Quantum fluctuation theorems in the strong damping limit
eepl draft
Quantum fluctuation theorems in the strong damping limit
Sebastian Deffner, Michael Brunner and Eric Lutz
Department of Physics, University of Augsburg, D-86135 Augsburg, Germany
PACS – Fluctuation phenomena in statistical physics
PACS – Decoherence in quantum mechanics
Abstract. - We consider a driven quantum particle in the strong friction regime described by thequantum Smoluchowski equation. We derive Crooks and Jarzynski type relations for the reducedquantum system by properly generalizing the entropy production to take into account the non-Gibbsian character of the equilibrium distribution. In the case of a nonequilibrium steady state, weobtain a quantum version of the Hatano-Sasa relation. We, further, propose an experiment withdriven Josephson junctions that would allow to investigate nonequilibrium entropy fluctuations inoverdamped quantum systems.
Thermodynamic processes at the nanoscale are gov-erned by both thermal and quantum fluctuations. It haslately been recognized that for classical nanosystems thesecond law of thermodynamics has to be generalized inorder to include effects induced by thermal fluctuations.The latter are usually vanishingly small in macroscopicsystems and are, therefore, neglected in the traditionalformulation of thermodynamics [1]. These generalizationsof the second law take the form of fluctuation theoremsthat quantify the occurrence of negative fluctuations ofquantities like work, heat and entropy [2, 3]. A remark-able property of these new thermodynamic identities istheir general validity arbitrarily far from equilibrium. Animportant example of a fluctuation theorem is the one de-rived by Crooks [4]: it relates the probability distributionsof work, ρ F ( W ) and ρ R ( W ), along forward and reversed transformations of a system according to, ρ R ( − W ) = ρ F ( W ) exp ( − β ( W − ∆ F )) . (1)Here ∆ F is the free energy difference between final andinitial states. Equation (1) indicates that large negativework fluctuations are exponentially suppressed and, hence,not observable in macroscopic systems. In its integratedform, the Crooks relation reduces to an equality previouslyobtained by Jarzynski [5], connecting the equilibrium freeenergy difference ∆ F to the nonequilibrium work W via, (cid:104) exp ( − β ( W − ∆ F )) (cid:105) = 1 . (2)In the above equation, the average (cid:104) ... (cid:105) is taken over theforward work distribution. It is essential to realize thatEqs. (1) and (2) only apply for systems that are initially in an equilibrium Gibbs state. Extensions of these expres-sions for different nonequilibrium initial distributions havebeen introduced by Hatano and Sasa [6] and by Seifert [7].Fluctuation theorems have been investigated experimen-tally in various nonequilibrium situations [8–13], where thecanonical example consists of a highly damped Brownianparticle in a driven potential. Due to the experimentaland theoretical importance of the strongly damped regime,the overdamped Langevin equation, and the equivalentSmoluchowski equation, have become the tool of choicefor the analysis of classical fluctuation theorems.In this paper, we derive quantum generalizations of theclassical Crooks and Jarzynski relations, Eqs. (1) and (2),in the strong friction regime. Previous studies on iso-lated or weakly coupled quantum systems can be found inRefs. [14–17], while an extension to the strongly coupledregime has been recently put forward in Ref. [18]. In thefollowing, we use the quantum generalization of the Smolu-chowski equation to treat both thermal and quantum fluc-tuations. Using the Onsager-Machlup path-integral rep-resentation, we show that the free energy difference fora driven quantum system can be obtained from its re-duced semiclassical density operator. We, moreover, pro-pose an experiment involving a driven Josephson junctionthat would allow to test our predictions. Quantum Smoluchowski equation. –
