumping is prohibited for the case of non-interacting electrons if we modulate only temperatures and chemical potentials of reservoirs, while the geometrical pumping occurs in the presence of an interaction between electrons.


I. INTRODUCTION
When a quantum system is slowly and periodically modulated by two or more control parameters such as gate voltages, a net number of particles can be transported per period of the modulation even in the absence of dc driving force (e.g., bias voltage). This phenomenon is known as an adiabatic pump. The original idea of the adiabatic pump was proposed by Thouless 1 , where a pump current of a closed quantum system is related to the Berry phase 2 of the ground state of the Hamiltonian 1,3,4 . Since then, experimental and theoretical studies of the adiabatic pump have been developed [5][6][7] .
Later, the idea of the adiabatic pump has been applied to open quantum systems both in experiments 8 and theories [9][10][11][12] . Theoretically, the average pump current can be expressed by the Berry phase associated with the scattering matrix 10 . The cumulant generating function of pump current can also be described by geometrical quantities 12 . For this reason, the adiabatic pump is referred to as the geometrical pump. The abovementioned theories are based on the time-dependent scattering theory 13 .
Similar phenomena have been studied in stochastic systems described by classical master equation [14][15][16][17][18][19][20][21] , which is referred to as the adiabatic stochastic pump. The pump current in the classical stochastic pump has also geometrical properties; the cumulant generating function of the pump current is expressed by a Berry-phase-like quantity that is associated with the generator of the classical master equation. We shall refer to this quantity as the Berry-Sinitsyn-Nemenman (BSN) phase 16,17 . There have also been several works on the adiabatic pump for quantum open systems described by quantum master equation (QME) [22][23][24] . However, the geometrical effects in the adiabatic pump described by QME have not been fully understood.
In this paper, we investigate the quantum adiabatic pump on the basis of QME, and derive general formulae of the cumulant generating function and average of the pump. These formulae are geometrical and expressed by a quantum analogous of the BSN phase which is associated with the generator of the QME. Since our theoretical framework is independent of the details of the system, it is useful for analyzing a large variety of applications of the adiabatic pump such as a qubit rotation in quantum dots and a spin pumping. We note that the master equation approach is suitable to analyze systems that include interaction between particles 24 , dissipations, and decoherences 25 .
This paper is organized as follows. In Sec. II A, the QME approach for an open system is discussed. The full counting statistics in the QME approach is discussed in Sec. II B. For the adiabatic pump, the geometrical formulae (BSN phase expressions) for the cumulant generating function and the average of the pumped quantity are derived in Sec. II C. In this formulation, the temperatures and chemical potentials of reservoirs are the control parameters. This is in contrast to most of the conventional studies on the adiabatic pump, where only the parameters in the system Hamiltonian or in the coupling with the reservoirs are considered. In Sec. III, we demonstrate our theory by the spinless electron transport in quantum dot systems. For non-interacting cases with (Sec. III A) and without (Sec. III B) the rotating wave approximation (RWA), we find that no geometrical pump occurs if we control only the temperatures and chemical potentials of reservoirs. In contrast, the geometrical pump occurs for an interacting electron system even when we control only the reservoir parameters (Sec. III C). Section IV is devoted to the discussion and conclusion. In Appendix A, we describe the details of the QME in the framework of the full counting statistics. In Appendix B, we show the detailed derivation of Eq. (17). In Appendix C, we verify the equivalence between without and within the RWA concerning the unit-time generating function of the cur- 1. (Color online) Illustration of our setup with two reservoirs (L and R). rent between the system and reservoirs in a steady state.

