Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework

Abstract

A central topic in the emerging field of quantum thermodynamics is the definition of thermodynamic work in the quantum regime. One widely used solution is to define work for a closed system undergoing non-equilibrium dynamics according to the two-point energy measurement scheme. However, due to the invasive nature of measurement the two-point quantum work probability distribution cannot describe the statistics of energy change from the perspective of the system alone. We here introduce the quantum histories framework as a method to characterise the thermodynamic properties of the unmeasured, closed dynamics. Constructing continuous power operator trajectories allows us to derive an alternative quantum work distribution for closed quantum dynamics that fulfils energy conservation and is time-reversal symmetric. This opens the possibility to compare the measured work with the unmeasured work, contrasting with the classical situation where measurement does not affect the work statistics. We find that the work distribution of the unmeasured dynamics leads to deviations from the classical Jarzynski equality and can have negative values highlighting distinctly non-classical features of quantum work.

Quantum Fluctuation Theorems, Contextuality, and Work Quasiprobabilities

“…(1) and related equalities provide refined statements of the second law of thermodynamics beyond averages, are valid at the microscopic scale far from the thermodynamic limit, recover results from linear response theory and give a way to measure the free energy from non-equilibrium measurements of work [7]. Much effort has been devoted to finding analogous results characterising work fluctuations of quantum processes and in the presence of quantum coherence [6,[8][9][10][11][12][13][14][15][16][17]. One of the main challenges is the definition of work for quantum systems (see [18] and references therein).…”

Quantum work in the Bohmian framework

“…In contrast to proposals based on trajectories in Hilbert space, such as the quantum jump approach [16,46,47,59,60], consistent histories framework [25] or continuous measurements [17,23], the Hamilton-Jacobi approach is based on trajectories in phase space where the system has a well defined energy at all times. These trajectories have been reconstructed using weak measurements [63][64][65], from which the distribution (5) can be inferred.…”

“…Other approaches include using a power or work operator [4,12,13,22,53,54], measuring the system by coupling it weakly to a * rui.ferreirasampaio@aalto.fi † t.g.philbin@exeter.ac.uk detector [17], the full counting statistics (FCS) approach [26][27][28], and work definitions based on entropic principles [19][20][21][55][56][57][58]. Work definitions based on trajectories in Hilbert space include the quantum jump approach [46,59,60] and the consistent histories framework [25]. None of these approaches yields a positive work distribution based on trajectories in phase space.…”

Contributions to single-shot energy exchanges in open quantum systems

“…To study energy fluctuations we need first to provide a link between the CWF of the system and its energy. Different proposals can be found in the literature to study fluctuations in open quantum systems [5][6][7][36][37][38][39][40]. For descriptions based on stochastic states (pure or mixed), a typical approach to estimate the energy of the system is to take the expectation value of the system Hamiltonian with respect to the stochastic state (see, e.g., Refs.…”