Abstract

We analytically investigate the Wigner function (WF) and the optical tomography for the two-variable Hermite polynomial state (THPS) and the effect of decoherence on the THPS via the entangled-state representations. The nonclassicality of the THPS is investigated in terms of the partial negativity of the WF, which depends much on the polynomial orders m, n and the squeezing parameter r. We also extend recent theoretical studies of optical tomography and introduce the radon transformation between the Wigner operator and the projection operator of the entangled state |η,τ1,τ2 to derive the tomogram of the THPS. Furthermore, we investigate how the WF for the THPS evolves undergoing the thermal channel. The results show that quantum dissipation in the decoherence channel can thoroughly deteriorate the nonclassicality of the THPS, and thermal noise leads to much quicker decoherence than amplitude damping.

© 2013 Optical Society of America

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  1. V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
    [CrossRef]
  2. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  3. G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I: mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970).
    [CrossRef]
  4. 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]
  5. H. J. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields (Springer, 2007).
  6. W. P. Schleich, Quantum Optics in Phase Space (Wiley, 2001).
  7. V. V. Dodonov and V. I. Man’ko, Theory of Nonclassical States of Light (Taylor & Francis, 2003).
  8. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
    [CrossRef]
  9. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [CrossRef]
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    [CrossRef]
  11. V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B 7, S490–S499 (2005).
    [CrossRef]
  12. V. V. Dodonov and L. A. de Souza, “Decoherence of multicomponent symmetrical superpositions of displaced quantum states,” J. Phys. A 40, 13955–13974 (2007).
    [CrossRef]
  13. J. A. Bergou, M. Hillery, and D. Q. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
    [CrossRef]
  14. V. Buzek, “SU(1, 1) squeezing of SU(1, 1) generalized coherent states,” J. Mod. Opt. 37, 303–316 (1990).
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  15. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
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  16. P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynôme d’ermite (Gauthier-Villards, 1926).
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    [CrossRef]
  18. X. G. Meng, J. S. Wang, and Y. L. Li, “Wigner function and tomogram of the Hermite polynomial state,” Chin. Phys. B 16, 2415–2421 (2007).
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  22. H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
    [CrossRef]
  23. H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles’ relative position and total momentum,” Phys. Rev. A 49, 704–707 (1994).
    [CrossRef]
  24. A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of nonclassicality,” J. Opt. B 6, 396–404 (2004).
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  25. M. G. Raymer and M. Beck, “Experimental quantum state tomography of optical fields and ultrafast statistical sampling,” Lect. Notes Phys. 649, 235–295 (2004).
    [CrossRef]
  26. V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
    [CrossRef]
  27. V. V. Dodonov and V. I. Man’ko, “Invariants and the evolution of nonstationary quantum system,” in Proceedings of the Lebedev Physical Institute (Nova Science, 1989), pp. 3–101.
  28. A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
    [CrossRef]
  29. Y. A. Korennoy and V. I. Man’ko, “Optical tomography of photon-added coherent states, even and odd coherent states, and thermal states,” Phys. Rev. A 83, 053817 (2011).
    [CrossRef]
  30. C. L. Mehta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
    [CrossRef]
  31. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
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    [CrossRef]

2012 (3)

2011 (3)

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
[CrossRef]

Y. A. Korennoy and V. I. Man’ko, “Optical tomography of photon-added coherent states, even and odd coherent states, and thermal states,” Phys. Rev. A 83, 053817 (2011).
[CrossRef]

2009 (1)

A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
[CrossRef]

2007 (2)

X. G. Meng, J. S. Wang, and Y. L. Li, “Wigner function and tomogram of the Hermite polynomial state,” Chin. Phys. B 16, 2415–2421 (2007).
[CrossRef]

V. V. Dodonov and L. A. de Souza, “Decoherence of multicomponent symmetrical superpositions of displaced quantum states,” J. Phys. A 40, 13955–13974 (2007).
[CrossRef]

2006 (1)

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

2005 (1)

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B 7, S490–S499 (2005).
[CrossRef]

2004 (2)

A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of nonclassicality,” J. Opt. B 6, 396–404 (2004).
[CrossRef]

M. G. Raymer and M. Beck, “Experimental quantum state tomography of optical fields and ultrafast statistical sampling,” Lect. Notes Phys. 649, 235–295 (2004).
[CrossRef]