A. Setup
We consider a quantum system S that is weakly coupled to reservoirs {R b }, where b is an index of reservoirs (see Fig. 1 for a schematic). The total Hamiltonian of the coupled system isĤ tot =Ĥ S + b (Ĥ b +Ĥ Sb ), whereĤ S is the system Hamiltonian,Ĥ b is the Hamiltonian of the bth reservoir R b , andĤ Sb is the interaction Hamiltonian between S and R b . If the interaction between the system S and the reservoirs is weak, the dynamics of S can be described by a QME for the reduced density matrix of S, which is denoted asρ. Suppose that the initial state of the system S is decoupled with the reservoirs. Then, up to the second order in the system-reservoir coupling (Born approximation) with the Markov approximation 26 , the QME for the system S reads where Here the symbol "ˇ" stands for the interaction picture with respect toĤ S + bĤ b , Tr b represents the trace over the bth reservoir, andρ b = e −β b (Ĥ b −µ bNb ) /Z b is the grandcanonical distribution with the inverse temperature β b , chemical potential µ b , and the particle number opera-torN b of the bth reservoir. The time-evolution generator K of the QME depends on several parameters; the system parameters inĤ S andĤ Sb such as the energy levels of quantum dots and the tunnel barriers between them, and the reservoir parameters, {β b } and {µ b }. We write the set of these parameters as α. The right eigenvalue equation for K is written as where λ n (α) is an eigenvalue of K, n is a label of the eigenvalues, andρ n (α) is the corresponding right eigenvector. By introducing the Hilbert-Schmidt inner product of linear operatorsÂ andB on H S as Tr S (Â †B ), we define the adjoint K † of the QME generator such that Tr S [(K †Â ) †B ] = Tr S (Â † KB) holds for anyÂ,B. We then have the left eigenvalue equation for K: wherel n (α) is the left eigenvector corresponding to the eigenvalue λ n (α). In the following, we assume that K has the zero eigenvalue λ 0 = 0 without degeneracy, so that Kρ 0 = 0 and K †l 0 = 0 hold. This implies that the QME has a unique steady solutionρ ss =ρ 0 for fixed α. We note thatl 0 (α) =1 (identity operator) holds for any α.

B. Full counting statistics
We consider the statistics of a quantity ∆q transferred from the reservoirs to the system S during a time interval τ . The measurement scheme of ∆q is as follows. First, at t = 0 we perform a projection measurement of a reservoir variableQ to obtain a measurement outcome q 0 . We assume [Q,N b ] = 0 and [Q,Ĥ b ] = 0 for any b. For t > 0, the system S undergoes the time evolution with interacting with the reservoirs. At t = τ we again perform a projection measurement ofQ to obtain another measurement outcome q τ . Then ∆q is defined as ∆q = q τ − q 0 .
The cumulant generating function of the statistics is given by S τ (χ) = ln P τ (∆q)e iχ∆q d∆q, where P τ (∆q) is the probability of ∆q during τ . χ is called the counting field, and the derivatives of S τ (χ) give the cumulants of P τ (∆q); e.g., ∆q τ = ∂S τ (χ)/∂(iχ)| χ=0 . Note that if Q is the bth reservoir's particle numberN b (Hamiltonian H b ), then ∆q τ /τ is the average of the particle (energy) current from the bth reservoir into the system S.
For calculating S τ (χ), we employ a method developed in the context of the full counting statistics 27 . In this method S τ (χ) is obtained from the solution of the modified equation of motion which is governed by the χ-dependent Hamiltonian. In the QME approach, this reads S τ (χ) = ln Tr Sρ χ (τ ), where Tr S is the trace over the system S andρ χ (τ ) is a solution of the generalized QME (GQME): Here the modified generator K χ is given by where [Ô,P ] χ ≡Ô χP −PÔ −χ andÔ χ ≡ e −iχQ/2Ô e iχQ/2 . See Appendix A for details.

C. Geometrical pump
We slowly modulate the parameters α along a curve C in the parameter space during a time interval τ . If the system is in the instantaneous steady state for the value of α t at each time t in the whole of the process, the cumulant generating function for ∆q for this process is equal to the time integral of the unit-time cumulant generating function λ χ 0 (α t ) of the instantaneous steady state. In general, however, there exists additional (pumped) con- . We call the latter contribution the excess part. The excess part is intrinsic in the transitions between the steady states, and is of our interest.
We now derive the geometrical expression of the excess part of the generating function by using the method similar to those in Refs 16,28 . First, to solve the GQME for a given curve C of α in the parameter space, we expand ρ χ (t) asρ where Λ χ n (t) ≡ t 0 dt ′ λ χ n (α t ′ ). Inserting this equation into Eq. (5) and taking the Hilbert-Schmidt inner product withl χ 0 (α t ), we obtain We assume that the modulation of the parameters α is sufficiently slow. Then we can approximate the sum on the right hand side of Eq. (8) by the contribution only from the term with n = 0. By solving this adiabatic approximation equation we obtain where dρ χ 0 (α) ≡ dα · ∂ρ χ 0 (α)/∂α. If the initial state of the system S isρ χ (0) =ρ ss (α 0 ), then c 0 (0) = Tr S l χ † 0 (α 0 )ρ ss (α 0 ) . We again use the adiabatic ap-proximation to obtain Thus we obtain the excess cumulant generating function S ex τ (χ) = S τ (χ) − Λ χ 0 (τ ) for the slow modulation: The right-hand side of Eq. (11) is analogous to the Berry phase, which depends only on the line integral along the curve C (not on time). We note that Λ χ 0 (τ ) corresponds to the dynamical phase.
By differentiating Eq. (11) with respect to iχ, we obtain a geometrical expression of the average excess in the quantity ∆q: (11) and (12) are regarded as quantum versions of the Berry-Sinitsyn-Nemenman (BSN) phases for the cumulant generating function and the average, respectively, in slow parametric modulation 16,21 . We refer to Tr S l ′ † 0 ∂ρ ss /∂α as the BSN vector potential for average excess quantity.
Equality (12) implies that a finite net quantity of ∆q can be transferred to the system S for a slow cyclic modulation of the parameters (i.e., for the case where the curve C is a closed loop) even if there is no dc driving force such as temperature and chemical potential differences. This is an adiabatic pump. By the Stokes theorem, Eq. (12) is rewritten as for a cyclic process, where ∧ is the wedge product, S C is a surface enclosed by C, and We refer to Eq. (14) as the BSN curvature for average pumped quantity. For some cyclic processes, the excess pump ∆q ex τc vanishes. A sufficient condition for the "no-pumping" is that F αmαn = 0 holds in S C for all (α m , α n ). We note that if the whole of a curve C (not necessarily closed) lies in a region of the no-pumping condition, then the average excess pump does not depend on the whole of C but depends on only the initial and final points of C.

