Abstract
The states of quantum oscillator with time-dependent frequency are described by the tomographic probability distributions. The integrals of motion, being linear in the position and momentum operators, are used to construct the Gaussian squeezed and correlated states of the oscillator associated with normal probability distributions of the quadrature determining the density matrices of the states. The even and odd coherent states of the oscillator and their symplectic tomograms are given in terms of probability distributions. Considering free particle as a partial case of the oscillator with zero frequency, we find tomograms of even and odd coherent states of free particle in the probability representation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Schrödinger, Ann. Phys., 384, 361, 489 (1926); https://doi.org/10.1002/andp.19263840404
L. Landau, Z. Phys., 45, 430 (1927); https://doi.org/10.1007/bf01343064
J. von Neumann, Gött. Nach., 245 (1927); J. von Neumann, Mathematical foundations of Quantum Mechanics, Princeton University Press, USA (1955); http://eudml.org/doc/59230
E. Wigner, Phys. Rev., 40, 749 (1932); https://doi.org/10.1103/PhysRev.40.749
K. Husimi, Proc. Phys. Math. Soc. Jpn., 22, 264 (1940); https://doi.org/10.11429/ppmsj1919.22.4_264
R. J. Glauber, Phys. Rev., 131, 2766 (1963); https://doi.org/10.1103/PhysRev.131.2766
E. C. G. Sudarshan, Phys. Rev. Lett., 10, 277 (1963); https://doi.org/10.1103/PhysRevLett.10.277
A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 88, 055003 (2013); https://doi.org/10.1088/0031-8949/88/05/055003
S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A, 213, 1 (1996); https://doi.org/10.1016/0375-9601(96)00107-7
O. V. Man’ko and V. I. Man’ko, J. Russ. Las. Res., 18, 407 (1997); https://doi.org/10.1007/BF02559668
O. V. Man’ko and V. I. Man’ko, Entropy, 23. 549 (2021); https://doi.org/10.3390/e23050549
M. Asorey, A. Ibort, G. Marmo, and F. Ventriglia, Phys. Scr., 90, 74031 (2015); https://doi.org/10.1088/0031-8949/90/7/074031
J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1989); https://doi.org/10.1007/BF00733376
K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989); https://doi.org/10.1103/PhysRevA.40.2847
G. M. d’Ariano, M. G. A. Paris, and M. F. Sacchi, Advances in Imaging and Electron Physics, Academic Press, San Diego, USA (1996), Vol. 128, p. 206; ISBN: 9780128246122
O. V. Man’ko, V. I. Man’ko, and G. Marmo, J. Phys. A Math. Gen., 35, 699 (2002); https://doi.org/10.1088/0305-4470/35/3/315
V. V. Dodonov, E. V. Kurmyshev, and V. I. Man’ko, Phys. Lett. A, 79, 150 (1980); https://doi.org/10.1016/0375-9601(80)90231-5
V. I. Man’ko and R. Vilela Mendes, Phys. Lett. A, 263, 53 (1999); https://doi.org/10.1016/S0375-9601(99)00688-X
V. V. Dodonov and V. I. Man’ko, Dynamical Symmetries and the Evolution of Nonstationary Quantum Systems, Proceedings of the Lebedev Physical Institute, Nova Science, Commack, New York (1989), Vol. 183; ISBN 0-941743-49-7
