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Laurent Bruneau

Publications and source records attributed to Laurent Bruneau.

17 recordsLinked to original sources
arXiv2025

Entropic Fluctuation Theorems for the Spin-Fermion Model

We study entropic fluctuations in the Spin-Fermion model describing an $N$-level quantum system coupled to several independent thermal free Fermi gas reservoirs. We establish the quantum Evans-Searles and Gallavotti-Cohen fluctuation theorems and identify their link with entropic ancilla state tomography and quantum phase space contraction of non-equilibrium steady state. The method of proof involves the spectral resonance theory of quantum transfer operators developed by the authors in previous works.

math-ph↗
arXiv2024

On the thermodynamic limit of two-times measurement entropy production

We provide a justification, via the thermodynamic limit, of the modular formula for entropy production in two-times measurement proposed in [Benoist, Bruneau, Jakšić, Panati and Pillet: Lett. Math. Phys. 114:32, (2023)]. We consider the cases of open quantum systems in which all thermal reservoirs are either (discrete) quantum spin systems or free Fermi gases.

math-ph↗
arXiv2024

Quantum trajectory of the one atom maser

The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.

math-ph↗
arXiv2024

A note on two-times measurement entropy production and modular theory

Recent theoretical investigations of the two-times measurement entropy production (2TMEP) in quantum statistical mechanics have shed a new light on the mathematics and physics of the quantum-mechanical probabilistic rules. Among notable developments are the extensions of entropic fluctuation relations to quantum domain and discovery of a deep link between 2TMEP and modular theory of operator algebras. All these developments concerned the setting where the state of the system at the instant of the first measurement is the same as the state whose entropy production is measured. In this work we consider the case where these two states are different and link this more general 2TEMP to modular theory. The established connection allows us to show that under general ergodicity assumptions the 2TEMP is essentially independent of the choice of the system state at the instant of the first measurement due to a decoherence effect induced by the first measurement. This stability sheds a new light on the concept of quantum entropy production, and, in particular, on possible quantum formulations of the celebrated classical Gallavotti--Cohen Fluctuation Theorem which will be studied in the continuation of this work.

math-ph↗
arXiv2015

Conductance and absolutely continuous spectrum of 1D samples

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators $h=-Δ+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-Büttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting $h$ to a finite interval $[1,L]\cap\mathbb{Z}_+$ and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval $I$ are non-vanishing in the limit $L\to\infty$ iff ${\rm sp}_{\rm ac}(h)\cap I=\emptyset$. We also discuss the relationship between this result and the Schrödinger Conjecture.

math-ph↗
arXiv2014

Landauer-Büttiker and Thouless conductance

In the independent electron approximation, the average (energy/charge/entropy) current flowing through a finite sample S connected to two electronic reservoirs can be computed by scattering theoretic arguments which lead to the famous Landauer-Büttiker formula. Another well known formula has been proposed by Thouless on the basis of a scaling argument. The Thouless formula relates the conductance of the sample to the width of the spectral bands of the infinite crystal obtained by periodic juxtaposition of S. In this spirit, we define Landauer-Büttiker crystalline currents by extending the Landauer-Büttiker formula to a setup where the sample S is replaced by a periodic structure whose unit cell is S. We argue that these crystalline currents are closely related to the Thouless currents. For example, the crystalline heat current is bounded above by the Thouless heat current, and this bound saturates iff the coupling between the reservoirs and the sample is reflectionless. Our analysis leads to a rigorous derivation of the Thouless formula from the first principles of quantum statistical mechanics.

math.SP↗
arXiv2013

Mixing properties of the one-atom maser

We study the relaxation properties of the quantized electromagnetic field in a cavity under repeated interactions with single two-level atoms, so-called one-atom maser. We improve the ergodic results obtained in [BP] and prove that, whenever the atoms are initially distributed according to the canonical ensemble at temparature T>0, all the invariant states are mixing. Under some non-resonance condition this invariant state is known to be thermal equilibirum at some renormalized temperature T* and we prove that the mixing is then arbitrarily slow, in other words that there is no lower bound on the relaxation speed.