III. APPLICATION TO SPINLESS ELECTRON TRANSPORT IN QUANTUM DOT
In this section, we apply the general framework obtained in the previous section to the transport of spinless electrons in quantum dot systems connected to two electron reservoirs (b = L, R). The spinless electron model can be used for analzying phenomena in which the spin degrees of freedom are irrelevant. The reservoir Hamiltonian is given byĤ Here Ω bk is the kth mode energy of the electron in the bth reservoir, andĉ † bk (ĉ bk ) is the corresponding creation (annihilation) operator, which satisfies We here take the quantity to be counted as the electron number transferred from the reservoir L to the system S, i.e.,Q =N L = kĉ † Lkĉ Lk .

A. Non-interacting electron model with RWA
First we consider a model of non-interacting spinless electrons in a series of N single-level quantum dots coupled to two reservoirs (b = L, R). The Hamiltonian of the system S in this model iŝ where ε i is the level energy of the ith dot,d † i (d i ) is the creation (annihilation) operator of the electron in the ith dot, and v is the transfer probability amplitude between the dots. This Hamiltonian can be diagonalized by a unitary transformation asĤ S = j ω jâ † jâ j . Here ω j is the jth mode energy of the electron in the system S, andâ † j (â j ) is the corresponding creation (annihilation) operator of the electron, which satisfies the canonical anti-commutation relations of fermion: The coupling Hamiltonian between the system S and the bth reservoir is given bŷ To derive the GQME for this system we use the Born (up to the second order inĤ Sb ) and the Markov approximations 26 . Furthermore, we adopt the RWA, which is a coarse-graining of the time evolution on the time scale longer than that of the system evolution without the coupling with the reservoirs 26,27 . For χ = 0 the RWA leads a Lindblad form of the QME and guarantees the complete positivity of the time evolution. We carry out the RWA by averaging over the rapidly oscillating terms in the Born-Markov GQME in the interaction picture (see Appendix A for the detail). Then we obtain the generator K χ of the GQME in the form of K χ = j K χ,j . Here the GQME generator K χ,j for the jth mode is given whereĤ Lamb . The control parameters α in this model can be both the system parameters, the levels {ε i } of the dots and the transfer v between the dots, and the reservoir parameters In this model within the RWA, we analytically obtain the BSN vector potential (see Appendix B for the derivation): is the spectral function of the bth reservoir. Also we can calculate the BSN curvature as From Eq. (18), we find that no net excess number of electrons flow per cycle if only the reservoir parameters (β L , µ L , β R , µ R ) are modulated with the system parameters fixed. This is because the spectral function is written as Γ bj (ω j ) = 2π k |V bjk | 2 δ(ω j − Ω bk ), so that Γ Lj /Γ j in Eq. (18) is independent of (β L , µ L , β R , µ R ). We note that this result is characteristic of fermion systems. Indeed, in a model of single two-level system connected to two bosonic heat reservoirs, there exists heat pump by cyclic modulations of the temperatures of the two reservoirs 21 . This difference between the results for the fermion and boson reservoirs comes from the particle statistics, which leads that c † c + cc † b is independent of (dependent on) the reservoir parameters for fermion (boson) reservoir. Because Φ + bj and Φ − bj are respectively proportional to c † c b and cc † b , the particle statistics determines whether Γ bj = Φ + bj +Φ − bj depends on the reservoir parameters.
Before closing this subsection, we make a remark on the validity of the RWA on the transport systems. The GQME gives the same result either with or without the RWA, as far as the transport between the system and the reservoirs is studied, whereas it is known that the internal current in the system vanishes in nonequilibrium steady states under the RWA 29 . In Appendix C, we analytically show that the unit-time cumulant generating function λ χ 0 (α) of the quantity transferred from the reservoirs to the system in the steady state for fixed α is equivalent between the GQMEs within and without the RWA. We also confirm numerically that the results of the adiabatic pump within the RWA quantitatively agree with those without the RWA in the next subsection.

