Linear Response Theory and Entropic Fluctuations in Repeated Interaction Quantum Systems
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Linear Response Theory and Entropic Fluctuations in Repeated Interaction Quantum Systems Jean-François Bougron1,2
· Laurent Bruneau1
Received: 3 April 2020 / Accepted: 18 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study linear response theory and entropic fluctuations of finite dimensional nonequilibrium Repeated Interaction Systems (RIS). More precisely, in a situation where the temperatures of the probes can take a finite number of different values, we prove analogs of the Green–Kubo fluctuation–dissipation formula and Onsager reciprocity relations on energy flux observables. Then we prove a Large Deviation Principle, or Fluctuation Theorem, and a Central Limit Theorem on the full counting statistics of entropy fluxes. We consider two types of non-equilibrium RIS: either the temperatures of the probes are deterministic and arrive in a cyclic way, or the temperatures of the probes are described by a sequence of i.i.d. random variables with uniform distribution over a finite set. Keywords Repeated interactions · Non-equilibrium quantum statistical mechanics · Entropy production
1 Introduction In this paper we are interested in the linear response theory and entropic fluctuation for a particular class of open quantum systems called Repeated Interaction Systems (RIS), see Sect. 3 for a precise description. Our study fits in the wider framework of non-equilibrium quantum statistical mechanics. In this context, linear response theory and entropic fluctuation have attracted lot of attention in the last decades, see e.g. [9,22–24,41–46,55] and references therein.
Communicated by Yoshiko Ogata.
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Laurent Bruneau [email protected] Jean-François Bougron [email protected]
1
Département de Mathématiques and UMR 8088, CNRS and CY Cergy Paris Université, 95000 Cergy-Pontoise, France
2
Univ. Grenoble Alpes, CNRS, Institut Fourier, 38000 Grenoble, France
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J.-F. Bougron, L. Bruneau
Repeated interaction systems consist of a small system S coupled to an environment made of a chain of independent probes En with which S will interact in a sequential way, i.e. S interacts with E1 during the time interval [0, τ1 [, then with E2 during the interval [τ1 , τ1 + τ2 [, etc. While S interacts with a given probe En the other ones evolve freely according to their intrinsic (uncoupled) dynamics. Formally, if HS and HEn denote the noninteracting hamiltonians of S and the En ’s and Vn denotes the coupling operator between S and En then the hamiltonian of the full system is the time-dependent, piecewise constant, operator HE p , t ∈ [τ1 + · · · + τn−1 , τ1 + · · · + τn [. (1.1) H (t) := HS + HEn + Vn + p =n
In the simplest case all the probes are identical (each En is a copy of the same E with the same initial state ρE , e.g. a thermal state) and interact with S by means of the same coupling operator V on S + E and for the same duration τ . The dynamics restricted to the small system is shown to be determined by a map L, see (2.2), which a
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