Group-theoretical approach to the calculation of quantum work distribution

Abstract

Usually the calculation of work distributions in an arbitrary nonequilibrium process in a quantum system, especially in a quantum many-body system is extremely cumbersome. For all quantum systems described by quadratic Hamiltonians, we invent a universal method for solving the work distribution of quantum systems in an arbitrary driving process by utilizing the group-representation theory. This method enables us to efficiently calculate work distributions where previous methods fail. In some specific models, such as the time-dependent harmonic oscillator, the dynamical Casimir effect, and the transverse XY model, the exact and analytical solutions of work distributions in an arbitrary nonequilibrium process are obtained. Our work initiates the study of quantum stochastic thermodynamics based on group-representation theory. PACS numbers:Introduction.-In the past two decades, a great breakthrough in the field of nonequilibrium thermodynamics is the discovery of fluctuation theorems [1] and the establishment of stochastic thermodynamics [2, 3]. It extends the usual definition of work, heat and entropy production in a thermodynamic process from ensembleaveraged quantities to trajectory-dependent quantities. Based on these extensions, the second law is sharpened and rewritten into equalities (fluctuation theorems) for arbitrary nonequilibrium processes. For an isolated quantum system, a proper definition of the trajectory work is defined through the so-called two-point measurement [4, 5], which preserves the non-negativity of the probabilities of trajectories and Jarzynski equality at the same time [6, 7]. As is known, the work distribution of a system provides a great deal of information about a nonequilibrium process [8][9][10], which is an analogue to the partition function encoding essential information about an equilibrium state. However, very few analytical results of work distributions of quantum systems have been obtained so far. In literature, the few exceptions are associated with either specific models (e.g., a forced harmonic oscillator [11][12][13], a harmonic oscillator with a time-dependent frequency [14,15], quenched Luttinger liquid at zero temperature [8], a diving scalar field [16] or a transverse Ising model at zero temperature [15]) or very special protocols (e.g., the sudden quench protocol [17][18][19][20][21][22][23] or the quantum adiabatic protocol [24,25]). Hence, to develop a universal method to efficiently calculate the work distribution for various quantum systems in an arbitrary nonequilibrium process is one of the most challenging problems in this field.

Theory of quantum work in metallic grains

“…Equation 10is one of the important results of this work; it establishes a connection between the work statistics of the many-body system and the time evolution of individual single-particle states [34,35]. In the following, we shall use this formula to study the properties of quantum quenches in generic fermionic nanosystems described by random matrix theory.…”

Exact thermal properties of free-fermionic spin chains

“…The complete expression for the partition function 3.1 was first derived with the aid of creation and annihilation operators by Katsura [63]. An alternative approach has been reported using Grassmann variables, without a numerical characterization [65]; see as well [45].…”

“…The universal dynamics of defect formation is not always desirable, and a variety of works have been devoted to circumvent it using diverse control protocols [32][33][34][35][36][37][38][39][40][41][42], beyond the use of nonlinear quenches and inhomogeneous driving. In addition, quasi-free fermion models have been discussed in the context of quantum thermodynamics, as a test-bed to explore work statistics and fluctuation theorems [43][44][45][46] and as a working substance in a quantum thermodynamic cycle [47].…”

“…Kapitonov and Il'inskii provided an alternative derivation of the closed form expression of the exact partition function using functional integrals over Grassmann variables [65]. More recently, Fei and Quan [45] used group theory methods to calculate the exact partition function and quantum work distribution.…”

Tensor-Network Approach to Work Statistics for 1D Quantum Lattice Systems

“…Previous studies mainly focus on analytical methods and are restricted to few exactly solvable models, and are studied case by case. Examples include harmonic oscillators [38][39][40][41][42][43], piston systems [44][45][46], 1D diatomic Toda lattice [47], 1D quantum gases [46,[48][49][50], quantum fields [51,52], and quantum systems of quadratic Hamiltonians [53][54][55][56][57]. Quantum Feynman-Kac equation [58] and phase-space formulation [59][60][61] have been proposed, but are practically constrained to single-particle systems and difficult to extend to many-body systems.…”