We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These “k-hypercats” can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, …, k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time.
REFERENCES
1.
E.
Schrödinger
, “Der stetige übergang von der mikro-zur makromechanik
,” Naturwissenschaften
14
, 664
–666
(1926
).2.
R.
Glauber
, “Coherent and incoherent states of the radiation field
,” Phys. Rev. A
131
, 2766
(1963
).3.
W.
Zhang
, D.
Feng
, and R.
Gilmore
, “Coherent states: Theory and some applications
,” Rev. Mod. Phys.
62
, 867
(1990
).4.
J. R.
Klauder
, “Coherent states for the hydrogen atom
,” J. Phys. A: Math. Gen.
29
, L293
(1996
).5.
A.
Vourdas
, “Analytic representations in quantum mechanics
,” J. Phys. A: Math. Gen.
39
, R65
–R141
(2006
).6.
S.
Twareque Ali
, J.-P.
Antoine
, F.
Bagarello
, and J.-P.
Gazeau
, “Special issue on coherent states: Mathematical and physical aspects
,” J. Phys. A: Math. Theor.
45
, 240201
(2011
).7.
J. R.
Klauder
and E. C. G.
Sudarshan
, Fundamentals of Quantum Optics
(W. A. Benjamen, Inc.
, New York
, 1968
).8.
A.
Perelomov
, Generalized Coherent States and Their Applications
(Springer-Verlag
, 1986
).9.
Theory of Nonclassical States of Light
, edited by V. V.
Dodonov
and V. I.
Man’ko
(Taylor & Francis
, 2003
).10.
J.-P.
Gazeau
, Coherent States in Quantum Physics
(Wiley-VCH
, Berlin
, 2009
).11.
R. J.
Glauber
, Quantum Theory of Optical Coherence: Selected Papers and Lectures
(Wiley-VCH
, Weinheim
, 2007
).12.
S. T.
Ali
, J.-P.
Antoine
, and J.-P.
Gazeau
, Coherent States, Wavelets and Their Generalizations
, 2nd ed. (Springer
, New York
, 2000
; 2013
).13.
R.
Gilmore
, “Geometry of symmetrized states
,” Ann. Phys.
74
, 391
–463
(1972
).14.
R.
Gilmore
, “On the properties of coherent states
,” Rev. Mex. Fis.
23
, 143
–187
(1974
), https://rmf.smf.mx/ojs/rmf/article/view/1046.15.
A. M.
Perelomov
, “Coherent states for arbitrary Lie group
,” Commun. Math. Phys.
26
, 222
–236
(1972
).16.
J. M.
Radcliffe
, “Some properties of coherent spin states
,” J. Phys. A: Gen. Phys.
4
, 313
–323
(1971
).17.
A. O.
Barut
and L.
Girardello
, “New “coherent” states associated with non compact groups
,” Commun. Math. Phys.
21
, 41
–55
(1971
).18.
C.
Aragone
, G.
Guerri
, S.
Salamo
, and J. L.
Tani
, “Intelligent spin states
,” J. Phys. A: Math., Nucl. Gen.
7
, L149
(1974
).19.
G.
Vanden-Bergh
and H.
DeMeyer
, “On the existence of intelligent states associated with the non-compact group SU(1,1)
,” J. Phys. A: Math. Gen.
11
, 1569
(1978
).20.
J.-P.
Gazeau
and J. R.
Klauder
, “Coherent states for systems with discrete and continuous spectrum
,” J. Phys. A: Math. Gen.
32
, 123
–132
(1999
).21.
V. I.
Man’ko
, G.
Marmo
, E. C. G.
Sudarshan
, and F.
Zaccaria
, “f-oscillators and nonlinear coherent states
,” Phys. Scr.
55
, 528
–541
(1997
).22.
J. R.
Klauder
, K. A.
Penson
, and J.-M.
Sixdeniers
, “Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems
,” Phys. Rev. A
64
, 013817
(2001
).23.
N. I.
Akhiezer
, The Classical Moment Problem and Some Related Questions in Analysis
(Oliver and Boyd
, London
, 1965
).24.
