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Journal of Physics A Mathematical and General

Compact and noncompact dynamics of theSU(1,1) coherent states driven by a coherence preserving Hamiltonian

A. Bechler

Published Sep 21, 2001• 5 citations

Abstract

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.

DOI: 10.1088/0305-4470/34/39/309

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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.

We study the time evolution of SU(1,1) coherent states driven by a damped harmonic oscillator as described by the Kanai-Caldirola Hamiltonian. The coherence of these states is preserved and their time evolution is described by nonlinear ``classical'' equations. The trajectories of all SU(1,1) coherent states in phase space asymptotically approach a common point as t\ensuremath{\rightarrow}\ensuremath{\infty}. In all cases, this approach is accompanied by a damping of the associated energy expectation value to zero.

We investigate the classical and quantum oscillator with time-dependent mass and frequency, in the framework of the Lie group theory. Our analysis shows that both systems possess five time-dependent Noether invariants, where only two of them are functionally independent. In the classical case, three of these invariants form the su(1,1) Lie algebra under the Poisson bracket operation, while in the quantum case the corresponding invariants form the su(1,1) Lie algebra under the commutator operation. Since this algebra underlies the noncompact group SU(1,1), we are thus able to define unambiguously the dynamical group related to the generalized quantum oscillator. Furthermore, the Hamiltonian of this system can be expressed in terms of the Noether invariant operators satisfying the su(1,1) Lie algebra. This form is useful in the evaluation of the energy spectrum and in the study of coherence and squeezing. Some particular cases are briefly illustrated. The other two Noether invariants can be interpreted as the quadrature-phase amplitudes involved in the quantum nondemolition measurements and in the problem of the generation of coherent and squeezed states.

Path integrals over coherent states of the dynamical group (noninvariance group) SU(1,1) are constructed. From the continuous limit the relevant classical dynamics is extracted and is shown to take place in a curved phase space of the form of a Lobachevskii plane. Applications are made to the harmonic oscillator, a model of superfluid helium, the Morse oscillator, and the hydrogen atom. It is shown that when SU(1,1) is the relevant dynamical group the motion will appear oscillator-like on the Lobachevskii plane.