Quantum Pumping in Closed Systems, Adiabatic Transport, and the Kubo Formula

PHYSICAL REVIEW B 68, 155303 ͑2003͒

Quantum pumping in closed systems, adiabatic transport, and the Kubo formula

Doron Cohen Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel ͑Received 8 April 2003; revised manuscript received 30 June 2003; published 3 October 2003͒ Quantum pumping in closed systems is considered. We explain that the Kubo formula contains all the physically relevant ingredients for the calculation of the pumped charge ͑Q͒ within the framework of linear response theory. The relation to the common formulations of adiabatic transport and ‘‘geometric magnetism’’ is clariﬁed. We distinguish between adiabatic and dissipative contributions to Q. On the one hand we observe that adiabatic pumping does not have to be quantized. On the other hand we deﬁne circumstances in which quantized adiabatic pumping holds as an approximation. The deviation from exact quantization is related to the Thouless conductance. As an application we discuss the following examples: classical dissipative pumping by conductance control, classical adiabatic ͑nondissipative͒ pumping by translation, and quantum pumping in the double barrier model. In the latter context we analyze a 3 site lattice Hamiltonian, which represents the simplest pumping device. We remark on the connection with the popular S matrix formalism which has been used to calculate pumping in open systems.

DOI: 10.1103/PhysRevB.68.155303 PACS number͑s͒: 03.65.Vf, 73.23.Ϫb, 05.45.Mt

I. INTRODUCTION the quasimomentum ␾ is a constant of motion. It follows that the mathematical treatment of a driven periodic structure Linear response theory1–3 ͑LRT͒ is the leading formalism reduces to an analysis of a driven ring system with flux. to deal with driven systems. Such systems are described by a Hamiltonian H(x) where x(t) is a set of time dependent B. Classification of pumps ͑ ͒ classical parameters ‘‘fields’’ . The Kubo formula is the cor- ‘‘Pumping’’ means that net charge ͑or maybe better to say ner stone of LRT. It allows the calculation of the response ‘‘net integrated probability current’’͒ is transported through coefficients, and in particular the conductance matrix (G)of the ring per cycle of a periodic driving. Using the common ͑ ͒ the system. If we know G, we can calculate the charge Q jargon of electrical engineering this can be described as which is transported through the system during one cycle of ac-dc conversion. We distinguish between pumping in open a periodic driving. This is called pumping. ͓ ͑ ͔͒ 4 systems such as in Fig. 1 e , pumping in closed systems Pumping of charge in mesoscopic and molecular size de- ͓such as in Fig. 1͑d͔͒, and pumping in periodic systems ͓such vices is regarded as a major issue in the realization of future as in Fig. 1͑f͔͒. quantum circuits or quantum gates, possibly for the purpose of quantum computing.

A. Model system In order to explain the motivation for the present work, and its relation to the published literature, we have to give a better definition of the problem. For presentation purpose we focus on a model system with a ring geometry ͑Fig. 1͒. The shape of the ring is controlled by some parameters x1 and x2. These parameters can be gate voltages that determine the location of some boundaries, or the height of some barriers. The third parameter is the flux through the ring: ϭ⌽ϵ ប ͒␾ ͑ ͒ x3 ͑ /e . 1 We shall use units such that the elementary charge is eϭ1. H Note that the Hamiltonian ͓x1(t),x2(t),x3(t)͔ has gauge -␾ϩ2␲. Another system with a ring toۋinvariance for ␾ ͑ ͒ pology is presents in Fig. 1 b , and its abstraction is repre- FIG. 1. Illustration of a ring system ͑a͒. The shape of the ring is sented in Fig. 2͑c͒. The ‘‘dot’’ can be represented by an S controlled by some parameters x1 and x2. The flux through the ring matrix that depends on x and x . In Fig. 1͑d͒ also the flux ϭ⌽ 1 2 is x3 . A system with equivalent topology, and abstraction of the x3 is regarded as a parameter of the dot. If we cut the wire in model are presented in ͑b͒ and ͑c͒. The ‘‘dot’’ can be represented by ͑ ͒ ͑ ͒ ͑ ͒ Fig. 1 d we get the open two lead geometry of Fig. 1 e . an S matrix that depends on x1 and x2.In d also the flux x3 is Finally we can put many such units in series ͑no flux͒, hence regarded as a parameter of the dot. If we cut the wire in ͑d͒ we get getting the periodic system of Fig. 1͑f͒. In the latter case the the open two lead geometry of ͑e͒. If we put many such units in Hamiltonian is invariant for unit translations, and therefore series we get the period system in ͑f͒.

and to explain how dissipation emerges in the quantum mechanical treatment. For one-parameter driving a unifying picture that bridges between the quantum mechanical adiabatic picture and LRT has been presented in Refs. 16–19. A previous attempt20 had ended in some confusion regarding the identification of the linear response regime, while Ref. 15 avoided the analysis of the mechanism that leads to dissipation in the quantum mechanical case. The presented ͑Kubo based͒ formulation of the pumping problem has a few advantages. It is not restricted to the adiabatic regime; it allows a clear distinction between dissipative and adiabatic contributions to the pumping; the classical limit is manifest in the formulation; it gives a level by level understanding of the pumping process; it allows the consid- eration of any type of occupation ͑not necessarily Fermi oc- cupation͒; it allows future incorporation of external environ- mental influences such as that of noise; and it regards the voltage over the pump as electromotive force, rather than adopting the conceptually more complicated view9 of having FIG. 2. Schematic illustration of quantum pumping in a closed a chemical potential difference. wire-dot system. The net charge via the third level ͑thick solid line on the right͒ is vanishingly small: As the dot potential is lowered an Of particular interest is the possibility to realize a pump- ͑ ͒ ing cycle that transfers exactly one unit of charge per cycle. electron is taken from the left side first avoided crossing , and then 7,8 ͑ ͒ In open systems this ‘‘quantization’’ holds only approxi- emitted back to the left side second avoided crossing . Assuming 7 that the bias is inverted before the dot potential is raised back, only mately, and it has been argued that the deviation from exact the second level carry a net charge QϭO(1). quantization is due to the dissipative effect. Furthermore it has been claimed7 that exact quantization would hold in the For a reason that was explained at the end of the previous strict adiabatic limit, if the system were closed. In this paper subsection we regard the last category5 as mathematically we would like to show that the correct picture is quite dif- equivalent to the second category. We also regard the first ferent. We shall demonstrate that the deviation from exact category6–9 as a special ͑subtle͒ limit of the second category: quantization is in fact of adiabatic nature. This deviation is in a follow up paper10 we demonstrate that in the limit of related to the so-called ‘‘Thouless conductance’’ of the de- open geometry the Kubo formula reduces to the S-matrix vice. formula of Bu¨ttiker, Pre´tre, and Thomas.6 There are works in the literature regarding ‘‘rectification’’ D. Examples 11 and ‘‘ratchets.’’ These can be regarded as studies of pump- We give several examples for the application of the Kubo ing in periodic systems with the connotation of having formula to the calculation of the pumped charge Q: classical damped non-Hamiltonian dynamics. These type of systems dissipative pumping, classical adiabatic pumping ͑by trans- are beyond the scope of the present paper. There is also a ͒ 12 lation , and quantum pumping in the double barrier model. recent interest in Hamiltonian ratchets, which is again a The last example is the main one. In the context of open synonym for pumping in periodic systems, but with the con- geometry it is known as ‘‘pumping around a resonance.’’8 notation of having a nonlinear pumping mechanism. We are We explain that this is in fact an diabatic transfer scheme, going to clarify what are the conditions for having a linear and we analyze a particular version of this model which is pumping mechanism. Only in case of linear pumping mecha- represented by a three site lattice Hamiltonian. This is defi- nism the Kubo formula can be used, which should be distin- nitely the simplest pump circuit possible, and we believe that guished from the nonlinear mechanism of Ref. 12. it can be realized as a molecular size device. It also can be regarded as an approximation for the closed geometry ver- C. Objectives sion of the two delta potential pump ͑Fig. 2͒.8 The purpose of this paper is to explain and demonstrate that the Kubo formula contains all the physically relevant E. Outline ingredients for the calculation of the charge ͑Q͒ which is In Sec. II we define the main object of the study, which is pumped during one cycle of a periodic driving. In the limit the conductance matrix G of Eq. ͑5͒. The conductance ma- of a very slow time variation ͑small x˙ ), the emerging picture trix can be written as the sum of a symmetric (␩) and an coincides with the adiabatic picture of Refs. 5,13–15. In this antisymmetric (B) matrices, which are later identified as the limit the response of the system is commonly described as a dissipative and the adiabatic contributions respectively. In nondissipative ‘‘geometric magnetism’’15 effect or as adia- the first part of the paper ͑Secs. II–VIII͒ we analyze the batic transport. adiabatic equation ͑Sec. III͒, and illuminate the distinction A major objective of this paper is to bridge between the between its zero order solution ͑Sec. IV͒ its stationary first adiabatic picture and the more general LRT/Kubo picture, order solution ͑Sec. V͒, and its nonstationary solution ͑Sec.