25.
B.
Simon
, “The classical moment problem as a self-adjoint finite difference operator
,” Adv. Math.
137
, 82
(1998
).26.
R. L.
de Matos Filho
and W.
Vogel
, “Nonlinear coherent states
,” Phys. Rev. A
54
, 4560
–4563
(1996
).27.
Y.
Yan
, J.-p.
Zhu
, and G.-x.
Li
, “Preparation of a nonlinear coherent state of the mechanical resonator in an optomechanical microcavity
,” Opt. Express
24
, 13590
–13609
(2016
).28.
T.
Appl
and D. H.
Schiller
, “Generalized hypergeometric coherent states
,” J. Phys. A: Math. Gen.
37
, 2731
(2004
).29.
A.
Dehghani
and B.
Mojaveri
, “New nonlinear coherent states based on hypergeometric-type operators
,” J. Phys. A: Math. Theor.
45
, 095304
(2012
).30.
D.
Popov
and M.
Popov
, “Some operatorial properties of the generalized hypergeometric coherent states
,” Phys. Scr.
90
, 035101
(2015
).31.
R.
Gilmore
, “The classical limit of quantum nonspin systems
,” J. Math. Phys.
20
, 891
–893
(1979
).32.
M.
Calixto
, Á.
Nagy
, I.
Paradela
, and E.
Romera
, “Signatures of quantum fluctuations in the Dicke model by means of Rényi uncertainty
,” Phys. Rev. A
85
, 053813
(2012
).33.
C.
Pérez-Campos
, J. R.
González-Alonso
, O.
Castaños
, and R.
López-Peña
, “Entanglement and localization of a two-mode Bose-Einstein condensate
,” Ann. Phys.
325
, 325
–344
(2010
).34.
H. J.
Lipkin
, N.
Meshkov
, and A. J.
Glick
, Nucl. Phys.
62
, 188
(1965
);H. J.
Lipkin
, N.
Meshkov
, and A. J.
Glick
, Nucl. Phys.
62
, 199
(1965
);H. J.
Lipkin
, N.
Meshkov
, and A. J.
Glick
, Nucl. Phys.
62
, 211
(1965
).35.
O.
Castaños
, E.
Nahmad-Achar
, R.
Lopez-Peña
, and J. G.
Hirsch
, Phys. Rev. A
83
, 051601
(2011
);O.
Castaños
, E.
Nahmad-Achar
, R.
Lopez-Peña
, and J. G.
Hirsch
, Phys. Rev. A
84
, 013819
(2011
).36.
E.
Romera
, M.
Calixto
, and Á.
Nagy
, “Entropic uncertainty and the quantum phase transition in the Dicke model
,” Europhys. Lett.
97
, 20011
(2012
).37.
O.
Castaños
, R.
López-Peña
, J. G.
Hirsch
, and E.
López-Moreno
, “Phase transitions and accidental degeneracy in nonlinear spin systems
,” Phys. Rev. B
72
, 012406
(2005
).38.
E.
Romera
, M.
Calixto
, and O.
Castaños
, “Phase space analysis of first-, second- and third-order quantum phase transitions in the Lipkin-Meshkov-Glick model
,” Phys. Scr.
89
, 095103
(2014
).39.
R. H.
Dicke
, “Coherence in spontaneous radiation processes
,” Phys. Rev.
93
, 99
(1954
).40.
M.
Calixto
, O.
Castaños
, and E.
Romera
, “Searching for pairing energies in phase space
,” Europhys. Lett.
108
, 47001
(2014
).41.
M.
Calixto
, E.
Romera
, and R.
del Real
, “Parity-symmetry-adapted coherent states and entanglement in quantum phase transitions of vibron models
,” J. Phys. A: Math. Theor.
45
, 365301
(2012
).42.
F.
Pérez-Bernal
and F.
Iachello
, “Algebraic approach to two-dimensional systems: Shape phase transitions, monodromy, and thermodynamic quantities
,” Phys. Rev. A
77
, 032115
(2008
).43.
M.
Calixto
, R.
del Real
, and E.
