Considerations of the Duality in the Generalized Coherent States Formalism

Dušan POPOV¹

Section:Research Paper, Product Type: Journal-Paper
Vol.13 , Issue.2 , pp.34-50, Apr-2025

Online published on Apr 30, 2025

Copyright © Dušan POPOV . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
In the paper, a pair of dual operators is introduced with which a dual pair of generalized displacement operators is constructed. With these entities it is shown that the Barut - Girardello and Klauder - Perelomov generalized coherent states are dual states. The characteristics of these coherent states are constructed, separately and comparatively. The dual characteristics are examined for both pure coherent states and mixed (thermal) ones, for the mixed ones there being no similar studies in the literature. Our study thus broadens the possibilities of applying the formalism of generalized coherent states related to different quantum systems (anharmonic oscillators, diatomic molecules, crystalline systems). Through the described procedure, the notion of duality extends, from the case of canonical states, to generalized coherent states associated with nonlinear oscillators.

Key-Words / Index Term :
duality, coherent states, density operator, displacement operator, distribution functions, thermal states.

References :
[1] Erwin Schrödinger, “Der stetige Übergang von der Mikro- zur Makromechanik”, Die Naturwissenschaften, Springer Science and Business Media LLC. Vol. 14, Isue 28, pp. 664–666, 1926.
[2] Asim O. Barut and Luciano Girardello, “New “Coherent” States associated with non-compact groups”, Communications in Mathematical Physics, Vol. 21, pp. 41-55, 1971.
[3] Roy J. Glauber, “Coherent and Incoherent States of the Radiation Field”, Physical Review, Vol. 131, Issue 6, pp. 2766–2788, 1963.
[4] John R. Klauder, “Continuous?representation theory. I. Postulates of continuous?representation theory”, Journal of Mathematical Physics, Vol. 4, Issue 8, pp. 1055-1058, 1963.
[5] Asim M. Perelomov, “Coherent states for arbitrary Lie groups”, Communications in Mathematical Physics,Vol. 26, Issue 3, pp. 222-236, 1972; arXiv: math-ph/0203002.
[6] David F. Walls, Gerard J. Milburn, “Quantum Optics“, Springer- Verlag, Berlin, pp. 13-15, 2008.
[7] Werner Vogel, Dirk-Gunnar Welsch, Quantum optics, Third Edition, Wiley-VCH, pp. 79-85, 2006.
[8] Rasoul Roknizadeh, Mohammad Kazem Tavassoly, “Construction of some important classes of generalized coherent states: the nonlinear coherent states method”, Journal of Physics A: Mathematical and General, Vol. 37, Issue 33, pp. 8111-8127, 2004.
[9] Ruynet Lima de Matos Filho, Werner Vogel, “Nonlinear coherent states”, Physical Review A, Vol. 54, Issue 5, pp. 4560-4563, 1996.
[10] Vladimir I. Man’ko, Giuseppe Marmo, Ennackel C. G. Sudarshan, Francesco Zaccaria, “f-oscillators and nonlinear coherent states method”, Physica Scripta, Vol. 55, Issue 5, pp. 528-541, 1997.
[11] S. Twareque Ali, Rasoul Roknizadeh, Mohammad Kazem Tavassoly, “Representations of coherent states in nonorthogonal bases”, Journal of Physics A: Mathematical and General, Vol. 37, Issue 15, pp. 4407-4422, 2004.
[12] Rasoul Roknizadeh, Mohammad Kazem Tavassoly, “Construction of the dual family of Gazeau-Klauder coherent states via temporally stable nonlinear coherent states”, Journal of Mathematical Physics, Vol. 46, Issue 042110, 2005.
[13] Omid Abbasi, Mohammad Kazem Tavassoly, “Superpositions of the dual family of nonlinear coherent states and their non-classical properties”, Optics Communications. Vol. 283, Issue 12, pp. 2566-2574, 2010.
