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arXiv:2303.10196

Three components of stochastic entropy production associated with the quantum Zeno and anti-Zeno effects

Ashutosh Kinikar, Ian J. Ford

Published Mar 17, 2023• 0 citations
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Abstract

We investigate stochastic entropy production in a two-level quantum system that performs Rabi oscillations while undergoing quantum measurement brought about by continuous random disturbance by an external measuring device or environment. The dynamics produce quantum Zeno and anti-Zeno effects for certain measurement regimes, and the stochastic entropy production is a measure of the irreversibility of the behaviour. When the strength of the measurement disturbance is time-dependent, the stochastic entropy production separates into three components. Two represent relaxational behaviour, one being specific to systems represented by coordinates that are odd under time reversal symmetry, and a third characterises the nonequilibrium stationary state arising from breakage of detailed balance in the dynamics. The study illustrates how the ideas of stochastic thermodynamics may be applied in similar ways to both quantum and classical systems.

DOI
10.48550/arxiv.2303.10196

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We investigate stochastic entropy production in a two-level quantum system that performs Rabi oscillations while undergoing quantum measurement brought about by continuous random disturbance by an external measuring device or environment. The dynamics produce quantum Zeno and anti-Zeno effects for certain measurement regimes, and the stochastic entropy production is a measure of the irreversibility of the behaviour. When the strength of the measurement disturbance is time-dependent, the stochastic entropy production separates into three components. Two represent relaxational behaviour, one being specific to systems represented by coordinates that are odd under time reversal symmetry, and a third characterises the nonequilibrium stationary state arising from breakage of detailed balance in the dynamics. The study illustrates how the ideas of stochastic thermodynamics may be applied in similar ways to both quantum and classical systems.
Abstract We investigate the total stochastic entropy production of a two-level bosonic open quantum system under protocols of time dependent coupling to a harmonic environment. These processes are intended to represent the measurement of a system observable, and consequent selection of an eigenstate, whilst the system is also subjected to thermalising environmental noise. The entropy production depends on the evolution of the system variables and their probability density function, and is expressed through system and environmental contributions. The continuous stochastic dynamics of the open system is based on the Markovian approximation to the exact, noise-averaged stochastic Liouville-von Neumann equation, unravelled through the addition of stochastic environmental disturbance mimicking a measuring device. Under the thermalising influence of time independent coupling to the environment, the mean rate of entropy production vanishes asymptotically, indicating equilibrium. In contrast, a positive mean production of entropy as the system responds to time dependent coupling characterises the irreversibility of quantum measurement, and a comparison of its production for two coupling protocols, representing connection to and disconnection from the external measuring device, satisfies a detailed fluctuation theorem.
The reduced density matrix that characterises the state of an open quantum system is a projection from the full density matrix of the quantum system and its environment, and there are many full density matrices consistent with a given reduced version. Without a specification of relevant details of the environment, the time evolution of a reduced density matrix is therefore typically unpredictable, even if the dynamics of the full density matrix are deterministic. With this in mind, we investigate a two-level open quantum system using a framework of quantum state diffusion. We consider the pseudorandom evolution of its reduced density matrix when subjected to an environment-driven process that performs a continuous quantum measurement of a system observable, invoking dynamics that asymptotically send the system to one of the relevant eigenstates. The unpredictability is characterised by a stochastic entropy production, the average of which corresponds to an increase in the subjective uncertainty of the quantum state adopted by the system and environment, given the underspecified dynamics. This differs from a change in von Neumann entropy, and can continue indefinitely as the system is guided towards an eigenstate. As one would expect, the simultaneous measurement of two non-commuting observables within the same framework does not send the system to an eigenstate. Instead, the probability density function describing the reduced density matrix of the system becomes stationary over a continuum of pure states, a situation characterised by zero further stochastic entropy production. Transitions between such stationary states, brought about by changes in the relative strengths of the two measurement processes, give rise to finite positive mean stochastic entropy production. The framework investigated can offer useful perspectives on both the dynamics and irreversible thermodynamics of measurement in quantum systems.
Abstract We investigate the quantum Zeno effect (QZE) in spin 1/2, spin 1 and spin 3/2 open quantum systems undergoing Rabi oscillations, revealing unexplored features for the spin 1 and spin 3/2 systems. The systems interact with an environment designed to perform continuous measurements of an observable, driving the systems stochastically towards one of the eigenstates of the corresponding operator. The system-environment coupling constant represents the strength of the measurement. Stochastic quantum trajectories are generated by unravelling a Markovian Lindblad master equation using the quantum state diffusion formalism. These are regarded as a more appropriate representation of system behaviour than consideration of the averaged evolution since the latter can mask the effect of measurement. Complete positivity is maintained and thus the trajectories can be considered as physically meaningful. The QZE is investigated over a range of measurement strengths. Increasing the strength leads to greater system dwell in the vicinity of the eigenstates of the measured observable and lengthens the time taken by the system to return to that eigenstate, thus the QZE emerges. For very strong measurement, the Rabi oscillations resemble randomly occurring near-instantaneous jumps between eigenstates. The trajectories followed by the quantum system are heavily dependent on the measurement strength which other than slowing down and adding noise to the Rabi oscillations, changes the paths taken in spin phase space from a circular precession into elaborate figures-of-eight. For spin 1 and spin 3/2 systems, the measurement strength determines which eigenstates are explored and the QZE is stronger when the system dwells in the vicinity of certain eigenstates compared to others.
Irreversible behavior in open stochastic dynamical systems is quantified by stochastic entropy production, a property that measures the difference in likelihoods of forward and subsequent backward system evolution. But for a closed system, governed by deterministic dynamics, such an approach is not appropriate. Instead, we can consider the difference in likelihoods of forward and "obverse" behavior: the latter being a backward trajectory initiated at the same time as the forward trajectory. Such a comparison allows us to define "dissipation production," an analog of stochastic entropy production. It quantifies the breakage of a property of the evolution termed "obversibility" just as stochastic entropy production quantifies a breakage of reversibility. Both are manifestations of irreversibility. In this study we discuss dissipation production in a quantum system. We consider a simple, deterministic, two-level quantum system characterized by a statistical ensemble of state vectors, and we provide numerical results to illustrate the ideas. We consider cases that both do and do not satisfy an Evans-Searles Fluctuation Theorem for the dissipation production, and hence identify conditions under which the system displays time-asymmetric average behavior: an arrow of time.
Modelling the evolution of a system using stochastic dynamics typically implies a greater subjective uncertainty in the adopted system coordinates as time progresses, and stochastic entropy production has been developed as a measure of this change. In some situations the evolution of stochastic entropy production can be described using an Itô process, but mathematical difficulties can emerge if diffusion in the system phase space is restricted to a subspace of lower dimension. This can arise if there are constants of the motion, for example, or more generally when there are functions of the coordinates that evolve without noise. We discuss such a case for an open three-level quantum system modelled within a framework of Markovian quantum state diffusion and show how the problem of computing the stochastic entropy production in such a situation can be overcome. We go on to illustrate how a nonequilibrium stationary state of the three-level system, with a constant mean production rate of stochastic entropy, can be established under suitable environmental couplings.