Romera
, “Husimi distribution and phase-space analysis of a vibron-model quantum phase transition
,” Phys. Rev. A
86
, 032508
(2012
).44.
M.
Calixto
and F.
Pérez-Bernal
, “Entanglement in shape phase transitions of coupled molecular benders
,” Phys. Rev. A
89
, 032126
(2014
).45.
M.
Calixto
and C.
Peón-Nieto
, “Husimi function and phase-space analysis of bilayer quantum Hall systems at ν = 2/λ
,” J. Stat. Mech.: Theory Exp.
2018
, 053112
.46.
M.
Calixto
, C.
Peón-Nieto
, and E.
Pérez-Romero
, “Hilbert space and ground-state structure of bilayer quantum Hall systems at ν = 2/λ
,” Phys. Rev. B
95
, 235302
(2017
).47.
V. V.
Dodonov
, I. A.
Malkin
, and V. I.
Man’ko
, “Even and odd coherent states and excitations of a singular oscillator
,” Physica
72
, 597
–615
(1974
).48.
O.
Castaños
, R.
Lopez-Peña
, and V.
Man’ko
, “Crystallized Schrödinger cat states
,” J. Russ. Laser Res.
16
, 477
(1995
).49.
M.
Nieto
and D.
Traux
, “Squeezed states for general systems
,” Phys. Rev. Lett.
71
, 2843
(1993
).50.
V.
Buzek
, A.
Viiella-Barranco
, and P.
Knight
, “Superpositions of coherent states: Squeezing and dissipation
,” Phys. Rev. A
45
, 6750
(1992
).51.
M.
Hillery
, “Amplitude-squared squeezing of the electromagnetic field
,” Phys. Rev. A
36
, 3796
(1987
).52.
S.
Mancini
, “Even and odd nonlinear coherent states
,” Phys. Lett. A
233
, 291
(1997
).53.
S.
Sivakumar
, 250
, 257
–262
(1998
);S.
Sivakumar
, J. Phys. A: Math. Gen.
33
, 2289
–2297
(2000
).54.
J.
Sun
, J.
Wang
, and C.
Wang
, “Orthonormalized eigenstates of cubic and higher powers of the annihilation operator
,” Phys. Rev. A
44
, 3369
–3372
(1991
).55.
J.
Sun
, J.
Wang
, and C.
Wang
, “Generation of orthonormalized eigenstates of the operator ak (for k ≥ 3) from coherent states and their higher-order squeezing
,” Phys. Rev. A
46
, 1700
–1702
(1992
).56.
R. A.
Fisher
, M. M.
Nieto
, and D.
Sandberg
, “Impossibility of naively generalizing squeezed coherent states
,” 29
, 1107
–1110
(1984
);Y.-Z.
Zhang
, “Solving the two-mode squeezed harmonic oscillator and the kth-order harmonic generation in Bargmann-Hilbert spaces
,” J. Phys. A: Math. Theor.
46
, 455302
(2013
).57.
U. M.
Titulaer
and R. J.
Glauber
, “Density operators for coherent states
,” Phys. Rev.
145
, 1041
–1050
(1966
).58.
Z.
Bialynicka-Birula
, “Properties of the generalized coherent state
,” Phys. Rev.
173
, 1207
–1209
(1968
).59.
D.
Stoler
, “Generalized coherent states
,” Phys. Rev. D
4
, 2309
–2312
(1971
).60.
B.
Yurke
and D.
Stoler
, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion
,” Phys. Rev. Lett.
57
, 13
–16
(1986
).61.
G.
Kirchmair
et al, “Observation of quantum state collapse and revival due to the single-photon Kerr effect
,” Nature
495
, 205
–209
(2013
).62.
A.
Ourjoumtsev
, R.
Tualle-Brouri
, J.
Laurat
, and P.
Grangier
, “Generating optical Schrödinger kittens for quantum information processing
,” Nature
312
, 83
–86
(2006
).63.
M.
Kira
, S. W.
Koch
, R. P.
Smith
, A. E.
Hunter
, and S. T.
