Abstract

Theoretical analysis is given of nonclassicality and decoherence of the field states generated by adding any number of photons to the squeezed thermal state (STS). Based on the fact that the squeezed number state can be considered as a single-variable Hermite polynomial excited state, the compact expression of the normalization factor is derived, a Legendre polynomial. The nonclassicality is investigated by exploring the sub-Poissonian and negative Wigner function (WF). The results show that the WF of single-photon–added STS (PASTS) always has negative values at the phase space center. The decoherence effect on PASTS is examined by the analytical expression of WF. It is found that a longer threshold value of decay time than in single-photon–subtraction STS is included in single PASTS.

© 2012 Optical Society of America

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    [CrossRef]
  40. L.-Y. Hu, X.-X. Xu, and H.-Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
    [CrossRef]
  41. C. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2000).
  42. L.-Y. Hu and H.-Y. Fan, “ Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
    [CrossRef]
  43. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
    [CrossRef]
  44. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
    [CrossRef]
  45. M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
    [CrossRef]

2011 (1)

J. S. Sales and N. G. de Almeida, “Robustness of superposition states evolving under the influence of a thermal reservoir,” Phys. Rev. A 83, 062121 (2011).
[CrossRef]

2010 (4)

L.-Y. Hu and H.-Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

L.-Y. Hu, X.-X. Xu, Z.-S. Wang, and X.-F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

L.-Y. Hu, X.-X. Xu, and H.-Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
[CrossRef]

2009 (3)

L.-Y. Hu and H.-Y. Fan, “Wigner functions of thermo number state, photon subtracted and added thermo vacuum state at finite temperature,” Mod. Phys. Lett. A 24, 2263–2274 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “ Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-added coherent state in a dissipative channel,” Phys. Scr. 79, 035004 (2009).
[CrossRef]

2008 (5)

H. Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528(2008).
[CrossRef]

P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. S. Kim, “Recent developments in photon-level operatoions on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

2007 (6)

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

2006 (3)

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

2005 (2)

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820–023828 (2005).
[CrossRef]

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

2004 (4)

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

2003 (1)

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

2002 (2)

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[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]

2001 (1)

A. Wunsche, “Hermite and Laguerre 2D polynomials,” J. Comput. Appl. Math. 133, 665–678 (2001).
[CrossRef]

2000 (1)

A. Wunsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math. Nucl. Gen. 33, 1603–1629 (2000).
[CrossRef]

1992 (1)

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485–488 (1992).
[CrossRef]

1991 (1)

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

1987 (2)

H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

H. Y. Fan and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
[CrossRef]

1979 (1)

Agarwal, G. S.

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485–488 (1992).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Albano, L.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2005), p. 743.

Banaszek, K.

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

Barbieri, M.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

Barnett, S. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon Press, 1997).

Bartlett, S. D.

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[CrossRef]

Bellini, M.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820–023828 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Blandino, R.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

Bouwmeester, D.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Browne, D. E.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Carmichael, H. J.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

Cerf, N. J.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Chefles, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Dantan, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

de Almeida, N. G.

J. S. Sales and N. G. de Almeida, “Robustness of superposition states evolving under the influence of a thermal reservoir,” Phys. Rev. A 83, 062121 (2011).
[CrossRef]

De Siena, S.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Dell’Anno, F.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Dodonov, V. V.

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

Eisert, J.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Ekert, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Fan, H. Y.

H. Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528(2008).
[CrossRef]

H. Y. Fan and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
[CrossRef]

Fan, H.-Y.

L.-Y. Hu, X.-X. Xu, and H.-Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-added coherent state in a dissipative channel,” Phys. Scr. 79, 035004 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “ Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Wigner functions of thermo number state, photon subtracted and added thermo vacuum state at finite temperature,” Mod. Phys. Lett. A 24, 2263–2274 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Fan, H-Y.

H-Y. Fan, Representation and Transformation Theory in Quantum Mechanics—Progress of Dirac’s Symbolic Method (Shanghai Scientific and Technical Press, 1997), p. 174.

Ferreyrol, F.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

Fiurášek, J.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Furusawa, A.

García-Patrón, R.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Gardiner, C.

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

Genoni, M. G.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

Grangier, P.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Grangier, Ph.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Hettich, C.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

Hu, L.-Y.

L.-Y. Hu, X.-X. Xu, and H.-Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

L.-Y. Hu, X.-X. Xu, Z.-S. Wang, and X.-F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-added coherent state in a dissipative channel,” Phys. Scr. 79, 035004 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “ Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Wigner functions of thermo number state, photon subtracted and added thermo vacuum state at finite temperature,” Mod. Phys. Lett. A 24, 2263–2274 (2009).
[CrossRef]

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

Illuminati, F.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Jeong, H.

