Abstract

We investigate the statistical properties of photon subtractions from a two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum, and the normalization of this state is the Jacobi polynomial of the squeezing parameter. A compact expression for the Wigner function (WF) is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the WF. The decoherence effect on this state is then discussed by deriving the time evolution of the WF. It is shown that the WF is always positive for any squeezing parameter and any photon-subtraction number if the decay time exceeds an upper bound (κt>12ln2n ¯+22n ¯+1).

© 2010 Optical Society of America

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2009 (5)

2008 (7)

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (2008).
[CrossRef]

S. Glancy and H. M. de Vasconcelos, “Methods for producing optical coherent state superpositions,” J. Opt. Soc. Am. B 25, 712-733 (2008).
[CrossRef]

H. Jeong, J. Lee, and H. Nha, “Decoherence of highly mixed macroscopic quantum superpositions,” J. Opt. Soc. Am. B 25, 1025-1030 (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]

H. Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV)--integrations within Weyl ordered product of operators and their applications,” Ann. Phys. (Paris) 323, 500-526 (2008).
[CrossRef]

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

P. Marek, H. Jeong, and M. S. Kim, “Generating 'squeezed' superposition of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811-063818 (2008).
[CrossRef]

2007 (2)

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104-032111 (2007).
[CrossRef]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502-030505 (2007).
[CrossRef] [PubMed]

2006 (6)

J. S. Neergaard-Nielsen, B. Melholt 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, 083604-083607 (2006).
[CrossRef] [PubMed]

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

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601-213604 (2006).
[CrossRef] [PubMed]

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, 042310-042321 (2006).
[CrossRef]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807-043824 (2006).
[CrossRef]

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. (Paris) 321, 480-494 (2006) and references therein.
[CrossRef]

2005 (6)

H. Y. Fan, J. S. Wang, “On the Weyl ordering invariant under general n-mode similar transformations,” Mod. Phys. Lett. A 20, 1525-1532 (2005).
[CrossRef]

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-substracted Gaussian field,” Phys. Rev. A 71, 043805-043809 (2005).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105-042116 (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, 013801-013812 (2005).
[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]

S. Olivares and Matteo G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
[CrossRef]

2004 (6)

S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112-032117 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection,” Phys. Rev. A 70, 053821-053826 (2004).
[CrossRef]

M. D' Angelo, A. Zavatta, V. Parigi, and M. Bellini, “Tomographic test of Bell's inequality for a time-delocalized single photon,” Phys. Rev. A 74, 052114-052119 (2004).
[CrossRef]

S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601-193604 (2004).
[CrossRef] [PubMed]

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] [PubMed]

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

2003 (1)

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,”Phys. Rev. A 67, 032314-032318 (2003).
[CrossRef]

2002 (2)

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements on the two-mode squeezed vacuum,” Phys. Rev. A 65, 062306-062311 (2002).
[CrossRef]

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304-042308 (2002).
[CrossRef]

2001 (2)

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

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402-050405 (2001).
[CrossRef] [PubMed]

2000 (4)

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302-1-7 (2000).
[CrossRef]

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A 33, 1603-1629 (2000).
[CrossRef]

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101-052105 (2000).
[CrossRef]

J. Lee, M. S. Kim, and H. Jeong, “Transfer of nonclassical features in quantum teleportation via a mixed quantum channel,” Phys. Rev. A 62, 032305 (2000).
[CrossRef]

1999 (1)

A. Wünsche, “About integration within ordered products in quantum optics,” J. Opt. Soc. Am. B 1, R11-R21 (1999).

1997 (1)

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

1992 (3)

G. S. Agarwal, “Negative binomial states of the field-operator representation and production by state reduction in optical processes,” Phys. Rev. A 45, 1787-1792 (1992).
[CrossRef] [PubMed]

H. Y. Fan, “Weyl ordering quantum mechanical operators by virtue of the IWWP technique,” J. Phys. A 25, 3443-3447 (1992).
[CrossRef]

M. S. Kim and V. Bužek, “Schrödinger-cat states at finit temperature: Influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
[CrossRef] [PubMed]

1990 (3)

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

C. T. Lee, “Many-photon anti-bunching in generalized pair coherent states,” Phys. Rev. A 41, 1569-1575 (1990).
[CrossRef] [PubMed]

V. Buzek, “SU(1,1) squeezing of SU(1,1) generalized coherent states,” J. Mod. Opt. 34, 303-316 (1990).
[CrossRef]

1987 (2)

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709-759 (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]

1975 (1)

Y. Takahashi and H. Umezawa, “Collective phenomena,” Int. J. Mod. Phys 2, 55-80 (1975).

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]

1963 (1)

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

1932 (1)

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

Agarwal, G. S.