155303-2 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒

VI͒. The outcome of the analysis in Sec. V is Eq. ͑26͒ for the Im͓␹kj͑␻͔͒ ϱ kjϭ ϭ ͵ ␣kj͑␶͒␶ ␶ ͑ ͒ conductance matrix G. This expression is purely adiabatic, G lim ␻ d . 4 and does not give any dissipation. In order to get dissipation ␻→0 0 we have to look for a nonstationary solution. Consequently, as explained further in Appendix B, the re- The standard textbook derivation of the Kubo formula sponse in the ‘‘dc limit’’ (␻→0) can be written as ͓Eq. ͑29͔͒ for the conductance matrix G implicitly assumes a nonstationary solution. We show how to get from it Eq. ͑30͒ ␩ ͑ ͒ k ϭϪ kj˙ ͑ ͒ for and Eq. 31 for B. The latter is shown to be identical ͗F ͘ ͚ G x j . 5 with the adiabatic result ͓Eq. ͑26͔͒. In Sec. VII we further j simplify the expression for ␩ leading to the fluctuation- As an example for the applicability of this formula note the dissipation relation ͓Eq. ͑33͔͒. following standard examples for one parameter driving: Let The disadvantages of the standard textbook derivation of ϭ ˙ ϭ Kubo formula make it is essential to introduce a different x wall or piston displacement, then x wall or piston route toward Eq. ͑33͒ for ␩. This route, which is discusssed velocity, Gϭ friction coefficient, and FϭϪGx˙ is the fric- in Sec. VIII, explicitly distinguishes the dissipative effect tion force. Another standard example is xϭ magnetic flux, from the adiabatic effect, and allows to determine the condi- Ϫx˙ ϭ electromotive force, Gϭ electrical conductance, and tions for the validity of either the adiabatic picture or LRT. In hence FϭϪGx˙ is the Ohm law. particular it is explained that LRT is based, as strange as it It is convenient to write the conductance matrix as Gkj sounds, on perturbation theory to infinite order. ϵ␩kjϩBkj, where ␩kjϭ␩ jk is the symmetric part of the In Sec. IX we clarify the general scheme of the pumping kjϭϪ jk ͓ ͑ ͔͒ conductance matrix, while B B is the antisymmetric calculation Eq. 39 . Secs. X and XI give two simple clas- part. In case of having three parameters we can arrange the sical examples. In Sec. XII we turn to discuss quantum ᠬ pumping, where the cycle is around a chain of degeneracies. elements of the antisymmetric part as a vector B ϭ 23 31 12 ͑ ͒ The general discussion is followed by presentation of the (B ,B ,B ). Consequently Eq. 5 can be written in ab- double barrier model ͑Sec. XIII͒. In order to get a quantita- stract notation as tive estimate for the pumped charge we consider a three site ͒ ͑ ˙ ͘ϭϪ␩ ˙Ϫ ٙ ͗ ͒ ͑ lattice Hamiltonian Sec. XIV . F •x B x, 6 The summary ͑Sec. XV͒ gives some larger perspective on the subject, pointing out the relation to the S-matrix formal- where the dot product should be interpreted as matrix-vector ism, and to the Born-Oppenheimer picture. In the Appen- multiplication, which involves summation over the index j. dixes we give some more details regarding the derivations, The wedge-product also can be regarded as a matrix-vector so as to have a self-contained presentation. multiplication. It reduces to the more familiar cross product in the case where we consider three parameters. The dissipation, which is defined as the rate in which energy is absorbed II. THE CONDUCTANCE MATRIX into the system, is given by Consider the Hamiltonian H͓x(t)͔, where x(t) is a set of d time dependent parameters ͑‘‘fields’’͒. For presentation, as W˙ ϭ H ϭϪ ˙ϭ ␩kj˙ ˙ ͑ ͒ ͗ ͘ ͗F͘ x ͚ xkx j . 7 well as for practical reasons, we assume later a set of three dt • kj ϭ time dependent parameters x(t) ͓x1(t),x2(t),x3(t)͔.We define generalized forces in the conventional way as Only the symmetric part contributes to the dissipation. The contribution of the antisymmetric part is identically zero. Hץ kϭϪ ͑ ͒ III. THE ADIABATIC EQUATION 2 . ץ F xk The adiabatic equation is conventionally obtained from 1 Note that if x1 is the location of a wall element, then F is the Schrodinger equation by expanding the wave function in the force in the Newtonian sense. If x2 is an electric field, the x-dependent adiabatic basis: 2 then F is the polarization. If x3 is the magnetic field or the 3 flux through a ring, then F is the magnetization or the cur- d i rent through the ring. ͉␺͘ϭϪ H͓x͑t͔͉͒␺͘, ͑8͒ dt ប In linear response theory ͑LRT͒ the response of the system is described by a causal response kernel, namely, ͉␺ ϭ ͉͒ ͒ ͑ ͒ ͘ ͚ an͑t n͓x͑t ͔͘, 9 ϱ n ͗ k͘ ϭ ͵ ␣kj͑ Ϫ Ј͒ ͑ Ј͒ Ј ͑ ͒ F t ͚ t t x j t dt , 3 j Ϫϱ da i i n ϭϪ ϩ ˙ j ͑ ͒ kj kj Enan ͚ ͚ x jAnmam , 10 where ␣ (␶)ϭ0 for ␶Ͻ0. The Fourier transform of ␣ (␶) dt ប ប m j is the generalized susceptibility ␹kj(␻). The conductance matrix is defined as where following Ref. 13 we define

155303-3 DORON COHEN PHYSICAL REVIEW B 68, 155303 ͑2003͒

Hץ ץ j ͑ ͒ϭ បͳ ͑ ͒ͯ ͑ ͒ʹ ͑ ͒ ͗ k͘ϭ͗ ͑ ͉͒Ϫ ͉ ͑ ͒͘ ͑ ͒ n x 22 ץ m x . 11 F n x ץ Anm x i n x‘ x j xk ϭ ͉ ץ Differentiation by parts of j͗n(x) m(x)͘ 0 leads to the ץ j conclusion that Anm is a hermitian matrix. Note that the ef- ϭϪ ͗ ͑ ͉͒H͑ ͉͒ ͑ ͒͘ ͑ ͒ n x x n x . 23 ץ fect of gauge transformation is xk Ϫ ⌳ ប ͒ ͑ ͒ ͑ ͉[ /(i[ n(x ۋ ͒ ͑ ͉ n x ͘ e n x ͘, 12 In the case of the standard examples that were mentioned ⌳ Ϫ⌳ ប previously this corresponds to conservative force or to per- ͒ ͑ ␦͒ ⌳ ץi[( n m)/ ] j ϩ͑ ۋ j Anm e Anm j n nm . 13 sistent current. From now on we ignore this trivial contribu- j ϵ j tion to ͗Fk͘, and look for the a first order contribution. Note that the diagonal elements An Ann are real, and trans- ⌳ ץj ϩ ۋ j form as An An j n . Associated with An(x) is the gauge invariant two form, V. THE STATIONARY ADIABATIC SOLUTION: which is defined as ADIABATIC TRANSPORT OR ‘‘GEOMETRIC MAGNETISM’’ ͒ ͑ i ץj Ϫ ץijϭ Bn iAn jAn 14 ͑ ͒ For linear driving unlike the case of a cycle the An(x) ͒ ͑ ץ͉ ץ ϭϪ ប 2 Im͗ in jn͘ 15 field can be gauged away. Assuming further that the adiabatic equation can be treated as parameter independent ͑that means 2 disregarding the parametric dependence of E and W on x) ϭϪ i j ͑ ͒ n Im͚ AnmAmn . 16 ͑ ͒ ប m one realizes that Eq. 20 possesses stationary solutions. To first order these are .This can be written in an abstract notation as BϭٌٙA Using standard manipulations, namely, via differentiation by W ͒ ͑ n(x)͉H͉m(x)͘ϭ0, we get for nÞm the expres- ͉␺͘ϭ͉ ͘ϩ mn ͉ ͗͘ ץ parts of j n ͚ Ϫ m . 24 sions m(Þn) En Em

H iបF j Note that in a fixed-basis representation the above stationaryץ iប j ͑ ͒ϭ ͗ ͉ ͉ ͘ϵϪ nm ͑ ͒ m Ϫ 17 solution is, in fact, time dependent. Hence the notations ץ Anm x Ϫ n Em En x j Em En ͉n͓x(t)͔͘, ͉m͓x(t)͔͘, and ͉␺(t)͘ are possibly more appro- and hence priate. With the above solution we can write ͗Fk͘ as a sum of Im͓Fi F j ͔ ijϭ ប nm mn ͑ ͒ zero order and first order contributions. From now on we Bn 2 ͚ . 18 Þ Ϫ ͒2 ignore the zero order contribution, and go on with the first m( n) ͑Em En order contribution IV. THE STRICTLY ADIABATIC SOLUTION, AND THE Hץ BERRY PHASE Wmn ͗Fk͘ϭϪ ͚ ͗n͉ ͉m͘ϩc.c. xץ Þ E ϪE We define the perturbation matrix as m( n) n m k

ϭ͚ ͩ i͚ Ak A j ϩc.c.ͪ x˙ ϭϪ͚ Bkjx˙ . ͑25͒ ϭϪ ˙ j Þ ͑ ͒ nm mn j n j Wnm ͚ x jAnm for n m 19 j m j j j ϭ ϭ and Wnm 0 for n m. Then the adiabatic equation can be For a general stationary preparation, either pure or mixed, rewritten as follows: one obtains Eq. ͑5͒ with

da i i n ϭϪ ͑ Ϫ ˙ ͒ Ϫ ͑ ͒ En xAn an ͚ Wnmam . 20 kjϭ ͑ ͒ kj ͑ ͒ dt ប ប m G ͚ f En Bn , 26 n If we neglect the perturbation W, then we get the strict adiabatic solution where f (En) are weighting factors, with the normalization ͚ ϭ n f (En) 1. For a pure state preparation f (En) distin- Ϫi/ប{͐t E [x(tЈ)]dtЈϪ͐x(t) A (x) dx} e 0 n x(0) n • ͉n x͑t͒ ͘. ͑21͒ guishes only one state n, while for canonical preparation „ … ϰ Ϫ f (En) exp( En /T), where T is the temperature. For a many- Due to A (x), we have the so called geometric phase. This n body system of non-interacting Fermions f (En) can be rein- can be gauged away unless we consider a closed cycle. For a ͚ terpreted as the Fermi occupation function, so that n f (En) ប ͛ ᠬ closed cycle, the gauge invariant phase (1/ ) A•dx is called is the total number of particles. Berry phase. Thus we see that the assumption of a stationary ﬁrst-order With the above zero-order solution we can obtain the fol- solution leads to a nondissipative ͑antisymmetric͒ conduc- lowing result: tance matrix. This is know as either adiabatic transport5,14 or