Cundiff
, “Quantum spectroscopy with Schrödinger-cat states
,” Nat. Phys.
7
, 799
–804
(2011
).64.
J.
Janszky
, P.
Domokos
, and P.
Adam
, “Coherent states on a circle and quantum interference
,” Phys. Rev. A
48
, 2213
–2219
(1993
).65.
P.
Domokos
, P.
Adam
, and J.
Janszky
, “One-dimensional coherent-state representation on a circle in phase space
,” Phys. Rev. A
50
, 4293
–4297
(1994
).66.
S.
Szabo
, P.
Adam
, J.
Janszky
, and P.
Domokos
, “Construction of quantum states of the radiation field by discrete coherent-state superpositions
,” Phys. Rev. A
53
, 2698
–2710
(1996
).67.
J. A.
González
and M. A.
del Olmo
, “Coherent states on the circle and quantization
,” J. Phys. A: Math. Gen.
31
, 8841
–8857
(1998
).68.
M.
Calixto
, J.
Guerrero
, and J. C.
Sánchez-Monreal
, “Sampling theorem and discrete Fourier transform on the Riemann sphere
,” J. Fourier Anal. Appl.
14
, 538
–567
(2008
).69.
M.
Calixto
, J.
Guerrero
, and J. C.
Sánchez-Monreal
, “Sampling theorem and discrete Fourier transform on the hyperboloid
,” J. Fourier Anal. Appl.
17
, 240
–264
(2011
).70.
M.
Calixto
, J.
Guerrero
, and J. C.
Sánchez-Monreal
, “Almost complete coherent state subsystems and partial reconstruction of wavefunctions in the Fock-Bargmann phase-number representation
,” J. Phys. A: Math. Theor.
45
, 244029
(2012
).71.
X.-M.
Liu
, “Orthonormalized eigenstates of
and their generation
,” J. Phys. A: Math. Gen.
32
, 8685
–8689
(1999
).72.
J.-S.
Wang
et al, “Quantum statistical properties of orthonormalized eigenstates of the operator
,” J. Phys. B: At., Mol. Opt. Phys.
35
, 2411
–2421
(2002
).73.
C.
Quesne
, “Generalized coherent states associated with the Cλ-extended oscillator
,” Ann. Phys.
293
, 147
(2001
).74.
O. I.
Marichev
, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables
(Ellis Harwood
, Chichester
, 1983
).75.
I. S.
Gradshteyn
and I. M.
Ryzhik
, Tables of Integrals, Series and Products
(Academic
, New York
, 1980
).76.
S. T.
Ali
et al, “Representations of coherent states in non-orthogonal bases
,” J. Phys. A: Math. Gen.
37
, 4407
(2004
).77.
K.
Zelaya
, O.
Rosas-Ortiz
, Z.
Blanco-Garcia
, and S.
Cruz y Cruz
, “Completeness and nonclassicality of coherent states for generalized oscillator algebras
,” Adv. Math. Phys.
2017
, 7168592
.78.
L.
Susskind
and J.
Glogower
, “Quantum mechanical phase and time operator
,” Physics
1
, 49
–61
(1964
).79.
M. M.
Nieto
, “Quantum phase and quantum phase operators: Some physics and some history
,” Phys. Scr.
T48
, 5
–12
(1993
).80.
D. T.
Pegg
and S. M.
Barnett
, “Unitary phase operator in quantum mechanics
,” Europhys. Lett.
6
, 483
–487
(1988
).81.
D. T.
Pegg
and S. M.
Barnett
, “Phase properties of the quantized single-mode electromagnetic field
,” Phys. Rev. A
39
, 1665
–1675
(1989
).82.
M. K.
Tavassoly
, “New nonlinear coherent states associated with inverse bosonic and f-deformed ladder operators
,” J. Phys. A: Math. Theor.
41
, 285305
(2008
).83.
W. H.
Louisell
, “Amplitude and phase uncertainty relations
,” Phys. Lett.
7
, 60
–61
(1963
).84.
R. G.
Newton
, “Quantum action-angle variables for harmonic oscillators
,” Ann. Phys.
