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Progress of Theoretical Physics

Dynamics of a Spin-Boson System

Toshio Tsuzuki

Published Feb 1, 1991• 0 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.85.205

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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.
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.
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.
A closed integral equation for the partition function of a spin-boson system is obtained by the use of the coherent state representation of boson. This is accomplished by the use of new method of taking the thermal average by differential operations. The dynamic compensation theorem for the thermodynamic fluctuation is proved without employing a series expansion of the partition function as done in our previous works. The zeroth approximation to the equation is shown to be the decoupling approximation introduced previously, as naturally expected.
A closed integral equation for the partition function of a spin-boson system is obtained by the use of the coherent state representation of boson. This is accomplished by the use of new method of taking the thermal average by differential operations. The dynamic compensation theorem for the thermodynamic fluctuation is proved without employing a series expansion of the partition function as done in our previous works. The zeroth approximation to the equation is shown to be the decoupling approximation introduced previously, as naturally expected.
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.