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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Long-time classical dynamics of the periodically driven system with a Kerr nonlinearity

T Lisowski

Published Feb 1, 1997• 1 citation
Publisher

Abstract

We examine the classical dynamics of a parametric amplifier driven by periodic `kicks' of the pump field inside a cavity filled with the Kerr medium. The time evolution of the initial SU(1,1) generalized coherent state is investigated, and the classification of the possible maps is presented. New features of these maps are observed if the evolution is continued after the initial few thousand iterations.

DOI
10.1088/1355-5111/9/1/007

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Journal of Physics A Mathematical and General•Sep 21, 2001•5 citations
We examine the classical dynamics of a parametric amplifier driven by periodic `kicks' of the pump field inside a cavity filled with the Kerr medium. The time evolution of the initial SU(1,1) generalized coherent state is investigated, and the classification of the possible maps is presented. New features of these maps are observed if the evolution is continued after the initial few thousand iterations.
We consider a parametric amplifier driven by a periodically pulsed pump field inside a cavity containing a Kerr nonlinearity. The dynamics of the device is modeled as a kicked nonlinear system. The pulsed parametric amplifier constitutes the kick. In between kicks the dynamics is determined by the Kerr nonlinearity and damping. In the absence of damping, a classical description of the device exhibits a rich phase-space structure including fixed points of multiple period and chaos. We contrast the classical behavior of the mean intensity with that predicted by quantum dynamics. The mean photon number inside the cavity is shown to undergo regular collapse and revival in the regular region of the phase space and irregular revivals in the chaotic region. When damping is included, the quantum recurrences are rapidly suppressed, and the classical behavior is restored. In this case a stable steady state is possible. The damping represents the effect of photon-number measurements on the system. We also discuss the photon statistics in the steady state.
In this paper we study the dynamics of a nonlinear quantum-optical system subjected to a periodic modulation in the form of kicks. The system consists of a nonabsorbing anharmonic medium related to optical bistability, a degenerate parametric amplifier with a classical pump field that is modulated by a periodic sequence of kicks, and the initial state is taken to be a squeezed vacuum state. Using the related SU(1,1) coherent states (which are squeezed vacuum states), we study and compare the quantum and classical dynamics of the system. The ``classical'' motion in this case means the motion projected onto the phase space defined by the parameter labeling the generalized coherent states. For certain values of the coupling constants, we obtain both regular and chaotic motion in the classical phase space. In the quantum dynamics of the system, the manifestation of the classical chaos turns out to be rather weak, at least in the regime numerically accessible to us. We study the time evolution of the initial-state population probability, the average photon number in the field, and the squeezing of the field. We also briefly discuss the level statistics of the evolution operator.
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 discuss a model of a cavity filled with a passive nonlinear ‚Kerr‘ medium and periodically kicked by a series of ultra-short laser pulses. The nonlinear medium is described by the (2q − 1)th nonlinearity X (2q−1). We find analytical formulas describing the field states inside the cavity. We show that such a system can produce, depending on the order of the nonlinearity, superpositions of several Fock states with the small photon numbers (0,1; 0,1,2; etc). In particular, the one-photon state can be approached during the evolution of the system with X (3) nonlinearity provided the cavity losses are negligible. The purity of states generated in this process, however, can be seriously degraded by the cavity damping. We perform numerical calculations to validate our analytical results.
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.