155303-4 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒

13 ‘‘geometric magnetism.’’ In the later sections we shall dis- 1 Im͓K˜ ij͑␻͔͒ cuss the limitations of the above result. ␩ijϭ 1 ͑ ijϩ ji͒ϭ ͑ ͒ 2 G G lim ␻ , 30 2␻→0 VI. THE NONSTATIONARY SOLUTION: THE KUBO FORMULA ϱ d␻ Re͓K˜ ij͑␻͔͒ ijϭ 1 ͑ ijϪ ji͒ϭϪ͵ ͑ ͒ B 2 G G . 31 The Kubo formula is an expression for the linear response Ϫϱ2␲ ␻2 of a driven system that goes beyond the stationary adiabatic solution of the previous section. The Kubo formula has many The antisymmetric part is identiﬁed15 ͑Appendix C͒ as cor- type of derivations. One possibility is to use the same proce- responding to the stationary solution Eq. ͑26͒ of the adiabatic dure as in Section V starting with equation. Ϫ ͉␺͑t͒͘ϭe iEnt͉n͘

t VII. THE EMERGENCE OF DISSIPATIVE RESPONSE, ͑ Ϫ ͒ Ј Ϫ ϩ ͚ ͫ ϪiW ͵ ei En Em t dtЈͬe iEmt͉m͘ AND THE FLUCTUATION-DISSIPATION RELATION mn m͑Þn͒ 0 The Kubo formula for the symmetric part of the conduc- For completeness we give in Appendix A the simplest ver- tance matrix (␩ij) can be further simplified. If we take Eq. sion of the standard derivation, which is based on a conven- ͑30͒ literally, then ␩ijϭ0 due to the simple fact that we have tional fixed-basis first-order-treatment of the perturbation. finite spacing between energy levels ͑see Ref. 21 for a sta- The disadvantages are: The standard derivation does not il- tistical point of view͒. But if we assume that the energy luminate the underlaying physical mechanisms of the re- levels have some finite width ⌫, then the smoothed version sponse, the stationary adiabatic limit is not manifest, the of K˜ ij(␻) should be considered. In common textbooks the fluctuation-dissipation relation is vague, and the validity con- introduction of ⌫ is ‘‘justified’’ by assuming some weak cou- ditions of the derivation are not clear: no identification of the pling to an environment, or by taking the limit of infinite regimes. For now we go on with the conventional approach, volume. But we are dealing with a strictly isolated finite but in a later section we refer to the more illuminating ap- ⌫ proach of Refs. 20 and 16–19. Then we clarify what is the system, and therefore the meaning of requires serious con- sideration. We postpone the discussion of this issue to the ͑ ˙ regime range of x) were the Kubo formula can be next section. trusted,18,19 and what is the subregime where the response If the smoothed version of K˜ ij(␻) should be used in Eq. can be described as nondissipative adiabatic transport. ͑ ͒ ␩ij In order to express the Kubo formula one introduces the 30 , then it is possible to obtain from power spectrum ij following definition: C˜ (␻) of the fluctuations. This is called the fluctuation- dissipation relation. The spectral function C˜ ij(␻) is defined i as the Fourier transform of the symmetrized correlation func- ij͑␶͒ϭ ͓͗ i͑␶͒ j͑ ͔͒͘ ͑ ͒ K ប F ,F 0 . 27 tion

We use the common interaction picture notation Fk(␶) iHt k ϪiHt ij͑␶͒ϭ͗ 1 ͓ i͑␶͒ j͑ ͒ϩ j͑ ͒ i͑␶͔͒͘ ͑ ͒ ϭe F e , where HϭH(x) with xϭconst. The expec- C 2 F F 0 F 0 F . 32 tation value assumes that the system is prepared in a stationary state ͑see previous section͒. It is also implicitly assumed We use again the interaction picture, as in the definition of that the result is not sensitive to the exact value of x. Note Kij(␶). Also this function has a well defined classical limit. that Kij(␶) has a well defined classical limit. Its Fourier There are several versions for the fluctuation-dissipation transform will be denoted K˜ ij(␻). relation. The microcanonical version15 has been derived us- The expectation value ͗Fk͘ is related to the driving x(t) ing classical considerations, leading to by the causal response kernel ␣ij(tϪtЈ). The Kubo expression for this response kernel, as derived in Appendix A, is 1 1 d ␩ij͉ ϭ ͓ ͑ ͒˜ ij͑␻→ ͔͒ ͑ ͒ E g E CE 0 . 33 ␣ij͑␶͒ϭ⌰͑␶͒Kij͑␶͒, ͑28͒ 2 g͑E͒ dE ⌰ ␶ where the step function ( ) cares for the upper cutoff of In Appendix C we introduce its quantum mechanical deriva- ͑ ͒ ␣ij ␶ the integration in Eq. 3 . The Fourier transform of ( )is tion. The subscript emphasizes that we assume a microca- ␹ij ␻ the generalized susceptibility ( ). The conductance ma- nonical state with energy E, and g(E) is the density of states. trix is defined as The traditional version of the fluctuation-dissipation relation assumes a canonical state. It can be obtained by canonical Im͓␹ij͑␻͔͒ ϱ ijϭ ϭ ͵ ij͑␶͒␶ ␶ ͑ ͒ averaging over the microcanonical version leading to G lim ␻ K d . 29 ␻→0 0

This can be split into symmetric and antisymmetric compo- 1 ␩ij͉ ϭ C˜ ij͑␻→0͒. ͑34͒ nents ͑see derivation in Appendix C͒ as follows: T 2T T

155303-5 DORON COHEN PHYSICAL REVIEW B 68, 155303 ͑2003͒

VIII. THE VALIDITY OF LINEAR RESPONSE THEORY The adiabaticity condition ͑36͒ can be explained in a AND BEYOND more illuminating way as follows: Let us assume that we prepare the system at time tϭ0 at the level ͉n͘. Using time The standard derivation of the dissipative part of the dependent FOPT we find out that a stationarylike solution is Kubo formula, leading to the fluctuation-dissipation relation reached after the Heisenberg time t ϭ2␲ប/⌬. This is of ͑33͒, is not very illuminating physically. More troubling is H course a valid description provided we do not have by then a the realization that one cannot tell from the standard deriva- breakdown of FOPT. The condition for this is easily found to tion what are the conditions for its validity. An alternate, ˙ Ӷ␦ ␦ ϭ⌬ ␴ physically appealing derivation,16–19 is based on the obser- be xtH xc , where xc / . This leads again to the adia- ͑ ͒ vation that energy absorption is related to having diffusion in baticity condition 36 . energy space.20 In Appendix E we outline the main ingredi- Another assumption in the derivation of Sec. V was that ents of this approach. we can ignore the parametric dependence of En and W on x. Ӷ␦ ˙ It should be clear that the diffusion picture of Appendix E The adiabaticity condition tH xc /x manifestly justify such holds only in case of chaotic systems. If this diffusion picture an assumption: We should think of tH as the transient time ij ␦ does not hold, then also the Kubo formula for ␩ does not for getting a stationarylike state, and we should regard xc as hold. Driven one-dimensional systems are the obvious ex- the parametric correlation scale. ample for the failure of linear response theory. As in the case As strange as it sounds, in order to have dissipation, it is of the kicked rotator ͑standard map͒͑Ref. 22͒ there is a essential to have a breakdown of FOPT. In the language of complicated route to chaos and stochasticity: By increasing perturbation theory this implies a required summation of dia- the driving amplitude the phase space structure is changed. If grams to infinite order, leading to an effective broadening of the amplitude is smaller than a threshold value, then the dif- the energy levels. By iterating FOPT, neglecting interference fusion is blocked by Kolmogorov-Arnold-Moser curves, and terms, we get a Markovian approximation for the energy consequently there is not dissipation. Therefore the Kubo spreading process. This leads to the diffusion equation of formula is not applicable in such cases. Appendix E. This diffusion can be regarded as arising from The following discussion of dissipative response assumes Fermi-golden-rule transitions between energy levels. A that we deal with a quantized chaotic system. We would like simple ad-hoc way to determine the energy level broadening to discuss the reason and the consequences of having an is to introduce ⌫ as a lower cutoff in the energy distribution energy scale ⌫. In the standard derivation the assumption of which is implied by Eq. ͑24͒: having level broadening as if comes out of the blue. As we already noted it is customary in textbooks to argue that either ͉ប ˙ ͉2 xFmn a continuum limit, or some small coupling to an environment ͉͗n͉␺͉͘2ϭ . ͑37͒ ͑ Ϫ ͒4ϩ͑⌫͒4 is essential in order to provide ⌫. But this is of course just a En Em way to avoid confrontation with the physical problem of having a driven isolated finite mesoscopic system. In fact the This constitutes a generalization of the well known proce- ⌫ ˙ dure used by Wigner in order to obtain the local density of energy scale is related to the rate (x) of the driving: states.16,17 However, in the present context we do not get a Lorentzian. The width parameter ⌫ is determined self- ប␴ 2/3 consistently from normalization, leading to Eq. ͑35͒͑disre- ⌫ϭͩ ͉x˙͉ͪ ϫ⌬, ͑35͒ ⌬2 garding numerical prefactor͒. We can summarize the above reasoning by saying that where for simplicity of presentation we assume one param- there is a perturbative regime that includes an adiabatic eter driving. We use ⌬ to denote the mean level spacing, and ͑FOPT͒ subregime. Outside of the adiabatic subregime we ␴ is the root mean square value of the matrix element Fnm need all orders of perturbation theory leading to Fermi- between neighboring levels. In order to derive the above ex- golden-rule transitions, diffusion in energy space, and hence pression for ⌫ we have used the result of Ref. 19 ͑Sec. 17͒ dissipation. Thus the dissipative part of Kubo formula ϭ ⌫Ͼ⌬ for the ‘‘core width’’ at the breaktime t tprt of perturbation emerges only in the regime , which is just the opposite theory. The purpose of the present section is to give an op- of the adiabaticity condition. The next obvious step is to tional ‘‘pedestrian derivation’’ for ⌫, and to discuss the determine the boundary of the perturbative regime. Follow- physical consequences. ing Refs. 18,19 we argue that the required condition is ⌫ ͑ ͒ Ӷ⌬ ⌬ ϰប Looking at the first order solution Eq. 24 of Sec. V one b . The bandwidth b is defined as the energy width ͉ ͉Ӷ⌬ ͉ Ϫ ͉ realizes that it makes sense provided Wmn . This leads En Em were the matrix elements Fnm are not vanishingly ⌫Ӷ⌬ to the adiabaticity condition small. If the condition b is violated we find ourselves in the nonperturbative regime where the Kubo formula cannot ⌬2 be trusted.18,19 ͉˙͉Ӷ ͑ ͒ x ប␴ . 36 We still have to illuminate why we can get in the perturbative regime a dissipative linear response in spite of the If this condition is not satisfied one should go beyond first breakdown of FOPT. The reason is having a separation of ⌬Ӷ⌫Ӷ⌬ order perturbation theory ͑FOPT͒, in a sense to be explained scales ( b). The nonperturbative mixing on the below. Note that this adiabaticity condition can be written as small energy scale ⌫ does not affect the rate of first-order ⌫Ӷ⌬ ⌫Ӷ͉ Ϫ ͉Ӷ⌬ . transitions between distant levels ( En Em b).

