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arXiv:adap-org/9508004

Ergodic Properties of the Spin - Boson System

Vojkan Jakšić, Claude‐Alain Pillet

Published Aug 17, 1995• 8 citations
arXivHTMLPDF

Abstract

We investigate the dynamics of a 2-level atom (or spin-1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared.

DOI
10.48550/arxiv.adap-org/9508004

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Physical review. B, Condensed matter•Nov 15, 1988•16 citations
We investigate the dynamics of a 2-level atom (or spin-1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared.
The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.
We present an application of the extended stochastic Liouville--von Neumann equation (ESLN) method introduced earlier [G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 95, 125124 (2017); G. M. G. McCaul, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 97, 224310 (2018)], which describes the dynamics of an exactly thermalized open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalized at finite temperature using an imaginary-time evolution procedure before the application of real-time evolution. This initializes the combined system in the correct canonical equilibrium state rather than being initially decoupled. Here we apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open-system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher-order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we first consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under nonequilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are nonzero, initially thermalizing the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than starting in a pure state.
The dynamics of a two-level system which is coupled to an environmental sea of infinitely many spin-1/2 particles is investigated by use of the resolvent operator approach. Only at zero temperature does this spin-spin-bath model exhibit identical behavior as the more familiar spin-boson model. It is found that increasing temperature favors coherent dynamics. At high temperatures, the spin-spin-bath model for an Ohmic spectral density sustains a coherent dynamics if the dissipation coefficient $\ensuremath{\alpha}$ is sufficiently small, i.e., $\ensuremath{\alpha}<1/2$; while the decoherence exhibits pure exponential decay if $\ensuremath{\alpha}>1/2$.
Abstract We study a concrete model of a confined particle in form of a Schrödinger operator with a compactly supported smooth potential coupled to a bosonic field at positive temperature. We show, that the model exhibits thermal ionization for any positive temperature, provided the coupling is sufficiently small. Mathematically, one has to rule out that zero is an eigenvalue of the self-adjoint generator of time evolution—the Liouvillian. This will be done by using positive commutator methods with dilations in the space of scattering functions. Our proof relies on a spatial cutoff in the coupling but does otherwise not require any unnatural restrictions.
A two-level system with a transverse linear infrared singular coupling to a boson gas is treated within a mode-coupling approximation for the dynamics. Static spin susceptibilities and spin polarizations are evaluated consistently with the dynamics by exploiting sum rules. The spin energy is obtained by integration over the excitation spectrum of the boson scattering matrix. In case of a high-frequency boson cutoff large compared to the level splitting, the results demonstrate pronounced precursor phenomena for the known zero-temperature phase transition of the model. In the opposite case the results are in semiquantitative agreement with previous numerical findings for the transition to self-trapping obtained by DeRaedt and DeRaedt.