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 Mat. Zametki, 2011, Volume 89, Issue 4, Pages 614–634 (Mi mz9096)

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

DOI: https://doi.org/10.4213/mzm9096

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

Bibliographic databases:

UDC: 517.958:530.145.6
Revised: 18.11.2010

Citation: A. M. Chebotarev, T. V. Tlyachev, A. A. Radionov, “Squeezed States and Their Applications to Quantum Evolution”, Mat. Zametki, 89:4 (2011), 614–634; Math. Notes, 89:4 (2011), 577–595

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz9096
• https://doi.org/10.4213/mzm9096
• http://mi.mathnet.ru/eng/mz/v89/i4/p614

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This publication is cited in the following articles:
1. A. M. Chebotarev, T. V. Tlyachev, A. A. Radionov, “Generalized Squeezed States and Multimode Factorization Formula”, Math. Notes, 92:5 (2012), 700–713
2. Tlyachev T.V., Chebotarev A.M., Chirkin A.S., “A New Approach to Quantum Theory of Multimode Coupled Parametric Processes”, Phys. Scr., T153 (2013), 014060
3. A. E. Teretenkov, “Quadratic Dissipative Evolution of Gaussian States”, Math. Notes, 100:4 (2016), 642–646
4. A. E. Teretenkov, “Dynamics of Moments for Quadratic GKSL Generators”, Math. Notes, 106:1 (2019), 151–155
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