Abstract

In this paper, we show that the Pegg–Barnett formalism accepts coherent states constructed as eigenstates of the annihilation operator, considering both the number and the phase. These operators are defined within a $ (s + 1) $-dimensional Hilbert space $ {{\cal H}_s} $ and with periodic conditions. The coherent states that we find are determined by the eigenvalue of the annihilation operator, which leads to a discrete spectrum. This approach allows calculation of the discrete-finite counterpart of the Wigner function in a phase space defined by the variables of number and phase.

© 2020 Optical Society of America

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References

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  1. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University, 1981), Vol. 27.
  2. P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
    [Crossref]
  3. F. London, “Winkelvariable und kanonische transformationen in der undulationsmechanik,” Z. Phys. 40, 193–210 (1927).
    [Crossref]
  4. L. Susskind and J. Glogower, “Quantum mechanical phase and time operator,” Physics Physique Fizika 1, 49–61 (1964).
    [Crossref]
  5. W. H. Louisell, “Amplitude and phase uncertainty relations,” Phys. Lett. 7, 60–61 (1963).
    [Crossref]
  6. P. Carruthers and M. M. Nieto, “Phase and angle variables in quantum mechanics,” Rev. Mod. Phys. 40, 411–440 (1968).
    [Crossref]
  7. J. Vaccaro and D. Pegg, “Physical number-phase intelligent and minimum-uncertainty states of light,” J. Mod. Opt. 37, 17–39 (1990).
    [Crossref]
  8. M. M. Nieto, “Quantum phase and quantum phase operators: some physics and some history,” Phys. Scripta T48, 5–12 (1993).
    [Crossref]
  9. I. Bialynicki-Birula, M. Freyberger, and W. Schleich, “Various measures of quantum phase uncertainty: a comparative study,” Phys. Scripta T48, 113–118 (1993).
    [Crossref]
  10. D. T. Pegg and S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).
    [Crossref]
  11. D. T. Pegg and S. M. Barnett, “Unitary phase operator in quantum mechanics,” Europhys. Lett. 6, 483 (1988).
    [Crossref]
  12. A. Perez-Leija, L. A. Andrade-Morales, F. Soto-Eguibar, A. Szameit, and H. M. Moya-Cessa, “The Pegg-Barnett phase operator and the discrete Fourier transform,” Phys. Scripta 91, 043008 (2016).
    [Crossref]
  13. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [Crossref]
  14. G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [Crossref]
  15. H.-Y. Fan and J.-H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. China 10, 1–6 (2015).
    [Crossref]
  16. N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
    [Crossref]
  17. J. V. der Jeugt, “A Wigner distribution function for finite oscillator systems,” J. Phys. A 46, 475302 (2013)..
    [Crossref]
  18. W. K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1–21 (1987).
    [Crossref]
  19. J. A. Vaccaro and D. T. Pegg, “Wigner function for number and phase,” Phys. Rev. A 41, 5156–5163 (1990).
    [Crossref]
  20. M. M. D. Galetti, “Discrete coherent states and probability distributions in finite-dimensional spaces,” Ann. Phys. 249, 454–480 (1996).
    [Crossref]
  21. G. Bjork, A. B. Klimov, and L. L. Sánchez-Soto, Progress in Optics (Elsevier B.V., 2008), Vol. 51, Chap. 7, pp. 469–516.
  22. A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267–320 (2004).
    [Crossref]
  23. A. B. Klimov, C. Muñoz, and L. L. Sánchez-Soto, “Discrete coherent and squeezed states of many-qudit systems,” Phys. Rev. A 80, 043836 (2009).
    [Crossref]
  24. A. Vourdas, Finite and Profinite Quantum Systems (Springer, 2017).
  25. T. Durt, “About mutually unbiased bases in even and odd prime power dimensions,” J. Phys. A 38, 5267–5283 (2005).
    [Crossref]
  26. M. Shalaby and A. Vourdas, “Tomographically complete sets of orthonormal bases in finite systems,” J. Phys. A 44, 345303 (2011).
    [Crossref]
  27. J. Schwinger, “Unitary operator bases,” Proc. Natl. Acad. Sci. USA 46, 570–579 (1960).
    [Crossref]
  28. J. Schwinger, “The special canonical group,” Proc. Natl. Acad. Sci. USA 46, 1401–1415 (1960).
    [Crossref]
  29. A. R. Urzúa, I. Ramos-Prieto, F. Soto-Eguibar, and H. Moya-Cessa, “Dynamical analysis of mass-spring models using Lie algebraic methods,” Physica A 540, 123193 (2019).
    [Crossref]
  30. E. Schrödinger, “Der stetige übergang von der mikro- zur makromechanik,” Naturwissenschaften 14, 664–666 (1926).
    [Crossref]
  31. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [Crossref]
  32. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
    [Crossref]
  33. J. R. Klauder, “The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers,” Ann. Phys. 11, 123–168 (1960).
    [Crossref]
  34. A. Barut and L. Girardello, “New coherent states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
    [Crossref]
  35. A. Barut and R. Raczka, Theory of Group Representations and Its Applications (World Scientific, 1977).
  36. W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
    [Crossref]
  37. J.-P. Gazeau, Coherent States in Quantum Physics (Wiley, 2009).
  38. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
    [Crossref]
  39. E. Schrödinger, “Die gegenwärtige situation in der quantenmechanik,” Naturwissenschaften 23, 823–828 (1935).
    [Crossref]
  40. V. Bužek, A. D. Wilson-Gordon, P. L. Knight, and W. K. Lai, “Coherent states in a finite-dimensional basis: their phase properties and relationship to coherent states of light,” Phys. Rev. A 45, 8079–8094 (1992).
    [Crossref]
  41. R. F. Bishop and A. Vourdas, “Displaced and squeezed parity operator: its role in classical mappings of quantum theories,” Phys. Rev. A 50, 4488–4501 (1994).
    [Crossref]
  42. H. Moya-Cessa and P. L. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
    [Crossref]
  43. A. Lukš and V. Peřinová, “Ordering of ‘ladder’ operators, the Wigner function for number and phase, and the enlarged Hilbert space,” Phys. Scripta T48, 94–99 (1993).
    [Crossref]
  44. H. Moya-Cessa, “A number-phase Wigner function,” J. Opt. B 5, S339–S341 (2003).
    [Crossref]

