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Journal of Mathematical Physics

Dynamics in the Diagonal Coherent-State Representation

Marvin M. Miller

Published Aug 1, 1969• 2 citations
Publisher

Abstract

The problem of following the dynamical behavior of a quantum-mechanical system in the diagonal coherent-state representation is examined for those systems whose time evolution is specified by equations of motion for the coherent-state weight functional which resemble Fokker-Planck equations but have non-positive-definite diffusion matrices. A particular equation of this type describing a linear parametric process is considered in detail and several proposed generalizations of the diagonal representation, which include dynamical effects of the atomic system coupled to the electromagnetic field in simple models of a unimodal laser, are also briefly discussed.

DOI
10.1063/1.1664981

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The problem of following the dynamical behavior of a quantum-mechanical system in the diagonal coherent-state representation is examined for those systems whose time evolution is specified by equations of motion for the coherent-state weight functional which resemble Fokker-Planck equations but have non-positive-definite diffusion matrices. A particular equation of this type describing a linear parametric process is considered in detail and several proposed generalizations of the diagonal representation, which include dynamical effects of the atomic system coupled to the electromagnetic field in simple models of a unimodal laser, are also briefly discussed.
It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the PPP representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and nnn-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative PPP representation for the density operator for a single mode of oscillation.
The dynamics of linear quantum systems is classical in the Wigner representation, yet linear problems are often analyzed using such general techniques as influence functionals and Bogoliubov transformations. In fact, the classical equations of motion provide a simpler and more intuitive formalism for these systems. As an important example, we show that quantum Brownian dynamics in the independent oscillator model is described directly and completely by a c-number Langevin equation. The corresponding Fokker-Planck equation is always local in time, regardless of the environmental spectrum.
An open quantum system is usually characterized by a reduced ensemble density matrix, the dynamics of which is governed by a generalized Master equation. Transforming this equation of motion into the instantaneous diagonal basis of the corresponding reduced density matrix, we can separate the coherent and incoherent part of the dynamics: The coherent dynamics is incorporated in the time development of the diagonal basis states, while the coupling to the reservoirs leads to simple rate equations. Interpreting these rate equations as a stochastic point process allows one to simulate the stochastic time evolution (random telegraph signals, ``quantum jumps'') of single-quantum systems. The diagonal representation can be considered as a generalization of the dressed-state picture of open quantum systems. Numerical simulations (``quantum Monte Carlo'') allow one to derive various dynamical properties (including correlation functions) of single-quantum systems. This concept is applied to different two- and three-level scenarios (\ensuremath{\lambda} and \ensuremath{\nu} configuration), and its limitations are discussed.
The analysis of the quantum-mechanical parametric amplifier begun in the preceding paper is extended to describe the joint quantum state of the two interacting modes of oscillation of the system. Equations are derived for the time evolution of the Schr\"odinger density operator. Comparison with classical theory is made by describing the state of the quantum-mechanical system by means of the PPP representation. The time evolution of the weight function $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ for this representation, and of the corresponding Wigner distribution function $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$, is found. A decoupled form for the equations of motion for the field variables is exhibited which simplifies the solutions for these functions. It is shown that the amplification process gives rise to a correlation between the two mode amplitudes which becomes exceedingly strong with increasing time. The relation of this correlation to the time development of the functions $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ and $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ is discussed.