math-ph↗
arXiv2013

Repeated interactions in open quantum systems

Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.

math-ph↗
arXiv2012

Landauer-Büttiker formula and Schrödinger conjecture

We study the entropy flux in the stationary state of a finite one-dimensional sample S connected at its left and right ends to two infinitely extended reservoirs at distinct temperatures T1, T2 and chemical potentials mu1, mu2. The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator -Δ+ v. The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system in terms of scattering data. We study the behavior of this steady state entropy flux in the limit L->infinity and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator h. A natural conjecture is that the set of energies at which transport persists in this limit is precisely the essential support of the absolutely continuous spectrum of h. We show that this conjecture is equivalent to the Schrödinger conjecture in spectral theory of one-dimensional Schrödinger operators, thus giving a physically appealing interpretation to the Schrödinger conjecture.

math-ph↗
arXiv2010

Scattering induced current in a tight-binding band

In the single band tight-binding approximation, we consider the transport properties of an electron in a homogeneous static electric field. We show that repeated interactions of the electron with two-level systems in thermal equilibrium suppress the Bloch oscillations and induce a steady current, the statistical properties of which we study.

math-ph↗
arXiv2009

Homogeneous Schrödinger operators on half-line

The differential expression $L_m=-\partial_x^2 +(m^2-1/4)x^{-2}$ defines a self-adjoint operator H_m on L^2(0;\infty) in a natural way when $m^2 \geq 1$. We study the dependence of H_m on the parameter m, show that it has a unique holomorphic extension to the half-plane Re(m) > -1, and analyze spectral and scattering properties of this family of operators.

math.FA↗
arXiv2009

Repeated and continuous interactions in open quantum systems

We consider a finite quantum system S coupled to two environments of different nature. One is a heat reservoir R (continuous interaction) and the other one is a chain C of independent quantum systems E (repeated interaction). The interactions of S with R and C lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S+R and show that it satisfies a second law of thermodynamics. We analyze a model where both S and E are two-level systems and obtain the asymptotic state explicitly (lowest order in the interaction strength). Even though R and C are not direcly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W*-dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We do not use master equation approximations.

math-ph↗
arXiv2008

Random repeated interaction quantum systems

We consider a quantum system S interacting sequentially with independent systems E_m, m=1,2,... Before interacting, each 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 S and all the E_m are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the E_m vary randomly. We analyze a model in which S is a spin and the 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.

math-ph↗
arXiv2008

Infinite Products of Random Matrices and Repeated Interaction Dynamics

Let $Ψ_n$ be a product of $n$ independent, identically distributed random matrices $M$, with the properties that $Ψ_n$ is bounded in $n$, and that $M$ has a deterministic (constant) invariant vector. Assuming that the probability of $M$ having only the simple eigenvalue 1 on the unit circle does not vanish, we show that $Ψ_n$ is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as $n\to\infty$. The fluctuating part converges in Cesaro mean to a limit that is characterized explicitly by the deterministic invariant vector and the spectral data of ${\mathbb E}[M]$ associated to 1. No additional assumptions are made on the matrices $M$; they may have complex entries and not be invertible. We apply our general results to two classes of dynamical systems: inhomogeneous Markov chains with random transition matrices (stochastic matrices), and random repeated interaction quantum systems. In both cases, we prove ergodic theorems for the dynamics, and we obtain the form of the limit states.

math.PR↗
arXiv2008

On the singularity of random matrices with independent entries

We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/sqrt{n}). The purpose of this note is to provide a short and elementary proof of this fact using a Bernoulli decomposition of arbitrary non degenerate random variables.

math.PR↗
arXiv2005

Asymptotics of repeated interaction quantum systems

A quantum system $\s$ interacts in a successive way with elements $\ee$ of a chain of identical independent quantum subsystems. Each interaction lasts for a duration $τ$ and is governed by a fixed coupling between $\s$ and $\ee$. We show that the system, initially in any state close to a reference state, approaches a {\it repeated interaction asymptotic state} in the limit of large times. This state is $τ$--periodic in time and does not depend on the initial state. If the reference state is chosen so that $\s$ and $\ee$ are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.

math-ph↗