2019 (1)

A. R. Urzúa, I. Ramos-Prieto, F. Soto-Eguibar, and H. Moya-Cessa, “Dynamical analysis of mass-spring models using Lie algebraic methods,” Physica A 540, 123193 (2019).
[Crossref]

2016 (1)

A. Perez-Leija, L. A. Andrade-Morales, F. Soto-Eguibar, A. Szameit, and H. M. Moya-Cessa, “The Pegg-Barnett phase operator and the discrete Fourier transform,” Phys. Scripta 91, 043008 (2016).
[Crossref]

2015 (1)

H.-Y. Fan and J.-H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. China 10, 1–6 (2015).
[Crossref]

2013 (1)

J. V. der Jeugt, “A Wigner distribution function for finite oscillator systems,” J. Phys. A 46, 475302 (2013)..
[Crossref]

2011 (1)

M. Shalaby and A. Vourdas, “Tomographically complete sets of orthonormal bases in finite systems,” J. Phys. A 44, 345303 (2011).
[Crossref]

2009 (1)

A. B. Klimov, C. Muñoz, and L. L. Sánchez-Soto, “Discrete coherent and squeezed states of many-qudit systems,” Phys. Rev. A 80, 043836 (2009).
[Crossref]

2005 (1)

T. Durt, “About mutually unbiased bases in even and odd prime power dimensions,” J. Phys. A 38, 5267–5283 (2005).
[Crossref]

2004 (1)

A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267–320 (2004).
[Crossref]

2003 (1)