124
, 327
(1980
).85.
D.
Popov
, “Spin coherent states defined in the Barut-Girardello manner
,” Proc. Rom. Acad., Ser. A
17
, 328
–335
(2016
), https://academiaromana.ro/sectii2002/proceedings/doc2016-4/07ProcRoAcad-4-2016.pdf.86.
B.
Roy
and P.
Roy
, “Gazeau-Klauder coherent state for the Morse potential and some of its properties
,” Phys. Lett. A
296
, 187
–191
(2002
).87.
M.
Angelova
and V.
Hussin
, “Generalized and Gaussian coherent states for the Morse potential
,” J. Phys. A: Math. Theor.
41
, 304016
(2008
).88.
N.
Cotfas
, “Gazeau-Klauder type coherent states for hypergeometric type operators
,” Open Phys
.
7
, 147
–159
(2009
).89.
J.
Guerrero
and V.
Aldaya
, “Inconsistencies in the description of a quantum system with a finite number of bound states by a compact dynamical group
,” J. Phys. A: Math. Gen.
39
, L267
–L276
(2006
).90.
J.
Bajer
and A.
Miranowicz
, “Quantum versus classical descriptions of sub-Poissonian light generation in three-wave mixing
,” J. Opt. B: Quantum Semiclassical Opt.
2
, L10
(2000
).91.
L.
Mandel
, “Sub-Poissonian photon statistics in resonance fluorescence
,” Opt. Lett.
4
, 205
(1979
).92.
X.-Z.
Zhang
, Z.-H.
Wang
, H.
Li
, Q.
Wu
, B.-Q.
Tang
, F.
Gao
, and J.-J.
Xu
, “Characterization of photon statistical properties with normalized Mandel parameter
,” Chin. Phys. Lett.
25
, 3976
(2008
).93.
J.-P.
Antoine
, J.-P.
Gazeau
, P.
Monceau
, J. R.
Klauder
, and K. A.
Penson
, “Temporally stable coherent states for infinite well and Pöschl-Teller potentials
,” J. Math. Phys.
42
, 2349
(2001
).94.
M. B.
Harouni
and M.
Vaseghi
, “Preparation of vibrational quantum states in nanomechanical graphene resonator
,” Laser Phys.
26
, 115204
(2016
).95.
A.
Voje
, J. M.
Kinaret
, and A.
Isacsson
, “Generating macroscopic superposition states in nanomechanical graphene resonators
,” Phys. Rev. B
85
, 205415
(2012
).96.
E.
Díaz-Bautista
and D. J.
Fernández
, “Graphene coherent states
,” Eur. Phys. J. Plus
132
, 499
(2017
).97.
S.
de Bièvre
, “Coherent states over symplectic homogenous spaces
,” J. Math. Phys.
30
, 1401
–1407
(1989
).98.
C. J.
Isham
and J. R.
Klauder
, “Coherent states for n-dimensional Euclidean groups E(n) and their application
,” J. Math. Phys.
32
, 607
–620
(1991
).99.
B.
Torresani
, “Position-frequency analysis for signals defined on spheres
,” Signal Process.
43
, 341
–346
(1995
).100.
P. L.
García de León
and J. P.
Gazeau
, “Coherent state quantization and phase operator
,” Phys. Lett. A
361
, 301
–304
(2007
).101.
R.
Fresneda
, J. P.
Gazeau
, and D.
Noguera
, “Quantum localisation on the circle
,” J. Math. Phys.
59
, 052105
(2018
).102.
S.
De Bièvre
and J. A.
González
, in Quantization and Coherent States Methods
, Proceedings of XIth Workshop on Geometric Methods in Physics, Białowieza, Poland, 1992, edited by S.
Twareque Ali
, I. M.
Mladenov
, and A.
Odzijewicz
(World Scientific
, Singapore
, 1993
), p. 152
.103.
H. A.
Kastrup
, “Quantization of the canonically conjugate pair angle and orbital angular momentum
,” Phys. Rev. A
73
, 052104
(2006
).© 2019 Author(s).
2019
Author(s)
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