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Journal of Physics A Mathematical and General

Classical and quantum Liouville dynamics of SU(1,1) coherent states

Christopher C. Gerry, Jerry Kiefer

Published Aug 7, 1991• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/24/15/020

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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.
We consider the dynamics of a quantum joint phase-space probability density in an exactly solvable model. The density is defined to be a true (i.e., positive) probability distribution for approximate (and thus simultaneously measurable) position and momentum variables. The dynamics of the quantum density is governed by a second-order partial-differential equation with non-positive-definite second-order coefficients. The quantum dynamics is contrasted with the dynamics of a similar joint density in a classical description. The non-positive-definite second-order terms in the quantum evolution equation, not present in the classical case, are responsible for quantum recurrences and prevent the appearance of fine-scale-structure ``whorls'' predicted in a classical description. The generation of ``squeezing'' in the model is also discussed.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.
We investigate dynamics of the SU (1,1) coherent states with the use of the group transformations diagonalizing the coherence preserving Hamiltonian driving the physical system. The model physical system we consider may be viewed as a particular case of the generalized time-dependent harmonic oscillator, or as a generalization of the degenerate parametric amplifier, with the pumping field having modulated amplitude and a nonresonant phase. A Hamiltonian of such a system is given as a linear combination of the SU (1,1) generators with time-dependent coefficients, and the group transformations, mentioned above, transform this Hamiltonian to an expression containing only one generator, i.e. diagonalize the Hamiltonian. Trajectories of the complex coherent state parameter in the phase space (Lobachevskii plane) can be divided into two classes: compact trajectories never approaching the unit circle (boundary of the phase space) and noncompact trajectories approaching the unit circle from inside asymptotically, after sufficiently long time. The character of the dynamics is reflected by the time behaviour of the parameters of the group transformation diagonalizing the Hamiltonian. The main observation is that in the case of noncompact dynamics absolute values of the group parameters increase indefinitely, although in some cases they may exhibit a singular behaviour with regions of rapid variations, sudden changes of their values or cusplike singularities. For compact dynamics the group transformation parameters remain bounded and exhibit an oscillatory behaviour as functions of time.
In this paper we consider the time evolution of SU(1,1) coherent states driven by a coherence-preserving Hamiltonian containing periodic or quasiperiodic pulsing terms. This is a generalization of a system consisting of a two-level atom subjected to quasiperiodic pulsing that was recently studied by Milonni, Ackerhalt, and Goggin [Phys. Rev. A 35, 1714 (1987)]. The time-evolution operator in our case is given by a product of two finite group transformations of SU(1,1). Assuming an initial SU(1,1) coherent state, we determine the equivalent classical motion generated by a Poincar\'e map that is a M\"obius transformation on the Lobachevski plane, the interior of the unit circle in the complex plane. The quantum-mechanical evolution of the state vector is calculated exactly and in closed form even though the Hilbert space is infinite dimensional. We also study the autocorrelation function which, as in the work of Milonni, Ackerhalt, and Goggin, is found to decay in the case of quasiperiodic pulsing that may possibly be associated with a manifestation of chaos in a quantum-mechanical system.