155303-6 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒

Therefore Fermi golden rule picture applies to the descrip- The integration should be taken over a cylinder of vertical tion of the coarse grained energy spreading, and we get the height 2␲ប, and whose basis is determined by the projection linear response. of the pumping cycle onto the (x1 ,x2) plane. The existence of the adiabatic regime is obviously a quan- We already pointed out that the Berry phase tum mechanical effect. If we take the formal limit ប→0 the ប ͛ ᠬ (1/ ) An•dx is gauge invariant. Therefore from Stokes law adiabaticity condition ⌫Ӷ⌬ breaks down. In fact the proper ប ͐͐ ᠬٙᠬ it follows that (1/ ) B•dx dx is independent of the sur- classical limit is nonperturbative, because the weaker condi- face, and therefore (1/ប)͜B ᠬdxٙᠬdx with closed surface ⌫Ӷ⌬ ប→ • tion b does not survive the 0 limit. For further should be 2␲ϫinteger. Integrating over a cylinder, as in Eq. details see Refs. 16–19. In the nonperturbative regime the ͑40͒, is effectively similar to integrating over a closed sur- quantum mechanical derivation of Kubo formula is not valid. face ͑because of the 2␲ periodicity in the vertical direction͒. 18 Indeed we have demonstrated the failure of Kubo formula This means that the flux averaged Q of Eq. ͑40͒ has to be an in case of random-matrix models. But if the system has a integer. classical limit, then Kubo formula still holds in the nonper- The common interest is in pumping cycles in the ⌽ϭ0 turbative regime due to semiclassical ͑rather than quantum- plane. This means that the zero order conservative contribu- mechanical͒ reasons. tion to Q, due to a persistent current, does not exist. Further- The discussion of dissipation assumes a generic situation more, from the reciprocity relations ͑see Appendix B͒ it fol- such that the Schro¨dinger equation does not have a stationary lows that G31ϭϪG13 and G32ϭϪG23, which should be solution. This means that driven one-dimensional systems contrasted with G12ϭG21. This means that a pumping cycle are automatically excluded. Another nongeneric possibility is in the ⌽ϭ0 plane is purely adiabatic: there is no dissipative to consider a special driving scheme, such as translation, ᠬ ͓ ͑ ͔͒ 23 contribution to Q. Only the B field second term in Eq. 39 rotation, or dilation. In such case the time dependent is relevant to the calculation of the pumped charge, and its H͓ ͔ ͑ Hamiltonian x(t) possesses a stationary solution pro- vertical component B12 vanishes due to the time reversal vided the ‘‘velocity’’ x˙ is kept constant͒. Consequently we do symmetry. not have a dissipation effect. In Sec. XI we discuss the sim- The absence of dissipative contribution for a cycle in the plest example of pumping by translation, where the station- ⌽ϭ0 plane, does not imply that dissipation is not an issue. ary adiabatic solution of Sec. V is in fact exact, and no dis- The symmetric part of the conductance matrix ␩ij is in gen- sipation arises. eral nonzero, leading to an energy absorption rate which is proportional to x˙ 2. This implies that the energy absorption IX. APPLICATION TO PUMPING per cycle is proportional to ͉x˙͉. Therefore we are able to minimize the dissipation effect by making the pumping cycle So far we have discussed the response for driving in a very slow. Furthermore, if we get into the quantum- very general way. From now on we focus on a system with a mechanical adiabatic regime, then ␩ij becomes extremely ring geometry as described in the Introduction, and illus- small, and then we can neglect the dissipation effect as long trated in Fig. 1. The shape of the ring is controlled by some as quantum-mechanical adiabaticity can be trusted. parameters x and x , and x is the magnetic flux. The gen- 1 2 3 Whenever the dissipation effect cannot be neglected, one eralized force F3 which is conjugate to the flux is the current. should specify whether or how a stationary operation is The time integral over the current is the transported charge achieved. In case of pumping in open system the stationary operation is implicitly guaranteed by having equilibrated res- ervoirs, where the extra energy is dissipated to infinity. In Qϭ Ͷ ͗F3͘dt. ͑38͒ case of pumping in closed system the issue of stationary operation is more subtle: In the adiabatic regime, to the ex- In fact a less misleading terminology is to talk about ‘‘prob- tend that adiabaticity can be trusted, we have a stationary ability current’’ and ‘‘integrated probability current.’’ From a solution to the transport problem, as defined in Sec. V. But purely mathematical point of view it is not important outside of the adiabatic regime we have diffusion in energy whether the transported particle has an electrical charge. space ͑Appendix E͒ leading to a slow energy absorption ͑dis- Disregarding a possible persistent current contribution, sipation͒. Thus a driven system is heated up gradually the expression for the pumped charge is ͑though possibly very slowly͒. Strictly speaking a stationary operation is not achieved, unless the system is in ͑weak͒ thermal contact with some large bath. Another way to reach ,ϭϪͫ Ͷ ϩ Ͷ ٙ ͬ ͑ ͒ a stationary operation, that does not involve an external bath Q G•dx B dx . 39 kϭ3 is by having an effectively bounded phase space. This is the case with the mixed phase space example which is discussed If we neglect the first term, which is associated with the in Ref. 12. There the stochasticlike motion takes place in a dissipation effect, and average the second ͑‘‘adiabatic’’͒ term bounded chaotic region in phase space. over the flux, then we get X. CLASSICAL DISSIPATIVE PUMPING 1 Q͉ ϭϪ ͵͵B ᠬdxٙᠬdx. ͑40͒ Before we discuss the quantum mechanical pumping, it is adiabatic 2␲ប • instructive to bring simple examples for classical pumping.

155303-7 DORON COHEN PHYSICAL REVIEW B 68, 155303 ͑2003͒

In the following we consider one particle (r) in a two- This can be verified by calculation via Eq. ͑18͒. The singu- dimensional ring as in Fig. 1͑a͒. larity along the x3 axis is not of quantum-mechanical origin: The first example is for classical dissipative pumping. The It is not due to degeneracies, but rather due to the diverging ϰ ͱ 2ϩ 2 ץ Hץ ϭ 33 24 conductance G G can be calculated for this system current operator ( / x3 1/ x1 x2). leading to a mesoscopic variation of the Drude formula. The ϭϪ ϫ⌽˙ Ϫ⌽˙ current is given by Ohm law I G , where is the XII. QUANTUM PUMPING electromotive force. Consider now the following pumping cycle: Change the We turn now to the quantum-mechanical case. Consider ⌽ ⌽ ϭϪ an adiabatic cycle that involves a particular energy level n. flux from 1 to 2, hence pumping charge Q G(1) ϫ ⌽ Ϫ⌽ This level is assumed to have a degeneracy point at ( 2 1). Change the conductance from G(1) to G(2) ⌽ (0) (0) ⌽(0) by modifying the shape of the ring. Change the flux from 2 (x1 ,x2 , ). It follows that in fact there is a vertical ⌽ ϭϪ ϫ ⌽ back to 1, hence pumping charge Q(2) G(2) ( 1 chain of degeneracy points: Ϫ⌽ 2). Consequently the net pumping is chainϭ͑x(0) ,x(0) ,⌽(0)ϩ2␲បϫinteger͒. ͑46͒ ϭ ͒Ϫ ͒ ϫ ⌽ Ϫ⌽ ͒ ͑ ͒ 1 2 Q ͓G͑2 G͑1 ͔ ͑ 2 1 . 41 These degeneracy points are important for the geometrical Thus we have used the dissipative part of the conductance ᠬ ͑ ͒ matrix ͓first term in Eq. ͑39͔͒ in order to pump charge. In the understanding of the B field, as implied by Eq. 18 . Every quantum mechanical version of this example extra care degeneracy point is similar to a monopole charge. The total ␲ប should be taken with respect to the zero order contribution of flux that emerges from each monopole must be 2 ϫ ͑ ͒ the persistent current. integer for a reason that was explained after Eq. 40 . Thus the monopoles are quantized in units of ប/2. The Bᠬ field ͑ ͒ XI. CLASSICAL ADIABATIC PUMPING which is created so to say by a vertical chain of monopoles may have a different near field and far field behavior, which The second example is for classical adiabatic pumping. we discuss below. The idea is to trap the particle inside the ring by a potential The far field region exists if the chains are well isolated. Ӷ well, and then to make a translation of the trap along a circle. Later we explain that ‘‘far’’ means gT 1, where gT is the The result of such a cycle is evidently Qϭ1. We would like Thouless conductance. The far field is obtained by regarding to see how this trivial result emerges form the Kubo formula. the chain as a smooth line. This leads qualitatively to the Let (r,p) be the canonical coordinate of the particle in the same field as in Eq. ͑45͒. Consequently, for a ‘‘large radius’’ ring, while (x1 ,x2) are the center coordinate of a trapping pumping cycle in the ⌽ϭ0 plane, we get ͉Q͉Ϸ1. In the potential. The Hamiltonian is following we are interested in the deviation from exact quantization: If ␾(0)ϭ0 we expect to have ͉Q͉у1, while if ␾(0)ϭ␲ ͉ ͉р ␾ 1 2 1 we expect Q 1. Only for the averaged Q of H͓r,p;x͑t͔͒ϭ ͫ p ϩͩ pʈϪ ⌽͑t͒ͪͬ Ќ ͑ ͒ 2m 2␲ͱx2ϩx2 Eq. 40 we get exact quantization. 1 2 The deviation from ͉Q͉Ϸ1 is extremely large if we con- ϩ Ϫ ͒ Ϫ ͒ ͑ ͒ ␾(0)ϭ Utrap͓r1 x1͑t ,r2 x2͑t ͔, 42 sider a tight pumping cycle around a 0 degeneracy. After linear transformation of the shape parameters, the en- where pʈ and pЌ are the components of the momentum along ergy splitting ⌬ϭE ϪE of the energy level n from its the ring and in the perpendicular ͑transverse͒ directions. The n m neighboring ͑nearly degenerated͒ level m can be written as pumping is done simply by cycling the position of the trap. The translation of the trap is assumed to be along an inside ⌬ϭ͓͑x Ϫx(0)͒2ϩ͑x Ϫx(0)͒2ϩc2͑␾Ϫ␾(0)͒2͔1/2 ͑47͒ circle of radius R, 1 1 2 2 where c is a constant. The monopole field is accordingly x͑t͒ϭ͓R cos͑⍀t͒,R sin͑⍀t͒,⌽ϭconst͔. ͑43͒ ͑ Ϫ (0) Ϫ (0) Ϫ (0)͒ In this problem the stationary solution of Sec. V is an c x1 x1 ,x2 x2 ,x3 x3 Bᠬ ϭϮ , exact solution. Namely 2 c 2 3/2 ͫ͑ Ϫ (0)͒2ϩ͑ Ϫ (0)͒2ϩͩ ͪ ͑ Ϫ (0)͒2ͬ x1 x1 x2 x2 ប x3 x3 ͉␺͑ ͒͘ϭ imx˙ r͉ ͓ ͑ ͔͒͘ ͑ ͒ t e • n x t , 44 ͑48͒ -␺(n)(rϪx) are the eigenfunctions of a parۋwhere ͉n(x)͘ ͑ ͒ where the prefactor is determined by the requirement of hav- ticle in the trap. Equation 44 is just Galilei transformation ing a single (ប/2) monopole charge. Assuming a pumping from the moving ͑trap͒ frame to the laboratory frame. cycle of radius R in the ⌽ϭ0 plane we get from the second It is a priori clear that in this problem the pumped charge term of Eq. ͑39͒ per cycle is Qϭ1, irrespective of ⌽. Therefore the Bᠬ field must be QϭϪͫ Ͷ Bٙdxͬ ϭϯ␲ͱg , ͑49͒ ͒ T ͑x1 ,x2,0 3 Bᠬ ϭϪ . ͑45͒ ␲͑ 2ϩ 2͒ 2 x1 x2 where

