Abstract

We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek’s complex energy function, and Husimi and Glauber–Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.

© 2008 Optical Society of America

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  1. M. J. Bastiaans, The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, eds. (Elsevier,1997), pp. 375-426.
  2. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
    [CrossRef]
  3. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
    [CrossRef]
  4. H. Moya-Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1-41 (2006).
    [CrossRef]
  5. E. P. Wigner, “On the quantum correction for thermodynamic equiligrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  6. M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046-1049 (2003).
    [CrossRef]
  7. C. Gonzalo, J. Bescós, L. R. Berriel-Valdos, and P. Artal, “Optical digital implementations of the Wigner distribution function: use in space variant filtering of real images,” Appl. Opt. 29, 2569-2575 (1990).
    [CrossRef] [PubMed]
  8. L. R. Berriel-Valdos, J. Carranza, and J. L. Juarez-Perez, “The Wigner distribution function in the recovering of coherence from two mutually spatial incoherent source points,” in Proceedings of the 8th International Conference on Squeezed States and Uncertainty Relations, H. Moya-Cessa, R. Jauregui, S. Hacyan, and O. Castaños, eds. (Rinton, 2003) pp. 42-48.
  9. D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
    [CrossRef] [PubMed]
  10. P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
    [CrossRef] [PubMed]
  11. H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).
  12. H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
    [CrossRef]
  13. J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
    [CrossRef]
  14. A. N. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
    [CrossRef]
  15. L. Praxmeyer and K. Wódkiewicz, “Quantum interference in the Kirkwood-Rihaczek representation,” Opt. Commun. 223, 349-365 (2003).
    [CrossRef]
  16. L. Praxmeyer and K. Wódkiewicz, “Hydrogen atom in phase space: the Kirkwood-Rihaczek representation,” Phys. Rev. A 67, 054502 (2003).
    [CrossRef]
  17. H. Moya-Cessa and P. L. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479-2481 (1993).
    [CrossRef] [PubMed]
  18. P. M. Woodward, Probability and Information Theory with Applications to Radars (McGraw-Hill, 1953).
  19. S. Stenholm and N. Vitanov, “Ambiguity in quantum optics: the pure state,” J. Mod. Optics 46, 239-253 (1999).
  20. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).
  21. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766-2788 (1963).
    [CrossRef]
  22. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277-279 (1963).
    [CrossRef]
  23. L. Cohen, Time Frequency Analysis (Prentice-Hall, 1995).
  24. H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147-211 (1995).
    [CrossRef]
  25. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).
  26. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic,1985).
  27. R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).

2006 (1)

H. Moya-Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1-41 (2006).
[CrossRef]

2003 (4)

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046-1049 (2003).
[CrossRef]

L. Praxmeyer and K. Wódkiewicz, “Quantum interference in the Kirkwood-Rihaczek representation,” Opt. Commun. 223, 349-365 (2003).
[CrossRef]

L. Praxmeyer and K. Wódkiewicz, “Hydrogen atom in phase space: the Kirkwood-Rihaczek representation,” Phys. Rev. A 67, 054502 (2003).
[CrossRef]

L. R. Berriel-Valdos, J. Carranza, and J. L. Juarez-Perez, “The Wigner distribution function in the recovering of coherence from two mutually spatial incoherent source points,” in Proceedings of the 8th International Conference on Squeezed States and Uncertainty Relations, H. Moya-Cessa, R. Jauregui, S. Hacyan, and O. Castaños, eds. (Rinton, 2003) pp. 42-48.

2002 (1)

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

2001 (2)

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
[CrossRef]

R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).

1999 (3)

H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).

H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
[CrossRef]

S. Stenholm and N. Vitanov, “Ambiguity in quantum optics: the pure state,” J. Mod. Optics 46, 239-253 (1999).

1997 (1)

M. J. Bastiaans, The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, eds. (Elsevier,1997), pp. 375-426.

1996 (1)

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

1995 (2)

L. Cohen, Time Frequency Analysis (Prentice-Hall, 1995).

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147-211 (1995).
[CrossRef]

1993 (1)

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

1990 (1)

1985 (1)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic,1985).

1984 (1)

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

1973 (1)

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).

1968 (1)

A. N. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
[CrossRef]

1963 (2)

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]

1953 (1)

P. M. Woodward, Probability and Information Theory with Applications to Radars (McGraw-Hill, 1953).

1940 (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

1933 (1)

J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equiligrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic,1985).

