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Progress of Theoretical Physics

A Boson System Interacting with an External Harmonic Oscillator: A Possible Description of Statistically Mixed State

A. Kuriyama, João da Providência, Y. Tsue, M. Yamamura

Published Dec 1, 1998• 0 citations
Publisher

Abstract

A method for describing a boson system interacting with an external harmonic oscillator, which was proposed recently by the present authors, is extended to the case of a statistically mixed state. The basic idea comes from the su(2, 1)-boson model, the general form of which was recently investigated by the present authors. In terms of the interweaving of the phase space doubling and the su(1, 1)-spin-like behavior in the su(2)-spin system, the extension is performed in the framework of a canonical form of classical mechanics. The results obtained with regard to the present idea are interpreted from various viewpoints, for example, in relation to linear response theory of non-equilibrium statistical mechanics.

DOI
10.1143/ptp.100.1181

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A Boson System Interacting with an External Harmonic Oscillator: The su(1,1)-Spin Like Behavior in the su(2)-Spin System

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The Spin–Boson Model

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In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models: Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property! We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9.

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Concerning the energy of harmonic oscillator, a prescription is proposed for making the original form unchanged even after q-deformation. Applicability of the prescription is limited, but, it can be applied to various cases which are well known. 1 typeset using PTPTEX.sty <ver.1.0>An idea of q-deformation serves us an interesting viewpoint for the understanding of dynamics of many-body systems described by boson operators. For the q-deformation, the present authors recently gave a possible scheme in relation to the generalized boson coherent state. 1)-3) It is, in some sense, an extension of the idea given by Penson and Solomon. 4) Before entering the central part of this paper, first, we recapitulate the basic idea of Ref. 1). We treat a boson system described by one kind of boson (ĉ, ĉ ∗). As the deformation of (ĉ, ĉ ∗), the operator (ˆγ, ˆγ ∗ ) is introduced in the definition ˆγ = F ( ˆ N)ĉ, ˆγ ∗ = ĉ ∗ F ( ˆ N). (1) Here, F ( ˆ N) denotes a function of the boson number operator, ˆ N, defined by ˆN = ĉ ∗ ĉ. (2) The function F ( ˆ N) obeys a certain condition which is proper to each case under investigation. The present deformation is characterized by the function F ( ˆ N). The commutation relations [ĉ, ĉ ∗ ] and [ˆγ, ˆγ ∗ ] are as follows: [ĉ, ĉ ∗ ] = 1, [ˆγ, ˆγ ∗ ] = ( ˆ N + 1)F ( ˆ N) 2 − ˆ NF ( ˆ N − 1) 2. (3a) (3b) Later, we discuss the case in which the relation (3b) does not hold. If following our deformed boson scheme, any operator which is expressed in terms of a function (ĉ, ĉ ∗), Ô = O(ĉ, ĉ ∗), is deformed by replacing (ĉ, ĉ ∗ ) with (ˆγ, ˆγ ∗ ): Ôq = O(ˆγ, ˆγ ∗ ). (4) However, in the case of the Hamiltonian, a special consideration is necessary for the replacement

•Jan 1, 2003•0 citations

Deformed Boson Scheme including Conventional q-Deformation in Time-Dependent Variational Method. III: Deformation of the su(2,1)-Algebra in Terms of Three Kinds of Boson Operators

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The basic idea presented in Parts I and II for the deformed boson scheme is applied to the case of the su(2,1)-algebra for describing many-body systems consisting of three kinds of boson operators. A possible form of the coherent state for the su(2,1)-algebra is generalized, and the deformed su(2,1)-algebra is presented. As an application, the time evolution of the statistically mixed state in a certain boson system, which has already been investigated by the present authors, is reinvestigated.

Progress of Theoretical Physics•Jan 1, 2002•0 citations
A method for describing a boson system interacting with an external harmonic oscillator, which was proposed recently by the present authors, is extended to the case of a statistically mixed state. The basic idea comes from the su(2, 1)-boson model, the general form of which was recently investigated by the present authors. In terms of the interweaving of the phase space doubling and the su(1, 1)-spin-like behavior in the su(2)-spin system, the extension is performed in the framework of a canonical form of classical mechanics. The results obtained with regard to the present idea are interpreted from various viewpoints, for example, in relation to linear response theory of non-equilibrium statistical mechanics.
In order to avoid the introduction of a fictitious degree of freedom in the su(1,1)-algebraic approach for describing thermal effects in a boson system, the su(2)-algebraic approach is proposed in terms of a boson system interacting with an external harmonic oscillator. The relation between the two approaches is investigated from two points of view. One is a quantal point of view based on the MYT boson expansion. The other is a classical point of view based on one of three forms of coherent states for the su(2)-spin system proposed recently by the present authors. Solution of Hamilton's equation of motion is discussed in relation to that in the su(1,1)-algebraic approach.
In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models: Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property! We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9.
Concerning the energy of harmonic oscillator, a prescription is proposed for making the original form unchanged even after q-deformation. Applicability of the prescription is limited, but, it can be applied to various cases which are well known. 1 typeset using PTPTEX.sty <ver.1.0>An idea of q-deformation serves us an interesting viewpoint for the understanding of dynamics of many-body systems described by boson operators. For the q-deformation, the present authors recently gave a possible scheme in relation to the generalized boson coherent state. 1)-3) It is, in some sense, an extension of the idea given by Penson and Solomon. 4) Before entering the central part of this paper, first, we recapitulate the basic idea of Ref. 1). We treat a boson system described by one kind of boson (ĉ, ĉ ∗). As the deformation of (ĉ, ĉ ∗), the operator (ˆγ, ˆγ ∗ ) is introduced in the definition ˆγ = F ( ˆ N)ĉ, ˆγ ∗ = ĉ ∗ F ( ˆ N). (1) Here, F ( ˆ N) denotes a function of the boson number operator, ˆ N, defined by ˆN = ĉ ∗ ĉ. (2) The function F ( ˆ N) obeys a certain condition which is proper to each case under investigation. The present deformation is characterized by the function F ( ˆ N). The commutation relations [ĉ, ĉ ∗ ] and [ˆγ, ˆγ ∗ ] are as follows: [ĉ, ĉ ∗ ] = 1, [ˆγ, ˆγ ∗ ] = ( ˆ N + 1)F ( ˆ N) 2 − ˆ NF ( ˆ N − 1) 2. (3a) (3b) Later, we discuss the case in which the relation (3b) does not hold. If following our deformed boson scheme, any operator which is expressed in terms of a function (ĉ, ĉ ∗), Ô = O(ĉ, ĉ ∗), is deformed by replacing (ĉ, ĉ ∗ ) with (ˆγ, ˆγ ∗ ): Ôq = O(ˆγ, ˆγ ∗ ). (4) However, in the case of the Hamiltonian, a special consideration is necessary for the replacement