B. Non-interacting electron in double quantum dot without RWA
Next we consider a non-interacting double quantum dot system coupled to two reservoirs, as illustrated in Fig. 2 (a). We assume the wide band limit: Γ bj (ω) = Γ 0 bj = const. In this subsection we use three different methods for calculating ∆q ex τ in modulations of the control parameters of the model. In the first method (denoted by RWA), we apply our results within the RWA [Eqs. (17) and (18)]. In the second one (denoted by Non-RWA1), we numerically solve the eigenvalue problem of the GQME generator K χ without RWA, and use Eq. (12). In the third method (denoted by NonRWA2), we numerically solve the time evolution differential equation of the GQME without RWA, and use S τ (χ) = Tr Sρ χ (τ ). In Fig. 2 (b), we plot the excess electron number ∆q ex τ transferred from the reservoir L to the system for noncyclic modulations of β L , µ L , and µ R along the curves illustrated in the inset of Fig. 2 (b). We plot the dependence of ∆q ex τ on the final value of the right reservoir chemical potential µ fin R of the modulations. We see that all the results agree within the numerical precision. This implies that the no-pumping condition described below Eq. (18) within the RWA still holds without the RWA. We have also confirmed that the absolute value of the BSN curvature computed without the RWA is less than 10 −6 in the space of the reservoir parameters, which is zero within the numerical precision.
In Fig. 2 (c), we plot the BSN curvature F ε1ε2 calculated by the method of NonRWA1 as a function of ε 1 and ε 2 . We see that the curvature takes the non-zero values in this case. Therefore we have a finite geometrical pump for the slow periodic modulation of the dot levels (system parameters). We note that this result of the BSN curvature also agrees with the RWA result given by Eq. (18), although not shown in the figure.
We also calculate ∆q ex τc for the cyclic process depicted in Fig. 2 (c). In Fig. 2 (d), we plot ∆q ex τc calculated by NonRWA2 as a function of the cycle period τ c for various values of the amplitude Γ 0 bj of the spectral function. We see that for large τ c the asymptotic results by NonRWA2 agree with the results by RWA and NonRWA1. This supports the validity of the adiabatic approximation used in deriving Eq. (12) for slow modulations. The characteristic time for the validity of the adiabatic approximation becomes shorter as Γ 0 bj increases, which implies that τ c ≫ /Γ bj is an adiabatic condition.
We note that all the results by RWA and NonRWA1 agree with each other not only qualitatively but also quantitatively. This implies that the rotating wave approximation is valid in discussing transport between the system and the reservoirs under slow modulations of the parameters.

C. Interacting electron model
We next consider an interacting spinless electron system in a double quantum dot. We take the system Hamil- tonian aŝ In this model, an electron in one dot interacts with an electron in the other dot. We investigate the excess electron transfer ∆q ex τc with the modulation of the chemical potentials (µ L and µ R ). We also assume the wide band limit: Γ bj (ω) = Γ 0 bj = const. In Fig. 3 (a)-(c), we plot the BSN curvature F µLµR for various values of the interaction strength U obtained from the numerical diagonalization method without the RWA. We observe the two positive and two negative peaks. In Fig. 3 (d), we plot the position of one of the positive peaks and that of one of the negative peaks. We see that the positions of the negative peaks move as U increases, whereas those of the positive ones do not. Moreover we find that these peak positions are located around at the energies necessary to add one electron; the approximate position of the positive peaks are (µ L , µ R ) = ( ω 1 , ω 2 ) and ( ω 2 , ω 1 ), and those of the negative ones are (µ L , µ R ) = ( ω 1 + U, ω 2 + U ) and ( ω 2 + U, ω 1 + U ). This implies that the positive and negative peaks merge for non-interacting system (U = 0), and thus the curvature F µLµR becomes zero in (µ L , µ R )space to achieve the no-pumping condition mentioned below Eq. (18). This result also implies that, for U > 0, an adiabatic pump can occur even if only the reservoir parameters are modulated.
Indeed, Fig. 3 (e) shows the U -dependence of the excess electron transfer ∆q ex τc for cyclic processes [cycle 1 and 2 depicted in Fig. 3 (c)], where ∆q ex τc is non-zero for U > 0. Note that the direction of the cycle 2 is opposite to that of the cycle 1. We also note that ∆q ex τc for the cycle 1 becomes a constant for U 1[meV], because the positive peak positions of F µLµR do not move as U increases. In contrast, ∆q ex τc for the cycle 2 has a peak at U ≃ 0.5[meV] and becomes nearly zero for U 1[meV]. This results from the fact that the negative peak positions of F µLµR move away from the cycle 2 as U increases.

