This article is cited in 9 papers

Squeezed States and Their Applications to Quantum Evolution

A. M. Chebotarev, T. V. Tlyachev, A. A. Radionov

M. V. Lomonosov Moscow State University

Abstract: In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell–Baker–Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant $(E,H)$-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.

Keywords: squeezed state, Bogolyubov formula, Campbell–Baker–Hausdorff formula, Schrödinger equation, carbon films in $(E,H)$-fields.

UDC: 517.958:530.145.6

Received: 17.10.2010
Revised: 18.11.2010

DOI: 10.4213/mzm9096

 English version: Mathematical Notes, 2011, 89:4, 577–595

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