155303-8 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒ c2 and then from right to left. As long as the bias is positive ⌬2ץ 1 g ϭ ϭ ͑50͒ (x Ͼ0) the net charge being pumped is very small (͉Q͉ T ⌬ 2 2 1 ␾ R Ӷ1). Only the lowest wire level that participate in theץ ϭO is a practical definition for the Thouless conductance in this pumping cycle carries Q (1) net charge: It takes an elec- context. It is used here simply as a measure for the sensitiv- tron from the left side, and after the bias reversal it emits it ⌽ into the right side. Thus the pumping process in this model ity of an energy level to the magnetic flux . 14 What we want to do in the next sections is to interpolate can be regarded as a particular example of an adiabatic between the near field result, which is QϭO(ͱg ), and the transfer scheme: The electrons are adiabatically transferred T from state to state, one by one, as in ‘‘musical chair game.’’ far field result, which is QϭO(1). For this purpose it is For a single occupied level the net Q is the sum of charge convenient to consider a particular model that can be solved transfer events that take place in four avoided crossings ͑two exactly. avoided crossings in case of the lowest level͒. For many particle occupation the total Q is the sum over the net Q’s XIII. THE DOUBLE BARRIER MODEL which are carried by individual levels. For a dense zero tem- A simple example for quantum pumping is the double perature Fermi occupation the summation over all the net barrier model. An open geometry version of this model has Q’s is a telescopic sum, leaving noncanceling contributions been analyzed in Ref. 8 using the S matrix approach. The only from the first and the last adiabatic crossings. The latter analogous closed geometry version is obtained by consider- involve the last occupied level at the Fermi energy. ing a one-dimensional ring with two delta barriers. As we are going to explain below, the pumping process in this model XIV. THE THREE SITE LATTICE HAMILTONIAN can be regarded as a particular example of an adiabatic transfer scheme: The electrons are adiabatically transferred from Rather than analyzing the two-delta-barriers version of state to state one by one as in ‘‘musical chair game.’’ the double barrier model, we consider below a simplified The two delta barriers version of the double barrier model version that still contains the same essential ingredients. This is illustrated in Fig. 2. The length of the ring is L, with is obtained by considering a three site lattice Hamiltonian. periodic boundary conditions on Ϫ(L/2)ϽrϽ(L/2). A dot The advantage is obviously the possibility to make an exact region ͉Q͉Ͻa/2 is defined by the potential analytical treatment that does not involve approximations. The middle site in the three site lattice Hamiltonian sup- 1 a 1 a ports a single dot state, while the two other sites support two ͑ ͒ϭ ␦ͩ ϩ ͪ ϩ ␦ͩ Ϫ ͪ ͑ ͒ U r;c1 ,c2 r r . 51 wire states. The Hamiltonian is c1 2 c2 2 i␾ It is assumed that c1 and c2 are small enough so one can 0 c1 e classify the ring eigenstates into two categories: wire states c1 uc2 ͪ . ͑52͒ ͩۋH and dot states. The latter are those states that are localized in Ϫi␾ the dot region ͉Q͉Ͻa/2 in the limit of infinitely high barri- e c2 0 ers. We define the Fermi energy as the energy of the last ϭ Ϫ The three parameters are the bias x1 c1 c2, the dot energy occupied wire level in the limit of infinitely high barriers. ϭ ϭ⌽ϭប␾ x2 u, and the flux x3 . For presentation purpose The three parameters that we can control are the flux x3 we assume that 0Ͻc ,c Ӷ1, and characterize the wire-dot ϭ⌽ϭប␾, the bias x ϭc Ϫc , and the dot potential x 1 2 1 1 2 2 coupling by the parameter cϭͱc c . ϭE which is related to c ϩc . The energy E corre- 1 2 dot 1 2 dot The eigenstates are E . Disregarding the interaction with spond to the dot state which is closest to the Fermi energy E n F the dot (cϭ0) we have two wire states with EϭϮ1. This from above. We assume that the other dot levels are much implies degeneracies for x ϭuϭϯ1. Once we switch on the further away from the Fermi energy, and can be ignored. 2 coupling (cϾ0), the only possible degeneracies are between Note that another possible way to control the dot potential, is the even dot state and the odd wire state of the mirror sym- simply by changing a gate voltage: That means to assume metric Hamiltonian (x ϭ0). The flux should be either inte- that there is a control over the potential floor in the region 1 ger ͑for degeneracy of the dot level with the lower wire ͉Q͉Ͻa/2. level͒, or half integer ͑for degeneracy of the dot level with The pumping cycle is assumed to be in the ⌽ϭ0 plane, the upper wire level͒. Thus we have two vertical chains of so there is no issue of conservative persistent current contri- degeneracies: bution. The pumping cycle is defined as follows: We start Ͼ with a positive bias (x1 0) and lower the dot potential from ϭ͑ Ϫ ϩ 2 ␲បϫ ͒ Ͼ Ͻ the negative chain 0, 1 c ,2 integer , a large x2 EF value to a small x2 EF value. As a result, one electron is transfered via the left barrier into the dot Ͻ the positive chainϭ͑0,ϩ1Ϫc2,␲ϩ2␲បϫinteger͒. region. Then we invert the bias (x1 0) and raise back x2. As a result the electron is transferred back into the wire via the right barrier. In order to calculate the Bᠬ field and pumped charge Q,we A closer look at the above scenario ͓Fig. 2͑b͔͒ reveals the have to find the eigenvalues and the eigenvectors of the following: As we lower the dot potential across a wire level, Hamiltonian matrix. The secular equation for the eigenvalues an electron is adiabatically transferred once from left to right is

155303-9 DORON COHEN PHYSICAL REVIEW B 68, 155303 ͑2003͒

3Ϫ 2Ϫ͑ ϩ 2ϩ 2͒ ϩ Ϫ ͑␾͒ϭ E uE 1 c1 c2 E u 2c1c2cos 0. Using the notations

1 1 Qϭ u2ϩ ͑1ϩc2ϩc2͒, 9 3 1 2

1 1 1 Rϭ u3ϩ ͑1ϩc2ϩc2͒uϪ uϩc c cos͑␾͒cos͑␪͒ 27 6 1 2 2 1 2 R ϭ ͱQ 3 FIG. 3. The first component of the Bᠬ field for a particle in the middle level of the three site lattice model. It is plotted as a function the roots of the above cubic equation are ϭ ␾ϭ of the dot potential x2 u. The other parameters are 0, and ϭ ϭ ϭ ␲ c1 0.1, while c2 0.04 for the thick line and c2 0.02 for the thin 1 1 2 → ϭ ϩ ͱQ ͩ ␪ϩ ͪ ͑ ͒ line. In the limit c2 0, all the charge that is transferred from the En u 2 cos n , 53 3 3 3 left side into the dot during the first avoided crossing, is emitted where nϭ0,Ϯ1. The corresponding eigenstates are back into the left side during the second avoided crossing. Inset: ϭ The eigenenergies En(x) for the c2 0.04 calculation. i␾ϩ c2e c1En A similar calculation of the pumped charge for a planar cycle 1 Ϫ 2 En ͪ , ͑54͒ 1 ͩ ۋn͑x͉͒͘ around the positive chain leads to ͱS c eϪi␾ϩc E 1 2 n c Ϫc ϭϪ 1 2 ϭϪͱ Ϫ ͑ ͒ where S is the normalization, namely, Q ϩ 1 2gT 61 c1 c2 ϭ͑ Ϫ 2͒2ϩ͑ ϩ ͒2ϩ͑ ϩ ͒2 ͑ ͒ ϭ ϩ 2 S 1 En c1 c2En c2 c1En . 55 with gT 2c1c2 /(c1 c2) . In both cases we have approxi- ϭϮ ϩO Ӷ For the calculation of the pumped charge in the next para- mate quantization Q 1 (gT) for gT 1, while for a →ϱ → graph it is useful to notice that for EϭϮ1 the normalization tight cycle either Q or Q 0 depending on which line of is Sϭ2(c Ϯc )2, while for Eϭ0 the normalization is S degeneracies is being encircled. If the pumping cycle en- 1 2 ϭ 2Ϫ 2 Ϸ1. circles both chains then we get Q 4c1c2 /(c1 c2). In the ϭO Ӷ After some algebra we find that the first component of the latter case Q (gT) for gT 1, with no indication for quantization. Bᠬ field in the ⌽ϭ0 plane is