In thelimit of high friction, the off-diagonal matrix elements (cid:104) x | ˆ ρ ( t ) | x (cid:48) (cid:105) of the system density operator in the positionrepresentation are strongly suppressed over a time scaleof the order of 1 /γ , where γ is the friction coefficient. Asa result, a coarse-grained description of the dynamics ofp-1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov ebastian Deffner, Michael Brunner and Eric Lutzthe system in terms of the diagonal part of the positiondistribution alone, p ( x, t ) = (cid:104) x | ˆ ρ ( t ) | x (cid:105) , becomes possible[19–21]. In this semiclassical picture, the quantum systemfollows a classical trajectory and quantum effects mani-fest themselves through quantum fluctuations that act inaddition to the thermal fluctuations induced by the heatbath. A notable advantage of this description is that theusual classical definitions of work and heat are valid, incontrast to the full quantum regime [22]. The quantumSmoluchowski equation can be written as [19–21], ∂ t p ( x, t ) = 1 γm ∂ x (cid:20) V (cid:48) ( x ) + 1 β ∂ x D e ( x ) (cid:21) p ( x, t ) , (3)where V (cid:48) ( x ) is the derivative of the external potential withrespect to position and β the inverse temperature. Theeffective diffusion coefficient D e ( x ) is given by, D e ( x ) = 1 / [1 − λβV (cid:48)(cid:48) ( x )] , (4)with the parameter, λ = ( (cid:126) /πγm ) [ c + Ψ ( γ (cid:126) β/ π + 1)] (5)measuring the magnitude of the quantum fluctuations.Here m denotes the mass of the system, c = 0 . ... isthe Euler constant and Ψ the digamma function [30]. Itshould be noted that quantum corrections depend explic-itly on the position of the system through the curvatureof the potential V (cid:48)(cid:48) ( x ). When λ = 0, Eq. (3) reducesto the classical Smoluchowski equation with constant dif-fusion coefficient. The stationary equilibrium solution ofEq. (3), with natural boundary conditions, is p s ( x ) = 1 Z exp (cid:0) − βV ( x ) + λβ V (cid:48) ( x ) / (cid:1) [1 − λβV (cid:48)(cid:48) ( x )] , (6)where Z is the normalization constant. The above equi-librium expression is in general non-Gibbsian when λ (cid:54) = 0.The quantum Smoluchowski equation (3) with the ef-fective diffusion coefficient (4) is valid in the semiclassi-cal range of parameters, γ/ω (cid:29) ( (cid:126) β, /γ ), (cid:126) γ (cid:29) /β and | λβV (cid:48)(cid:48) ( x ) | <
1, where ω is a characteristic frequency, i.e.curvature at a potential minimum of the system [19–21].In the present analysis, we consider a time-dependentproblem where the potential V ( x, α t ) is driven by someexternal parameter α t = α ( t ). The driving rate should besmaller than the relaxation rate, ˙ α t /α t (cid:28) γ , to ensure thatthe non-diagonal elements of the density operator remainnegligibly small at all times [23]. Note that this conditionis not restrictive in the limit of very large γ . Quantum fluctuation theorems. –
We derive ex-tensions of the Crooks and Jarzynski relations, Eqs. (1)and (2), by using a path integral representation of the so-lution of the quantum Smoluchowski equation followingRef. [24]. For the sake of generality, we consider a genericdriven Fokker-Planck equation, with position-dependentdrift and diffusion coefficients D and D , of the form, ∂ t p ( x, α, t ) = L α p ( x, α, t ) , (7) where the linear operator L α is given by, L α = − ∂ x D ( x, α ) + ∂ x D ( x, α ) . (8)The quantum Smoluchowski equation (3) correspondsto the particular choice D ( x, α ) = − V (cid:48) ( x, α ) /γm and D ( x, α ) = 1 / [1 − λβV (cid:48)(cid:48) ( x, α )] γmβ . For any fixed value ofthe driving parameter α , we write the stationary solutionof Eq. (7) as, p s ( x, α ) = exp ( − ϕ ( x, α )) , (9)where the function ϕ ( x, α ) is explicitly given by, ϕ ( x, α ) = (cid:90) x dy ∂ y D ( y, α ) − D ( y, α ) D ( y, α ) . (10)We denote by X = { x } + τ − τ a trajectory of the system thatstarts at t = − τ and ends at t = + τ . We further de-fine a forward