[14] Barnana Roy, Pinaki Roy, “New nonlinear coherent states and some of their nonclassical properties”, Journal of Optics B, Quantum and Semiclassical Optics, Vol. 2, Issue 1, pp. 65-68, 2000.
[15] Kazuyuki Fujii, Tatsuo Suzuki, “A new symmetric expression of Weyl ordering”, Modern Physics Letters, Vol. A 19, Issue 11, pp.827-840, 2004.
[16] Hong-yi Fan, “Operator ordering in quantum optics theory and the development of Dirac`s symbolic method”, Journal of Optics B, Quantum and Semiclassical Optics, Vol. 5, Issue 4, pp. R147-R163, 2003.
[17] Dusan Popov, Miodrag Popov, “Some operatorial properties of the generalized hypergeometric coherent states”, Physica Scripta, Vol.90, Issue 3, 035101, pp. 1-16, 2015.
[18] Hong-yi Fan, “The development of Dirac’s symbolic method by virtue of IWOP technique”, Communications in Theoretical Physics (Beijing, China), Vol. 31, Issue 2, pp. 285-290, 1999.
[19] Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika, “Even and odd generalized hypergeometric coherent states”, arXiv:1406.3004v1[math-ph], https://doi.org/10.48550/arXiv.1406.3004
[20] Dusan Popov, Romeo Negrea, “Some integrals involving Meijer’s and hypergeometric functions using the coherent state technique”, Mathematical Problems in Engineering, Vol. 2022, Article ID 1284378, pp. 1-13, 2022,
[21] Arak M. Mathai, Rachit K. Saxena, “Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences”, Lecture Notes in Mathematics, Vol. 348, Springer-Verlag, Berlin, pp.78-79, 1973.
[22] Jean–Pierre Gazeau, John R. Klauder, “Coherent states for systems with discrete and continuous spectrum”, Journal of Physics A: Mathematical and General, Vol. 32, Issue 1, pp. 123-132, 1999.
[23] Dusan Popov, Romeo Negrea, Miodrag Popov, “Gazeau-Klauder coherent states examined from the viewpoint of diagonal ordering operation technique”, Chinese Physics B, Vol. 25, Issue 7, 070301, pp. 1-10, 2016.
[24] John R. Klauder, Karol A. Penson, Jean M. Sixdeniers, “Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems”, Physical Review A, Vol. 64, Issue 013817, pp. 1-18, 2001.
[25] Dusan Popov, “Gazeau-Klauder quasi-coherent states for the Morse oscillator”, Physics Letters A, Vol. 316, Issue 6, pp. 369-381,2003.
[26] Sharmishtha Ghoshal, Ashok Chatterjee, “Phonon distribution in a model polariton system”, Physical Review B, Vol. 52, Issue 2, pp.982-986, 1995.
[27] Andrea Laforgia, Pierpaolo Natalini, “Some inequalities for modified Bessel functions”, Journal of Inequalities and Applications, Vol. 2010, Article number 253035, pp. 1-10, 2010.
[28] Ennackal Chandy George Sudarshan, “Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams”, Physical Review Letters, Vol 10, Issue 7, pp. 277–279, 1963.
[29] Robert Gilmore, “On properties of coherent states”, Revista Mexicana de Física, Vol. 23, Issue 1–2, pp. 143–187, 1974.
[30] John R. Klauder, Bo Skagerstam, “Coherent States”, World Scientific, Singapore, pp. 10-24, 1985.
[31] Dusan Popov, “A unitary coherent states approach for linear spectra”, Journal of Advanced in Physics, Vol.6, Issue.4, pp. 434-440, 2017. http://www.aspbs.com/jap/contents_jap2017.htm
[32] D. Popov, “On some properties of the k-coherent states”, Proceedings of Romanian Academy, Series A, Vol. 184, Issue 4, pp. 323-330, 2017.
[33] Dusan Popov, Coherent states with non-equidistant energy levels, Theoretical Physics, Vol. 2, Issue 2, pp. 97-107, 2017.