2002 (1)

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

1998 (1)

H. Y. Fan and Y. Fan, “New representation of thermal states in thermal field dynamics,” Phys. Lett. A 246, 242–246 (1998).
[CrossRef]

1994 (1)

H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles’ relative position and total momentum,” Phys. Rev. A 49, 704–707 (1994).
[CrossRef]

1993 (2)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef]

H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
[CrossRef]

1991 (1)

J. A. Bergou, M. Hillery, and D. Q. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

1990 (1)

V. Buzek, “SU(1, 1) squeezing of SU(1, 1) generalized coherent states,” J. Mod. Opt. 37, 303–316 (1990).
[CrossRef]

1989 (1)

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef]

1987 (1)

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

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]

1970 (1)

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I: mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970).
[CrossRef]

1967 (1)

C. L. Mehta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

1932 (1)

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

Agarwal, G. S.

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I: mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186 (1970).
[CrossRef]

Appell, P.

P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynôme d’ermite (Gauthier-Villards, 1926).

Beck, M.

M. G. Raymer and M. Beck, “Experimental quantum state tomography of optical fields and ultrafast statistical sampling,” Lect. Notes Phys. 649, 235–295 (2004).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef]

Bergou, J. A.

J. A. Bergou, M. Hillery, and D. Q. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[CrossRef]

Buzek, V.

V. Buzek, “SU(1, 1) squeezing of SU(1, 1) generalized coherent states,” J. Mod. Opt. 37, 303–316 (1990).
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields (Springer, 2007).

Carranza, R.

Castellanos-Beltran, M. A.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Cerf, N. J.

C. Navarrete-Benlloch, R. Garcia-Patron, J. H. Shapiro, and N. J. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A 86, 012328 (2012).
[CrossRef]

de Souza, L. A.

V. V. Dodonov and L. A. de Souza, “Decoherence of multicomponent symmetrical superpositions of displaced quantum states,” J. Phys. A 40, 13955–13974 (2007).
[CrossRef]

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B 7, S490–S499 (2005).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov and L. A. de Souza, “Decoherence of multicomponent symmetrical superpositions of displaced quantum states,” J. Phys. A 40, 13955–13974 (2007).
[CrossRef]

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B 7, S490–S499 (2005).
[CrossRef]

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

V. V. Dodonov and V. I. Man’ko, Theory of Nonclassical States of Light (Taylor & Francis, 2003).

V. V. Dodonov and V. I. Man’ko, “Invariants and the evolution of nonstationary quantum system,” in Proceedings of the Lebedev Physical Institute (Nova Science, 1989), pp. 3–101.

Fan, H. Y.

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

H. Y. Fan and Y. Fan, “New representation of thermal states in thermal field dynamics,” Phys. Lett. A 246, 242–246 (1998).
[CrossRef]

H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles’ relative position and total momentum,” Phys. Rev. A 49, 704–707 (1994).
[CrossRef]

H. Y. Fan and X. Ye, “Hermite polynomial states in two-mode Fock space,” Phys. Lett. A 175, 387–390 (1993).
[CrossRef]

Fan, Y.

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

H. Y. Fan and Y. Fan, “New representation of thermal states in thermal field dynamics,” Phys. Lett. A 246, 242–246 (1998).
[CrossRef]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef]

Garcia-Patron, R.

C. Navarrete-Benlloch, R. Garcia-Patron, J. H. Shapiro, and N. J. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A 86, 012328 (2012).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Gerry, C. C.

Glancy, S.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Hillery, M.

J. A. Bergou, M. Hillery, and D. Q. Yu, “Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states,” Phys. Rev. A 43, 515–520 (1991).
[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]

Hilton, G. C.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Ibort, A.

A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
[CrossRef]

Irwin, K. D.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Kampé de Fériet, J.

P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynôme d’ermite (Gauthier-Villards, 1926).

Kenfack, A.

A. Kenfack and K. Zyczkowski, “Negativity of the Wigner function as an indicator of nonclassicality,” J. Opt. B 6, 396–404 (2004).
[CrossRef]

Klauder, J. R.

H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles’ relative position and total momentum,” Phys. Rev. A 49, 704–707 (1994).
[CrossRef]

Knight, P. L.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

Knill, E.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Korennoy, Y. A.

