The Tasaki-Crooks quantum fluctuation theorem

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.Nn Submitted to: J. Phys. A Work 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 β until a time t0. At time t0 the contact to the heat bath is then either kept at this weak level, or may even be switched off altogether. A classical time dependent force solely acts on the system according to a prescribed protocol until time tf . A protocol defines a family of Hamiltonians {H(t)}tf ,t0 which govern the time evolution of the system during the indicated interval of time [t0, tf ] 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 tf − t0 [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 state ρ(t0) = Z(t0) exp{−βH(t0)} (1) which is a mixed state for all finite β. Here, Z(t0) = Tr exp{−βH(t0)} denotes the partition function. As a random quantity, the work is characterized by a probability density ptf ,t0(w) or equivalently by the corresponding characteristic function Gtf ,t0(u), which is defined as the Fourier transform of the probability density, i.e. Gtf ,t0(u) = ∫ dw eiuwptf ,t0(w). (2) ∗ Corresponding author: [email protected] The Tasaki-Crooks quantum fluctuation theorem 2 In a recent work [11] we have demonstrated that the characteristic function Gtf ,t0(u) of the work can be expressed as quantum correlation function of the two exponential operators exp{iuH(tf)} and exp{−iuH(t0)}. It explicitly reads: Gtf ,t0(u) = 〈e iuH(tf e〉t0 ≡ Z(t0)Tr U + tf ,t0 ef )Utf ,t0e 0e0 , (3) where the index at the bracket signifies the fact that the average is taken over the initial density matrix ρ(t0). For a protocol consisting of Hamiltonians H(t), each of which is bounded from below and has a purely discrete spectrum, the characteristic function Gtf ,t0(u) is an analytic function of u in the strip S = {u|0 ≤ Iu ≤ β,−∞ < Ru < ∞} [12] where Ru and Iu denote the real and imaginary part of u, respectively. Collecting the two exponential factors e0 and e0 into one, and introducing the complex parameter v = −u+ iβ ∈ S we find Z(t0)Gtf ,t0(u) = Tr U + tf ,t0 ef ) Utf ,t0 e ivH(t0) = Tr ef ) ef ) Utf ,t0 e ivH(t0) U tf ,t0 = Tr ef ) ef ) U t0,tf e ivH(t0) Ut0,tf = Tr U t0,tf e ivH(t0) Ut0,tf e −ivH(tf ) ef ) = Z(tf)Gt0,tf (v) (4) where we used the unitarity of the time evolution operator, i.e. U tf ,t0 = U −1 tf ,t0 = Ut0,tf . We hence obtain Gtf ,t0(u) = Z(tf ) Z(t0) Gt0,tf (−u+ iβ). (5) The ratio of the canonical partition functions can be expressed in terms of the difference of free energies ∆F between the two thermal equilibrium systems as Z(tf)/Z(t0) = exp{−β∆F}. The quantity Gt0,tf (v) coincides with the characteristic function of the work performed on a system that is initially prepared in the thermal equilibrium state Z(tf) −1 exp{−βH(tf) under the influence of the time-reversed protocol {H(t)}t0.tf . Applying the inverse Fourier transform on both sides of eq. (5) we obtain the following fluctuation theorem ptf ,t0(w) pt0,tf (−w) = Z(tf) Z(t0) e = e . (6) It relates the probability density of performed work for a given protocol to that of the work for the time-reversed process. This process can in principle be realized by preparing the Gibbs state Z(tf ) exp{−βH(tf)} as the initial density matrix and letting run the time-reversed protocol {H(t)}t0,tf . In the classical context this fluctuation theorem was proved by Gavin Crooks [4], its quantum version goes back to Hal Tasaki [6]. The Tasaki-Crooks quantum fluctuation theorem 3 Acknowledgments. This work has been supported by the Deutsche Forschungsgemeinschaft 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.