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Progress of Theoretical Physics

A Generalized Equation of Motion for the Linear Response. III: Simple Application to a Quantal Spin

Mizuhiko Saeki

Published Feb 1, 1987• 6 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.77.262

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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 time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal oscillator. This system is assumed to be interacting weakly with its surroundings (heat bath). The two cases for the initial state of the total system are considered. One case is that the oscillator and heat bath are in thermal equilibrium, and another is that these two systems are uncoupled and in each thermal equilibrium at the same temperature. The operator forms of equations are transcribed into the c-number ones 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.
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.
A time-convolutionless expression for the quantal equation of motion for the linear response of a system in contact with a heat bath to an external, time-varying driving field is derived from the quantal Liouville equation of motion for the total system. This is done by renormalizing the memory effect by the method of Shibata, Takahashi and Hashitsume. It has a form convenient for the perturbational expansion. It is shown that this equation coincides with the time-convolution equation of Argyres and Kelley in their Markoffian approximation in the narrowing limit and in the lowest Born approximation for the system-bath interaction. This equation is expanded up to fourth order in powers of the system-bath interaction.
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.
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.
A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.