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Progress of Theoretical Physics

Rigorous Quantum Mechanics and Dynamics of an Inversion Symmetric Spin-Boson System

T. Tsuzuki

Published Aug 1, 1994• 0 citations
Publisher

Abstract

Dynamic compensation theory of an inversion symmetric spin-boson system is developed rigorously, describing the existence of two branches of localization and delocalization, the transition of ground state between them, excited states, thermodynamics and dynamics. This is achieved by introducing two kinds of displaced bosons. The oscillation of quantum coherence is described.

DOI
10.1143/ptp.92.465

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Dynamic compensation theory of an inversion symmetric spin-boson system is developed rigorously, describing the existence of two branches of localization and delocalization, the transition of ground state between them, excited states, thermodynamics and dynamics. This is achieved by introducing two kinds of displaced bosons. The oscillation of quantum coherence is described.
The physics of an inversion-symmetric (IS) spin-boson system is first solved rigorously in the white limit of boson system on the basis of the concept of dynamic compensation which the author proposed previously. Two kinds of displaced bosons are introduced. They are defined freely out of the infrared divergence due to the dynamic compensation. It is shown that this IS system allows both IS-preserving and IS-breaking states when the bare zero-point contribution of bosons brings about the infrared divergence. The ground state and the excited state are determined explicitly in the white limit. Using them, after the brief study of thermodynamics, we construct the density matrix and study the problem of the persistence of oscillation of quantum coherence under the influence of environment. We describe the persistence of it up to a limited strength of interaction, taking a reasonable initial condition. An explicit formula is given.
We present a dynamic theory of the ground state of a spin boson system. We show the infrared divergence due to the zero-point fluctuation of displaced bosons is completely compensated by that due to the dynamical fluctuation of coherently excited bosons for arbitrary positive power sof the frequency dependence of the spectral density of interaction. We propose this as the theorem of dynamic compensation of the infrared divergence. In this sense our theory is a dynamic extension of the Anderson orthogonality theorem. Solving the eigenvalue equation in the lowest approximation we find that (1) there exists a critical value αc of the interaction strenght below which the ground state (GS) is the tunneling state for arbitrary positive s, (2) for s>1/2 the GS is the localized state above αc but (3) for 1/2>s>0 we find the localized GS above another critical value α'c>αc and we have found nothing definite, for the moment, for αc≪α≪α'c and (4) the tunneling gap tends to zero continuously when α approaches αc.
The existence of the delocalized-localized quantum phase transition (QPT) in the ohmic spin-boson model has been commonly recognized. While the physics in the localized regime is relatively simple, the delocalized regime shows many interesting behaviors. Here we reveal that in this regime there exists a novel QPT: namely, from a phase without a bound state to a phase with a bound state, which leads to completely different dynamical behaviors in these two phases. If the reservoir is initially in the displaced vacuum state (i.e., the coherent state), the spin dynamics exhibits lossless oscillation when the bound state exists; otherwise, the oscillation will decay completely. This result is compatible with the coherence-incoherence transition occurring in the small-tunneling limit. Our work indicates that the QPT physics in the spin-boson model needs further exploration.
We study the non-adiabatic effect of the bosonic bath on the spin dynamics by the light of the dynamic compensation theorem. We examine the time-evolutions of the probability of finding the up-state of spin and the thermal average of σz(t). The dynamics is formulated in competition between the tunneling coherence of spin and the non-adiabatic effect of bath in the equitable treatment. We prove the complete compensation of the infrared divergences due to the zero-point and thermodynamic fluctuations by the dynamical fluctuation of the coherent boson (the dynamic compensation). We further present the exact expressions of the above two quantities in the series form.