process α Ft , in which the driving parameteris varied from an initial value α F − τ = α to a final value α F + τ = α , as well as its time reversed process , α Rt = α F − t .The conditional probability of observing a trajectory start-ing at x − τ for the forward process can then be written as, P F [ X | x − τ ] = exp − + τ (cid:90) − τ dt S (cid:0) x t , ˙ x t , α Ft (cid:1) , (11)with a similar expression for the reversed process.In Eq. (11) the generalized Onsager-Machlup function S ( x t , ˙ x t , α t ) is taken to be of the form [25], S ( x t , ˙ x t , α t ) = [ ˙ x t − ( D ( x t , α t ) − ∂ x D ( x t , α t ))] D ( x t , α t ) . (12)The last term in the numerator of Eq. (12) is included toguarantee that thermodynamic potentials are independentof the state representation [26], and follows from the Itˆo-formula. By assuming that the system is initially in anequilibrium state given by the solution (9) of the Fokker-Planck equation (7), we obtain that the net probability ofobserving the trajectory X for the forward process is, P F [ X ] = p s ( x − τ , α ) P F [ X | x − τ ] . (13)In complete analogy, we find that the corresponding un-conditional probability for the reversed process reads, P R [ X ] = p s ( x τ , α ) P R (cid:2) X † | x τ (cid:3) , (14)where we have introduced the time-reversed trajectory, X † = { x † t } + τ − τ with x † t = x − t . We next compare the prob-ability of having the trajectory X during the forward pro-cess with that of having the trajectory X † during the re-versed process. We have P R (cid:104) X † | x †− τ (cid:105) = exp − + τ (cid:90) − τ dt S (cid:16) x † t , ˙ x † t , α Rt (cid:17) = exp − + τ (cid:90) − τ dt S † (cid:0) x t , ˙ x t , α Ft (cid:1) , (15)p-2uantum fluctuation theorems in the strong damping limitwhere we have defined the conjugate Onsager-Machlupfunction, S † ( x t , ˙ x t , α t ) = S ( x t , − ˙ x t , α t ). The ratio of theconditional probabilities (11) and (15) is simply deter-mined by the difference of S and S † . Using the definition(12), we thus obtain, P F [ X | x − τ ] P R (cid:104) X † | x †− τ (cid:105) = exp + τ (cid:90) − τ dt D (cid:0) x t , α Ft (cid:1) D (cid:0) x t , α FT (cid:1) ˙ x t ) × exp − + τ (cid:90) − τ dt ∂ x D (cid:0) x t , α Ft (cid:1) D (cid:0) x t , α Ft (cid:1) ˙ x t . (16)The ratio of the forward and reversed probabilities,Eqs. (13) and (14), follows directly as, P F [ X ] P R [ X † ] = p s ( x − τ , α ) P F [ X | x − τ ] p s ( x τ , α ) P R (cid:104) X † | x †− τ (cid:105) = exp ∆ ϕ + + τ (cid:90) − τ dt D (cid:0) x t , α Ft (cid:1) D (cid:0) x t , α Ft (cid:1) ˙ x t × exp − + τ (cid:90) − τ dt ∂ x D (cid:0) x t , α Ft (cid:1) D (cid:0) x t , α Ft (cid:1) ˙ x t , (17)where ∆ ϕ = + τ (cid:90) − τ dt (cid:0) ˙ α Ft ∂ α ϕ + ˙ x t ∂ x ϕ (cid:1) . (18)By using the explicit expression (10) of the stationary so-lution ϕ ( x, α ), we finally arrive at P F [ X ] P R [ X † ] = exp + τ (cid:90) − τ dt ˙ α Ft ∂ α ϕ . (19)We are now in the position to derive generalized fluctu-ation theorems for stochastic processes described by thegeneric Fokker-Planck equation (7). We begin by definingthe generalized entropy production Σ as,Σ = τ (cid:90) − τ dt ˙ α Ft ∂ α ϕ . (20)The entropy production Σ in Eq. (20) is similar to theentropy production introduced by Hatano and Sasa forsystems initially in a nonequilibrium steady state [6]. Inthe present situation, however, it corresponds to a non-Gibbsian equilibrium state. We note, in addition, thatthe entropy production, as defined in Eq. (20), is odd un-der time-reversal, Σ R (cid:2) X † (cid:3) = − Σ F [ X ]. The distributionof the entropy production, ρ F (Σ), for an ensemble of re-alizations of forward processes can then be defined as, ρ F (Σ) = (cid:90) D X P F [ X ] δ (cid:0) Σ − Σ F [ X ] (cid:1) == exp (Σ) (cid:90) D X † P R (cid:2) X † (cid:3) δ (cid:0) Σ + Σ R (cid:2) X † (cid:3)(cid:1) (21) where we have used Eq. (19) in the last line. Here, (cid:82) D X =lim N →∞ (4 πs ) − N/ N − (cid:81) i =1 (cid:82) dx is D ( x is , α is ) − / , s = 2 τ /N , de-notes the product of integrals over all possible paths X .The continuous integral in Eq. (21) is interpreted as thelimit of a discrete sum. Equation (21) can be recast inthe form of a generalized Crooks relation for the entropyproduction, ρ R ( − Σ) = ρ F (Σ) exp ( − Σ) . (22)By, moreover, integrating Eq. (22) over Σ, we obtain anextended version of the Jarzynski equality, (cid:104) exp ( − Σ) (cid:105) = 1 . (23)Expression (20) for the entropy production, together withthe fluctuation theorems (22) and (23), constitutes ourmain result. Combined, they represent the quantumgeneralizations of the Crooks and Jarzynski equalities,Eqs. (1) and (2), in the limit of strong damping. In theclassical limit λ = 0, ϕ ( x, α ) = β ( V ( x, α ) − F ( α )), andthe entropy production (20) takes the familiar form, Σ = β (cid:82) dt ˙ α Ft ∂ α V ( x t , α Ft ) − β ∆ F . The inequality (cid:104) Σ (cid:105) ≥ Parametric harmonic oscillator. –
Let us illus-trate our results with the example of a harmonic oscilla-tor with time-dependent frequency, V ( x, ω t ) = mω t x / D e ( x ) = 1 / (1 − λβmω ), and quantum fluc-tuations therefore renormalize the width of the stationarydistribution of the oscillator, which is no longer given bythe temperature of the bath as in the classical regime. Weassume that the driving parameter α t = ω t is changedfrom ω to ω during time 2 τ . We moreover define thepartition function Z of the system as the normalizationconstant of the stationary distribution (6) of the quan-tum Smoluchowski equation. The free energy differencebetween final and initial state is then, β ∆ F = − ln( Z /Z ) = ln ω ω + ln (cid:115) − λβmω − λβmω . (24)In the limit (cid:126) γβ (cid:29)
1, the quantum parameter (5) simpli-fies to λ = ( (cid:126) /πγm )[ln( γ/ν ) + c ] with ν = 2 π/ (cid:126) β . Thefree energy difference (24) reduces accordingly to β ∆ F (cid:39) ln ω ω + ln( γ/ν ) νγ (cid:0) ω − ω (cid:1) + cνγ (cid:0) ω − ω (cid:1) . (25)The generalized Jarzynski equality (23) can now be usedto evaluate the free energy difference for the parametricp-3ebastian Deffner, Michael Brunner and Eric Lutz (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) Β (cid:119) Ρ (cid:72) Β (cid:119) (cid:76) Fig. 1: (color online) Work distribution ρ ( βw ) for a quantumoscillator with time-dependent frequency, ω t = ω + ( ω − ω ) t/ τ , for slow driving, 2 τ = 300 τ r (blue dots), and fastdriving, 2 τ = τ r (red squares), with τ r = 1 /γ the relaxationtime of the oscillator. In both cases, the free energy differenceevaluated numerically using the Jarzynski equality (23) (green,solid vertical line) agrees with the analytical expression (24)(black, dashed vertical line). Parameters are (cid:126) = 1 . m = 1, γ = 6000, β = 103 . ω = 5 and ω = 11, for an ensemble of2 · trajectories. quantum oscillator from nonequilibrium work measure-ments. For simplicity, we consider a linear variation ofthe square frequency, ω t = ω + ( ω − ω ) t/ τ . We de-fine the semiclassical work as βw = (cid:82) τ dt ˙ α t ∂ α ˜ ϕ with˜ ϕ = βV ( x, α ) − ( λβ / V (cid:48) ( x, α ) − ln(1 − λβV (cid:48)(cid:48) ( x, α )); thelatter is related to the entropy production (20) via βw =Σ + β ∆ F and reduces to the classical expression βW ofthe work in the limit λ = 0. We numerically determine theprobability distribution ρ ( βw ) of the work from an ensem-ble of identical driving realizations with the help of the Itˆo-Langevin equation, mγ ˙ x + V (cid:48) ( x, α t ) = (cid:112) D e ( x, t ) F ( t ),corresponding to the quantum Smoluchowski equation (3).Here F ( t ) denotes a Gaussian random force with zeromean and variance (cid:104) F ( t ) F ( t (cid:48) ) (cid:105) = mγ/β δ ( t − t (cid:48) ). Figure 1shows the work distribution ρ ( βw ) for two different driv-ing times: a slow driving (2 τ = 300 τ r ) and a fast driving(2 τ = τ r ), where τ r = 1 /γ is the relaxation time of theoscillator. We observe that equality (23) leads