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Progress of Theoretical Physics

Stochastic Liouville Equation for Weakly Driven System. II: Application to a Quantal Spin

Mizuhiko Saeki

Published Sep 1, 1988• 3 citations
Publisher

Abstract

The time-convolutionless stochastic equation derived in a previous paper is applied to a quantal spin interacting weakly with its surroundings under decoupled initial condition, up to second order in powers of random perturbations. The transverse and longitudinal susceptibilities are derived from the stationary solutions. The transverse and longitudinal spectra are displayed and are compared with the ones derived from the time-convolution equation in the conventional Markoffian approximation, and with the ones by the method of second-order spin relaxation theory and Kubo's theory. It is shown that the line shape of transverse spectrum is enhanced over the Lorentzian at the resonance region, due to the effects of interference of the external driving field and the random interaction.

DOI
10.1143/ptp.80.433

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Time-convolutionless forms of the quantal master equations for driven open system are derived from the Liouville equation for the total system under an arbitrary initial condition for the two cases of weak external driving fields and of arbitrary ones. They have forms convenient for the perturbational expansions, and are respectively compared with the time-convolution forms of equations in the lowest Born approximation for the interaction H SB of the system with its surroundings. It is shown that the time-convolutionless forms of equations prevail over the time-convolution ones, and that the conventional Markoffian approximation for the latter in the narrowing limit leads to the incorrect conclusion. The time-convolutionless form of master equation is expanded up to n -th order in powers of the external driving fields in the lowest Born approximation for H SB .
Time-convolution and time-convolutionless expressions for the quantal equation of motion for the linear response of a system under random perturbations to an external, time-varying driving field are derived from the stochastic quantal Liouville equation for the total system. They have forms convenient for the perturbational expansions in powers of the random perturbations. One is compared with another for two initial conditions. It is shown that the time-convolutionless form of equation prevails over the time-convolution form, especially when the effects of the random perturbations are discussed except in the narrowing limit or when the effects of interference of the external driving field and the random perturbations are discussed. The higher order expansions of these equations are mentioned.
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 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.
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 time-convolutionless master equation for driven open system derived in a previous paper, is applied to a simple system of a driven quantal oscillator. This system is assumed to be interacting weakly with its surroundings (heat bath). The operator form of equation is transcribed into the c -number one by the phase-space method. The equations of motion for the expectation values of the boson operator and boson number operator are derived from the c -number equations and are solved for the case of a non-adiabatic oscillator-bath interaction. The results are compared with the ones in the conventional Markoffian approximation for the time-convolution equation in the narrowing limit.