Artal, P.

Auffeves, A.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Bastiaans, M. J.

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046-1049 (2003).
[CrossRef]

M. J. Bastiaans, The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, eds. (Elsevier,1997), pp. 375-426.

Berriel-Valdos, L. R.

L. R. Berriel-Valdos, J. Carranza, and J. L. Juarez-Perez, “The Wigner distribution function in the recovering of coherence from two mutually spatial incoherent source points,” in Proceedings of the 8th International Conference on Squeezed States and Uncertainty Relations, H. Moya-Cessa, R. Jauregui, S. Hacyan, and O. Castaños, eds. (Rinton, 2003) pp. 42-48.

C. Gonzalo, J. Bescós, L. R. Berriel-Valdos, and P. Artal, “Optical digital implementations of the Wigner distribution function: use in space variant filtering of real images,” Appl. Opt. 29, 2569-2575 (1990).
[CrossRef] [PubMed]

Bertet, P.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Bescós, J.

Brune, M.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Carranza, J.

L. R. Berriel-Valdos, J. Carranza, and J. L. Juarez-Perez, “The Wigner distribution function in the recovering of coherence from two mutually spatial incoherent source points,” in Proceedings of the 8th International Conference on Squeezed States and Uncertainty Relations, H. Moya-Cessa, R. Jauregui, S. Hacyan, and O. Castaños, eds. (Rinton, 2003) pp. 42-48.

Cohen, L.

L. Cohen, Time Frequency Analysis (Prentice-Hall, 1995).

Dutra, S. M.

H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
[CrossRef]

H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).

Glauber, R. J.

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

Gonzalo, C.

Haroche, S.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Hillery, M.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Husimi, K.

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

Itano, W. M.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

Juarez-Perez, J. L.

L. R. Berriel-Valdos, J. Carranza, and J. L. Juarez-Perez, “The Wigner distribution function in the recovering of coherence from two mutually spatial incoherent source points,” in Proceedings of the 8th International Conference on Squeezed States and Uncertainty Relations, H. Moya-Cessa, R. Jauregui, S. Hacyan, and O. Castaños, eds. (Rinton, 2003) pp. 42-48.

King, B. E.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

Kirkwood, J. G.

J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Knight, P. L.

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

Lee, H.-W.

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147-211 (1995).
[CrossRef]

Leibfried, D.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).

Maioli, P.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Meekhof, D. M.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

Meunier, T.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Monroe, C.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

Moya-Cessa, H.

H. Moya-Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1-41 (2006).
[CrossRef]

H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
[CrossRef]

H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).

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

O'Connell, R. F.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Osnaghi, S.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Praxmeyer, L.

L. Praxmeyer and K. Wódkiewicz, “Hydrogen atom in phase space: the Kirkwood-Rihaczek representation,” Phys. Rev. A 67, 054502 (2003).
[CrossRef]

L. Praxmeyer and K. Wódkiewicz, “Quantum interference in the Kirkwood-Rihaczek representation,” Opt. Commun. 223, 349-365 (2003).
[CrossRef]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).

Raimond, J. M.

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

Rihaczek, A. N.

A. N. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
[CrossRef]

Roversi, J. A.

H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).

H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
[CrossRef]

Schleich, W. P.

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
[CrossRef]

Scully, M. O.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Stenholm, S.

S. Stenholm and N. Vitanov, “Ambiguity in quantum optics: the pure state,” J. Mod. Optics 46, 239-253 (1999).

Sudarshan, E. C. G.

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

Vidiella-Barranco, A.

H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).

H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
[CrossRef]

Vitanov, N.

S. Stenholm and N. Vitanov, “Ambiguity in quantum optics: the pure state,” J. Mod. Optics 46, 239-253 (1999).

Wigner, E. P.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

E. P. Wigner, “On the quantum correction for thermodynamic equiligrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wineland, D. J.

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

Wódkiewicz, K.

L. Praxmeyer and K. Wódkiewicz, “Quantum interference in the Kirkwood-Rihaczek representation,” Opt. Commun. 223, 349-365 (2003).
[CrossRef]

L. Praxmeyer and K. Wódkiewicz, “Hydrogen atom in phase space: the Kirkwood-Rihaczek representation,” Phys. Rev. A 67, 054502 (2003).
[CrossRef]

Wolf, K. B.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radars (McGraw-Hill, 1953).