in bothcases to the free energy difference (24), whose value is in-dicated by the vertical line. It is worth noticing that anaive application of the classical Jarzynski equality (2) tothe quantum oscillator would result in an apparent viola-tion of the latter [27]; this deviation is of course due to thenon-Gibbsian property of the stationary distribution (6).In the approach of Ref. [18], the free energy F S of thesystem is defined as the difference between the total freeenergy of system plus bath and the free energy of the bathalone. The corresponding free energy difference can be (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) Β (cid:119) Ρ (cid:72) Β (cid:119) (cid:76) Fig. 2: (color online) Work distribution ρ ( βw ) for a drivenJosephson junction with potential V ( φ, α t ) = − E J (cos( φ ) + α t φ ) and driving parameter α t = α + ( α − α ) t/ τ , forfast driving, 2 τ = 0 . γ/E J (red squares), and slow driving,2 τ = γ/E J (blue dots). The free energy difference evaluatednumerically using the Jarzynski equality (23) (green, solid ver-tical line) agrees with the one determined via the normalizationconstant of the stationary distribution (30) (black, dashed ver-tical line). Parameters are (cid:126) = 1 . m = 1, γ = 22 . β = 0 . α = 0 and α = 0 . E J = 50 .
64, for an ensemble of 2 · trajectories. The latter correspond to T = 1, λ = 0 .
026 andΘ = 0 . evaluated exactly and reads [28, 29], β ∆ F S = ln ω Γ (cid:16) λ ( ω ) ν (cid:17) Γ (cid:16) λ ( ω ) ν (cid:17) ω Γ (cid:16) λ ( ω ) ν (cid:17) Γ (cid:16) λ ( ω ) ν (cid:17) , (26)where Γ( x ) is the Gamma function and λ , the charac-teristic frequencies of the damped oscillator; in the limitof large bath cutoff frequency, they are given by λ , ( ω ) = γ/ ± (cid:112) γ / − ω [28]. Using the asymptotic expansionsof the Gamma function, Γ( x ) (cid:39) / ( x + cx )( x (cid:28)
1) andΓ( x ) (cid:39) √ π x x − / exp( − x )( x (cid:29)
1) [30], expression (26)is seen to reduce to the free energy difference (25) ob-tained from the quantum Smoluchowski equation in thelimit γ (cid:29) (cid:126) βω i and γ (cid:29) ω i . Experimental verification in driven Josephsonjunctions. –
No experimental investigation of quantumfluctuation theorems has been performed so far. A schemeto study the Crooks and Jarzynski relations in isolatedand weakly damped quantum systems using modulatedion traps has recently been put forward in Ref. [31]. Herewe propose to test the predictions for the quantum fluctu-ation theorems, Eqs. (22) and (23), in the strong dampinglimit using driven Josephson junctions [32]. The Joseph-son relations for the current I s ( t ) across the junction andthe phase difference φ ( t ) between left and right supercon-ductors are I s = I c sin ( φ ) and ˙ φ = 2 e/ (cid:126) U ( t ) , (27)p-4uantum fluctuation theorems in the strong damping limitwhere U ( t ) is the voltage drop across the junction. Themaximal current I c is given by I c = (2 e/ (cid:126) ) E J , where E J is the coupling energy (Josephson energy). An externallyshunted Josephson junction can be described via an equiv-alent circuit consisting of an ideal junction, a capacitance C and a resistance R (Resistively Shunted Junction (RSJ)model) [32]. In this model, the Josephson junction is inter-preted as describing the diffusive motion of a particle withposition φ ( t ) and mass m = ( (cid:126) / e ) C , the friction coeffi-cient being given by γ = 1 /RC . An important quantity inthe RSJ model is the dimensionless capacitance (Stewart-McCumber parameter), β c = (2 π/ Φ ) I c R C , where Φ = h/ e is the magnetic flux quantum. In the overdampedregime, β c <