H. Moya-Cessa, “A number-phase Wigner function,” J. Opt. B 5, S339–S341 (2003).
[Crossref]

1998 (1)

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[Crossref]

1996 (1)

M. M. D. Galetti, “Discrete coherent states and probability distributions in finite-dimensional spaces,” Ann. Phys. 249, 454–480 (1996).
[Crossref]

1994 (2)

G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

R. F. Bishop and A. Vourdas, “Displaced and squeezed parity operator: its role in classical mappings of quantum theories,” Phys. Rev. A 50, 4488–4501 (1994).
[Crossref]

1993 (4)

H. Moya-Cessa and P. L. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[Crossref]

A. Lukš and V. Peřinová, “Ordering of ‘ladder’ operators, the Wigner function for number and phase, and the enlarged Hilbert space,” Phys. Scripta T48, 94–99 (1993).
[Crossref]

M. M. Nieto, “Quantum phase and quantum phase operators: some physics and some history,” Phys. Scripta T48, 5–12 (1993).
[Crossref]

I. Bialynicki-Birula, M. Freyberger, and W. Schleich, “Various measures of quantum phase uncertainty: a comparative study,” Phys. Scripta T48, 113–118 (1993).
[Crossref]

1992 (1)

V. Bužek, A. D. Wilson-Gordon, P. L. Knight, and W. K. Lai, “Coherent states in a finite-dimensional basis: their phase properties and relationship to coherent states of light,” Phys. Rev. A 45, 8079–8094 (1992).
[Crossref]

1990 (3)

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[Crossref]

J. A. Vaccaro and D. T. Pegg, “Wigner function for number and phase,” Phys. Rev. A 41, 5156–5163 (1990).
[Crossref]

J. Vaccaro and D. Pegg, “Physical number-phase intelligent and minimum-uncertainty states of light,” J. Mod. Opt. 37, 17–39 (1990).
[Crossref]

1989 (1)

D. T. Pegg and S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).
[Crossref]

1988 (1)

D. T. Pegg and S. M. Barnett, “Unitary phase operator in quantum mechanics,” Europhys. Lett. 6, 483 (1988).
[Crossref]

1987 (1)

W. K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1–21 (1987).
[Crossref]

1971 (1)

A. Barut and L. Girardello, “New coherent states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[Crossref]

1968 (1)

P. Carruthers and M. M. Nieto, “Phase and angle variables in quantum mechanics,” Rev. Mod. Phys. 40, 411–440 (1968).
[Crossref]

1964 (1)

L. Susskind and J. Glogower, “Quantum mechanical phase and time operator,” Physics Physique Fizika 1, 49–61 (1964).
[Crossref]

1963 (3)

W. H. Louisell, “Amplitude and phase uncertainty relations,” Phys. Lett. 7, 60–61 (1963).
[Crossref]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

1960 (3)

J. R. Klauder, “The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers,” Ann. Phys. 11, 123–168 (1960).
[Crossref]

J. Schwinger, “Unitary operator bases,” Proc. Natl. Acad. Sci. USA 46, 570–579 (1960).
[Crossref]

J. Schwinger, “The special canonical group,” Proc. Natl. Acad. Sci. USA 46, 1401–1415 (1960).
[Crossref]

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

E. Schrödinger, “Die gegenwärtige situation in der quantenmechanik,” Naturwissenschaften 23, 823–828 (1935).
[Crossref]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

1927 (2)

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
[Crossref]

F. London, “Winkelvariable und kanonische transformationen in der undulationsmechanik,” Z. Phys. 40, 193–210 (1927).
[Crossref]

1926 (1)

E. Schrödinger, “Der stetige übergang von der mikro- zur makromechanik,” Naturwissenschaften 14, 664–666 (1926).
[Crossref]

Agarwal, G.

G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Andrade-Morales, L. A.

A. Perez-Leija, L. A. Andrade-Morales, F. Soto-Eguibar, A. Szameit, and H. M. Moya-Cessa, “The Pegg-Barnett phase operator and the discrete Fourier transform,” Phys. Scripta 91, 043008 (2016).
[Crossref]

Atakishiyev, N. M.

