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Physical Review A

Quantum transitions in the center-of-mass tomographic probability representation

A. S. Arkhipov, Vladimir I. Man’ko

Published Jan 4, 2005• 18 citations
Publisher

Abstract

Propagators for quantum evolution equation of multipartite systems in the center-of-mass probability representation are introduced. Properties of these propagators and their relation to the Green function of Schr\"odinger equation and propagator of Moyal evolution equation are studied. Examples of quadratic systems are considered and explicit expressions for the center of mass tomographic propagator are obtained.

DOI
10.1103/physreva.71.012101

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Propagators for quantum evolution equation of multipartite systems in the center-of-mass probability representation are introduced. Properties of these propagators and their relation to the Green function of Schr"odinger equation and propagator of Moyal evolution equation are studied. Examples of quadratic systems are considered and explicit expressions for the center of mass tomographic propagator are obtained.
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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.
The tomographic map of quantum state of a system with several degrees of freedom which depends on one random variable, analogous to center of mass considered in rotated and scaled reference frame in the phase space, is constructed. Time evolution equation of the tomogram for this map is given in explicit form. Properties of the map like the transition probabilities between the different states and relation to the star product formalism are elucidated. Example of multimode oscillator is considered in details. Symmetry properties in respect to identical particles permutations are discussed in the framework of proposed tomography scheme.
In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, equations for the Green function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green function of the Schr\"odinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.