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Physical Review

Coherent States and Irreversible Processes in Anharmonic Crystals

Peter Carruthers, K. S. Dy

Published Jul 8, 1966• 43 citations
Publisher

Abstract

Irreversible phenomena involving phonons in a slightly anharmonic crystal are investigated using coherent states as a basis. This basis retains the advantages of the classical theory, in which phase relations are clearly exhibited, for a quantum-mechanical system. In the coherent-state basis the equation of motion for the density matrix has an obvious correspondence with the classical Liouville equation. In particular, the connection with the action-angle variables of Brout and Prigogine is elucidated. A new, rapid proof of the Brout-Prigogine equation is given using the method of semi-invariants. This technique exhibits higher-order corrections in a usable form. The quantum corrections to the classical equation of motion for the density function are shown to be due to extra terms involving second derivatives in the action-angle variables. The Peierls master equation is then derived from the Brout-Prigogine equation. The important problem of elastic scattering of phonons is treated by our method in an appendix.

DOI
10.1103/physrev.147.214

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Irreversible phenomena involving phonons in a slightly anharmonic crystal are investigated using coherent states as a basis. This basis retains the advantages of the classical theory, in which phase relations are clearly exhibited, for a quantum-mechanical system. In the coherent-state basis the equation of motion for the density matrix has an obvious correspondence with the classical Liouville equation. In particular, the connection with the action-angle variables of Brout and Prigogine is elucidated. A new, rapid proof of the Brout-Prigogine equation is given using the method of semi-invariants. This technique exhibits higher-order corrections in a usable form. The quantum corrections to the classical equation of motion for the density function are shown to be due to extra terms involving second derivatives in the action-angle variables. The Peierls master equation is then derived from the Brout-Prigogine equation. The important problem of elastic scattering of phonons is treated by our method in an appendix.
The coherent-state formalism is used to develop a theory of fluctuations and irreversible processes in a superfluid Bose system. Equations of motion for the density matrix are derived both for the case of an isolated system and for the system in interaction with a heat bath. The formalism allows one to go easily to the classical limit and thus to obtain equations for the probability distribution of the superfluid order parameter. The results are in a form suitable for the calculation of the decay of superflow near the $\ensuremath{\lambda}$ point and other similar phenomena.
Using the unsymmetrized self-consistent field method the quasi-classical expansion of the one-particle density matrices is considered. The integral equations for the first quantum corrections to the classical one-particle probability densities are derived. The exact solution for the perfect crystals with the strong anharmonicity of the fourth order is found and used to calculate the first quantum correction to the effective amplitudes of the strongly anharmonic vibrations of atoms. The solution is obtained also for the semi-infinite linear chain as the simplest model of the anharmonic crystal with the surface.
The authors contrast the classical and quantum dynamics of SU(1,1) coherent states by the use of a positive-definite quasiprobability distribution, a Q-function, defined over these states. For Hamiltonians that are linear in the SU(1,1) generators, therefore coherence preserving, the quantum and classical Liouville equations are identical. This is illustrated for a degenerate parametric amplifier. For an anharmonic oscillator, the quantum Liouville equation contains additional terms containing second-order derivatives with non-positive-definite coefficients. These terms give rise to the quantum recurrences not seen in the classical evolution of this system. It is pointed out that the quantum equation for the Q-function goes over to the classical equation as the Bargmann index, k, becomes large, in agreement with previous semiclassical considerations.
The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.
Starting from the quantum-Boltzmann equation derived in a previous paper, we study the irreversible evolution of an electron gas in the one-particle phase space. The connection with phase space is established by expressing one-electron states in terms of the overcomplete and nonorthogonal generating system of coherent states. By using the generalized closure relation for coherent states, as well as the fact that a one-particle operator is completely determined by the ensemble of expectation values for all coherent states, we obtain the master equations in a form that allows us to follow the evolution in phase space. This form of the master equations provides a direct link between the quantum-statistical approach and the semi-classical Boltzmann equation. The latter is obtained after a coarse-graining procedure in the one-particle phase space and by using the fact that the electron-electron interaction, as well as the interactions between the electron gas and the bath subsystems provided by phonons or photons, are local in real space.
We use the so-called Liouville--von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr\"odinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal, and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Because of the spinodal instability of long wavelength modes, the two-point correlation functions lead to the t1/4{t}^{1/4}t1/4-scaling relation during the quench and the Cahn-Allen scaling relation t1/2{t}^{1/2}t1/2 after completion of the quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of the quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.