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arXiv:2003.01556 · Journal of Russian Laser Research

Coherent States of Parametric Oscillators in the Probability Representation of Quantum Mechanics†

Vladimir N. Chernega, Olga V. Man’ko

Published Jan 1, 2020• 7 citations
arXivHTMLPDF
DOI
10.1007/s10946-020-09844-2

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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.
Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin–systems with discrete variables. New entropy–information inequalities are obtained for Franck–Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.