Abstract
We consider a quantum system \({\mathcal{S}}\) interacting sequentially with independent systems \({\mathcal{E}_m}\) , m = 1,2,... Before interacting, each \({\mathcal{E}_m}\) is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics.
We solve exactly a model in which \({\mathcal{S}}\) and all the \({\mathcal{E}_m}\) are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the \({\mathcal{E}_m}\) vary randomly. We analyze a model in which \({\mathcal{S}}\) is a spin and the \({\mathcal{E}_m}\) are thermal fermion baths and obtain the asymptotic state by rigorous perturbation theory, for random interaction times varying slightly around a fixed mean, and for small values of a coupling constant.
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Communicated by I.M. Sigal
Partly supported by the Ministère français des affaires étrangères through a séjour scientifique haut niveau.
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Bruneau, L., Joye, A. & Merkli, M. Random Repeated Interaction Quantum Systems. Commun. Math. Phys. 284, 553–581 (2008). https://doi.org/10.1007/s00220-008-0580-8
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DOI: https://doi.org/10.1007/s00220-008-0580-8