Derivation of quantum work equalities using a quantum Feynman-Kac formula

Abstract

On the basis of a quantum mechanical analogue of the famous Feynman-Kac formula and the Kolmogorov picture, we present a novel method to derive nonequilibrium work equalities for isolated quantum systems, which include the Jarzynski equality and Bochkov-Kuzovlev equality. Compared with previous methods in the literature, our method shows higher similarity in form to that deriving the classical fluctuation relations, which would give important insight when exploring new quantum fluctuation relations. Kac [2] establishes an important connection between partial differential equations and classical stochastic processes. Briefly, assuming that in a continuous diffusion process the probability of a stochastic trajectory X started from a statehas a concise path integral representation [3]:where L + is the Markovian generator of the diffusion. This is the famous FK formula in classical stochastic processes. Pioneered by Lebowitz and Sphon [4], this formula was also found very useful in studying fluctuation relations [5][6][7][8][9]. In the past two decades, these important relations have greatly deepened our understanding about the second law of thermodynamics and nonequilibrium physics of small systems [4,[10][11][12][13][14][15][16][17][18][19][20]. Recently, finding quantum fluctuation relations is attracting intensive interests. Fruitful theoretical and experimental results [21][22][23][24][25][26][27][28][29][30] have been obtained. To our best knowledge, however, there is no work explicitly using the FK formula. At first glance, the reason is very obvious, because the classical trajectory picture on which the FK is based is not available in quantum physics. Contrary to the intuition, in this Rapid Communication we use an isolated quantum system as an example to show that there indeed exists a quantum mechanical analogue of the classical FK formula and it is very useful to derive the quantum nonequilibrium work relations including the Jarzynski [21,23,24] and Bochkov-Kuzovlev equalities [29].Kolmogorov picture and backward invariable. We start by introducing essential notations and a new picture that is a quantum-mechanical analogue of Kolmogorov's idea [31] in classical stochastic theory. Although the picture is virtually equivalent to other well-known pictures, e.g. the Heisenberg picture, we will see later that it is very relevant with the time reversal concept. We assume that the closed quantum system is described by a time-dependent Hamiltonian H(t). The system's density operators ρ at two different times t and t ′ (< t) are connected by the time-evolution operator U (t ′ ), i.e.,Given an arbitrary observable F that does not depend explicitly on time, we define its Kolmogorov picture aswhere the superscript H denotes the Heisenberg picture: F H (t)=U † (t)F U (t). On the basis of Eqs. (3) and (4), the expectation value F (t) at time t in the picture is

Characteristic functions based on a quantum jump trajectory

“…Although this new form does not bring any advantages in computing over the original ones, Eqs. (29) and (34), it indeed corresponds to the celebrated Feynman-Kac formula in classical stochastic processes [10,11,15,16,94,110]. Analogously, the concise expressions of the CFs of the heat and entropy production are given by…”

“…This is essential for the validity of a variety of fluctuation relations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][21][22][23]. It is worthwhile to emphasize that either the seemingly complex descriptions of H(t) or the detailed balance condition have nothing to do with our formalism for the CFs.…”

Verification of the quantum nonequilibrium work relation in the presence of decoherence

“…Other studies of the work relation overcome this issue by defining work at the system level without referencing an environment. In this vein there are several equivalent formalisms for treating quantum detailed balance master equations [50,51,52,53,54,55,56] of which we focus on the quantum jump trajectory method [57,58,59,60,52,53,35]. Originally developed in the field of quantum optics [61], this approach treats a system's density operator as an average over pure states evolving according to stochastic trajectories.…”

Calculating work in adiabatic two-level quantum Markovian master equations: A characteristic function method

“…(1). We do not pursue the further details here [16,53]. Finally, Talkner et al [12] have used an another characteristic function method to prove the validity of the JE and Crooks' equality for very general open quantum systems.…”