Y. A. Korennoy and V. I. Man’ko, “Optical tomography of photon-added coherent states, even and odd coherent states, and thermal states,” Phys. Rev. A 83, 053817 (2011).
[CrossRef]

Ku, H. S.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Lehnert, K. W.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Li, S. X.

Li, Y. L.

X. G. Meng, J. S. Wang, and Y. L. Li, “Wigner function and tomogram of the Hermite polynomial state,” Chin. Phys. B 16, 2415–2421 (2007).
[CrossRef]

Loudon, R.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

Lu, H. L.

H. Y. Fan, H. L. Lu, and Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480–494 (2006).
[CrossRef]

Mallet, F.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Man’ko, V. I.

V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
[CrossRef]

Y. A. Korennoy and V. I. Man’ko, “Optical tomography of photon-added coherent states, even and odd coherent states, and thermal states,” Phys. Rev. A 83, 053817 (2011).
[CrossRef]

A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
[CrossRef]

V. V. Dodonov and V. I. Man’ko, “Invariants and the evolution of nonstationary quantum system,” in Proceedings of the Lebedev Physical Institute (Nova Science, 1989), pp. 3–101.

V. V. Dodonov and V. I. Man’ko, Theory of Nonclassical States of Light (Taylor & Francis, 2003).

Marmo, G.

V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
[CrossRef]

A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
[CrossRef]

Mehta, C. L.

C. L. Mehta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

Meng, X. G.

X. G. Meng, J. S. Wang, and Y. L. Li, “Wigner function and tomogram of the Hermite polynomial state,” Chin. Phys. B 16, 2415–2421 (2007).
[CrossRef]

Navarrete-Benlloch, C.

C. Navarrete-Benlloch, R. Garcia-Patron, J. H. Shapiro, and N. J. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A 86, 012328 (2012).
[CrossRef]

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]

Porzio, A.

V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
[CrossRef]

Raymer, M. G.

M. G. Raymer and M. Beck, “Experimental quantum state tomography of optical fields and ultrafast statistical sampling,” Lect. Notes Phys. 649, 235–295 (2004).
[CrossRef]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef]

Risken, H.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef]

Schleich, W. P.

W. P. Schleich, Quantum Optics in Phase Space (Wiley, 2001).

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]

Shapiro, J. H.

C. Navarrete-Benlloch, R. Garcia-Patron, J. H. Shapiro, and N. J. Cerf, “Enhancing quantum entanglement by photon addition and subtraction,” Phys. Rev. A 86, 012328 (2012).
[CrossRef]

Simoni, A.

A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
[CrossRef]

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef]

Solimeno, S.

V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
[CrossRef]

Tang, H. Q.

Vale, L. R.

F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
[CrossRef]

Ventriglia, F.

V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno, and F. Ventriglia, “Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation,” Phys. Scr. 83, 045001 (2011).
[CrossRef]

A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, “An introduction to the tomographic picture of quantum mechanics,” Phys. Scr. 79, 065013 (2009).
[CrossRef]

Vogel, K.

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989).
[CrossRef]

Wang, J. S.

X. G. Meng, J. S. Wang, and Y. L. Li, “Wigner function and tomogram of the Hermite polynomial state,” Chin. Phys. B 16, 2415–2421 (2007).
[CrossRef]

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 equilibrium,” Phys. Rev. 40, 749–759 (1932).
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Figures (4)

Fig. 1.
Fig. 1.

WF W(ρ,γ) in (Reρ, Reγ, Imρ=0, and Imγ=0) phase space when (a) m=n=0, r=0.2; (b) m=0, n=3, r=0.2; (c) m=1, n=3, r=0.2; (d) m=3, n=3, r=0.2; (e) m=4, n=3, r=0.2; and (f) m=1, n=3, r=1.

Fig. 2.
Fig. 2.

Evolution of negative volume of W(ρ,γ) for the THPS versus r with n=3 for m=4, 0, 1, and 3 (from upper to lower curves).

Fig. 3.
Fig. 3.

Time evolutions of the WF distributions in (Reα, Reβ, Imα=0, and Imβ=0) phase space under the TC are depicted with given n=3, m=1, and r=0.2 for (a) n¯=0 and κt=0.1, (b) n¯=0 and κt=0.5, (c) n¯=0.8 and κt=0.1, and (d) n¯=0.8 and κt=0.5.

Fig. 4.
Fig. 4.