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[Crossref]

Barnett, S. M.

D. T. Pegg and S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).
[Crossref]

D. T. Pegg and S. M. Barnett, “Unitary phase operator in quantum mechanics,” Europhys. Lett. 6, 483 (1988).
[Crossref]

Barut, A.

A. Barut and L. Girardello, “New coherent states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[Crossref]

A. Barut and R. Raczka, Theory of Group Representations and Its Applications (World Scientific, 1977).

Bialynicki-Birula, I.

I. Bialynicki-Birula, M. Freyberger, and W. Schleich, “Various measures of quantum phase uncertainty: a comparative study,” Phys. Scripta T48, 113–118 (1993).
[Crossref]

Bishop, R. F.

R. F. Bishop and A. Vourdas, “Displaced and squeezed parity operator: its role in classical mappings of quantum theories,” Phys. Rev. A 50, 4488–4501 (1994).
[Crossref]

Bjork, G.

G. Bjork, A. B. Klimov, and L. L. Sánchez-Soto, Progress in Optics (Elsevier B.V., 2008), Vol. 51, Chap. 7, pp. 469–516.

Bužek, V.

V. Bužek, A. D. Wilson-Gordon, P. L. Knight, and W. K. Lai, “Coherent states in a finite-dimensional basis: their phase properties and relationship to coherent states of light,” Phys. Rev. A 45, 8079–8094 (1992).
[Crossref]

Carruthers, P.

P. Carruthers and M. M. Nieto, “Phase and angle variables in quantum mechanics,” Rev. Mod. Phys. 40, 411–440 (1968).
[Crossref]

Chen, J.-H.

H.-Y. Fan and J.-H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. China 10, 1–6 (2015).
[Crossref]

Chumakov, S. M.

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[Crossref]

der Jeugt, J. V.

J. V. der Jeugt, “A Wigner distribution function for finite oscillator systems,” J. Phys. A 46, 475302 (2013)..
[Crossref]

Dirac, P. A. M.

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
[Crossref]

P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University, 1981), Vol. 27.

Durt, T.

T. Durt, “About mutually unbiased bases in even and odd prime power dimensions,” J. Phys. A 38, 5267–5283 (2005).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Fan, H.-Y.

H.-Y. Fan and J.-H. Chen, “On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations,” Front. Phys. China 10, 1–6 (2015).
[Crossref]

Feng, D. H.

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[Crossref]

Freyberger, M.

I. Bialynicki-Birula, M. Freyberger, and W. Schleich, “Various measures of quantum phase uncertainty: a comparative study,” Phys. Scripta T48, 113–118 (1993).
[Crossref]

Galetti, M. M. D.

M. M. D. Galetti, “Discrete coherent states and probability distributions in finite-dimensional spaces,” Ann. Phys. 249, 454–480 (1996).
[Crossref]

Gazeau, J.-P.

J.-P. Gazeau, Coherent States in Quantum Physics (Wiley, 2009).

Gilmore, R.

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[Crossref]

Girardello, L.

A. Barut and L. Girardello, “New coherent states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Glogower, J.

L. Susskind and J. Glogower, “Quantum mechanical phase and time operator,” Physics Physique Fizika 1, 49–61 (1964).
[Crossref]

Klauder, J. R.

J. R. Klauder, “The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers,” Ann. Phys. 11, 123–168 (1960).
[Crossref]

Klimov, A. B.

A. B. Klimov, C. Muñoz, and L. L. Sánchez-Soto, “Discrete coherent and squeezed states of many-qudit systems,” Phys. Rev. A 80, 043836 (2009).
[Crossref]

G. Bjork, A. B. Klimov, and L. L. Sánchez-Soto, Progress in Optics (Elsevier B.V., 2008), Vol. 51, Chap. 7, pp. 469–516.

Knight, P. L.

