Quantum Brownian motion is a frontier area of research in quantum optics (Louisell 1973; Sargent, Scully, and Lamb 1974; Meystre and Sargent 1991), condensed matter (Grabert, Schramm, and Ingold 1988; Weiss 1999), and chemical physics (Hänggi, Talkner, and Borkovec 1990; Zwanzig 1973) over the last several decades. The fundamental paradigm is the irregular movement of a particle in a thermal bath kept at finite temperature. The bath constitutes many, in principle, infinite degrees of freedom interacting with the particle. In general, the bath is considered to be a set of harmonic oscillators of varied frequency ranges and is therefore bosonic in character. The particle follows the reduced stochastic dynamics obtained after appropriate elimination of these reservoir degrees of freedom under suitable approximation schemes. The dynamical evolution is characterized by quantum dissipation and bosonic noise related to each other by fluctuation-dissipation (FD) relation, ensuring a detailed balance in the system. This chapter concerns a system-reservoir model where the reservoir is constituted by a set of two-level atoms or spin-1/2 systems.
Quantum Brownian motion is a frontier area of research in quantum optics (Louisell 1973; Sargent, Scully, and Lamb 1974; Meystre and Sargent 1991), condensed matter (Grabert, Schramm, and Ingold 1988; Weiss 1999), and chemical physics (Hänggi, Talkner, and Borkovec 1990; Zwanzig 1973) over the last several decades. The fundamental paradigm is the irregular movement of a particle in a thermal bath kept at finite temperature. The bath constitutes many, in principle, infinite degrees of freedom interacting with the particle. In general, the bath is considered to be a set of harmonic oscillators of varied frequency ranges and is therefore bosonic in character. The particle follows the reduced stochastic dynamics obtained after appropriate elimination of these reservoir degrees of freedom under suitable approximation schemes. The dynamical evolution is characterized by quantum dissipation and bosonic noise related to each other by fluctuation-dissipation (FD) relation, ensuring a detailed balance in the system. This chapter concerns a system-reservoir model where the reservoir is constituted by a set of two-level atoms or spin-1/2 systems.
We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins (the Brownian motion limit), the spin–spin model can be mapped on the spin–boson model (although with a temperature dependent spectral density). We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime.
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.
The motion of a free quantum particle in a thermal environment is usually described by the quantum Langevin equation, where the effect of the bath is encoded through a dissipative and a noise term, related to each other via the fluctuation dissipation theorem. The quantum Langevin equation can be derived starting from a microscopic model of the thermal bath as an infinite collection of harmonic oscillators prepared in an initial equilibrium state. The spectral properties of the bath oscillators and their coupling to the particle determine the specific form of the dissipation and noise. Here we investigate in detail the well-known Rubin bath model, which consists of a one-dimensional harmonic chain with the boundary bath particle coupled to the Brownian particle. We show how in the limit of infinite bath bandwidth, we get the Drude model, and a second limit of infinite system-bath coupling gives the Ohmic model. A detailed analysis of relevant equilibrium correlation functions, such as the mean squared displacement, velocity autocorrelation functions, and response function are presented, with the aim of understanding the various temporal regimes. In particular, we discuss the quantum-to-classical crossover time scales where the mean square displacement changes from a ∼lnt to a ∼t dependence. We relate our study to recent work using linear response theory to understand quantum Brownian motion.
A binary collision model for phenomena of quantum dissipation is developed. Unlike the harmonic oscillator model, widely used for over thirty years, this model assumes nonlinear coupling between system and environment, and is applicable to both bosonic and fermionic baths. The system interacts with an ideal bath through binary collisions only. Solutions for the classical and quantum-mechanical problems in the case of free Brownian motion are presented, and the quantum-classical correspondence for nonequilibrium processes is established. It is shown that in the Brownian motion limit the two models lead to identical dynamical behavior, provided the coupling coefficients in the harmonic oscillator Hamiltonian are temperature dependent. For cases of bath particles of finite mass and number the two models lead to different results. Linear response theory for the model is developed, and the results are compared with those for the harmonic oscillator model. At the end, possible applications of the model are suggested.
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.
This problem in lieu of thesis is a discussion of two topics: Brownian movement and quantum computers. Brownian movement is a physical phenomenon in which the particle velocity is constantly undergoing random fluctuations. Chapters 2, 3 and 4, describe Brownian motion from three different perspectives. The next four chapters are devoted to the subject of quantum computers, which are the signal of a new era of technology and science combined together. In the first chapter I present to a reader the two topics of my problem in lieu of thesis. In the second chapter I explain the idea of Brownian motion, its interpretation as a stochastic process and I find its distribution function. The next chapter illustrates the probabilistic picture of Brownian motion, where the statistical averages over trajectories are related to the probability distribution function. Chapter 4 shows how to derive the Langevin equation, introduced in chapter 1, using a Hamiltonian picture of a bath with infinite number of harmonic oscillators. The chapter 5 explains how the idea of quantum computers was developed and how step-by-step all the puzzles for the field of quantum computers were created. The next chapter, chapter 6, discus the basic quantum unit of information namely, the so called qubit and its properties. Chapter 7 is devoted to quantum logic gates, which are important for conducting logic operation in quantum computers. This chapter explains how they were developed and how they are different from classical ones. Chapter 8 is about the quantum algorithm, Shor's algorithm. Quantum algorithm in quantum computers enables one to solve problems that are hard to solve on digital computers. The last chapter contains conclusions on Brownian movement and the field of quantum computers.