Abstract
Glauber coherent states of quantum systems are reviewed. We construct the tomographic probability distributions of the oscillator states. The possibility to describe quantum states by tomographic probability distributions (tomograms) is presented on an example of coherent states of parametric oscillator. The integrals of motion linear in the position and momentum are used to explicitly obtain the tomogram evolution expressed in terms of trajectories of classical parametric oscillator.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Schrödinger, Naturwissenchaften, 14, 664 (1926).
R. J. Glauber, Phys. Rev. Lett., 10, 84 (1963).
C. E. Cahill and R. J. Glauber, Phys. Rev., 177, 1882 (1969).
R. J. Glauber, Phys. Rev., 131, 2766 (1963).
R. Glauber and V. I. Man’ko, “Damping and fluctuations in the systems of two entangled quantum oscillators,” in: A. A. Komar (Ed.), Group Theory, Gravitation, and Physics of Elementary Particles, Proceedings of the Lebedev Physical Institute, Nauka, Moscow (1986), Vol. 167 [English translation by Nova Science, Commack, New York (1987), Vol. 167].
R. Glauber and V. I. Man’ko, Zh. Éksp. Teor. Fiz., 87, 790 (1984) [Sov. Phys. JETP, 60, 450 (1984)].
R. J. Glauber, “The quantum mechanics of trapped wavepackets,” in: E. Arimondo, W. D. Philips, and F. Sttrumia (Eds.), Proceedings of the International Enrico Fermi School, Course 118 (Varenna) Italy, July 1-19, 1992, North Holland, Amstertdam (1992), p. 643.
R. J. Glauber, Quantum Theory of Optical Coherence. Selected Papers and Lectures, Wiley-VCH (2007).
G. Schrade, V. I. Man’ko, W. Schleich, and R. Glauber, Quantum Semiclass. Opt., 7, 307 (1995).
I. Ya. Doskoch and M. A. Man’ko, J. Russ. Laser Res., 40, 1 (2019).
I. Ya. Doskoch and M. A. Man’ko, Quantum Rep., 1(2) 130 (2019).
M. A. Man’ko, Phys. Scr., 87, 038013 (3013).
M. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).
W. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001).
W. Heisenberg, Z. Phys., 43, 172 (1927).
E. C. G. Sudarshan, Phys. Rev. Lett., 10, 277 (1863).
K. Husimi, Proc. Phys. Math. Soc. Jpn., 22 264 (1940).
V. Ermakov, “Second-order differential equation. Conditions of complete integrability,” Kiev University Izvestia, Series III, 9, 1 (1980) [English translation: A. O. Harin, Appl. Anal. Discrete Math., 2, 123 (2008)].
H. R. Lewis and W. B. Reisenfeld, J. Math. Phys., 10, 1458 (1969).
I. A. Malkin, V. I. Man’ko, and D. A. Trifonov, J. Math. Phys., 14, 576 (1973).
V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, Int. J. Theor. Phys., 14, 37 (1975).
D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett., 70, 1244 (1993).
J. Radon, Ber. Sach. Akad. Wiss. Leipzig, 29 262(1917).
E. Wigner, Phys. Rev., 40, 749 (1932).
J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1987).
K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).
S. Mancini, V. I. Man’ko, and P. Tombesi, Phys. Lett. A, 213, 1 (1996).
G. Marmo and G. F. Volkert, Phys. Scr., 82, 038117 (2010).
Ya. A. Korennoy and V. I. Man’ko, J. Russ. Laser Res., 32, 74 (2011).
G. G. Amosov, Ya. A. Korennoy, and V. I. Man’ko, Phys. Rev. A85 052119 (2012).
S. Mancini, V. I. Man’ko, and P. Tombesi, Found. Phys., 27, 801 (1997).
S. Mancini, O. V. Man’ko, V. I. Man’ko, and P. Tombesi, J. Phys. A: Math. Gen, 34, 3461 (2001).
S. V. Kuznetsov, O. V. Man’ko, and N. V. Tcherniega, J. Opt. B: Quantum Semiclass. Opt., 5, S503 (2003).
O. V. Man’ko and N. V. Tcherniega, Proc. SPIE, 6256, 62560W-1 (2006).
O. V. Man’ko, and N. V. Tcherniega, J. Russ. Laser Res., 22, 201 (2001).
E. Pinney, Proc. Amer. Math. Soc., 1, 681 (1950).
D. Schuch, Int. J. Quantum Chem.24 767 (1990).
O. Rosas-Ortiz, O. Castaños, and D. Schuch, J. Phys. A: Math. Theor., 48, 445302 (2015).
J. A. Lopez-Saldivar, O. Castaños, E. Nahmad-Achar, et al., Entropy, 20(9), 630 (2018).
R. R. Ancheyta, M. Berrondo, and J. Récamier, J. Opt. Soc. Am B, 32(8) 1651 (2015).
K. Zelaya and O. Rosas-Ortiz, J. Phys.: Conf. Ser., 839, 012018 (2017).
Z. Blanco-Garcia, O. Rosas-Ortiz, and K. Zelaya, “Interplay between Riccati, Ermakov and Schrödinger equations to produce complex-valued potentials with real energy spectrum,” Math. Methods Appl. Sci. (2018), p. 1.
D. J. Fernández and V. Hussin, J. Phys. A: Math. Gen., 32, 3603 (1999).
V. I. Man’ko and R. V. Mendes, Phys. Lett. A, 263, 53 (1999).
E. Schrödinger, Ber. Kgl. Akad. Wiss. Berlin, 296 (1930).
H. P. Robertson, Phys. Rev. A, 35, 667 (1930).
V. V. Dodonov, E. V. Kurmushev, and V. I. Man’ko, Phys. Lett. A, 79 150, (1980).
M. A. Man’ko and V. I. Man’ko, Entropy, 20, 692:1 (2018).
O. V. Man’ko and V. I. Man’ko, Laser Phys., 19, 1804 (2009).
O. V. Man’ko and V. I. Man’ko, Fortschritte Phys., 57, 1054 (2009).
O. V. Man’ko and V. I. Man’ko, Phys. Scr., T140, 014028 (21010).
Author information
Authors and Affiliations
Corresponding author
Additional information
†We dedicate this paper to the memory of Roy Jay Glauber, the great scientist and Nobel Prize Winner, on his first death anniversary, December 26, 2019. Ad Memoriam of Roy Glauber and George Sudarshan is published in [10, 11] and is also available on springerlink.bibliotecabuap.elogim.com/article/10.1007/s10946-019-09805-4 and www.mdpi.com/2624-960X/1/2/13.
Rights and permissions
About this article
Cite this article
Chernega, V.N., Man’ko, O.V. Coherent States of Parametric Oscillators in the Probability Representation of Quantum Mechanics†. J Russ Laser Res 41, 11–22 (2020). https://doi.org/10.1007/s10946-020-09844-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-020-09844-2