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arXiv:0912.3424 · Physica Scripta

Moyal and tomographic probability representations for f-oscillator quantum states

Vladimir I. Man’ko, Giuseppe Marmo, F. Zaccaria

Published Mar 12, 2010• 12 citations
arXivHTMLPDF

Abstract

States of nonlinear quantum oscillators (f-oscillators) are considered in the Weyl–Wigner–Moyal representation and the tomographic probability representation, where the states are described by standard probability distributions instead of wave functions or density matrices. The evolving integrals of motion for classical and quantum f-oscillators are found and the solution for the Liouville equation associated with the probability distribution on the phase space for this oscillator is obtained along with the solution of the Moyal equation for the quantum f-oscillator, which provide solutions for the partial case of f-nonlinearity existing in Kerr media. Nonlinear coherent states and the thermodynamics of nonlinear oscillators are studied.

DOI
10.1088/0031-8949/81/04/045004

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Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase-space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for an irreducible set of observables. These Moyal solutions incorporate the asymptotics of the classical limit in a simple, explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase-space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.
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The quantizer–dequantizer method is employed. Using the construction of probability distributions describing density operators of a quantum system states, the connection between the Feynman path integral and the time evolution of the density operator (Landau density matrix) as well as the wave function of the stateconsidered. For single–mode systems with continuous variables, a tomographic propagator is introduced in the probability representation of quantum mechanics. An explicit expression for the probability in terms of the Green function of the Schrödinger equation is obtained. Equations for the Green functions defined by arbitrary integrals of motion are derived. Examples of probability distributions describing the evolution of state of a free particle, as well as states of systems with integrals of motion that depend on time (oscillator type) are discussed.