Appl. Opt. (1)

IEEE Trans. Inf. Theory (1)

A. N. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
[CrossRef]

J. Mod. Opt. (1)

H. Moya-Cessa, S. M. Dutra, J. A. Roversi, and A. Vidiella-Barranco, “Quantum state reconstruction in the presence of dissipation,” J. Mod. Opt. 46, 555-558 (1999).

J. Mod. Optics (1)

S. Stenholm and N. Vitanov, “Ambiguity in quantum optics: the pure state,” J. Mod. Optics 46, 239-253 (1999).

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

L. Praxmeyer and K. Wódkiewicz, “Quantum interference in the Kirkwood-Rihaczek representation,” Opt. Commun. 223, 349-365 (2003).
[CrossRef]

Phys. Rep. (3)

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147-211 (1995).
[CrossRef]

H. Moya-Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1-41 (2006).
[CrossRef]

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Phys. Rev. (3)

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

E. P. Wigner, “On the quantum correction for thermodynamic equiligrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Phys. Rev. A (2)

L. Praxmeyer and K. Wódkiewicz, “Hydrogen atom in phase space: the Kirkwood-Rihaczek representation,” Phys. Rev. A 67, 054502 (2003).
[CrossRef]

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

Phys. Rev. A. (1)

H. Moya-Cessa, J. A. Roversi, S. M. Dutra, and A. Vidiella-Barranco, “Recovering coherence from decoherence: a method of quantum state reconstrucion,” Phys. Rev. A. 60, 4029-4033 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Phys. Rev. Lett. 77, 4281-4285(1996).
[CrossRef] [PubMed]

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[CrossRef] [PubMed]

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

Proc. Phys. Math. Soc. Jpn. (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

Other (8)

L. R. Berriel-Valdos, J. Carranza, and J. L. Juarez-Perez, “The Wigner distribution function in the recovering of coherence from two mutually spatial incoherent source points,” in Proceedings of the 8th International Conference on Squeezed States and Uncertainty Relations, H. Moya-Cessa, R. Jauregui, S. Hacyan, and O. Castaños, eds. (Rinton, 2003) pp. 42-48.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic,1985).

R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).

M. J. Bastiaans, The Wigner Distribution--Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, eds. (Elsevier,1997), pp. 375-426.

P. M. Woodward, Probability and Information Theory with Applications to Radars (McGraw-Hill, 1953).

L. Cohen, Time Frequency Analysis (Prentice-Hall, 1995).

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).
[CrossRef]

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Equations (58)