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104-032111 (2007).
[CrossRef]

G. S. Agarwal, “Negative binomial states of the field-operator representation and production by state reduction in optical processes,” Phys. Rev. A 45, 1787-1792 (1992).
[CrossRef] [PubMed]

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]

Aichele, T.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402-050405 (2001).
[CrossRef] [PubMed]

Anhut, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

Appel, J.

S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601-193604 (2004).
[CrossRef] [PubMed]

Babichev, S. A.

S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601-193604 (2004).
[CrossRef] [PubMed]

Banaszek, K.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105-042116 (2005).
[CrossRef]

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, 042304-042308 (2002).
[CrossRef]

Bellini, M.

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, “Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection,” Phys. Rev. A 70, 053821-053826 (2004).
[CrossRef]

M. D' Angelo, A. Zavatta, V. Parigi, and M. Bellini, “Tomographic test of Bell's inequality for a time-delocalized single photon,” Phys. Rev. A 74, 052114-052119 (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] [PubMed]

Benson, O.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402-050405 (2001).
[CrossRef] [PubMed]

Biswas, A.

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104-032111 (2007).
[CrossRef]

Bonifacio, R.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,”Phys. Rev. A 67, 032314-032318 (2003).
[CrossRef]

Bouwmeester, D.

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

Buzek, V.

V. Buzek, “SU(1,1) squeezing of SU(1,1) generalized coherent states,” J. Mod. Opt. 34, 303-316 (1990).
[CrossRef]

Bužek, V.

M. S. Kim and V. Bužek, “Schrödinger-cat states at finit temperature: Influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
[CrossRef] [PubMed]

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] [PubMed]

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, 042310-042321 (2006).
[CrossRef]

Cochrane, P. T.

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M. D' Angelo, A. Zavatta, V. Parigi, and M. Bellini, “Tomographic test of Bell's inequality for a time-delocalized single photon,” Phys. Rev. A 74, 052114-052119 (2004).
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M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
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A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502-030505 (2007).
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L. Y. Hu and H. Y. Fan, “Statistical properties of photon-added coherent state in a dissipative channel,” Phys. Scr. 79, 035004-035011 (2009).
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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).
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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).
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H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (2008).
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H. Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV)--integrations within Weyl ordered product of operators and their applications,” Ann. Phys. (Paris) 323, 500-526 (2008).
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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. (Paris) 321, 480-494 (2006) and references therein.
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H. Y. Fan, J. S. Wang, “On the Weyl ordering invariant under general n-mode similar transformations,” Mod. Phys. Lett. A 20, 1525-1532 (2005).
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H. Y. Fan, “Weyl ordering quantum mechanical operators by virtue of the IWWP technique,” J. Phys. A 25, 3443-3447 (1992).
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H. Y. Fan and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303-307 (1987).
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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. (Paris) 321, 480-494 (2006) and references therein.
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W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867-927 (1990).
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C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

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W. M. Zhang, D. F. Feng, and R. Gilmore, “Coherent state: theory and some applications,” Rev. Mod. Phys. 62, 867-927 (1990).
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A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502-030505 (2007).
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A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601-213604 (2006).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
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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).
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L. Y. Hu and H. Y. Fan, “Statistical properties of photon-added coherent state in a dissipative channel,” Phys. Scr. 79, 035004-035011 (2009).
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H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (2008).
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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).
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M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-substracted Gaussian field,” Phys. Rev. A 71, 043805-043809 (2005).
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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, 013801-013812 (2005).
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P. Marek, H. Jeong, and M. S. Kim, “Generating 'squeezed' superposition of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811-063818 (2008).
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J. Lee, M. S. Kim, and H. Jeong, “Transfer of nonclassical features in quantum teleportation via a mixed quantum channel,” Phys. Rev. A 62, 032305 (2000).
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H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101-052105 (2000).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
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H. Jeong, J. Lee, and H. Nha, “Decoherence of highly mixed macroscopic quantum superpositions,” J. Opt. Soc. Am. B 25, 1025-1030 (2008).
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H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101-052105 (2000).
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J. Lee, M. S. Kim, and H. Jeong, “Transfer of nonclassical features in quantum teleportation via a mixed quantum channel,” Phys. Rev. A 62, 032305 (2000).
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Li, F. L.

