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Journal of the Physical Society of Japan

Theory of Nonlinear Spin Relaxation

Fumiaki Shibata

Published Jul 1, 1980• 40 citations
Publisher

Abstract

A rigorous quantum theory is developed on the nonlinear spin relaxation process. With the use of the phase space method, the quantum mechanical equation for a density matrix is mapped onto a c -number space. A quasi-probability function is then expanded in terms of spherical harmonics. It is clarified that the nonlinear relaxation process is characterized by 2 J ( J being the magnitude of a spin) different relaxation times. For J =1, time-evolutions of the first moment, fluctuation and probability function are exactly solved. An enhancement of the fluctuation is observed in the course of time development: This is similar to the phenomenon called “anomalous fluctuation” found in certain nonequilibrium systems.

DOI
10.1143/jpsj.49.15

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A rigorous quantum theory is developed on the nonlinear spin relaxation process. With the use of the phase space method, the quantum mechanical equation for a density matrix is mapped onto a c -number space. A quasi-probability function is then expanded in terms of spherical harmonics. It is clarified that the nonlinear relaxation process is characterized by 2 J ( J being the magnitude of a spin) different relaxation times. For J =1, time-evolutions of the first moment, fluctuation and probability function are exactly solved. An enhancement of the fluctuation is observed in the course of time development: This is similar to the phenomenon called “anomalous fluctuation” found in certain nonequilibrium systems.
The spin system with Hamiltonian (ξ and σ are external and internal field parameters) is treated as an example of the phase space description of spin dynamics using a master equation for the quasiprobability distribution function of spin orientations in the representation (phase) space of the polar angles (analogous to the Wigner phase space distribution for translational motion). The master equation yields (via the Wigner–Stratonovich transformation of the density matrix) the solution as a Fourier series in the spherical harmonics with Fourier coefficients given by the statistical moments in a manner analogous to the classical distribution. The relaxation time of the longitudinal component of the spin can be estimated using a quantum generalization of the classical integral relaxation time via the stationary distribution and diffusion coefficient of the master equation.
Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).
The approach to thermal equilibrium of a free particle and a harmonic oscillator interacting with a heat bath can be understood from a semiclassical point of view in terms of the time evolution of a quantum mechanical state function appropriate to each system. The state functions are parameterized such that expectation values comply with the predictions of the classical theory of Brownian motion, with the consequence that the Fokker-Planck equation for a free particle in velocity space, the Montroll-Shuler equation for a quantum oscillator in discrete energy space, and their solutions are arrived at by a common procedure. These relaxation processes become visualizable as the progressive development from a coherent, relatively ordered state to that of equilibrium, so that the loss of information as equilibrium is reached and the consequences of random phasing can be traced dynamically in some detail. The unity of approach is reinforced by the demonstration that since both equations represent Markoffian processes, the Montroll-Shuler equation predicts distributions in coordinate space and momentum space which satisfy a Fokker-Planck equation.
A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.
An improved density-matrix theory is developed that can treat both spectral hole burning and carrier heating self consistently. Various intraband and interband relaxation terms characterized by different relaxation times and quasi-equilibrium distributions are introduced into the density-matrix equations within a relaxation-time approximation. Conservation of total number and energy densities of carrier systems in each band is considered to determine the quasi-equilibrium distributions. Formalism is applied to the calculation of the transient optical nonlinearity of highly carrier-injected semiconductors. Spectral hole burning and carrier-heating effects on the spectral and temporal characteristics are then clarified. In particular, the significant four-wave-mixing effect due to carrier heating is pointed out. An experiment that can be used to directly prove the existence of the carrier-heating effects on gain nonlinearity is also proposed.>