XV. SUMMARY AND DISCUSSION ץ ץ 1ϭϪ ͑ ͒ͯ ͑ ͒ ͑ ͒ ␾ n x ʹ 56ץ n x ץ B 2Imͳ u We have shown how the Kubo formalism can be used in order to derive both classical and quantum-mechanical re- Sץ 1 ϭϪ͑ 2Ϫ 2͒ ͑ ͒ sults for the pumped charge Q in a closed system. In this c1 c2 , 57 -u formulation the distinction between dissipative and nondissiץ S2 pative contributions is manifest. which is illustrated in Fig. 3. From here it follow that if we Within the framework of the Kubo formalism ͑disregard- ϭ keep constant bias, and change only x2 u, then the pumped ing nonlinear corrections͒ we have made a distinction be- charge is tween the following levels of treatment: strict adiabaticity ͑ ͒ ͑ final outcome of zero order treatment , adiabatic transport out- 1 ͒ ͑ QϭϪ͵ B1dx ϭϪ͑c2Ϫc2͒ ͯ . ͑58͒ come of stationary first order treatment , and dissipation the 2 1 2 S ͒ initial result of first order transitions . ⌽ϭ In the adiabatic regime one can assume a stationary solu- For a planar ( 0) pumping cycle around the negative ver- tion to the adiabatic equation, which implies no dissipation tical chain the main contribution to Q comes from the two ϷϪ effect. This leads to the picture of adiabatic transport, where crossings of the x2 1 line. Hence we get the Berry phase is the outcome of a zero order treatment, ͑ ͒ c ϩc while the ‘‘geometric magnetism’’ of Eq. 26 is the outcome ϭ 1 2 ϭͱ ϩ ͑ ͒ of a first order treatment of the interlevel couplings. Q Ϫ 1 2gT, 59 c1 c2 In some very special cases ͑translations, rotations and di- where the Thouless conductance in this context refers to the lations͒ this assumption ͑of having a stationary solution͒ is in avoided crossing, and is defined as fact exact, but in generic circumstance this assumption is an approximation. Outside of the adiabatic regime the stationary .2c c solution cannot be trusted ⌬2ץ 1 ϵ ͯ ϭ 1 2 ͑ ͒ gT . 60 Assuming quantized chaotic dynamics one argues that ␾2 ͑c Ϫc ͒2ץ ⌬ ␾ϭ0 1 2 Fermi-golden-rule transitions between levels lead to ͑slow͒

155303-10 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒

͓ ͑ ͔͒ diffusion in energy Eq. E1 . This leads to the emergence of is the study of diatomic molecules. In such a case x j are the the dissipative part in the Kubo formula. We have obtained locations of the nuclei. The total Hamiltonian is assumed to an expression ͓Eq. ͑35͔͒ for the energy scale ⌫ϰ͉x˙͉2/3 that be of the general form controls the dissipative effect. We have explained that the 1 dissipative contribution to the Kubo formula is valid only in H ϭ ͚ p2ϩH͑x͒, ͑63͒ ⌬Ͻ⌫Ӷ⌬ total j the regime b . Otherwise the dynamics is either of 2M j adiabatic nature (⌫Ӷ⌬) or nonperturbative (⌫Ͼ⌬). where H is the Hamiltonian of the ‘‘fast’’ degrees of freedom In order to calculate the pumped charge Q we have to ͑in the context of molecular physics these are the electrons͒. perform a closed line integral over the conductance ͓Eq. Rather than using the standard basis, one can use the Born- ͑39͔͒. This may have in general both adiabatic and dissipa- Oppenheimer basis ͉x,n(x)͘ϭ͉x͘ ͉n(x)͘. Then the Hamil- tive contributions. For the common pumping cycle in the tonian can be written as ⌽ϭ0 plane, only the adiabatic contribution exists. This fol- ͑ ͒ lows from the reciprocity relations Sec. IX . Still we have 1 ͑ ͒ H ϭ ͓ Ϫ j ͑ ͔͒2ϩ␦ ͑ ͒ emphasized without any contradiction that in the same cir- total ͚ p j Anm x nmEn x , cumstances a dissipation effect typically accompanies the 2M j pumping process. where the interaction term is consistent with Eq. ͑19͒. Thus it The quantum adiabatic contribution to the pumping is de- is evident that the theory of driven systems is a special limit ᠬ termined by a line integral over a B field which is created by of this problem, which is obtained if we treat the x j as clas- monopoles. The monopoles, which are related to the degen- sical variables. eracies of the Hamiltonian, are located along vertical chains in x space ͓Eq. ͑46͔͒. The three site model provides the sim- ACKNOWLEDGMENT plest example for such vertical chains: By calculating the Bᠬ ͑ ͒ It is my pleasure to thank Yshai Avishai ͑Ben-Gurion Uni- field which is created so to say by these chains, we were ͒ ͑ ͒ ͑ able to determine the charge which is pumped during a cycle versity , Yosi Avron Technion , Thomas Dittrich Colom- ͓ ͑ ͔͒ bia͒, Shmuel Fishman ͑Technion͒, Tsampikos Kottos ͑Got- e.g., Eq. 59 . ͒ ͑ ͒ The ͑monopoles of the͒ vertical chains have near field tingen , and Holger Schantz Gottingen for useful ͓ ͑ ͔͒ discussions. This research was supported by the Israel Sci- regions Eq. 48 . If the chains are well isolated in x space, ͑ ͒ then there are also far field regions. The far field regions are ence Foundation Grant No. 11/02 , and by a grant from the defined as those where the Thouless conductance is very GIF, the German-Israeli Foundation for Scientific Research Ӷ and Development. small (gT 1). Pumping cycles that are contained in the far field region of a given chain lead to an approximately quan- ϭ ϩO APPENDIX A: THE KUBO FORMULA: STANDARD tized pumping Q integer (gT). It is important to realize that the existence of far field regions in x space is associated DERIVATION with having a low dimensional system far away from the In this appendix we present an elementary textbook-style classical limit. In a quantized chaotic system it is unlikely to derivation of the Kubo formula. For notational simplicity we Ӷ ប→ have gT 1 along a pumping cycle. As we take the 0 Hϭ Ϫ write the Hamiltonian as H0 f (t)V. It is assumed that limit the vertical chains become very dense, and the far field the system, in the absence of driving, is prepared in a sta- regions disappear. ␳ tionary state 0. In the presence of driving we look for a first In the subtle limiting case of open geometry we expect to ␳ ϭ␳ ϩ˜␳ ˜␳ get agreement with the S-matrix formula of Bu¨ttiker, Pre´tre, order solution (t) 0 (t). The equation for (t)is and Thomas ͑BPT͒.6 Using the notations of the present Paper ␳˜͑t͒ץ the BPT formula for the current that comes out of ͑say͒ the ϷϪ ˜␳ ͒ ϩ ͒ ␳ ͑ ͒ i͓H0 , ͑t ͔ if͑t ͓V, 0͔. A1 tץ right lead can be written as This equation can be rewritten as Sץ e 3 jϭ † ͑ ͒ ץ S ͪ , 62 ץ G ␲ traceͩ P 2 i x j ͓U ͑t͒Ϫ1˜␳͑t͒U ͑t͔͒Ϸif͑t͓͒U ͑t͒Ϫ1V U ͑t͒,␳ ͔, t 0 0 0 • 0 0ץ 33 where P is the projector on the right lead channels. For G where U (t) is the evolution operator which is generated by the above reduces to the Landauer formula. The details re- 0 H . The solution of the latter equation is garding the relation between the Kubo formula and the BPT 0 10 formula will be published in a separate paper. Here we just t ˜␳͑ ͒Ϸ ͵ ͕ ͓Ϫ͑ Ϫ Ј͔͒ ␳ ͖ ͑ Ј͒ Ј ͑ ͒ note that the derivation is based on a generalization of the t i V t t , 0 f t dt , A2 Fisher-Lee approach.3,25,26 Finally it is important to remember that the theory of where we use the usual definition of the ‘‘interaction picture’’ driven systems is the corner stone for the analysis of inter- ␶ ϭ ␶ Ϫ1 ␶ operator V( ) U0( ) V•U0( ). action between ‘‘slow’’ and ‘‘fast’’ degrees of freedom. As- Consider now the time dependence of the expectation ϭ ␳ sume that the x j are in fact dynamical variable, and that the value ͗F͘t tr͓F (t)͔ of an observable. Disregarding the conjugate momenta are p j . The standard textbook example zero order contribution, the first order expression is