I. A. Malkin, V. I. Man’ko, and D. Trifonov, Phys. Rev. D, 2, 1371 (1970); https://doi.org/10.1103/PhysRevD.2.1371
E. Schrödinger, Ber. Kgl. Akad. Wiss. Berlin, 24, 296 (1930).
H. P. Robertson, Phys. Rev. A, 35, 667 (1930); https://doi.org/10.1103/PhysRev.34.163
W. Heisenberg, Z. Phys., 43, 172 (1927); https://doi.org/10.1007/BF01397280
D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett., 70, 1244 (1993); https://doi.org/10.1103/PhysRevLett.70.1244
V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, Physica, 72, 597 (1974); https://doi.org/10.1016/0031-8914(74)90215-8
L. Ferraria, Eur. Phys. J. Plus, 134, 156 (2019); https://doi.org/10.1140/epjp/i2019-12625-y
V. N. Chernega and O. V. Man’ko, J. Russ. Laser Res., 41, 11 (2020); https://doi.org/10.1007/s10946-020-09844-2
S. Iqbal, J. Russ. Laser Res., 43, 96 (2022); https://doi.org/10.1007/s10946-022-10027-4
O. V. Man’ko, J. Russ. Laser Res., 43, 90 (2022); https://doi.org/10.1007/s10946-022-10026-5
S. V. Kuznetsov, O. V. Man’ko, and N. V. Tcherniega, J. Opt. B: Quantum Semiclass. Opt, 5, S5503 (2003); https://doi.org/10.1088/1464-4266/5/4/357
A. Pathak, J. K\( \overset{\sim }{\mathrm{r}} \)epelka, and J. Pe\( \overset{\sim }{\mathrm{r}} \)ina, Phys. Lett. A, 377, 2692(2013); https://doi.org/10.1016/j.physleta.2013.07.046
S. Kumar Giri, B. Sen, A. Pathak, and P. Chandra Jana, Phys. Rev. A, 93, 012340 (2016); https://doi.org/10.1103/PhysRevA.93.012340
O. V. Man’ko, AIP Conf. Proc., 1424, 221 (2012); https://doi.org/10.1063/1.3688974
S. V. Kuznetsov, A. V. Kyusev, O. V. Man’ko, and N. V. Tcherniega, Bull. Russ. Acad. Sci. Phys., 68, 1239 (2004).
O. V. Man’ko, Phys. Lett. A, 228, 29 (1997); https://doi.org/10.1016/S0375-9601(97)00091-1
P. Vasil’ev, J. Russ. Laser Res., 42, 730 (2021); https://doi.org/10.1007/s10946-021-10015-0
K. Thapliyal, S. Banerjee, and A. Pathaka, Ann. Phys., 366, 148 (2018); https://doi.org/10.1016/j.aop.2016.01.010
S. Abdel-Khalek, E. M. Khalil, B. Alsubei, et al., J. Russ. Laser Res., 41, 30 (2020); https://doi.org/10.1007/s10946-020-09854-0
A. N. Khedr, A.-B. A. Mohamed, A.-H. Abdel-Aty, et al., Entropy, 23, 1595 (2021); https://doi.org/10.3390/e23121595
M. A. Man’ko and V. I. Man’ko, AIP Conf. Proc., 1488, 110 (2012); https://doi.org/10.1063/1.4759389
A. K. Fedorov and S. O. Yurchenko, J. Phys.: Conf. Ser., 414, 012040 (2013); https://doi.org/10.1088/1742-6596/414/1/012040
C. Stornaiolo, Int. J. Geom. Meth. Mod. Phys., 17, 2050167 (2020); https://doi.org/10.1142/S0219887820501674
J. Berra-Montiel and A. Molgado, Eur. Phys. J. Plus, 137, 283 (2022); https://doi.org/10.1140/epjp/s13360-022-02504-1
J. Berra-Montiel and R. Cartas Int. J. Geom. Meth. Mod. Phys., 17, 2050217 (2020); https://doi.org/10.1142/S0219887820502175
O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 22, 149 (2001); https://doi.org/10.1023/A:1011360006073
P. Facchi and M. Ligabó, AIP Conf. Proc., 1260, 3 (2010); 10.1063/1.3479322
H.-T. Elze, G. Gambarotta, and F. Vallone, J. Phys: Conf. Ser., 306, 012010 (2011); https://doi.org/10.1088/1742-6596/306/1/012010
S. Avner, Entropy, 23, 1338 (2021); https://doi.org/10.3390/e23101338
A. Khrennikov, J. Russ. Laser Res., 43, 48 (2022); https://doi.org/10.1007/s10946-022-10022-9
A. Khrennikov, Found. Phys., 51, 16 (2021); https://doi.org/10.1007/s10701-021-00430-3
Y. V. Przhiyalkovskiy, J. Phys. A: Math. Gen., 55, 085301 (2022); https://doi.org/10.1088/1751-8121/ac4b15
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chernega, V.N., Man’ko, O.V. Squeezed and Correlated States of Parametric Oscillator and Free Particle in the Probability Representation of Quantum Mechanics. J Russ Laser Res 43, 280–289 (2022). https://doi.org/10.1007/s10946-022-10050-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-022-10050-5