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W ( q , p ) = 1 2 π d u e i u p q + u 2 | ρ | q u 2 .
W ( α ) = 1 4 π 2 d 2 β exp ( α β * α * β ) C ( β ) ,
a = q ^ + i p ^ 2 = q + d d q 2 a q ^ i p ^ 2 = q d d q 2 ,
C ( β ) = Tr { ρ exp ( β a β * a ) } ,
Q ( α ) = 1 π 2 d 2 β exp ( α β * α * β ) Tr { ρ exp ( β * a ) exp ( β a ) } ,
Q ( α ) = 1 π α | ρ | α .
F ( α , s ) = 1 π d 2 β C ( β , s ) exp ( α β * α * β ) ,
C ( β , s ) = Tr { D ( β ) ρ } exp ( s | β | 2 / 2 ) ,
Q ( α ) = d 2 β G ( β ) exp ( α β * α * β ) ,
W ( α ) = d 2 β G ( β ) exp ( α β * α * β ) exp ( | β | 2 / 2 ) ,
G ( β ) = 1 π 2 Tr { D ( β ) ρ } exp ( | β | 2 / 2 ) ,
W ( α ) = n = 0 2 n n ! d 2 β G ( β ) exp ( α β * α * β ) | β | 2 n .
α α * exp ( α β * α * β ) = | β | 2 exp ( α β * α * β ) ,
W ( α ) = n = 0 2 n n ! ( α α * ) n Q ( α ) ,
W ( α ) = exp ( 1 2 α α * ) Q ( α ) .
ρ = 1 π 2 d 2 α d 2 β α | ρ | β | α β | .
ρ = d 2 α P ( α ) | α α | ,
P ( α ) = 1 π 2 d 2 β exp ( α β * α * β ) Tr { ρ exp ( β a ) exp ( β * a ) } ,
P ( α ) = F ( α , 1 ) = 1 π d 2 β C ( β , 1 ) exp ( α β * α * β ) .
W C ( x , u ) = 1 2 π d y d x d u ϕ ( y + 1 2 x ) ϕ * ( y 1 2 x ) k ( y , u , x , u ) e i ( u x u x + u y ) ,
K ( β ) = d 2 α e β α * β * α e α 2 α * 2 4 C ( α ) .
K ( q , p ) = d u d v e i u p e i v q Tr { ρ e i v q ^ e i u p ^ }
K ( β ) = e 1 4 2 2 β e 1 4 2 2 β * W ( β ) .
W ( β ) = Tr [ ( 1 ) n ^ D ( β ) ρ D ( β ) ] ,
W ( β ) = Tr [ ( 1 ) n ^ ρ D ( 2 β ) ] ,
W ( β ) = Tr [ ( 1 ) n ^ ρ e 2 | β | 2 e 2 β a e 2 β * a ] .
K ( β , β * ) = e 1 4 2 2 β e 1 4 2 2 β * W ( β , β * ) = Tr [ ( 1 ) n ^ ρ e 1 4 2 2 β e 1 4 2 2 β * D ( 2 β ) ] .
e 1 4 2 2 β * D ( 2 β ) = e β 2 e 2 β ( a + a β * ) e a 2 e 2 β * a .
e t 2 + 2 t x = k = 0 H k ( x ) t k k ! ,
e 1 4 2 2 β * D ( 2 β ) = k = 0 H k ( a + a β * ) β k k ! e a 2 e 2 β * a ,
2 n β 2 n k = 0 H k ( x ) β k k ! = k = 0 H k + 2 n ( x ) β k k ! ,
e 1 4 2 2 β e 1 4 2 2 β * D ( 2 β ) = n = 0 k = 0 ( 1 4 ) n n ! H k + 2 n ( a + a β * ) ( β ) k k ! e a 2 e 2 β * a .
H p ( x ) = 2 p π d t ( x + i t ) p e t 2 ,
K ( β , β * ) = e β * 2 e 2 β β * π d x d t e ( 2 2 x + 2 β * + 2 β ) i t e 2 x 2 e 2 2 x ( β * + β ) x | e a 2 e 2 β * a ( 1 ) n ^ ρ | x
d t e i k t = 2 π δ ( k )
K ( β , β * ) = 2 π e β * 2 e 2 β β * d x δ ( 2 2 x 2 β * 2 β ) e 2 x 2 e 2 2 x ( β * + β ) ,
× x | e a 2 e 2 β * a ( 1 ) n ^ ρ | x .
K ( β , β * ) = e 2 β * 2 e 2 β β * π 2 e 2 ( β * + β 2 ) 2 β * + β 2 | e ( a β * ) 2 ( 1 ) n ^ ρ | β * + β 2 ,
K ( β , β * ) = π 2 e β 2 β * 2 X | D ( β * ) e a 2 D ( β * ) ( 1 ) n ^ ρ | X ,
ψ n ( x ) = π 1 / 4 2 n n ! e x 2 / 2 H n ( x ) ,
f ( x ) = n = 0 c n ψ n ( x ) ,
c n = d x f ( x ) ψ n ( x ) ,
f = n = 0 c n | n ,
m | n = d x ψ m * ( x ) ψ n ( x ) = δ n m
p | p = 1 2 π d x e i ( p p ) q = δ ( p p ) ,
e i p q / 2 π = d p δ ( p p ) e i p q / 2 π ,
| p = d p δ ( p p ) | p = d p p | p | p ,
| p = ( d p | p p | ) | p = 1 | p ,
d p | p p | = 1 .
q ^ | q = q | q
q | q = q | q ^ .
q | q ( q q ) = 0 ,
q | q = δ ( q q ) .
1 = d q | q q | ,
| Ψ = 1 | Ψ = d q | q q | | Ψ = d q Ψ ( q ) | q ,
Ψ | A ^ | Ψ = Ψ | A ^ 1 | Ψ = Ψ | A ^ d q | q q | | Ψ = d q Ψ | A ^ | q q | Ψ ,
Ψ | A ^ | Ψ = d q q | Ψ Ψ | A ^ | q .
Ψ | A ^ | Ψ = n = 0 n | Ψ Ψ | A ^ | n Tr { ρ A ^ } .

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