1, the dynamics of the Josephson phase φ can be described by the quantum Smoluchowski equation(3) with the potential V ( φ ) = − E J cos ( φ ) − E I φ [33, 34].The energy E I = ( (cid:126) / e ) I is determined by the bias cur-rent I and the effective diffusion coefficient (4) reads, D e ( φ ) = 1 / [1 − Θ cos ( φ )] . (28)The constant Θ = λβE J is the crucial parameter whichgoverns the magnitude of quantum effects in a Josephsonjunction. It is directly proportional to the quantum pa-rameter λ , Eq. (5), which in the context of the RSJ modelcan be reexpressed as, λ = 2 r (cid:2) c + Ψ (cid:0) βE c / π r + 1 (cid:1)(cid:3) , (29)where E c = 2 e /C is the charging energy, r = R/R Q thedimensionless resistance and R Q = h/ e the resistancequantum. The stationary solution of the quantum Smolu-chowski equation (3), with periodic boundary conditions,with V (cid:48) ( φ ) = V (cid:48) ( φ + L ), can be written as [35], p stat ( φ ) = p s ( φ ) Z J (cid:90) φ + Lφ dy [ D e ( y ) p s ( y )] − = 1 Z J e − ˜ ϕ , (30)where p s ( φ ) is given by Eq. (6) and Z J is the normaliza-tion constant. In Fig. 2 we have plotted the work distribu-tion ρ ( βw ) for a driven Josephson junction with potential V ( φ, α t ) = − E J (cos( φ ) + α t φ ) and linear driving param-eter α t = E I ( t ) /E J = α + ( α − α ) t/ τ , for two drivingtimes, 2 τ = 0 . γ/E J and 2 τ = γ/E J . As in the case ofthe parametric quantum oscillator, the free energy differ-ence determined numerically using the generalized Jarzyn-ski equality (23) agrees with the one determined via thenormalization constant of the stationary distribution (30), β ∆ F = − ln( Z J, /Z J, ), of the quantum Smoluchowskiequation.The nonequilibrium entropy production (20) can be ex-perimentally determined in a Josephson junction by ap-plying the following measurement procedure. The phase φ can be directly deduced from a measurement of theJosephson current once the current–phase relation of thejunction has been determined [36]. The system is thenfirst prepared in a given initial state and let to relax to itsstationary state (6). After the latter has been attained, T [ K ] λ [10 − ] ΘQuantum 0.98 0.21 0.99Classical 4.2 0.087 0.097 Table 1: Typical Θ values for circle shaped Josephson junctionswith C = 1 . pF , R = 0 . I c = 0 . mA and β c = 0 . the Josephson potential V ( φ ) is modified according to aspecific driving protocol α t with the help of an externalmagnetic field. The entropy production Σ during sucha protocol (corresponding to either a forward or reversed transformation) can be evaluated via Eq. (20) from therecorded values of the current. The distribution functionof the entropy can eventually be reconstructed by repeat-ing the above measurement sequence, and the validity ofthe quantum fluctuation theorems (22) and (23) in thestrong friction regime can be tested. In Tab. 1 we list typ-ical parameter values for niobium-based Josephson junc-tions [37]. By varying the temperature, both the classical,Θ (cid:28)
1, and the quantum regime, Θ (cid:46)
1, can be exploredwith the same junctions.
Conclusion. –
We have analyzed quantum fluctua-tion theorems in the strong coupling limit with the helpof the quantum Smoluchowski equation. We have shownthat quantum Crooks and Jarzynski type relations canbe derived in this regime when the entropy production isproperly modified to take into account the non-Gibbsianproperty of the initial equilibrium state. In the case of aninitial nonequilibrium steady state, a similar calculationleads to a quantum Hatano-Sasa relation. By investigat-ing a parametric harmonic oscillator and a driven Joseph-son junction, we have additionally shown that the free en-ergy difference can be directly obtained from the reduceddensity operator of the quantum system. We have, finally,proposed an experiment based on a driven Josephson junc-tion that would enable to study quantum nonequilibriumentropy fluctuations in overdamped systems. ∗ ∗ ∗
We thank P. Talkner for bringing Ref. [26] to our at-tention, C. Schneider and R. Held, as well as C. Kaiserand R. Sch¨afer for their experimental advice on Joseph-son junctions. This work was supported by the EmmyNoether Program of the DFG (contract No LU1382/1-1)and the cluster of excellence Nanosystems Initiative Mu-nich (NIM).
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