After developing the concept of displaced squeezed vacuum states in the non- unitary approach and establishing the connection to the unitary approach we calculate their quasiprobabilities and expectation values in general form. Then we consider the displacement of the squeezed vacuum states and calculate their photon statistics and their quasiprobabilities. The expectation values of the displaced states are related to the expectation values of the undisplaced states and are calculated for some simplest cases which are sufficient to discuss their categorization as sub-Poissonian and super-Poissonian statistics. A large set of these states do not belong to sub- or to super-Poissonian states but are also not Poissonian states. We illustrate in examples their photon distributions. This shows that the notions of sub- and of super-Poissonian statistics and their use for the definition of nonclassicality of states are problematic. In Appen dix A we present the most important relations for SU (1,1) treatment of squeezing and the disentanglement of their operators. Some initial members of sequences of expectation values for squeezed vacuum states are collected in Appen dix E .
Keywords<i>SU</i>(11) Group of Squeezing and RotationWigner QuasiprobabilityUnitary Approach to SqueezingNonclassical StatesUncertainty MatrixDistance of StatesJacobiUltrasphericalLegendre and Hermite Polynomials
Schrödinger, E. (1926) Der stetige übergang von der Mikro zur Makromechanik. Naturwiss. 14, 664-666. https://doi.org/10.1007/BF01507634
Pauli, W. (1958) Die allgemeinen Prinzipien der Wellenmechanik. Handbuch der Physik, Band V, Teil 1, Herausgegeben von S. Flügge. Springer, Berlin, 1-168. (This Is a Republication of the Same Article with Small Changes in Old “Handbuch der Physik”, Band XXIV, Teil 1, herausgegeben von Geiger und Scheel, 1933). https://doi.org/10.1007/978-3-642-80539-4_1
Louisell, W.H. (1973) Quantum Statistical Properties of Radiation. John Wiley & Sons, New York.
Walls, D.F. (1983) Squeezed States of Light. Nature, 306, 141. https://doi.org/10.1038/306141a0
Kimble, J. and Walls, D.F. (1987) Squeezed States of the Electromagnetic Field. Journal of the Optical Society of America (JOSA), B4, 10.
Loudon, R. and Knight, P.L. (1987) Squeezed Light. Journal of Modern Optics, 34, 709. https://doi.org/10.1080/09500348714550721
Perelomov, A.M. (1977) Generalized Coherent States and Some of Their Applications. Uspekhi Fizicheskih Nauk, 123, 23-55 (1977) (In Russian). Soviet Physics Uspekhi, 20, 703 (In English). https://doi.org/10.3367/UFNr.0123.197709b.0023
Perelomov, A. (1986) Generalized Coherent States and Their Application. Springer, Berlin. https://doi.org/10.1007/978-3-642-61629-7
Perelomov, A.M. (1987) Obobstschonnyje kogerentnyje sostoyanya i ikh primenyenyje. Nauka, Moskva (Is Not Fully Identical with [8]).
Mandel, L. and Wolf, E. (1995) Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9781139644105
Scully, M.O. and Zubairy, M.S. (1997) Quantum Optics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511813993
Schleich, W. (2001) Quantum Optics in Phase Space. Wiley-VCH, Berlin. https://doi.org/10.1002/3527602976
Leonhardt, U. (1997) Measuring the Quantum State of Light. Cambridge University Press, Cambridge.
Fan, H.-Y. (1990) Squeezed States for Two Types of One- and Two-Mode Squeezing Transformations. Physical Review A, 41, 1526-1532. https://doi.org/10.1103/PhysRevA.41.1526
Poisson Statistics
Buzek, V. (1989) Time Evolution of an Anharmonic Oscillator in an Initial Holstein-Primakoff SU(1,1) Coherent State. Physical Review A, 39, 5432. https://doi.org/10.1103/PhysRevA.39.5432
Vourdas, A. (1993) Phase States: An Analytic Approach in the Unit Disc. Physica Scripta T, 48, 84. https://doi.org/10.1088/0031-8949/1993/T48/012
Wünsche, A. (2003) Squeezed States. In: Dodonov, V.V., Man’ko, V.I., Taylor and Francis, Eds., Theory of Nonclassical States of Light, London and New York, 95-152.
Dodonov, V.V. (2002) Nonclassical States in Quantum Optics: A “Squeezed” Review of First 75 Years. Journal of Optics B: Quantum and Semiclassical Optics, 4, No. 1. https://doi.org/10.1088/1464-4266/4/1/201
Wünsche, A. (1995) The Distance to Poissonian Statistics as a Supplementary Measure in Quantum Optics. Journal of Applied Physics, 60, 119-122.
Dodonov, V.V., Man’ko, O.V., Man’ko, V.I. and Wünsche, A. (2000) Hilbert-Schmidt Distance and Nonclassicality of States in Quantum Optics. Journal of Modern Optics, 47, 633-654. https://doi.org/10.1080/09500340008233385
Wünsche, A. (2017) About Classical to Quantum Weyl Correspondence. Applied Mathematics, 07, No. 10. https://doi.org/10.4236/apm.2017.710034
Dodonov, V.V., Man’ko, O.V. and Man’ko, V.I. (1994) Photon Distribution for One-Mode Mixed Light with a Generic Gaussian Wigner Function. Physical Review A, 49, 2993. https://doi.org/10.1103/PhysRevA.49.2993
Wünsche, A. (2015) Quantum-Mechanical Cumulant Expansions and Their Application to Phase-Space and to Phase Distributions. Physica Scripta, 90, Article ID: 074063. https://doi.org/10.1088/0031-8949/90/7/074063
Klauder, J.R. and Sudarshan, E.C.G. (1968) Fundamentals of Quantum Optics. W.A. Benjamin, Inc., New York.
Klauder, J.R. and Skagerstam, B.-S. (1985) Coherent States, Chapter I: A Coherent-State Primer. World Scientific, Singapore.
Wigner, E. (1932) On the Quantum Correction For Thermodynamic Equilibrium. Physical Review, 40, 749 (Republished in [28]). https://doi.org/10.1103/PhysRev.40.749
Kim, Y.S. and Noz, M.E. (1991) Phase Space Picture of Quantum Mechanics. World Scientific, Singapore (With Republication of Article [27]). https://doi.org/10.1142/1197
Wünsche, A. (1996) The Complete Gaussian Class of Quasiprobabilities and Its Relation to Squeezed States and Their Excitations, Quantum Semiclass. Optical, 8, 343-379.
Szegö, G. (1959) Orthogonal Polynomials. 2nd Edition, American Mathematical Society, New York (1st Edition 1939).
Bateman, H. and Erdélyi, A. (1953) Higher Transcendental Functions, Vol. II. McGraw-Hill, New York.
Wünsche, A. (2015) Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials. Applied Mathematics, 7, 213-261.
Akhiezer, N.I. and Glazman, I.M. (2013) Teoriya lineinykh operatorov v gilbertovym prostranstve, Nauka, Moskva 1966. English Translation: Theory of Linear Operators in Hilbert Space, Dover, New York.
Vourdas, A. and Wünsche, A. (1998) Resolutions of the Identity in Terms of Line Integrals of Coherent States. Journal of Physics A: Mathematical and General, 31, 9341-9352. https://doi.org/10.1088/0305-4470/31/46/024
Wünsche, A. (1997) Radon Transform and Pattern Functions in Quantum Tomography. Journal of Modern Optics, 44, 2293-2331. https://doi.org/10.1080/09500349708231885