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Journal of the Physical Society of Japan

Linear Response of a Quantal Spin (S=1/2) Interacting Weakly with Heat Bath

Mizuhiko Saeki

Published May 15, 2000• 7 citations
Publisher

Abstract

For a quantal spin S (magnitude 1/2) interacting weakly with heat bath, the fluctuation-dissipation theorem, dispersion relation and sum rules are discussed by the method of TCL equations with external driving terms (the TCLE method). The transverse susceptibility of the quantal spin is numerically calculated using the TCLE method for the case of a boson-like bath including a quickly damped subsystem, comparing with the results obtained using the relaxation method under decoupled initial conditions of the spin and bath. It is shown that the interference terms in the TCLE method produce the effects which enhance the peaks of line shapes and abase the line shapes at the small frequency region. It is also shown that the peaks of line shapes are enhanced as the temperature becomes low, as shown for the case of a spin-like bath. For the case of a boson-like bath including a subsystem with finite damping constants, the transverse susceptibility is calculated using the TCLE method numerically.

DOI
10.1143/jpsj.69.1327

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For a quantal spin S (magnitude 1/2) interacting weakly with heat bath, the fluctuation-dissipation theorem, dispersion relation and sum rules are discussed by the method of TCL equations with external driving terms (the TCLE method). The transverse susceptibility of the quantal spin is numerically calculated using the TCLE method for the case of a boson-like bath including a quickly damped subsystem, comparing with the results obtained using the relaxation method under decoupled initial conditions of the spin and bath. It is shown that the interference terms in the TCLE method produce the effects which enhance the peaks of line shapes and abase the line shapes at the small frequency region. It is also shown that the peaks of line shapes are enhanced as the temperature becomes low, as shown for the case of a spin-like bath. For the case of a boson-like bath including a subsystem with finite damping constants, the transverse susceptibility is calculated using the TCLE method numerically.
The transverse susceptibility of a quantal spin (magnitude 1/2) interacting weakly with its surroundings (quantal heat bath) is discussed using the method of the TCL equation with external driving terms (the TCLE method) analytically and numerically. The susceptibility obtained by the TCLE method under decoupled initial conditions is shown to coincide with that obtained under thermal equilibrium initial conditions. The results obtained with the TCLE method are compared with those obtained with the relaxation method. It is shown that the susceptibility in the relaxation method includes the second term of the interference terms in the TCLE method, but not its first term. With numerical calculations for the case of a spin-like bath including a quickly damped subsystem, it is shown that the interference terms in the TCLE method give larger effects as the temperature becomes lower, that its first term enhances the peaks of line shapes slightly, and that the peaks of line shapes are enhanced as the temperature becomes low. A model which shows that the TCLE method is simpler than the relaxation method is investigated numerically using the TCLE method.
A general form of admittance is derived using the relaxation method under thermal equilibrium initial conditions from a second-order TCL equation with no external driving terms. Its admittance is shown to include the two interference terms in the TCLE method (method of TCL equations with external driving terms). Effects of the first and second interference terms in the TCLE method are numerically discussed for a quantal spin (magnitude 1/2) interacting with a spin-like bath and with a boson-like bath, respectively. It is shown that the first interference term which represents effects of the initial correlation and which is neglected in the relaxation method under decoupled initial conditions of the spin and bath, produces the effects which enhance the peaks of line shapes slightly and which abase the line shapes at the small frequency region, and becomes larger as the characteristic frequency of the heat bath becomes smaller.
The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.
The TCLE method (a method in which the admittance of a system is directly calculated from time-convolutionless equations with external driving terms) formulated in terms of thermo-field dynamics, is applied to a boson system interacting with its surroundings, referred to as the “heat reservoir”, and with an external driving field. For a boson system interacting with its heat reservoir, the fluctuation-dissipation theorem, a dispersion relation, and sum rules are studied through application of the TCLE method. The effects of the interference between the external driving field and the heat reservoir, which give the effects of the initial correlation and memory, as well as the effects of the boson-boson interaction, are numerically investigated for a non-degenerate boson system interacting with its heat reservoir, up to first order in powers of the boson-boson interaction.
The time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal spin interacting weakly with its surroundings (heat bath). The operator form of equation is derived and is transcribed into the c-number one by the generalized phase-space method for spin operators. The equations of motion for the longitudinal and transverse expectation values of the spin are derived and are solved for the case of S=1/2 (S: the magnitude of spin). The longitudinal and transverse susceptibilities are derived from the stationary solutions, and are compared with the ones derived by the method of spin relaxation theory and Kubo theory for the linear response.
The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.