H. Moya-Cessa and P. L. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[Crossref]

V. Bužek, A. D. Wilson-Gordon, P. L. Knight, and W. K. Lai, “Coherent states in a finite-dimensional basis: their phase properties and relationship to coherent states of light,” Phys. Rev. A 45, 8079–8094 (1992).
[Crossref]

Lai, W. K.

V. Bužek, A. D. Wilson-Gordon, P. L. Knight, and W. K. Lai, “Coherent states in a finite-dimensional basis: their phase properties and relationship to coherent states of light,” Phys. Rev. A 45, 8079–8094 (1992).
[Crossref]

London, F.

F. London, “Winkelvariable und kanonische transformationen in der undulationsmechanik,” Z. Phys. 40, 193–210 (1927).
[Crossref]

Louisell, W. H.

W. H. Louisell, “Amplitude and phase uncertainty relations,” Phys. Lett. 7, 60–61 (1963).
[Crossref]

Lukš, A.

A. Lukš and V. Peřinová, “Ordering of ‘ladder’ operators, the Wigner function for number and phase, and the enlarged Hilbert space,” Phys. Scripta T48, 94–99 (1993).
[Crossref]

Moya-Cessa, H.

A. R. Urzúa, I. Ramos-Prieto, F. Soto-Eguibar, and H. Moya-Cessa, “Dynamical analysis of mass-spring models using Lie algebraic methods,” Physica A 540, 123193 (2019).
[Crossref]

H. Moya-Cessa, “A number-phase Wigner function,” J. Opt. B 5, S339–S341 (2003).
[Crossref]

H. Moya-Cessa and P. L. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[Crossref]

Moya-Cessa, H. M.

A. Perez-Leija, L. A. Andrade-Morales, F. Soto-Eguibar, A. Szameit, and H. M. Moya-Cessa, “The Pegg-Barnett phase operator and the discrete Fourier transform,” Phys. Scripta 91, 043008 (2016).
[Crossref]

Muñoz, C.

A. B. Klimov, C. Muñoz, and L. L. Sánchez-Soto, “Discrete coherent and squeezed states of many-qudit systems,” Phys. Rev. A 80, 043836 (2009).
[Crossref]

Nieto, M. M.

M. M. Nieto, “Quantum phase and quantum phase operators: some physics and some history,” Phys. Scripta T48, 5–12 (1993).
[Crossref]

P. Carruthers and M. M. Nieto, “Phase and angle variables in quantum mechanics,” Rev. Mod. Phys. 40, 411–440 (1968).
[Crossref]

Pegg, D.

J. Vaccaro and D. Pegg, “Physical number-phase intelligent and minimum-uncertainty states of light,” J. Mod. Opt. 37, 17–39 (1990).
[Crossref]

Pegg, D. T.

J. A. Vaccaro and D. T. Pegg, “Wigner function for number and phase,” Phys. Rev. A 41, 5156–5163 (1990).
[Crossref]

D. T. Pegg and S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).
[Crossref]

D. T. Pegg and S. M. Barnett, “Unitary phase operator in quantum mechanics,” Europhys. Lett. 6, 483 (1988).
[Crossref]

Perez-Leija, A.

A. Perez-Leija, L. A. Andrade-Morales, F. Soto-Eguibar, A. Szameit, and H. M. Moya-Cessa, “The Pegg-Barnett phase operator and the discrete Fourier transform,” Phys. Scripta 91, 043008 (2016).
[Crossref]

Perinová, V.

A. Lukš and V. Peřinová, “Ordering of ‘ladder’ operators, the Wigner function for number and phase, and the enlarged Hilbert space,” Phys. Scripta T48, 94–99 (1993).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Raczka, R.

A. Barut and R. Raczka, Theory of Group Representations and Its Applications (World Scientific, 1977).

Ramos-Prieto, I.

A. R. Urzúa, I. Ramos-Prieto, F. Soto-Eguibar, and H. Moya-Cessa, “Dynamical analysis of mass-spring models using Lie algebraic methods,” Physica A 540, 123193 (2019).
[Crossref]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Sánchez-Soto, L. L.

