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arXiv:1106.3788 · Journal of Statistical Mechanics Theory and Experiment

Microscopic analysis of the microscopic reversibility in quantum systems

Tatsuro Kawamoto

Published Nov 25, 2011• 4 citations
arXivHTMLPDF

Abstract

We investigate the robustness of the microscopic reversibility in open quantum systems which is discussed by Monnai in a recent article. We derive an exact relation between the forward transition probability and the reversed transition probability in the case of a general measurement basis. We show that the microscopic reversibility acquires some corrections in general and discuss the physical meaning of the corrections. Under certain processes, some of the correction terms vanish and we numerically confirmed that the remaining correction term becomes negligible; for such processes, the microscopic reversibility almost holds even when the local system cannot be regarded as macroscopic.

DOI
10.1088/1742-5468/2011/11/p11019

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We investigate the robustness of the microscopic reversibility in open quantum systems which is discussed by Monnai in a recent article. We derive an exact relation between the forward transition probability and the reversed transition probability in the case of a general measurement basis. We show that the microscopic reversibility acquires some corrections in general and discuss the physical meaning of the corrections. Under certain processes, some of the correction terms vanish and we numerically confirmed that the remaining correction term becomes negligible; for such processes, the microscopic reversibility almost holds even when the local system cannot be regarded as macroscopic.
The transition probability of an isolated system for a time-dependent unitary evolution is invariant under the reversal of protocols. In this paper, we generalize the expression of microscopic reversibility to externally perturbed large quantum open systems, which provides a model-independent equality between time forward and reversed joint transition probabilities. A time-dependent external perturbation acts on the subsystem during a transient duration, and subsequently the perturbation is switched off so that the total system would thermalize. We concern ourselves with the net transition probability for the subsystem from the initial to final states after a time evolution during which the energy is irreversibly exchanged between the subsystem and reservoir. The time-reversed probability is given by the reversal of the forcing protocol and the initial ensemble. Microscopic reversibility equates the time forward and reversed probabilities, and therefore appears as a thermodynamic symmetry for open quantum systems.
The principle of microscopic reversibility is a fundamental element in the formulation of fluctuation relations and Onsager reciprocal relations. As such, a clear description of whether and how this principle is adapted to the quantum-mechanical scenario might be essential for a better understanding of nonequilibrium quantum processes. Here, we propose a quantum generalization of this principle which highlights the role played by coherence in the symmetry relations involving the probability of observing a quantum transition and that of the corresponding time-reversed process. We study the implications of our findings in the framework of a qubit system interacting with a thermal reservoir and implement an optical experiment that simulates the dynamics. Our theoretical and experimental results show that the influence of coherence is more decisive at low temperatures and that the maximum departure from the classical case does not take place for maximally coherent states. Classical predictions are recovered in the appropriate limits.
The relation between macroscopic irreversibility and microscopic reversibility is a present unsolved problem. Constructal law is introduced to develop analytically the Einstein's, Schrödinger's, and Gibbs' considerations on the interaction between particles and thermal radiation (photons). The result leads to consider the atoms and molecules as open systems in continuous interaction with flows of photons from their surroundings. The consequent result is that, in any atomic transition, the energy related to the microscopic irreversibility is negligible, while when a great number of atoms (of the order of Avogadro's number) is considered, this energy related to irreversibility becomes so large that its order of magnitude must be taken into account. Consequently, macroscopic irreversibility results related to microscopic irreversibility by flows of photons and amount of atoms involved in the processes.
Quantitative studies of irreversibility in statistical mechanics often involve the consideration of a reverse process, whose definition has been the object of many discussions, in particular for quantum mechanical systems. Here we show that the reverse channel very naturally arises as Bayesian retrodiction, both in classical and quantum theories. Fluctuation relations then become expressions of the statistical divergence between the forward (i.e., predictive) and the reverse (i.e., retrodictive) process. With the reverse process constructed on consistent logical inference, rather than on particular balancing mechanisms in microscopic physics, fluctuation relations acquire a much broader form and scope. Previous paradigmatic results like Jarzynski's and Crooks' fluctuation relations are recovered as special cases.
Quantitative studies of irreversibility in statistical mechanics often involve the consideration of a reverse process, whose definition has been the object of many discussions, in particular for quantum mechanical systems. Here we show that the reverse channel very naturally arises from Bayesian retrodiction, in both classical and quantum theories. Previous paradigmatic results, such as Jarzynski's equality, Crooks' fluctuation theorem, and Tasaki's two-measurement fluctuation theorem for closed driven quantum systems, are all shown to be consistent with retrodictive arguments. Also, various corrections that were introduced to deal with nonequilibrium steady states or open quantum systems are justified on general grounds as remnants of Bayesian retrodiction. More generally, with the reverse process constructed on consistent logical inference, fluctuation relations acquire a much broader form and scope.
Quantum mechanics must be regarded as open systems. On one hand, this is due to the fact that, like in classical physics, any realistic system is subjected to a coupling to an uncontrollable environment which influences it in a non-negligible way. The theory of open quantum systems thus plays a major role in many applications of quantum physics since perfect isolation of quantum system is not possible and since a complete microscopic description or control of the environment degrees of freedom is not feasible or only partially so [1]. Practical considerations therefore force one to seek for a simpler, effectively probabilistic description in terms of an open system. There is a close physical and mathematical connection between the evolution of an open system, the state changes induced by quantum measurements, and the classical notion of a stochastic process. The paper provides a bibliographic review of this interrelations, it shows the mathematical equivalence between markovian master equation and generalized piecewise deterministic processes [1] and it introduces the open system in an open observed environment model.