Quantum fluctuation theorems and power measurements

Abstract

Work in the paradigm of the quantum fluctuation theorems of Crooks and Jarzynski is determined by projective measurements of energy at the beginning and end of the force protocol. In analogy to classical systems, we consider an alternative definition of work given by the integral of the supplied power determined by integrating up the results of repeated measurements of the instantaneous power during the force protocol. We observe that such a definition of work, in spite of taking account of the process dependence, has different possible values and statistics from the work determined by the conventional two energy measurement approach (TEMA). In the limit of many projective measurements of power, the system's dynamics is frozen in the power measurement basis due to the quantum Zeno effect leading to statistics only trivially dependent on the force protocol. In general the Jarzynski relation is not satisfied except for the case when the instantaneous power operator commutes with the total Hamiltonian at all times. We also consider properties of the joint statistics of powerbased definition of work and TEMA work in protocols where both values are determined. This allows us to quantify their correlations. Relaxing the projective measurement condition, weak continuous measurements of power are considered within the stochastic master equation formalism. Even in this scenario the power-based work statistics is in general not able to reproduce qualitative features of the TEMA work statistics.

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

“…We note that the derivation of inequality (26) assumes an initial thermal state, contrasting with previous discussions of quantum signatures in the average exponentiated work by Allhaverdyan, Solinas et al and Elouard et al that have focussed on nonthermal initial states [10,12,35]. The properties of p(w) are listed in table 1 and will be compared with another power-operator-based work distribution [32,34] in section 5. Before that we will first discuss the connection to the TPM definition of work.…”

“…The histories framework and the resulting work distribution p(w) for the unmeasured dynamics differ from previous proposals to define fluctuating work using the power operator [32][33][34]48]. In those proposals the power is continuously measured leading to a work probability distribution given as which is nonlinear in the class operators C n  assigned to each power trajectory n  .…”

“…Using the histories approach to quantum mechanics [26][27][28][29][30] we construct the work distribution function associated with the continuous dynamics of the system in the absence of any experimental monitoring. Instead operator trajectories are introduced that describe the evolution in time through the set of instantaneous eigenvalues of a time-dependent power operator [31,32]. We follow Goldstein and Page [29] in assigning linear probabilities to general interfering trajectories, or histories.…”

“…where A and B are time-independent Hermitian operators while t l ( ) is a real, timedependent scalar function[32], then the power operator in the Heisenberg picture is X t t At each discrete point in time, labelled again by j, the operator B b…”

Perspective on quantum thermodynamics

“…The celebrated Jarzynski [25,26] and Crooks [27,28] relations link the free energy difference between states with the work done in transforming between them, and have been experimentally verified in several systems [29][30][31][32][33][34][35][36]. These relations also hold unmodified in closed quantum systems [37][38][39], with slight modifications when the system is open [40][41][42].…”