A. B. Klimov, C. Muñoz, and L. L. Sánchez-Soto, “Discrete coherent and squeezed states of many-qudit systems,” Phys. Rev. A 80, 043836 (2009).
[Crossref]

G. Bjork, A. B. Klimov, and L. L. Sánchez-Soto, Progress in Optics (Elsevier B.V., 2008), Vol. 51, Chap. 7, pp. 469–516.

Schleich, W.

I. Bialynicki-Birula, M. Freyberger, and W. Schleich, “Various measures of quantum phase uncertainty: a comparative study,” Phys. Scripta T48, 113–118 (1993).
[Crossref]

Schrödinger, E.

E. Schrödinger, “Die gegenwärtige situation in der quantenmechanik,” Naturwissenschaften 23, 823–828 (1935).
[Crossref]

E. Schrödinger, “Der stetige übergang von der mikro- zur makromechanik,” Naturwissenschaften 14, 664–666 (1926).
[Crossref]

Schwinger, J.

J. Schwinger, “Unitary operator bases,” Proc. Natl. Acad. Sci. USA 46, 570–579 (1960).
[Crossref]

J. Schwinger, “The special canonical group,” Proc. Natl. Acad. Sci. USA 46, 1401–1415 (1960).
[Crossref]

Shalaby, M.

M. Shalaby and A. Vourdas, “Tomographically complete sets of orthonormal bases in finite systems,” J. Phys. A 44, 345303 (2011).
[Crossref]

Simon, R.

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Figures (4)

Fig. 1.
Fig. 1. Plots of the real and imaginary parts of the eigenvalues of the number and phase annihilation operator, $ {\hat a_s} $ and $ {\hat \phi _s} $.
Fig. 2.
Fig. 2. On the left side, we show a discrete mesh created by the argument of each coherent state projected on the number states. The column on the right shows the distribution of four coherent states ($ k = 0,10,15,20 $) projected on the phase states $ |{\theta _n}\rangle $. Both cases are built in a Hilbert space of dimension 23 ($ s = 22 $).
Fig. 3.
Fig. 3. Wigner function in a rectangular and polar mesh of a number state. The first row represents the Wigner function of the number states $ |5\rangle $, $ \frac{1}{{\sqrt 2 }}( {|1\rangle + |5\rangle } ) $, and $ \frac{1}{{\sqrt 3 }}( {|0\rangle + |4\rangle + |6\rangle } ) $. In the second row, we plot the Wigner function where, depending on the Hilbert space, the number of points changes; for example, if the dimension is 11, we obtain phase endecagonal bars.
Fig. 4.
Fig. 4. First we plot a coherent state with $ {\alpha _0} $, and then the superposition of two coherent states with opposite phases, where the Hilbert space dimension is $ s + 1 = 31 $.

Equations (37)

