Fluctuation theorems for continuous quantum measurements and absolute irreversibility

Abstract

Fluctuation theorems are relations constraining the out-of-equilibrium fluctuations of thermodynamic quantities like the entropy production that were initially introduced for classical or quantum systems in contact with a thermal bath. Here we show, in the absence of thermal bath, the dynamics of continuously measured quantum systems can also be described by a fluctuation theorem, expressed in terms of a recently introduced arrow of time measure. This theorem captures the emergence of irreversible behavior from microscopic reversibility in continuous quantum measurements. From this relation, we demonstrate that measurement-induced wave-function collapse exhibits absolute irreversibility, such that Jarzynski and Crooks-like equalities are violated. We apply our results to different continuous measurement schemes on a qubit: dispersive measurement, homodyne and heterodyne detection of a qubit's fluorescence.The emergence of macroscopic irreversibility from microscopic time-reversal invariant physical laws has been a long-standing issue, well described by the formalism of statistical thermodynamics [1,2]. In this framework, the small system under study follows stochastic trajectories in its phase-space, where the randomness models the uncontrolled forces exerted on the system by its thermal environment. Although these trajectories are microscopically reversible, one direction of time is more probable than the other and a arrow of time emerges for the ensemble of trajectories. In this framework, the thermodynamic variables like the work, the heat and the entropy produced during a process appear as random variables, defined for a single realization (i.e. a single trajectory), whose averages comply with the first and second law of thermodynamics. Furthermore, the fluctuations of these quantities are constrained beyond the second law, as captured by the so-called Fluctuation Theorems (FT) [3][4][5], which can be written under the form e −σ(Γ) = 1, where σ(Γ) is the entropy production along a single trajectory Γ. We denote · , the ensemble average over the realizations of the studied process (or equivalently, over the possible trajectories). The entropy production σ(Γ) fulfilling the FT is equal to the ratio of the probability of the (forward in time) trajectory Γ and the probability of the time-reversed (or backward in time) trajectory corresponding to Γ. During the last decades, these results have been investigated in the quantum regime where the system and the thermal bath can be quantum systems, allowing the proof of quantum extensions of the FTs [6][7][8][9][10][11][12][13][14][15][16][17][18]. Experiments have demonstrated the validity of these FTs in both classical and quantum regimes [19][20][21][22][23][24].However, it was shown that the form of the FTs must be modified for special processes [25][26][27][28][29][30][31][32][33][34], which are such that some theoretically allowed backward trajectories do not have a forward-in-time counterpart. A canonical example is the free expansion of a single particle gas ini...

Operational approach to quantum stochastic thermodynamics

“…The control operations A(r i |r i−1 ) are either measurements or feedback operations (which can be emitative or absorbative). Its effect on the cavity field, can be described by the conditional probabilities (52), (53) and (54). Due to the time-delay, we can set A(r i |r i−1 ) = A(r i |r i−d−1 ).…”

“…Given such a measurement scheme, the system dynamics can be 'unraveled' by describing it in terms of a stochastic Schrödinger or master equation. Combined with this dynamical description, researchers recently applied the ideas of stochastic thermodynamics to such quantum systems [49][50][51][52][53][54][55]; a completely general picture is, however, still missing. For instance, a trajectory dependent system entropy was never introduced making it hard to study entropy production along a single trajectory or on average (specific fluctuation theorems based on a particular choice of the backward dynamics were studied in Refs.…”

Characterizing a Statistical Arrow of Time in Quantum Measurement Dynamics

“…In this letter, we characterize the entropy production of an open quantum system with individual quantum measurement trajectories [54,57,[64][65][66], using information entropy measures to characterize a statistical arrow of time in quantum measurement. We show how a statistical arrow of time is revealed by path probabilities of forward versus time reversed quantum trajectories [67][68][69][70]. As in the case of classical trajectories, these probability densities satisfy a fluctuation theorem that is consistent with the correspondence between microscopic dynamics and ensemble behavior.…”

“…Here, the initial state imposes a lower bound on the possible values of Q [68]. This sensitivity to initial conditions results from the 'un'-likelihood of a particular initial state, quantified by an absolute irreversibility [87][88][89][90]. As presented in Figure 4b, the absolute irreversibility is quantified by the integral fluctuation theorem, which gives a deviation from unity resulting from the ensemble of trajectories containing a surplus of state updates that have a positive statistical arrow of time.…”

Entropy production in continuously measured Gaussian quantum systems

“…The ways such modifications occur have been the focus of some attention recently. Elouard et al and Manikandan et al 29,30 tackled the problem by focusing on the stochastic energy fluctuations that occur during measurements, while refs. [31][32][33][34][35][36][37][38][39] addressed the case of weak quantum measurements of a system, which allowed for the introduction of trajectory-dependent work and heat quantifiers (cf.…”