Asymptotics of repeated interaction quantum systems

Bruneau, Laurent; Joye, Alain; Merkli, Marco

doi:10.1016/j.jfa.2006.02.006

Cited by 41 publications

(121 citation statements)

References 15 publications

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“…In this case however, the (random) channels Φ βn do not have in general a common invariant state, so one has to look at ergodic limits. We use here the machinery developed by L. Bruneau, A. Joye and M. Merkli in [6] (see [7,5] for additional results in this direction). For the sake of completeness, let us state their main result.…”

Section: Repeated Interactions With Random Auxiliary Statesmentioning

confidence: 99%

Random repeated quantum interactions and random invariant states

Nechita

Pellegrini

2010

Probab. Theory Relat. Fields

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Abstract. We consider a generalized model of repeated quantum interactions, where a system H is interacting in a random way with a sequence of independent quantum systems Kn, n 1. Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between H and Kn. The other involves random quantum states describing each copy Kn. In the limit of a large number of interactions, we present convergence results for the asymptotic state of H. This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble. IntroductionInitially introduced in [2] as a discrete approximation of Langevin dynamics, the model of repeated quantum interactions has found since many applications (quantum trajectories, stochastic control, etc.). In this work we generalize this model by allowing random interactions at each time step. Our main focus is the long-time behavior of the reduced dynamics.Our viewpoint is that of Quantum Open Systems, where a "small" system is in interaction with an inaccessible environment (or an auxiliary system). We are interested in the reduced dynamics of the small system, which is described by the action of quantum channels. When repeating such interactions, under some mild conditions on the spectrum of the quantum channel, we show that the successive states of the small system converge to the invariant density matrix of the channel.These considerations motivated us to consider random invariant states, and we introduce a new probability measure on the set of density matrices. There exists extensive literature [8,28,19,3] on what is a "typical" density matrix. There are two general categories of such probability measures on M 1,+ d (C): measures that come from metrics with statistical significance and the so-called "induced measures", where density matrices are obtained as partial traces of larger, random pure states. Our construction from Section 4 falls into the second category, since our model involves an open system in interaction with a chain of "auxiliary" systems.Next, we introduce two models of random quantum channels. In the first model, we allow for the states of the auxiliary system to be random. In the second one, the unitary matrices acting on the coupled system are assumed random, distributed along the Haar invariant probability on the unitary group, and independent between different interactions. Since the (random) state of the system fluctuates, almost sure convergence does not hold, and we state results in the ergodic sense.2000 Mathematics Subject Classification. Primary 15A52; Secondary 94A40, 60F15.

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Section: Repeated Interactions With Random Auxiliary Statesmentioning

confidence: 99%

Random repeated quantum interactions and random invariant states

Nechita

Pellegrini

2010

Probab. Theory Relat. Fields

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“…Such models have been analyzed in [BJM1,WBKM] (see also [BJM2] for a random setting). It was shown in [BJM1] that the RI dynamics gives rise to a Markovian effective dynamics on the system S and drives the latter to an asymptotic state, at an exponential rate (provided S has a finite dimensional Hilbert space). The limit τ → 0 with appropriate rescaling of the interaction Hamiltonian H SE was studied in [APa,AJ2].…”

Section: Repeated Interactionsmentioning

confidence: 99%

Thermal Relaxation of a QED Cavity

Bruneau

Pillet

2008

J Stat Phys

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Abstract. We study repeated interactions of the quantized electromagnetic field in a cavity with single two-levels atoms. Using the Markovian nature of the resulting quantum evolution we study its large time asymptotics. We show that, whenever the atoms are distributed according to the canonical ensemble at temperature T > 0 and some generic non-degeneracy condition is satisfied, the cavity field relaxes towards some invariant state. Under some more stringent non-resonance condition, this invariant state is thermal equilibrium at some renormalized temperature T * . Our result is non-perturbative in the strength of the atom-field coupling. The relaxation process is slow (non-exponential) due to the presence of infinitely many metastable states of the cavity field.

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“…They have shown that these "deterministic" dynamics give rise to quantum stochastic differential equations in the continuous limit. Since that result, some interest has been found in the repeated quantum interaction model in itself (cf [AJ1], [AJ2], [BJM1], [BJM2], [BJM3]) and several physical works are in progress on that subject (for example [AKP]). These repeated interaction models are interesting for several reasons: -they provide a quantum dynamics which is at the same time Hamiltonian and Markovian, -they allow to implement easily the dissipation for a quantum system, in particular they are practical models for simulation.…”

Section: Introductionmentioning

confidence: 99%

Repeated Quantum Interactions and Unitary Random Walks

Attal

Dhahri²

2010

J Theor Probab

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Among the discrete evolution equations describing a quantum system H S undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in R N . The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equations are then random walks on the group U (H 0 ) of unitary operators on H 0 .

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Asymptotics of repeated interaction quantum systems

Cited by 41 publications

References 15 publications

Random repeated quantum interactions and random invariant states

Random repeated quantum interactions and random invariant states

Thermal Relaxation of a QED Cavity

Repeated Quantum Interactions and Unitary Random Walks

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