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F ^ s = 1 s + 1 k , n = 0 s exp ( i 2 π k n s + 1 ) | k n | ,
F ^ s | m = | θ m = 1 s + 1 k = 0 s exp ( i θ m k ) | k ,
a ^ s = m = 0 s 1 m + 1 | m m + 1 | + s + 1 | s 0 | ,
a ^ s = m = 0 s 1 m + 1 | m + 1 m | + s + 1 | 0 s | .
[ a ^ s , a ^ s ] = m = 0 s | m m | ( s + 1 ) | 0 0 | .
ϕ ^ s = m = 0 s 1 θ m + 1 | θ m θ m + 1 | + 2 π | θ s θ 0 | ,
ϕ ^ s = m = 0 s 1 θ m + 1 | θ m + 1 θ m | + 2 π | θ 0 θ s | .
N ^ s ( C ) = a ^ s a ^ s = m = 0 s m | m m | + ( s + 1 ) | 0 0 | ,
Φ ^ s ( C ) = ϕ ^ s ϕ ^ s = m = 0 s θ m | θ m θ m | + 2 π | θ 0 θ 0 | .
Φ ^ s ( P B ) = m = 0 s θ m | θ m θ m |
V ^ s = e i Φ ^ s ( C ) = m = 0 s 1 | m m + 1 | + | s 0 | ,
V ^ s = e i Φ ^ s ( C ) = m = 0 s 1 | m + 1 m | + | 0 s | .
U ^ s = e i 2 π s + 1 N ^ s ( C ) = m = 0 s 1 | θ m θ m + 1 | + | θ s θ 0 | ,
U ^ s = e i 2 π s + 1 N ^ s ( C ) = m = 0 s 1 | θ m + 1 θ m | + | θ 0 θ s | .
V ^ s | m = | m 1 , V ^ s | 0 = | s , V ^ s | m = | m + 1 , V ^ s | s = | 0 ,
U ^ s | θ m = | θ m 1 , U ^ s | θ 0 = | θ s , U ^ s | θ m = | θ m + 1 , U ^ s | θ s = | θ 0 .
a ^ s | α k = α k | α k ,
ϕ ^ s | β k = β k | β k .
| α k = C j = 0 s α k j j ! | j , k = 0 , 1 , 2 , 3 , , s ,
C = exp ( i μ ) e η s ! Γ { s + 1 , η } ,
α k = [ ( s + 1 ) ! ] 1 s + 1 ω k , k = 0 , 1 , 2 , , s ,
| β k = C j = 0 s β k j θ j ! | θ j , k = 0 , 1 , 2 , 3 , , s ,
β k = 2 π s + 1 [ ( s + 1 ) ! ] 1 s + 1 ω k , k = 0 , 1 , 2 , , s .
P ( n | α k ) = | n | α k | 2 = | C | 2 [ ( s + 1 ) ! ] n s + 1 n ! .
P ( θ n | α k ) = | θ n | α k | 2 = | C | 2 s + 1 j , l = 0 s ( s + 1 ) ! j + l s + 1 j ! l ! × exp [ i 2 π s + 1 ( j l ) ( k n ) ] ,
P ( n | β k ) = | n | β k | 2 = | C | 2 s + 1 j , l = 0 s ( s + 1 ) ! j + l s + 1 j ! l ! × exp [ i 2 π s + 1 ( j l ) ( k + n ) ] .
( V ^ s ) l | m = { | m + l i f m + l s | l 1 i f m + l > s ,
( U ^ s ) l | θ m = { | θ m + l i f m + l s | θ l 1 i f m + l > s ,
D ^ s ( n , k ) = 1 s + 1 exp ( i π n k s + 1 ) ( U ^ s ) n ( V ^ s ) k .
D ^ s ( n , k ) | m = 1 s + 1 exp [ i π s + 1 ( n k + 2 n m ) ] | m + k ,
D ^ s ( n , k ) | θ m = 1 s + 1 exp [ i π s + 1 ( n k + 2 k m ) ] | θ m + n ;
P ^ ( n , k ) = D ^ s ( n , k ) F ^ s 2 D ^ s ( n , k ) .
W n , k ( ρ ^ s ) = T r [ P ^ ( n , k ) ρ ^ s ] ,
W n , k ( ρ ^ s ) = 1 s + 1 m = 0 s exp ( i 4 π m n s + 1 ) k m | ρ ^ s | k + m = 1 s + 1 m = 0 s exp ( i 4 π m k s + 1 ) θ n m | ρ ^ s | θ n + m .
W n , k ( | α l α l | ) = | C | 2 s + 1 m = 0 s exp ( i 4 π n m s + 1 ) × α l m o d ( k m , s + 1 ) α l m o d ( k + m , s + 1 ) m o d ( k m , s + 1 ) ! m o d ( k + m , s + 1 ) ! ,
| ψ ± s = 1 2 ( | α l ± | α l ) , l l ,
W n , k ( ρ ^ s ± ) = [ W n , k ( | α l α l | ) + W n , k ( | α l α l | ) ± W n , k ( | α l α l | ) ± W n , k ( | α l α l | ) ] .

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