IV. DISCUSSION AND CONCLUSION
By using a QME approach, we have derived a geometrical expressions of the cumulant generating function and the average of the pumped (excess) quantity transferred from a reservoir to the system under slow modulation of control parameters, where the BSN phases and the BSN curvature of the QME play crucial roles.
For non-interacting electrons in quantum dot systems, there is no pump current when only the temperatures and chemical potentials of the reservoirs are modulated. In contrast, for an interacting system, the pump current can be observed even in this situation. We note that the modulations of only the chemical potentials of the reservoirs are required for the pump. This has an advantage for the control of the pump in experiments, since the modulation of chemical potential is easier than that of the temperatures. As shown in Fig. 3 (e), for a cyclic modulation of the chemical potentials, the pump current depends not only on the difference of the chemical potentials but also on their average. This implies that the average number of electrons in the system S is important for the pump. For example, when we modulate the chemical potentials as µ L (t) = µ c + µ r cos(2πt/τ c ) and µ R (t) = µ c + µ r sin(2πt/τ c ), µ c affects the quantity of the pump. This fact may be applicable for switching the pump by the change of µ c or the electron density, which can be controlled by a gate voltage.
Since we applied the method of the full counting statistics, we can calculate the fluctuation of the pump variable. It is a future issue to analyse the detailed properties of the fluctuation in the adiabatic pump. It is also interesting to investigate the relation between the present geometrical expression of the adiabatic pump based on the QME and the conventional geometrical expressions based on the scattering theory 9-12 . The investigations of non-adiabatic pump, non-Markovian situation, and spin effect are also future issues.
We here derive the concrete form of the GQME (5)- (6) without and within the RWA.
We start from the total Hamiltonian (system plus reservoirs) given aŝ Here, for simplicity, we assume that the reservoir Hamil-tonianĤ b and the coupling HamiltonianĤ Sb between the system and the reservoir are respectively written aŝ whereâ i is a single-particle mode annihilation operator in the system S, i b is the index of the system mode that couples to the bth reservoir, andĉ bk is the kth mode annihilation operator in the bth reservoir. We denote the eigenenergy of the system HamiltonianĤ S as E x , and the corresponding energy eigenstate as |E x . We also assume that all the eigenenergies ofĤ S are non-degenerate.
We consider the quantityQ = b k q bkĉ † bkĉ bk to be counted, and define the current ofQ from the reservoirs to the system S as positive.
For the derivation of the GQME, it is convenient to introduce the eigenoperators 26 Then the modified coupling Hamiltonian in the interaction picture is written aš We assume that the initial state of the total system is written asρ tot (0) =ρ 0 ⊗ρ res , whereρ 0 is a initial state of the system S,ρ res = bρ G b , andρ G b = e −β b (Ĥ b −µ bNb ) /Z b is the grand-canonical state of the bth reservoir. Then substituting Eq. (A6) into Eq. (5), we obtain the GQME d dtρ where we used ∞ 0 dt ′ e iωt ′ = πδ(ω) + i P ω .
Here, q(Ω bk ) ≡ q bk , and Equation (A7) is the concrete form of the GQME without the RWA. When we transform Eq. (A7) in the interaction picture, we see that rapidly oscillating terms proportional to exp[±i(ω S −ω ′ S )t] appear. In the RWA we neglect these terms 26 . Thus we obtain the GQME with the RWA by leaving only the terms with ω ′ S = ω S in Eq. (A7). For the non-interacting models in Secs. III A and B, because the eigenoperators are the mode operators themselves, the GQMEs for these models, in particular Eq. (16) for the RWA case, are derived.
Appendix B: Derivation of Eq. (17) In the model in Sec. III A within the RWA, the eigenvalues and the eigenvectors of K χ can be decomposed into those of K χ,j . That is, λ χ 0 = j λ χ 0,j ,l χ 0 = jl χ 0,j , and ρ χ 0 = jρ χ 0,j , where λ χ 0,j is the eigenvalue of K χ,j with maximum real part, andl χ 0,j andρ χ 0,j are respectively the corresponding left and right eigenvectors, which are operators on the jth mode Hilbert space.