Evolution of negative volume of W(α,β,t) versus (a) κt with n¯=0.8, n=3, and r=0.2 for m=0, 1, 4, and 3 (from upper to lower curves) and (b) n¯ with κt=0.1, n=3, and r=0.2 for m=0, 4, 1, and 3 (from upper to lower curves).

Equations (44)

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|m,ϵ=Cm(ϵ)S(r)Hm(iξ(ϵ)a)|0,
[Cm(ϵ)]2=n=0[m/2](m!)2[4|ξ(ϵ)|2]m2n(m2n)!(n!)2.
Z1=12(ab+ab),Z2=i2(abab),
(Z1+iεZ2)|n,m,ε=β|n,m,ε,
|n,m,ε=Cm,n(ε)S2(r)Hm,n(fa,fb)|00,
Cm,n2(ε)=l=0min(m,n)(ml)(nl)m!n!|f|2(m+n2l)
Hm,n(fa,fb)=l=0min(m,n)(1)lm!n!fm+n2ll!(lm)!(ln)!amlbnl
Δ(ρ,γ)=d2ηπ3|γηγ+η|exp(ηρ*η*ρ),
|η=exp[|η|2/2+ηaη*b+ab]|00,η=η1+iη2,
W(ρ,γ)=Cm,n2(ε)00|Hm,n(f*a,f*b)S21(r)Δ(ρ,γ)S2(r)Hm,n(fa,fb)|00.
S21(r)Δ(ρ,γ)S2(r)=μ2d2ηπ3|μ(γη)μ(γ+η)|exp(ηρ*η*ρ)=Δ(ρ/μ,μγ),
W(ρ,γ)=Cm,n2(ε)00|Hm,n(f*a,f*b)Δ(ρ/μ,μγ)Hm,n(fa,fb)|00.
Hm,n(x,x*)=m+nτmτnexp(ττ+xτ+x*τ)|τ=τ=0,
μ(γ+η)|Hm,n(fa,fb)|00=m+nτmτnμ(γ+η)|d2αd2βπ2|α,βα,β|exp(ττ+faτ+fbτ)|00|τ=τ=0=m+nτmτnd2αd2βπ2exp{μ2|γ+η|22+μ(γ+η)*αμ(γ+η)β+αβττ+fα*τ+fβ*τ|α|2|β|2}|τ=τ=0=m+nτmτnexp{μ2|γ+η|22+(f21)ττ+μf(γ+η)*τμf(γ+η)τ}|τ=τ=0=(1f2)m+nHm,n[R(γ+η)*,R(γ+η)]exp(μ22|γ+η|2),
00|Hm,n(f*a,f*b)|μ(γη)=(1f2)m+nHm,n[R*(γη),R*(γη)*]exp(μ22|γη|2).
W(ρ,γ)=Cm,n2(ε)π2k=0ml=0n(m!n!)2(|f|2)k+lk!l![(mk)!(nl)!]2|Hmk,nl(B,C)|2exp(μ2|γ|2|ρ|2/μ2),
k+lxkx*lHm,n(x,x*)=m!n!(mk)!(nl)!Hmk,nl(x,x*),
W(α,β;m=n=0)=1π2exp{2[(αβ+α*β*)sinh2r(|α|2+|β|2)cosh2r]},
W(ρ,γ;m=0)=(1)nπ2Ln(|ρ/μμγ|2)exp(μ2|γ|2|ρ|2/μ2),
δ=12dqdp[|W(q,p)|W(q,p)]
d2γΔ(ρ,γ)=1π|ηη|η=ρ,d2ρΔ(ρ,γ)=1π|ττ|τ=γ,
d2γW(ρ,γ)=1π|η|n,m,ε|η=ρ2,d2ρW(ρ,γ)=1π|τ|n,m,ε|τ=γ2.
η|S2(r)Hm,n(fa,fb)|00=μ(1f2)m+nHm,n(Rη*,Rη)exp(μ2|η|2/2),
d2γW(ρ,γ)=Cm,n2(ε)μ2(1f2)m+nπ|Hm,n(Rρ*,Rρ)|2exp(μ2|ρ|2).
d2ρW(ρ,γ)=Cm,n2(ε)(1+f2)m+nπμ2|Hm,n(Sγ*,Sγ)|2exp(|γ|2/μ2),
T(η,τ1,τ2)=πd2ρd2γδ(η1μ1γ1ν1ρ2)δ(η2ν2γ2μ2ρ1)W(ρ,γ),
