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Physical Review A

Brownian motion of a quantum oscillator

Salvatore Carusotto

Published Apr 1, 1975• 15 citations
Publisher

Abstract

The theory of Brownian motion of a quantum oscillator is presented. The equation of motion for the density operator of the oscillator is obtained by a model Hamiltonian which characterizes the interaction between this oscillator and a reservoir. By applying iteration methods the equation of motion is solved and the time development of the density operator is obtained. The solution is then used to write the normally ordered generating functional for the field of the oscillator executing Brownian motion. This field appears to be the superposition of a thermal field in equilibrium with the reservoir, a chaotic field, and a field statistically similar to the one before the interaction. Next, it is shown that for large times the oscillator approaches equilibrium; moreover, it is shown that, if the reservoir is at zero temperature and initially the state of the oscillator is a coherent state, then it remains in a coherent state for later time. In order to obtain by differentiation two-time modal densities, a suitable two-time generating functional is introduced. This functional allows two-time averages to be computed from one-time averages before the interaction. Finally, some time-dependent statistical properties of the oscillator are studied.

DOI
10.1103/physreva.11.1397

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The theory of Brownian motion of a quantum oscillator is presented. The equation of motion for the density operator of the oscillator is obtained by a model Hamiltonian which characterizes the interaction between this oscillator and a reservoir. By applying iteration methods the equation of motion is solved and the time development of the density operator is obtained. The solution is then used to write the normally ordered generating functional for the field of the oscillator executing Brownian motion. This field appears to be the superposition of a thermal field in equilibrium with the reservoir, a chaotic field, and a field statistically similar to the one before the interaction. Next, it is shown that for large times the oscillator approaches equilibrium; moreover, it is shown that, if the reservoir is at zero temperature and initially the state of the oscillator is a coherent state, then it remains in a coherent state for later time. In order to obtain by differentiation two-time modal densities, a suitable two-time generating functional is introduced. This functional allows two-time averages to be computed from one-time averages before the interaction. Finally, some time-dependent statistical properties of the oscillator are studied.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
The quantum theory of a driven damped harmonic oscillator is presented. By using iteration methods the time development of the density operator for the oscillator is obtained and its normally ordered generating functional is written. The field of the oscillator appears to be the superposition of three fields. The first field, which shows the damping of the oscillator, is statistically similar to the one before the interaction; the second one is a coherent field that depends on the driving field; and the third describes a thermal field. In order to evaluate two-time averages by differentiation, a two-time generating functional of the oscillator is obtained. This functional allows two-time averages to be computed from one-time averages before the interaction.
The Brownian motion of a harmonic oscillator is treated quantum-mechanically and the same method is applied to the relaxation of a spin. The method is quite general and applicable to a system displaced from the equilibrium state by an arbitrary amount initially. The oscillator or the spin is interacting with a thermostat composed of a great number of harmonic oscillators. The equation of motion is solved in operator form taking average as to the thermostat to the second order of the interaction parameter. The Langevin equation, the Kramers-Chandra-sekhar equation for a harmonic oscillator and the Bloch equation for a static field are derived.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
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.
We study the quantum fluctuational properties of a parametric oscillator with and without coupling to an Ohmic environment. After considering the momentum and coordinate variances as a function of initial squeezing for the undamped dynamics, we invoke the functional integral method to derive the fully exact reduced density matrix for parametric dissipative quantum Brownian motion, covering the whole temperature regime from T=0 up to the classical limit at room temperatures. Moreover, we present the exact result for the quantum master equation for both the density matrix and the corresponding Wigner function. The time evolution of the covariance matrix elements of damped quantum fluctuations is studied numerically. These variances undergo within the regime of global stability asymptotic, periodic oscillations. As an interesting result, we find that the minima of these oscillations fall below the corresponding thermal equilibrium values.