P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

Kim, M. S.

P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

M. S. Kim, “Recent developments in photon-level operatoions on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

Kitagawa, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Klauder, J. R.

H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Lund, A. P.

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

Mandel, L.

Marek, P.

P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

Mølmer, K.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

Neergaard-Nielsen, J. S.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

Nha, H.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

Nielsen, B. M.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Parigi, V.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

Paris, M. G. A.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

Plenio, M. B.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Polzik, E. S.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001), Appendix A.

Radmore, P. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon Press, 1997).

Ralph, T. C.

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

Sales, J. S.

J. S. Sales and N. G. de Almeida, “Robustness of superposition states evolving under the influence of a thermal reservoir,” Phys. Rev. A 83, 062121 (2011).
[CrossRef]

Sanders, B. C.

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[CrossRef]

Sasaki, M.

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Scheel, S.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1998).

Spagnolo, N.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

Takahashi, H.

Takeoka, M.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Tara, K.

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485–488 (1992).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Tualle-Brouri, R.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820–023828 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Wakui, K.

Wang, Z.-S.

L.-Y. Hu, X.-X. Xu, Z.-S. Wang, and X.-F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2005), p. 743.

Wenger, J.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

Wunsche, A.

A. Wunsche, “Hermite and Laguerre 2D polynomials,” J. Comput. Appl. Math. 133, 665–678 (2001).
[CrossRef]

A. Wunsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math. Nucl. Gen. 33, 1603–1629 (2000).
[CrossRef]

Xu, X.-F.

L.-Y. Hu, X.-X. Xu, Z.-S. Wang, and X.-F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Xu, X.-X.

L.-Y. Hu, X.-X. Xu, Z.-S. Wang, and X.-F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

L.-Y. Hu, X.-X. Xu, and H.-Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299 (2010).
[CrossRef]

Zaidi, H. R.

H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

H. Y. Fan and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
[CrossRef]

Zavatta, A.

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820–023828 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Zeilinger, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Zoller, P.

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

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1998).

Ann. Phys. (1)

H. Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528(2008).
[CrossRef]

J. Comput. Appl. Math. (1)

A. Wunsche, “Hermite and Laguerre 2D polynomials,” J. Comput. Appl. Math. 133, 665–678 (2001).
[CrossRef]

J. Mod. Opt. (1)

L.-Y. Hu and H.-Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Opt. 57, 1344–1354 (2010).
[CrossRef]

J. Opt. B (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]

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Nucl. Gen. (1)

A. Wunsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math. Nucl. Gen. 33, 1603–1629 (2000).
[CrossRef]

J. Phys. B (1)

M. S. Kim, “Recent developments in photon-level operatoions on travelling light fields,” J. Phys. B 41, 133001–133018 (2008).
[CrossRef]

Mod. Phys. Lett. A (1)

L.-Y. Hu and H.-Y. Fan, “Wigner functions of thermo number state, photon subtracted and added thermo vacuum state at finite temperature,” Mod. Phys. Lett. A 24, 2263–2274 (2009).
[CrossRef]

Opt. Commun. (1)

L.-Y. Hu and H.-Y. Fan, “ Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

H. Y. Fan and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
[CrossRef]

Phys. Rev. A (15)

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820–023828 (2005).
[CrossRef]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[CrossRef]

L.-Y. Hu, X.-X. Xu, Z.-S. Wang, and X.-F. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[CrossRef]

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46, 485–488 (1992).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

P. Marek, H. Jeong, and M. S. Kim, “Generating ‘squeezed’ superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[CrossRef]

J. S. Sales and N. G. de Almeida, “Robustness of superposition states evolving under the influence of a thermal reservoir,” Phys. Rev. A 83, 062121 (2011).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[CrossRef]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Measure of the non-Gaussian character of a quantum state,” Phys. Rev. A 76, 042327 (2007).
[CrossRef]

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833(2010).
[CrossRef]

Phys. Rev. D (1)

H.-Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831–1834 (1987).
[CrossRef]

Phys. Rev. Lett. (5)

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[CrossRef]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free Bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef]

Phys. Scr. (1)

L.-Y. Hu and H.-Y. Fan, “Statistical properties of photon-added coherent state in a dissipative channel,” Phys. Scr. 79, 035004 (2009).
[CrossRef]

Science (3)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Other (7)

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

H-Y. Fan, Representation and Transformation Theory in Quantum Mechanics—Progress of Dirac’s Symbolic Method (Shanghai Scientific and Technical Press, 1997), p. 174.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001), Appendix A.