155303-11 DORON COHEN PHYSICAL REVIEW B 68, 155303 ͑2003͒

t Gij͑ϪB͒ϭ͓Ϯ͔G ji͑B͒, ͑B6͒ ͗ ͘ Ϸ ͵ ͕ ͓ Ϫ͑ Ϫ Ј͒ ␳ ͔͖ ͑ Ј͒ Ј F t i tr F V„ t t …, 0 f t dt where the plus ͑minus͒ applies if the signs of Fi and F j t transform ͑not͒ in the same way under time reversal. These ϭ ͵ ␣͑tϪtЈ͒f ͑tЈ͒dtЈ, reciprocity relations follow from the Kubo formula ͓Eq. ͑28͔͒, using Kij(Ϫ␶,ϪB)ϭϪ͓Ϯ͔Kij(␶,B), together with where the response kernel ␣(␶) is defined for ␶Ͼ0as the trivial identity Kij(Ϫ␶,B)ϭϪK ji(␶,B). In Sec. IX we discuss the implications of the reciprocity relations in the ␣͑␶͒ϭ ͕ ͓ ͑Ϫ␶͒ ␳ ͔͖ϭ ͕͓ ͑Ϫ␶͔͒␳ ͖ i tr F V , 0 i tr F,V 0 context of pumping. ϭi͓͗F,V͑Ϫ␶͔͒͘ϭi͓͗F͑␶͒,V͔͘. ͑A3͒ APPENDIX C: EXPRESSIONS FOR B AND G We have used above the cyclic property of the trace opera- ␳ Ϫ1ϭ␳ The functions Cij(␶) and Kij(␶) are the expectation val- tion; the stationarity U0 0U0 0 of the unperturbed state; and the definition F(␶)ϭU (␶)Ϫ1F U (␶). ues of hermitian operators. Therefore they are real functions. 0 • 0 It follows that the real part of their Fourier transform is a symmetric function with respect to ␻, while the imaginary APPENDIX B: REMARKS REGARDING THE part of their Fourier transform is anti symmetric with respect GENERALIZED SUSCEPTIBILITY to ␻. By definition they satisfy Cij(␶)ϭC ji(Ϫ␶) and In this appendix we would like to further illuminate the Kij(␶)ϭϪK ji(Ϫ␶). It is convenient to regard them as the relation between the generalized susceptibility and the con- real and imaginary parts of one complex function ⌽ij(␶). ductance matrix. The generalized susceptibility ␹kj(␻) is the Namely, Fourier transform of the causal response kernel ␣kj(␶). ប ⌽ij͑␶͒ϭ͗Fi͑␶͒F j͑0͒͘ϭCij͑␶͒Ϫi Kij͑␶͒, ͑C1͒ Therefore it is an analytic function in the upper half of the 2 complex ␻ plan, whose real and imaginary parts are related by Hilbert transforms ͑Kramers-Kronig relations͒ 1 Cij͑␶͒ϭ ͓⌽ij͑␶͒ϩ⌽ ji͑Ϫ␶͔͒, ͑C2͒ 2 ϱ Im͓␹kj͑␻Ј͔͒ d␻Ј ␹kj͑␻͒ϵ ͓␹kj͑␻͔͒ϭ ͵ ͑ ͒ i 0 Re . B1 ij͑␶͒ϭ ͓⌽ij͑␶͒Ϫ⌽ ji͑Ϫ␶͔͒ ͑ ͒ Ϫϱ ␻ЈϪ␻ ␲ K ប . C3 ␹kj ␻ The imaginary part of ( ) is the sine transforms of It is possible to express the decomposition Gijϭ␩ij ␣kj ␶ ␻ ( ), and therefore it is proportional to for small fre- ϩ ij ˜ ij ␻ ͑ ͒ quencies. Consequently it is convenient to write the Fourier B in terms of K ( ). Using the definition 29 we get transformed version of Eq. ͑3͒ as ϱ Gijϭ ͵ Kij͑␶͒␶d␶ 0 ͓͗ k͔͘ ϭ ␹kj͑␻͓͒ ͔ Ϫ␮kj͑␻͓͒ ˙ ͔ ͑ ͒ F ␻ ͚ 0 x j ␻ x j ␻ , B2 j ϱ Re͓K˜ ij͑␻͔͒ d␻ 1 Im͓K˜ ij͑␻͔͒ ϭϪ͵ ϩͫ ͬ . ͑C4͒ Ϫϱ ␻2 2␲ 2 ␻ where the dissipation coefficient is defined as ␻ϭ0 Im͓␹kj͑␻͔͒ ϱ sin͑␻␶͒ The first term is antisymmetric with respect to its indexes, ␮kj͑␻͒ϭ ϭ ͵ ␣kj͑␶͒ ␶ ͑ ͒ ij ␻ ␻ d . B3 and is identified as B . The second term is symmetric with 0 respect to its indexes, and is identified as ␩ij. The last step in In this paper we ignore the first term in Eq. ͑B2͒ which the above derivation involves the following identity that hold signify the nondissipative in-phase response. Rather we put for any real function f (␶) ␻→ the emphasis on the ‘‘dc limit’’ ( 0) of the second term. ϱ ϱ d␻ ϱ Thus the conductance matrix Gkjϭ␮kj(␻→0) is just a syn- ͵ f ͑␶͒␶d␶ϭ ͵ ˜f ͑␻͒ ͵ eϪi␻␶␶d␶ Ϫϱ2␲ onym for the term ‘‘dissipation coefficient.’’ However, ‘‘con- 0 0 ductance’’ is a better ͑less misleading͒ terminology: it does not have the ͑wrong͒ connotation of being specifically asso- ϱ d␻ 1 ϭ ͵ ˜f ͑␻͒ͩ Ϫ ϩi␲␦Ј͑␻͒ͪ ciated with dissipation, and consequently it is less confusing Ϫϱ2␲ ␻2 to say that it contains a ͑nondissipative͒ adiabatic component. ϱ d␻ Re͓˜f ͑␻͔͒ For systems where time reversal symmetry is broken due ϭ ͵ ͩ Ϫ Ϫ␲ Im͓˜f ͑␻͔͒␦Ј͑␻͒ͪ Ϫϱ2␲ ␻2 to the presence of a magnetic field B, the response kernel, and consequently the generalized susceptibility and the con- ϱ Re͓˜f ͑␻͔͒ d␻ 1 Im͓˜f ͑␻͔͒ ductance matrix satisfies the Onsager reciprocity relations ϭϪ͵ ϩͫ ͬ . ͑C5͒ Ϫϱ ␻2 2␲ 2 ␻ ␻ϭ0 ␣ij͑␶,ϪB͒ϭ͓Ϯ͔␣ ji͑␶,B͒, ͑B4͒ Note that Im͓˜f (␻)͔ is the sine transform of f (␶), and there- ␹ij͑␻,ϪB͒ϭ͓Ϯ͔␹ ji͑␻,B͒, ͑B5͒ fore it is proportional to ␻ is the limit of small frequencies.

155303-12 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒

͑ ͒ It is of practical value to rederive Eq. C4 by writing Im͓Fi F j ͔ ⌽ij ␶ i ijϭ ប ͒ nm mn ͑ ͒ ( ) using the energies En and the matrix elements Fnm . B 2 ͚ f ͑En ͚ . C10 Þ ͑ Ϫ ͒2 Then we can get from it straightforwardly ͑using the defini- n m( n) Em En tions͒ all the other expressions. Namely, One observes that the expression for Bij coincides with the adiabatic transport result Eq. ͑26͒. Alternatively this identifi- E ϪE ͑ ͒ ⌽ij͑␶͒ϭ ͑ ͒ i j ͩ Ϫ m n ͪ cation can be obtained by expressing the sum in Eq. C10 as ͚ f En ͚ FnmFmnexp i t , n m ប an integral, getting form it the first term in Eq. ͑C4͒: ͑C6͒ 2 ϱ Im͓⌽˜ ij͑␻͔͒ d␻ ij Ϫ B ϭ ͵ Em En ប Ϫϱ ␻2 2␲ ⌽˜ ij͑␻͒ϭ ͑ ͒ i j ␲␦ͩ ␻Ϫ ͪ ͚ f En ͚ FnmFmn2 , n m ប ͑ ͒ ប C7 Im͓C˜ ij͑␻͔͒Ϫ Re͓K˜ ij͑␻͔͒ 2 ϱ 2 d␻ i j ϭ ͵ ϪF F ប 2 ␲ ␹ij͑␻͒ϭ ͑ ͒ͩ nm mn Ϫϱ ␻ 2 ͚ f En ប␻Ϫ Ϫ ͒ϩ n,m ͑Em En i0 ϱ Re͓K˜ ij͑␻͔͒ d␻ F j Fi ϭϪ͵ . ͑C11͒ ϩ nm mn ͪ ͑ ͒ Ϫϱ ␻2 2␲ ប␻ϩ Ϫ ͒ϩ , C8 ͑Em En i0 ˜ ␻ ˜ ␻ APPENDIX D: EXPRESSING K„ … USING C„ … ␩ijϭϪ ␲ប ͑ ͒ ͓ i j ͔␦Ј͑ Ϫ ͒ 2 ͚ f En ͚ Re FnmFmn Em En , n m(Þn) We can use the following manipulation in order to relate ͑C9͒ K˜ ij(␻)toC˜ ij(␻):

i ˜ ij͑␻͒ϭ ͑ ͒˜ ij͑␻͒ϭ ␲ ͑ ͓͒ i j ␦͑␻ϩ␻ ͒Ϫ j i ␦͑␻Ϫ␻ ͔͒ K ͚ f En Kn 2 ͚ f En FnmFmn nm FnmFmn nm n ប nm i ϭ ␲ ͑ ͓͒Ϫ i j ␦͑␻ϩ␻ ͒ϩ j i ␦͑␻Ϫ␻ ͔͒ 2 ͚ f Em FnmFmn nm FnmFmn nm ប nm i f ͑E ͒Ϫ f ͑E ͒ ϭ ␲ n m ͓ i j ␦͑␻ϩ␻ ͒Ϫ j i ␦͑␻Ϫ␻ ͔͒ 2 ͚ FnmFmn nm FnmFmn nm ប nm 2 f ͑E ͒Ϫ f ͑E ͒ ϭϪ ␻␲ n m ͓ i j ␦͑␻ϩ␻ ͒ϩ j i ␦͑␻Ϫ␻ ͔͒ϭϪ ␻ Ј͑ ͒ ij͑␻͒ ͑ ͒ i ͚ Ϫ FnmFmn nm FnmFmn nm i ͚ f En Cn , D1 nm En Em n

