Partial factorization of wave functions for a quantum dissipative system

Abstract

The microscopic approach quantum dissipation process presented by Yu and Sun [Phys. Rev., A49(1994) 592, A51(1995)1845] is developed to analyze the wave function structure of dynamic evolution of a typical dissipative system, a single mode boson soaked in a bath of many bosons. In this paper, the wave function of total system is explicitly obtained as a product of two components of the system and the bath in the coherent state representation. It not only describes the influence of the bath on the variable of the system through the Brownian motion, but also manifests the back-action of the system on the bath and the effects of the mutual interaction among the bosons of the bath. Due to the back-action, the total wave function can only be partially factorizable even for the Brownian motion can be ignored in certain senses, such as the cases with weak coupling and large detuning.PACS numbers: 03.65, 32.80, 42.50 1 1. Introduction: The typical microscopic treatment for the model of quantum dissipation is to consider a harmonic oscillator interacting with a many-oscillator bath (e.g., in [1][2][3][4][5][6]). This simple and rather conventional model has been used to completely clarify the relation between two different approaches for quantum dissipation process frequently appearing in the literature, i.e., the system plus bath model and the time-dependent effective Hamiltonian by Kanai and Calderora [7,8], since Yu and one (Sun) of the authors wrote down the total wave function explicitly in a form of direct product of the bath component and the system component [1,2]. In the discussion, because the mixed variables were chosen to describe the system and the bath, the wave function only manifested the influence of the bath on the system through the Brownian broadening of the width of the wave function for the system, but the back-action of the system on the bath was not discussed. If there indeed exists the back-action of the system on the bath, it is reasonable to expect that, for the individual particles constituting the bath, the mutual couplings among them can indirectly appear in a second order through coupling the system as an intermediate process.Another question is the relation between quantum and classical systems. In many real situations, the classical or macroscopic states can be represented by coherent states in the quantum optics and the macroscopic quantum mechanics. Therefore it is significant to study how the system with an initial coherent state evolves if it really has a macroscopic or classical meanings and to test if it can move in the classical orbits.In this paper, the back-action and mutual couplings with a simple model is studied and manifested in the coherent representation. In the presence of both the back-action and the indirect mutual coupling, we also consider the meaning of the wave function of the dissipative system governed by the effective Hamiltonian, which is also to be determined in this paper. A very interesting result is that a factorizable evolution represented by a prod...

Normal Ordering Solution to Quantum Dissipation and Its Induced Decoherence

“…(13), due to the bath fluctuation and the back-action of system on the bath, the state vector evolved from factorized initial state becomes fully entangled. If the Brownian effect caused by the terms j β j v j (t) can be ignored, the total state vector can be partially factorized [11]. We can further consider T = 0K for the bath, that is, all the oscillator modes of the reservoir are in vacuum state initially.…”

“…The model described in eq. (1) can be exactly solved as long as the coupling constant g j and the spectrum density of the bath are specified explicitly [11]. Here we adopt the spirit of the normal ordering method to study the dynamic evolution of a single mode oscillator with dissipation.…”

“…The rate of the decoherence is proportional to the distance between the two states, which has been demonstrated in cavity QED experiment [8]. The wave function structure of a single mode boson system plus a bath of many bosons is studied for both factorization case [9,10] and partial factorization case [11]. It shows that when the Brownian motion effect is ignored in certain conditions the total wave function of the total system can be written in a form of a product of the bath and system components.…”

“…It shows that when the Brownian motion effect is ignored in certain conditions the total wave function of the total system can be written in a form of a product of the bath and system components. Based on the result [11] , we have studied the decoherence process of excitons in an idealized quantum well placed in a lossy cavity [14]. The result shows that the coherence of excitonic superposition is reduced in an oscillating form due to the dissipation of the optical cavity.…”

“…As far as our knowledge is concerned, the dynamic evolution of a damped oscillator system is only investigated by implementing the Heisenberg-Schrödinger picture transformation (HSPT) [11][12][13]. Can we solve the Schrödinger equation directly and get the state vector of a damped harmonic oscillator system without HSPT?…”