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R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709-759 (1987).
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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. (Paris) 321, 480-494 (2006) and references therein.
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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, 013801-013812 (2005).
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W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).

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P. Marek, H. Jeong, and M. S. Kim, “Generating 'squeezed' superposition of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811-063818 (2008).
[CrossRef]

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J. S. Neergaard-Nielsen, B. Melholt 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, 083604-083607 (2006).
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Milburn, G. J.

P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements on the two-mode squeezed vacuum,” Phys. Rev. A 65, 062306-062311 (2002).
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A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402-050405 (2001).
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Mølmer, K.

J. S. Neergaard-Nielsen, B. Melholt 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, 083604-083607 (2006).
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J. S. Neergaard-Nielsen, B. Melholt 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, 083604-083607 (2006).
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W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).

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S. Olivares and Matteo G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
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C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105-042116 (2005).
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S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112-032117 (2004).
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S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,”Phys. Rev. A 67, 032314-032318 (2003).
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M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
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T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302-1-7 (2000).
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A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502-030505 (2007).
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A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601-213604 (2006).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
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Parigi, V.

M. D' Angelo, A. Zavatta, V. Parigi, and M. Bellini, “Tomographic test of Bell's inequality for a time-delocalized single photon,” Phys. Rev. A 74, 052114-052119 (2004).
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C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photo detection,” Phys. Rev. A 72, 042105-042116 (2005).
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S. Olivares and M. G. A. Paris, “Enhancement of nonlocality in phase space,” Phys. Rev. A 70, 032112-032117 (2004).
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S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,”Phys. Rev. A 67, 032314-032318 (2003).
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Paris, Matteo G. A.

S. Olivares and Matteo G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
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M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-substracted Gaussian field,” Phys. Rev. A 71, 043805-043809 (2005).
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Park, J.

Polzik, E. S.

J. S. Neergaard-Nielsen, B. Melholt 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, 083604-083607 (2006).
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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, 013801-013812 (2005).
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P. T. Cochrane, T. C. Ralph, and G. J. Milburn, “Teleportation improvement by condition measurements on the two-mode squeezed vacuum,” Phys. Rev. A 65, 062306-062311 (2002).
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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, 042310-042321 (2006).
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A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402-050405 (2001).
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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, 042310-042321 (2006).
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A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502-030505 (2007).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
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A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601-213604 (2006).
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H. Y. Fan, J. S. Wang, “On the Weyl ordering invariant under general n-mode similar transformations,” Mod. Phys. Lett. A 20, 1525-1532 (2005).
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T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302-1-7 (2000).
[CrossRef]

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schröinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
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H. Y. Fan and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303-307 (1987).
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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).
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A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection,” Phys. Rev. A 70, 053821-053826 (2004).
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M. D' Angelo, A. Zavatta, V. Parigi, and M. Bellini, “Tomographic test of Bell's inequality for a time-delocalized single photon,” Phys. Rev. A 74, 052114-052119 (2004).
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A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-662 (2004).
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C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Ann. Phys. (Paris) (2)

H. Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV)--integrations within Weyl ordered product of operators and their applications,” Ann. Phys. (Paris) 323, 500-526 (2008).
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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. (Paris) 321, 480-494 (2006) and references therein.
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Int. J. Mod. Phys (1)

Y. Takahashi and H. Umezawa, “Collective phenomena,” Int. J. Mod. Phys 2, 55-80 (1975).

J. Comput. Appl. Math. (1)

A. Wünsche, “Hermite and Laguerre 2D polynomials,” J. Comput. Appl. Math. 133, 665-678 (2001).
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S. Olivares and Matteo G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
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J. Opt. Soc. Am. B (7)