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

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier Academic Press, 2005), p. 743.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon Press, 1997).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1998).

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Figures (5)

Fig. 1.
Fig. 1.

The Q parameter as the function of the squeezing parameter r for different m=0, 1, 2, 3, 4, 19, 20 with a small nc value.

Fig. 2.
Fig. 2.

Photon-number distributions of PASTS with n¯=1 for λ=0.3, m=0 (blue bar); λ=0.3, m=1 (red bar); λ=0.3, m=5 (yellow bar); and λ=0.8, m=1 (green bar).

Fig. 3.
Fig. 3.

Wigner function distributions W(α,α*) of PASTS with λ=0.3 for different nc and m values (a) nc=0.1, m=1; (b) nc=0.5, m=1; (c) nc=0.1, m=2; (d) nc=0.1, m=3.

Fig. 4.
Fig. 4.

Wigner function distributions W(α,α*) of PASTS with m=1, nc=0.3 for different N, λ, and κt values: (a) N=0.2, λ=0.3, κt=0.05; (b) N=0.2, λ=0.3, κt=0.2; (c) N=0.2, λ=0.8, κt=0.05; (d) N=2, λ=0.3, κt=0.05.

Fig. 5.
Fig. 5.

The fidelity F between PASTS (PSSTS) and STS as the function of squeezing parameter λ for different photon-addition number m=0, 1, 2, 3 (nc=0.2).

