Geodesic
Squeezed States and Their Applications to Quantum Evolution
Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 614-634.

Voir la notice de l'article provenant de la source Math-Net.Ru

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, Campbell–Baker–Hausdorff formula, Schrödinger equation, carbon films in $(E,H)$-fields.
Mots-clés : Bogolyubov formula 
@article{MZM_2011_89_4_a13,
     author = {A. M. Chebotarev and T. V. Tlyachev and A. A. Radionov},
     title = {Squeezed {States} and {Their} {Applications} to {Quantum} {Evolution}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {614--634},
     publisher = {mathdoc},
     volume = {89},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a13/}
}
TY  - JOUR
AU  - A. M. Chebotarev
AU  - T. V. Tlyachev
AU  - A. A. Radionov
TI  - Squeezed States and Their Applications to Quantum Evolution
JO  - Matematičeskie zametki
PY  - 2011
SP  - 614
EP  - 634
VL  - 89
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a13/
LA  - ru
ID  - MZM_2011_89_4_a13
ER  - 
%0 Journal Article
%A A. M. Chebotarev
%A T. V. Tlyachev
%A A. A. Radionov
%T Squeezed States and Their Applications to Quantum Evolution
%J Matematičeskie zametki
%D 2011
%P 614-634
%V 89
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a13/
%G ru
%F MZM_2011_89_4_a13
A. M. Chebotarev; T. V. Tlyachev; A. A. Radionov. Squeezed States and Their Applications to Quantum Evolution. Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 614-634. http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a13/

[1] C. M. Caves, “Quantum mechanical noise in an interferometer”, Phys. Rev. D, 23:8 (1981), 1693–1708 | DOI

[2] M. Hillery, M. Zubairy, “Path-integral approach to problems in quantum optics”, Phys. Rev. A (3), 26:1 (1982), 451–460 | DOI | MR

[3] G. D'Ariano, S. Morosi, M. Rasetti, J. Katriel, A. I. Solomon, “Squeezing versus photon-number fluctuations”, Phys. Rev. D, 36:8 (1987), 2399–2407 | DOI

[4] V. V. Dodonov, ““Nonclassical” states in quantum optics: a “squeezed” review of the first 75 years”, J. Opt. B: Quantum Semiclass. Opt., 4:1 (2002), R1–R33 | MR

[5] M. O. Scully, M. S. Zubairy, Quantum Optics, Cambridge Univ. Press, Cambridge, 1997

[6] G. S. Agarwal, “Wigner-function description of quantum noise in interferometers”, J. Modern Opt., 34:6-7 (1987), 909–921 | DOI | MR | Zbl

[7] H. Sumei, G. S. Agarwal, Squeezing of a Nanomechanical Oscillator, arXiv: 0905.4234

[8] D. A. Kirzhnits, Polevye metody teorii mnogikh chastits, Gosatomizdat, M., 1963 | MR

[9] F. A. Berezin, Metod vtorichnogo kvantovaniya, Nauka, M., 1986 | MR | Zbl

[10] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1967 | MR | Zbl

[11] A. V. Polyakov, A. M. Chebotarev, “Metod Monte-Karlo dlya uravneniya Shredingera s periodicheskim asimmetrichnym potentsialom”, Zh. vychisl. matem. i matem. fiz., 44:10 (2004), 1898–1908 | MR | Zbl