The Kubo-Anderson model is a stochastic model of phase relaxation of an ensemble of systems in a fluctuating environment. This model is usually studied under the assumption that the system-environment interaction is a Gaussian stochastic process. This assumption only holds if the environment changes a very large number of times on the time scale of the system’s motion. This paper reviews our work on the Kubo-Anderson model for the case where this interaction is a continuous-time random walk. A continuous-time random walk is a simple model for a ‘slowly changing environment’, i.e., one which makes a relatively small number of changes on the time scale of the system’s motion. We present the key results from this model and show how to apply them to common problems in magnetic resonance spectroscopy and and non-linear optical spectroscopy.
The Kubo-Anderson model is a stochastic model of phase relaxation of an ensemble of systems in a fluctuating environment. This model is usually studied under the assumption that the system-environment interaction is a Gaussian stochastic process. This assumption only holds if the environment changes a very large number of times on the time scale of the system’s motion. This paper reviews our work on the Kubo-Anderson model for the case where this interaction is a continuous-time random walk. A continuous-time random walk is a simple model for a ‘slowly changing environment’, i.e., one which makes a relatively small number of changes on the time scale of the system’s motion. We present the key results from this model and show how to apply them to common problems in magnetic resonance spectroscopy and and non-linear optical spectroscopy.
Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions such as (exp(i ∫(0)(t) Q(s)ds)), where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets denote ensemble averaging. This paper gives an exact evaluation of these functions for the case where Q is a continuous-time random walk process. The continuous-time random walk describes an environment that undergoes slow steplike changes in time. It also has a well-defined Gaussian limit and so allows for non-Gaussian and Gaussian stochastic dynamics to be studied within a single framework. We apply the results to extract qubit-lattice interaction parameters from dephasing data of P-doped Si semiconductors (data collected elsewhere) and to calculate the two-dimensional spectrum of a three-level harmonic oscillator undergoing random frequency modulations.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's Q parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t). Using a semi-classical approach and linear response theory, we show that the Q parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime Q can be large even in the long time limit, while in the fast modulation regime it becomes small.
Introduction The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981). Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.
General discussion of stochastic models In the hydrodynamic theories, one writes equations which describe the system at hand on long time and length scales, using local conservation laws and the presence of broken symmetries to determine the identity of these slowly varying quantities. However, one would like a way to take account of the effects of the more rapidly varying degrees of freedom in such theories. Implicitly, these more rapid degrees of freedom are present in the integrals which determine the transport coefficients in the hydrodynamic equations by way of Kubo relations, but no method is presented to calculate the required integrands. One way to take account of the other degrees of freedom is to take on the entire many body problem all the way down to the atomic or electronic level, as one does in molecular dynamics simulations of various sorts. However, it is useful to have some approximate analytical ways to attack this problem as well. Here we review the basis for the most common such approach, the Langevin equation. For most of this discussion, we will assume that the identity of the slow variable or variables is known and that the separation of time scales is extreme so that the faster variables are essentially instantaneous in a sense we will discuss. These assumptions are not often particularly well justified. However, the resulting formulation has yielded very useful insights and is an important part of the subject.
A stochastic approach to the quantum dynamics randomly modulated in time by a discrete state non-Markovian noise, which possesses an arbitrary nonexponential distribution of the residence times, is developed. The formally exact expression for the Laplace-transformed quantum propagator averaged over the stationary realizations of such N -state non-Markovian noise is obtained. The theory possesses a wide range of applications. It includes some previous Markovian and non-Markovian theories as particular cases. In the context of the stochastic theory of spectral line shape and relaxation, the developed approach presents a non-Markovian generalization of the Kubo-Anderson theory of sudden modulation. In particular, the exact analytical expression is derived for the spectral line shape of optical transitions described by a Kubo oscillator with randomly modulated frequency which undergoes jumplike non-Markovian fluctuations in time.