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

ρad=Ca,m1amS1ρthS1am,
ρth=n=0ncn(nc+1)n+1|nn|=1nce1ncaa
ρth=1ncd2απe1nc|α|2|αα|,
S1=1μdq|qμq|,μ=eλ,
|q=π1/4exp{q22+2qaa22}|0.
q|n=12nn!πeq2/2Hn(q),
S1|n=dq2nn!μπeq2/2Hn(q)|qμ=sech1/2λ2nn!nτne2aτsechλ+(τ212a2)tanhλ|0|τ=0=(itanhλ)n2nn!Hn(asechλi2tanhλ)S1|0,
Hn(q)=nτnexp(2qττ2)|τ=0.
S1|1=asechλS1|0,S1|2=12(a2sech2λ+tanhλ)S1|0,
ρs=n=0ncn(nc+1)n+1S1(λ)|nn|S1(λ)=sechλnc+1n=0(nctanhλ)n2nn!(nc+1)nHn(asechλ2tanhλ)×exp[12(a2+a2)tanhλaa]Hn(asechλ2tanhλ),
n=0tn2nn!Hn(x)Hn(y)=11t2exp[2txyt2(x2+y2)1t2],
ρs=1Aexp[C2(a2+a2)+(B1)aa],
A=nc2+(2nc+1)cosh2λ,B=ncA(nc+1),C=2nc+12Asinh2λ.
ρs=1τ1τ2exp[Q22τ12P22τ22],
2τ12=(2nc+1)e2λ+1,2τ22=(2nc+1)e2λ+1.
ρad=Ca,m1τ1τ2amexp[C2(a2+a2)+(B1)aa]am.
Ca,m=1τ1τ2d2απ|α|2me(1B)|α|2+C2(α*2+α2)=2msmtmd2απτ1τ2.e(1B)|α|2+sα*+tα+C2(α*2+α2)|s=t=0=2msmtmexp[A(1B)st+AC2(s2+t2)]s=t=0,
d2zπexp(ζ|z|2+ξz+ηz*+fz2+gz*2)=1ζ24fgexp[ζξη+ξ2g+η2fζ24fg],
2mtmτmexp(t2τ2+2xτtx21)|t,τ=0=2mm!(x21)m/2Pm(x),
Ca,m=(AC)m2m2msmtmexp[2C(1B)sts2t2]s=t=0=m!Am/2Pm(B¯/A),
B¯=nccosh2λ+cosh2λ.
aa=Ca,m+1Ca,m1,
a2a2=Ca,m+2Ca,m4Ca,m+1Ca,m+2.
Q=Ca,m+24Ca,m+1+2Ca,mCa,m+1Ca,mCa,m+1Ca,mCa,m.
P(n)=n|S1ρthS1|n.
P(n)=sechλ2nn!nc2ntnτnexp[(t2+τ2)tanhλ]×d2απexp[2(αt+α*τ)sechλnc+1nc|α|2]×exp[tanhλ2(α2+α*2)]τ=t=0=sechλ2nn!A2ntnτnexp[2Btτ+C(t2+τ2)]τ=t=0.
P(n)=Dn/2APn(B/D),
D=nc2(2nc+1)sinh2λnc2+(2nc+1)cosh2λ.
112xt+t2=n=0Pn(x)tn;
P2(n)=Ca,m1n|amρsam|n=n!Ca,m1D(nm)/2(nm)!APnm(B/D).
W(α,α*)=e2|α|2d2βπ2β|ρ|βe2(αβ*α*β),
W(α,α*)=Fm(α,α*)W0(α,α*),
W0(α,α*)=1π(2nc+1)exp[2cosh2r2nc+1|α|2+sinh2r2nc+1(α2+α*2)],
Fm(α,α*)=(m!)2Cam1sinhm2λ22m(2nc+1)m×l=0m(1)l22l(nc+cosh2λ)ll![(ml)!]2sinhl2λ|Hml(γ¯)|2,
W(α,α*)=(1)me2|α|22n¯+1π(2nc+1)m+1Lm(4(nc+1)2nc+1|α|2),
W1(α,α*)=F1(α,α*)W0(α,α*),
F1(α,α*)=sinh2λ(2nc+1)B¯[|γ¯|2nc+cosh2λsinh2λ].
dρdt=κ(N+1)(2aρaaaρρaa)+κN(2aρaaaρρaa),
W(η,η*,t)=2(2N+1)Td2απW(α,α*,0)e2|ηαeκt|2(2N+1)T,
W(η,η*,t)=Fm(η,η*,t)W0(η,η*,t),
W0(η,η*,t)=2/(2nc+1)π(2N+1)TG×exp[Δ1|η|2+g2g32G(η*2+η2)],
Fm(η,η*,t)=Cam1l=0m(m!)2χlΔ2mll![(ml)!]2|Hml(iω/2Δ2)|2,
g0=cosh2λ2nc+1,g1=nc+cosh2λ2nc+1,g2=sinh2λ2nc+1,g3=2eκt(2N+1)T,
G=(2g0+g3eκt)24g22,Δ1=g3eκtg32G(2g0+g3eκt),Δ2=g2G(g3eκt/21)2,ω=2g3g3eκt2(2Δ2η*+χη),χ=2g3eκtG(g0+g1g3eκt+1(2nc+1)2).
W1(η,η*,t)=Ca11W0(η,η*,t)(|ω|2+χ).
κt<κtc=12ln2N+22N+1,
κtcs=12ln[12nc+12N+1ncsinh2λnccosh2λ+sinh2λ],
e2κtce2κtcs=2nc(nc+1)(2N+1)(nccosh2λ+sinh2λ),
F=tr(ρsρ)/tr(ρs2),
F=m!Ca,mK2m/2Pm(K1K2)=(K2A)m/2Pm(K1/K2)Pm(B¯/A),
K1=nc(nc+1)2nc+1cosh2λ,K2=nc2(nc+1)2(2nc+1)2sinh22λ4.
FFs=(ZA)m/2Pm(H/Z)Pm(B¯/A)=Cs,mCa,m,
FFs=nccosh2λ+sinh2λnccosh2λ+cosh2λ<1,
W(α,α*)=Cam1e2|α|2τ1τ22msmtmd2βπ2exp[(1+B)|β|2.+(2α+s)β*(2α*+t)β+C2(β*2+β2)]s=t=0=W0(α,α*)Fm(α,α*),
W0(α,α*)=A1πτ1τ2exp[A2(α2+α*2)2A3|α|2],
Fm(α,α*)=Cam12msmtmexp[A24(s2+t2)A42st.+(A2α*A4α)t+(A2αA4α*)s]s=t=0,
A1=1(1+B)2C2=A(2nc+1)2,A2=2C(1+B)2C2=sinh2λ2nc+1,A3=2(B+1)(1+B)2C21=cosh2λ2nc+1,A4=2(B+1)(1+B)2C2=A3+1=2nc+cosh2λ2nc+1.
Hn(x)=ntnexp(2xtt2)|t=0,
ntnexp(At+Bt2)|t=0=(iB)nHn[A/(2iB)]=(iB)nHn[A/(2iB)];
Fm(α,α*)=Cam1l=0(A4)l2ll!2lγlγ*l2msmtm×exp[A24(s2+t2)+γt+γ*s]s=t=0=A2m22mCam1l=0(A4)l2ll!2lγlγ*l|Hm(γ¯)|2,
γ¯=α*sinh2λ2α(cosh2λ+nc)i(2nc+1)sinh2λ,
ddxlHn(x)=2ln!(nl)!Hnl(x),
Fm(α,α*)=A2m22mCam1l=0m(m!)2(2A4/A2)ll![(ml)!]2|Hml(γ¯)|2.

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