The Langevin Equation for a Quantum Heat Bath

Université de Lyon, Université Lyon 1, U.M.R. 5208, 21, av. Claude Bernard, 69622 Villeurbanne Cedex, France
Journal of Functional Analysis (Impact Factor: 1.32). 01/2007; 247(2):253-288. DOI: 10.1016/j.jfa.2006.09.019
Source: arXiv


We compute the quantum Langevin equation (or more exactly, the quantum stochastic differential equation) representing the action of a quantum heat bath at thermal equilibrium on a simple quantum system. These equations are obtained by taking the continuous limit of the Hamiltonian description for repeated quantum interactions with a sequence of quantum systems at a given density matrix state. In particular we specialise these equations to the case of thermal equilibrium states. In the process, new quantum noises are appearing: thermal quantum noises. We discuss the mathematical properties of these thermal quantum noises. We compute the Lindblad generator associated with the action of the heat bath on the small system. We exhibit the typical Lindblad generator that provides thermalization of a given quantum system.

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Available from: Stéphane Attal, Sep 10, 2014
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    • "Motivated by their work we generalise their result by giving the necessary and sufficient conditions when QS cocycles (obtained as a continuoustime limit from repeated interactions) induces quasifree representations of the CCR algebra. In contrast to [5], we obtain the representations which induce a gauge-invariant quasifree state, as well as, a squeezed quasifree state. "
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    ABSTRACT: In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree representations and the corresponding quasifree stochastic calculus are also described. In particular, we study in detail gauge-invariant and squeezed quasifree states. We describe repeated quantum interactions between a `small' quantum system and an environment consisting of an infinite chain of particles. We study different cases of interaction, in particular those which occur in weak coupling limits and low density limits. Under different choices of scaling of the interaction part we show that random walks, which are generated by the associated unitary evolutions of a repeated interaction system, strongly converge to unitary quantum stochastic cocycles. We provide necessary and sufficient conditions for such convergence. Furthermore, under repeated quantum interactions, we consider the situation of an infinite chain of identical particles where each particle is in an arbitrary faithful normal state. This includes the case of thermal Gibbs states. We show that the corresponding random walks converge strongly to unitary cocycles for which the driving noises depend on the state of the incoming particles. We also use conditional expectations to obtain a simple condition, at the level of generators, which suffices for the convergence of the associated random walks. Limit cocycles, for which noises depend on the state of the incoming particles, are also obtained by investigating what we refer to as `compressed' random walks. Lastly, we show that the cocycles obtained via the procedure of repeated quantum interactions are quasifree, thus the driving noises form a representation of the relevant CCR algebra. Both gauge-invariant and squeezed representations are shown to occur.
    05/2014, Degree: PhD, Supervisor: Alexander C. R. Belton, J. Martin Lindsay
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    • "In this work, we shall use a different approach, introduced recently by the second author in [25] [26] [27]. This discrete-time model of indirect measurement, called Quantum Repeated Measurements is based on the model of Quantum Repeated Interactions [3] [4] [5] introduced by S. Attal and Y. Pautrat. The setup is the following: a small system H is in contact with an infinite chain, ∞ k=1 E k , of identical and independent quantum systems, that is E k = E for all k. "
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    ABSTRACT: In this article, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in \cite{A1P1}, where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
    Confluentes Mathematici 08/2009; 01(02). DOI:10.1142/S1793744209000109
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    • "This theory allows to consider the state µ l k ⊗ β as a state of the form |X 0 X 0 | in a biggest Hilbert space. The G.N.S representation modifies then the expression of operator U k , and the control expressed in µ l k ⊗β is again expressed in the new expression of U k (see [2] for more details). In our case, we do not use such theory because it is more explicit to make directly computations to reach the discrete equation in asymptotic form. "
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    ABSTRACT: "Quantum trajectories" are solutions of stochastic differential equations of non-usual type. Such equations are called "Belavkin" or "Stochastic Schr\"odinger Equations" and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called "Quantum Repeated Measurements". In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics.
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