η,τ1,τ2η,τ1,τ2|=πd2ρd2γδ(η1μ1γ1ν1ρ2)δ(η2ν2γ2μ2ρ1)Δ(ρ,γ),
|η,τ1,τ2=Dexp[D1+D2a+D3b+D4abD5a2D5b2]|00,
D=1|τ1τ2|,D1=η122|τ1|2η222|τ2|2,D2=η1τ1*+η2τ2*,D3=η1τ1*+η2τ2*,D4=12(ei2θ1ei2θ2),D5=14(ei2θ1+ei2θ2).
T(η,τ1,τ2)=|η,τ1,τ2|n,m,ε|2,
S2(r)=sechrexp[(1sechr)(aa+bb)+tanhr(abab)],
η,τ1,τ2|n,m,ε=Cm,n(ε)Dsechr00|exp[D1+D2*a+D3*b+D4*abD5*a2D5*b2]exp[(1sechr)(aa+bb)+tanhr(abab)]Hm,n(fa,fb)|00.
η,τ1,τ2|n,m,ε=Cm,n(ε)DK0exp(D1+K1)l=0min(m,n)()lm!n!fm+n2ll!(ml)!(nl)!m+n2lλmlσnlexp[K2λ2K3σ2+K4λσ+K5λ+K6σ],
G=coshrD4*sinhr,K0=G24D5*2tanh2r,K1=G(η22τ12η12τ22)sinhr2D5*(η22τ12+η12τ22)sinh2rK0τ12τ22,K2=D5*cosh2r+[2G(D4*tanhr)cschr+(D4*tanhr)2+4D5*2]D5*sinh22r4K0,K3=D5*K0,K4=G(D4*tanhr)coshr+2D5*2sinh2rK0,K5=D2*coshr+[2D5*(2D2*D5*sinhrGD3*)+(D4*tanhr)(GD2*2D3*D5*sinhr)]sinh2rK0,K6=GD3*2D2*D5*sinhrK0.
η,τ1,τ2|n,m,ε=Cm,n(ε)DK0l=0min(m,n)k=0nl()lm!n!fm+n2lK2mlK3nll!k!(mlk)!(nlk)!(K4K2K3)kHmlk(K52K2)Hnlk(K62K3)exp(D1+K1),
dρdt=i=a,bκ(n¯+1)[2iρiiiρρii]+κn¯[2iρiiiρρii],
ρ(t)=m,n,r,s=0Mm,n,r,sρ0Mm,n,r,s,
Mm,n,r,s=1n¯T+1T1r+s+m+nm!n!r!s!(n¯+1n¯)m+narbse(aa+bb)lnT2ambn
W(α,β,t)=4(2n¯+1)2T2d2αd2βπ2W(α,β,0)exp[2(2n¯+1)T(|ααeκt|2+|ββeκt|2)],
W(α,β,t)=Cm,n2(ε)A12g1m+nπ2A2A3k=0ml=0n(m!n!)2k!l![(mk)!(nl)!]2(g2g1)k(g3g1)l|Hmk,nl(g4g1,g5g1)|2exp[A1(A1A3e2κt1)(|α|2+|β|2)+2A12sinh2rA2A3(αβ+α*β*)e2κt],
A1=2(2n¯+1)T,A2=A1e2κt+2(sinh2r+cosh2r),A3=A24sinh22rA2,A4=(4cosh2rA21)sinhr,A5=(14sinh2rA2)coshr,A6=2A1A2sinh2reκtα+A1eκtβ*
g1=1+2f2A2sinh2r4f2A4A5A3,g2=4|f|2A2cosh2r+4|f|2A42A3|f|2,g3=4|f|2A2sinh2r+4|f|2A52A3|f|2,g4=2fA1coshrA2A3(4A4sinhr+A3)α*eκt+2fA1A4A3βeκt,g5=2fA1sinhrA2A3(4A5coshrA3)αeκt+2fA1A5A3β*eκt.
W(α,β,t)=A12π2A2A3exp[A1(A1A3e2κt1)(|α|2+|β|2)+2A12sinh2rA2A3(αβ+α*β*)e2κt],
W(α,β,)=1π2(2n¯+1)2exp[2(2n¯+1)(|α|2+|β|2)],

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