The Tasaki-Crooks quantum fluctuation theorem

a r X i v :0705.1252v 1 [c o n d -m a t .s t a t -m e c h ] 9 M a y 2007NOTEThe Tasaki-Crooks quantum fluctuation theoremPeter Talkner ∗,Peter H¨a nggiInstitut f¨u r Physik,Universit¨a t Augsburg,Universit¨a tsstrasse 1,D-86135Augsburg,GermanyAbstract.Starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system [cond-mat/0703213]the quantum version of the Crooks fluctuation theorem is shown to emerge almost immediately by the mere application of an inverse Fourier transformation.PACS numbers:05.40.-a,87.16.-b,87.19.NnSubmitted to:J.Phys.AWork and fluctuation theorems have ignited much excitement during the recent decade [1–4].These theorems have prompted further theoretical investigations [5–8]as well as experimental research [9].We here consider a quantum system staying in weak thermal contact with a heat bath at the inverse temperature βuntil a time t 0.At time t 0the contact to the heat bath is then either kept at this weak level,or may even be switched offaltogether.A classical time dependent force solely acts on the system according to a prescribed protocol until time t f .A protocol defines a family of Hamiltonians {H (t )}t f ,t 0which govern the time evolution of the system during the indicated interval of time [t 0,t f ]in the presence of the external force.The weak action of the heat bath on the system can be neglected for any protocol of finite duration t f −t 0[10].The work performed by the force on the system is a random quantity because of the quantum nature of the considered system and because the system is prepared in the thermal equilibrium stateρ(t 0)=Z (t 0)exp {−βH (t 0)}(1)which is a mixed state for all finite β.Here,Z (t 0)=Tr exp {−βH (t 0)}denotes the partition function.As a random quantity,the work is characterized by a probability density p t f ,t 0(w )or equivalently by the corresponding characteristic function G t f ,t 0(u ),which is defined as the Fourier transform of the probability density,i.e.G t f ,t 0(u )=dw e iuw p t f ,t 0(w ).(2)∗Corresponding author:peter.talkner@physik.uni-augsburg.deIn a recent work[11]we have demonstrated that the characteristic function G tf,t0(u)of the work can be expressed as quantum correlation function of the two exponential operators exp{iuH(t f)}and exp{−iuH(t0)}.It explicitly reads:G tf,t0(u)= e iuH(t f)e−iuH(t0) t≡Z−1(t0)Tr U+tf,t0e iuH(t f)U tf,t0e−iuH(t0)e−βH(t0),(3)where the index at the bracket signifies the fact that the average is taken over the initial density matrixρ(t0).For a protocol consisting of Hamiltonians H(t),each of which is bounded frombelow and has a purely discrete spectrum,the characteristic function G tf,t0(u)is ananalytic function of u in the strip S={u|0≤ℑu≤β,−∞<ℜu<∞}[12] whereℜu andℑu denote the real and imaginary part of u,respectively.Collecting the two exponential factors e−iuH(t0)and e−βH(t0)into one,and introducing the complex parameter v=−u+iβ∈S wefindZ(t0)G tf,t0(u)=Tr U+tf,t0e i(−v+iβ)H(t f)U tf,t0e ivH(t0)=Tr e−ivH(t f)e−βH(t f)U tf,t0e ivH(t0)U+tf,t0=Tr e−ivH(t f)e−βH(t f)U+t0,t fe ivH(t0)U t0,t f=Tr U+t0,t fe ivH(t0)U t0,t fe−ivH(t f)e−βH(t f)=Z(t f)G t0,t f(v)(4)where we used the unitarity of the time evolution operator,i.e.U+tf,t0=U−1tf,t0=U t0,t f.We hence obtainG tf,t0(u)=Z(t f)p t0,t f (−w)=Z(t f)Acknowledgments.This work has been supported by the Deutsche Forschungs-gemeinschaft via the Collaborative Research Centre SFB-486,project A10.Financial support of the German Excellence Initiative via the Nanosystems Initiative Munich (NIM)is gratefully acknowledged as well.References[1]Evans DJ,Cohen EGD and Morriss GP1993Phys.Rev.Lett.712401[2]Gallavotti G and Cohen EGD1995Phys.Rev.Lett.742694[3]Jarzynski C1997Phys.Rev.Lett.782690[4]Crooks GE1999Phys.Rev.E602721[5]Kurchan J1998J.Phys.A:Math.Gen.313719[6]Tasaki H1999Jarzynski relations for quantum systems and some applications(Preprint cond-mat/0009244)[7]Mukamel S2003Phys.Rev.Lett.90170604[8]Seifert U2004J.Phys.A:Math.Gen.37L517[9]Bustamante C,Liphardt J and Ritort F2005Physics Today58(7)43[10]Spohn H1980Rev.Mod.Phys.52569[11]Talkner P,Lutz E and H¨a nggi P2007Phys.Rev.E75R17704(Preprint cond-mat/0703213)[12]This can be proved in the same way as the analyticity properties of equilibrium correlation functionsthat underly the KMS condition,cf.[13].[13]Haag R,Hugenholtz NM and Winnink M1967Commun.Math.Phys.5215。