␻ ϭ Ϫ ប ␻ where we use the notation nm (En Em)/ . The third line If we do not assume small , but instead assume canonical differs from the second line by permutation of the dummy state, then a variation on the last steps in Eq. ͑D1͒, using the Ϫ ϩ ϭ summation indexes, while the fourth line is the sum of the fact that ͓ f (En) f (Em)͔/͓ f (En) f (Em)͔ tanh͓(En Ϫ second and the third lines divided by 2. In the last equality Em)/(2T)͔ is an odd function, leads to the relation we assume small ␻. If the levels are very dense, then we can replace the summation by integration, leading to the relation 1 ប␻ K˜ ij͑␻͒ϭi␻ϫ tanhͩ ͪ Cij͑␻͒. ͑D4͒ T ប␻ 2T T ͵ ͑ ͒ ͑ ͒˜ ij͑␻͒ϭϪ ␻ ͵ ͑ ͒ Ј͑ ͒˜ ij͑␻͒ g E dEf E KE i g E dEf E CE , Upon substitution of the above expressions in the Kubo formula for ␩ij, one obtains the ﬂuctuation-dissipation relation. ͑D2͒

˜ ij ␻ ˜ ij ␻ APPENDIX E: THE KUBO FORMULA AND THE where KE ( ) and CE ( ) are microcanonically smoothed functions. Since this equality hold for any smoothed f (E), DIFFUSION IN ENERGY SPACE ͑ it follows that the following relation holds in the limit The illuminating derivation of Eq. ͑33͒ is based on the ␻→ 0): observation that energy absorption is related to having diffusion in energy space. Let us assume that the probability dis- 1 d ␳ ϭ K˜ ij͑␻͒ϭi␻ ͓g͑E͒Cij͑␻͔͒. ͑D3͒ tribution (E) g(E) f (E) of the energy satisﬁes the diffu- E g͑E͒ dE E sion equation

155303-13 DORON COHEN PHYSICAL REVIEW B 68, 155303 ͑2003͒

Assuming ͑for presentation purpose͒ that the rates x˙ are 1 ץ ץ ␳ץ ϭ ͫg͑E͒D ͩ ␳ͪͬ. ͑E1͒ k E g͑E͒ constant numbers, it follows that energy changes are relatedץ E Eץ tץ to the fluctuating Fk(t)as The energy of the system is ͗H͘ϭ͐E␳(E)dE. It follows that the rate of energy absorption is t ␦ ϭ͗H͘ Ϫ͗H͘ ϭϪ ˙ ͵ k͑ Ј͒ Ј ͑ ͒ ␳͑E͒ E t 0 ͚ xk F t dt . E6 ץ d ϱ ͗H͘ϭϪ͵ ͑ ͒ ͩ ͪ ͑ ͒ k 0 dEg E DE . E2 E g͑E͒ץ dt 0 For a microcanonical preparation we get Squaring this expression, and performing microcanonical averaging over initial conditions we obtain d 1 d H ϭ ͒ ͑ ͒ ͗ ͘ ͓g͑E DE͔. E3 dt g͑E͒ dE t t ␦ 2͑ ͒ϭ ˙ ˙ ͵ ͵ ij͑ ЉϪ Ј͒ Ј Љ ͑ ͒ E t ͚ xix j CE t t dt dt , E7 This diffusion-dissipation relation reduces immediately to ij 0 0 the fluctuation-dissipation relation if we assume that the diffusion in energy space due to the driving is given by ij ЉϪ Ј ϭ͗ i Ј j Љ ͘ where CE (t t ) F (t )F (t ) is the correlation func- 1 tion. For very short times this equation implies ‘‘ballistic’’ ϭ ˜ ij͑␻→ ͒˙ ˙ ͑ ͒ ␦ 2ϰ 2 DE ͚ CE 0 xix j . E4 spreading ( E t ) while on intermediate time scales it 2 ij ␦ 2 ϭ leads to diffusive spreading E (t) 2DEt, where Thus it is clear that a theory for linear response should es- tablish that there is a diffusion process in energy space due to 1 ϱ the driving, and that the diffusion coefficient is given by Eq. ϭ ˙ ˙ ͵ ij͑␶͒ ␶ ͑ ͒ DE ͚ xix j CE d . E8 ͑E4͒. More importantly, this approach also allows treating 2 ij Ϫϱ cases where the expression for DE is nonperturbative, while the diffusion-dissipation relation Eq. ͑E3͒ still holds. The latter result assumes a short correlation time. This is also A full exposition ͑and further reference͒ for this route of the reason that the integration over ␶ can be extended form derivation can be found in Refs. 16–19. Here we shall give Ϫϱ to ϩϱ. Hence we get Eq. ͑E4͒. We note that for long just the classical derivation of Eq. ͑E4͒, which is extremely times the systems deviates significantly from the initial mi- simple. We start with the identity crocanonical preparation. Hence, for long times, one should justify the use of the diffusion equation ͑E1͒. This leads Hץ d ͗H͘ϭͳ ʹ ϭϪ ˙ k͑ ͒ ͑ ͒ to the classical slowness condition which is discussed in ͚ xkF t . E5 .t k Ref. 19ץ dt

1 L.D. Landau and E.M. Lifshitz, Statistical Physics ͑Butterworth A: Mater Sci. Process 75 ͑2002͒. P. Reimann, M. Grifoni, and P. Heinemann, London, 2000͒. Hanggi, Phys. Rev. Lett. 79,10͑1997͒. 2 Y. Imry, Introduction to Mesoscopic Physics ͑Oxford University 12 H. Schanz, M.F. Otto, R. Ketzmerick, and T. Dittrich, Phys. Rev. Press, Oxford, 1997͒, and references therein. Lett. 87, 070601 ͑2001͒. 3 S. Datta, Electronic Transport in Mesoscopic Systems ͑Cambridge 13 M.V. Berry, Proc. R. Soc. London, Ser. A 392,45͑1984͒. University Press, Cambridge, 1995͒. 14 J.E. Avron and L. Sadun, Phys. Rev. Lett. 62, 3082 ͑1989͒; J.E. 4 L.P. Kouwenhoven et al., Proceedings of Advanced Study Insti- Avron and L. Sadun, Ann. Phys. ͑N.Y.͒ 206, 440 ͑1991͒; J.E. tute on Mesoscopic Electron Transport, edited by L.L. Sohn, Avron, A. Raveh, and B. Zur, Rev. Mod. Phys. 60, 873 L.P. Kouwenhoven, and G. Schon ͑Kluwer, Dordrecht, 1997͒. ͑1988͒. 5 D.J. Thouless, Phys. Rev. B 27, 6083 ͑1983͒. 15 J.M. Robbins and M.V. Berry, J. Phys. A 25, L961 ͑1992͒; M.V. 6 M. Bu¨ttiker et al., Z. Phys. B: Condens. Matter 94, 133 ͑1994͒; Berry and J.M. Robbins, Proc. R. Soc. London, Ser. A 442, 659 P.W. Brouwer, Phys. Rev. B 58, R10135 ͑1998͒; J.E. Avron ͑1993͒; M.V. Berry and E.C. Sinclair, J. Phys. A 30, 2853 et al., ibid. 62, R10618 ͑2000͒. ͑1997͒. 7 T.A. Shutenko, I.L. Aleiner, and B.L. Altshuler, Phys. Rev. B 61, 16 D. Cohen in New Directions in Quantum Chaos, Proceedings of 10366 ͑2000͒. the International School of Physics ‘‘Enrico Fermi,’’ Course 8 Y. Levinson, O. Entin-Wohlman, and P. Wolﬂe, CXLIII, edited by G. Casati, I. Guarneri, and U. Smilansky ͑IOS cond-mat/0010494 ͑unpublished͒; M. Blaauboer and E.J. Heller, Press, Amsterdam 2000͒. Phys. Rev. B 64, 241301͑R͒͑2001͒. 17 D. Cohen in Dynamics of Dissipation, Proceedings of the 38th 9 O. Entin-Wohlman, A. Aharony, and Y. Levinson, Phys. Rev. B Karpacz Winter School of Theoretical Physics, edited by P. Gar- 65, 195411 ͑2002͒. baczewski and R. Olkiewicz ͑Springer, Berlin, 2002͒. 10 D. Cohen, cond-mat/0304678 ͑unpublished͒. 18 D. Cohen, Phys. Rev. Lett. 82, 4951 ͑1999͒; D. Cohen and T. 11 P. Reimann, Phys. Rep. 361,57͑2002͒; Special issue, Appl. Phys. Kottos, ibid. 85, 4839 ͑2000͒.

155303-14 QUANTUM PUMPING IN CLOSED SYSTEMS... PHYSICAL REVIEW B 68, 155303 ͑2003͒

19 D. Cohen, Ann. Phys. ͑N.Y.͒ 283, 175 ͑2000͒. Casati, I. Guarneri, and U. Smilansky ͑North Holland, Amster- 20 M. Wilkinson, J. Phys. A 21, 4021 ͑1988͒; M. Wilkinson and E.J. dam, 1991͒. Austin, ibid. 28, 2277 ͑1995͒. 23 A. Barnett, D. Cohen, and E.J. Heller, Phys. Rev. Lett. 85, 1412 21 O.M. Auslaender and S. Fishman, Phys. Rev. Lett. 84, 1886 ͑2000͒. ͑2000͒; O.M. Auslaender and S. Fishman, J. Phys. A 33, 1957 24 A. Barnett, D. Cohen, and E.J. Heller, J. Phys. A 34, 413 ͑2000͒. ͑2001͒. 22 S. Fishman, in Quantum Chaos, Proceedings of the International 25 D.S. Fisher and P.A. Lee, Phys. Rev. B 23, 6851 ͑1981͒. School of Physics ‘‘Enrico Fermi,’’ Course CXIX, edited by G. 26 H.U. Baranger and A.D. Stone, Phys. Rev. B 40, 8169 ͑1989͒.

155303-15