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MARK OVIAN PROPERTIES OF THE PAULI-FIERZ MODEL

Ameur Dhahri

Published Jan 1, 2009• 2 citations
Publisher

Abstract

Abstract. The Hamiltonian approach of the Pauli-Fierz model is studied by Derezinski and Jaksic (cf. [7]). In this paper, we give the Markovian description of this model. Using the weak coupling limit, we derive the quan-tum Markovian semigroup of the Pauli-Fierz system from its Hamiltonian description. Also, we give the explicit form of the associated Lindblad gener-ator. As a consequence, we study the properties of the associated quantum master equation: Quantum detailed balance condition and return to equilib-rium. Finally, we give the associated quantum Langevin equation which can be obtained by a repeated quantum interaction Hamiltonian. 1.

DOI
10.31390/cosa.3.2.08

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Abstract. The Hamiltonian approach of the Pauli-Fierz model is studied by Derezinski and Jaksic (cf. [7]). In this paper, we give the Markovian description of this model. Using the weak coupling limit, we derive the quan-tum Markovian semigroup of the Pauli-Fierz system from its Hamiltonian description. Also, we give the explicit form of the associated Lindblad gener-ator. As a consequence, we study the properties of the associated quantum master equation: Quantum detailed balance condition and return to equilib-rium. Finally, we give the associated quantum Langevin equation which can be obtained by a repeated quantum interaction Hamiltonian. 1.
A comprehensive approach to modeling open quantum systems consistent with thermodynamics is presented. The theory of open quantum systems is employed to define system bath partitions. The Markovian master equation defines an isothermal partition between the system and bath. Two methods to derive the quantum master equation are described: the weak coupling limit and the repeated collision model. The role of the eigenoperators of the free system dynamics is highlighted, in particular, for driven systems. The thermodynamical relations are pointed out. Models that lead to loss of coherence, i.e., dephasing are described. The implication of the laws of thermodynamics to simulating transport and spectroscopy is described. The indications for self-averaging in large quantum systems and thus its importance in modeling are described. Basic modeling by the surrogate Hamiltonian is described, as well as thermal boundary conditions using the repeated collision model and their use in the stochastic surrogate Hamiltonian. The problem of modeling with explicitly time dependent driving is analyzed. Finally, the use of the stochastic surrogate Hamiltonian for modeling ultrafast spectroscopy and quantum control is reviewed.
While models of one qubit and two qubits open quantum systems are widely used in contemporary physical literature, especially one dealing with the problems of quantum information and computation, model of three qubits is rarely exploited. One reason, regarding dynamics of open quantum systems, lays in the fact that adding one qubit to state space complicates considerably computational tasks. By computational tasks here is meant obtaining analytical details of dynamics, which are preferable for fundamental physical considerations. In this paper, ab initio derivation of Markovian master equation, in regime of the weak coupling limit, for three interacting qubits is presented in pedagogical manner. In this way, it will become clearer what challenges in dealing with seemingly innocuous three-qubits model are. Master equation in hand can serve for further physical inquiries, not limited to the field of quantum information and quantum computation.
Abstract The quantum fluctuation-dissipation theorem is a central ingredient in the construction of quantum dynamics of Brownian motion which necessarily is non -Markovian. Yet, often Markovian approximations to quantum dynamics are studied in the literature. In this work, we investigate the limitations of the Markovian approximation within two paradigmatic models describing a single damped harmonic oscillator. These models are governed by distinct quantum Langevin equations, although both are constructed to satisfy the same set of phenomenological criteria: the canonical commutation relations between position and momentum, the Kubo response relation, the virial theorem, and the equilibrium quantum variance. The limitations of the Markovian approximation are underscored by the classical limit, violations of the Ehrenfest theorem, the breakdown of complete and simple positivity in the reconstructed master equations, and anomalies in thermalisation behaviour. Further phenomenological differences between the two models are illustrated through their quantum relaxation dynamics and phase diagrams, derived from their reinterpretation as mean-field approximations of a many-body interacting magnet. Our analysis explicitly reveals intrinsic inconsistencies introduced by the Markovian approximation, emphasising the need for non -Markovian frameworks for a consistent description of open quantum dynamics.
We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn–Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and the jump operators. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.
The subject of this thesis is the modelling of non-Markovian open quantum systems, in particular those that display memory effects due to time-delayed interactions. I study these systems in the context of a model of an open quantum system that interacts with a bosonic environment by way of particle exchanges. While the model is flexible enough to describe quite general memory effects, this work focuses to a large extent on a single prototypical system. This system emits into a coherent feedback loop with a time delay, and it captures the essential phenomena we aim to model. Within the context of this general model, I develop a general approach to spontaneous emission from singly-excited systems and show how to calculate the corresponding emission spectra. I also evaluate two commonly-used perturbative techniques in the context of spontaneous emission from singly-excited systems, coming to the conclusion that perturbation theory is of limited use in modelling open quantum systems with delayed feedback. I then turn attention to exact equations of motion for more general non-Markovian open quantum systems. I derive Heisenberg–Langevin equations that describe a large class of such systems, and solve these equations analytically for a series of simple bosonic systems. I also use the theory of cascaded open quantum systems to simulate systems with delayed coherent feedback in the limit of long time delay, and compare the results of these simulations with the analytic solutions obtained in the Heisenberg picture. Finally, I show that once we have a solution to a given Heisenberg–Langevin equation we can derive an alternative, time-local equation of motion that describes equivalent dynamics, and that these time-local Heisenberg–Langevin equations can in turn be used to derive corresponding time-local master equations. These master equations provide an exact description of the non-Markovian dynamics, but this treatment is limited to linear, bosonic systems—those systems for which the Heisenberg–Langevin equations can be solved in closed form. I discuss the prospect of generalising these master equations to take into account nonlinear and fermionic systems.