J. Phys. A (2)

H. Y. Fan, “Weyl ordering quantum mechanical operators by virtue of the IWWP technique,” J. Phys. A 25, 3443-3447 (1992).
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A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A 33, 1603-1629 (2000).
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J. Phys. B (1)

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001-133018 (2008).
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Mod. Phys. Lett. A (1)

H. Y. Fan, J. S. Wang, “On the Weyl ordering invariant under general n-mode similar transformations,” Mod. Phys. Lett. A 20, 1525-1532 (2005).
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Opt. Commun. (2)

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Figures (11)

Fig. 1
Fig. 1

Cross-correlation function between the two modes a and b as a function of λ for different parameters ( m , n ) . The numbers 1,2,3,4,5,6 in (a) denote that ( m , n ) are equal to (1,2), (3,4), (2,4),(6,8),(3,6), and (7,10), respectively.

Fig. 2
Fig. 2

Fluctuation variation of ( Δ P ) 2 with λ for several different parameter m , n values: (a) m = n = 1 , 2 , 8 , 35 from bottom to top; (b) (1) and (2) denote m = 0 and m = 1 , respectively, and n = 2 , 5 , 12 from bottom to top.

Fig. 3
Fig. 3

Photon number distribution P ( n a , n b ) in the Fock space ( n a , n b ) for some given m = n values: (a) m = n = 0 , λ = 1 ; (b) m = n = 1 , λ = 0.5 ; (c) m = n = 1 , λ = 1 ; (d) m = 2 , n = 5 , λ = 1 .

Fig. 4
Fig. 4

R a b as a function of λ and m , n . (1) and (2) in Fig. 4a denote m = 0 and m = 1 , respectively, and n = 2 , 3 , 12 from bottom to top.

Fig. 5
Fig. 5

Wigner function W ( α , β ) in phase space ( 0 , 0 , p 1 , p 2 ) for several different parameter values m = n with λ = 0.5 . (a) m = n = 0 , (b) m = n = 1 , and (c) m = n = 5 .

Fig. 6
Fig. 6

Wigner function W ( α , β ) in three different phase spaces for m = 0 , n = 1 with λ = 0.3 (first row) and λ = 0.5 (second row).

Fig. 7
Fig. 7

Wigner function W ( α , β ) in phase space ( 0 , 0 , p 1 , p 2 ) for several parameter values m , n with λ = 0.5 . (a) m 1 , = n = 2 ; (b) m = 1 , n = 3 ; and (c) m = 1 , n = 5 .

Fig. 8
Fig. 8

Time evolution of the WF ( m = 0 , n = 1 ) at ( q 1 , q 2 , 0 , 0 ) phase space for n ¯ = 1 , λ = 0.3 . (a) κ t = 0.05 , (b) κ t = 0.1 ,(c) κ t = 0.12 , (d) κ t = 0.2 .

Fig. 9
Fig. 9

Time evolution of the WF ( m = 0 , n = 1 ) in ( q 1 , q 2 , 0 , 0 ) phase space for λ = 0.3 and κ t = 0.05 with (a) n ¯ = 0 , (b) n ¯ = 1 , (c) n ¯ = 2 , (d) n ¯ = 7 .

Fig. 10
Fig. 10

Time evolution of the WF ( m = 0 , n = 1 ) in ( q 1 , q 2 , 0 , 0 ) phase space for n ¯ = 1 , and κ t = 0.05 with (a) λ = 0.03 , (b) λ = 0.5 , (c) λ = 0.8 , (d) λ = 1.2 .

Fig. 11
Fig. 11

Time evolution of the WF for m = 0 , n = 2 in ( q 1 , q 2 , 0 , 0 ) phase space with (a) κ t = 0 , (b) κ t = 0.05 , (c) κ t = 0.1 ,(d) κ t = 0.2 .

Equations (67)

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S 2 ( λ ) | 00 = sech λ exp ( a b tanh λ ) | 00 ,
| λ , m , n = a m b n S 2 ( λ ) | 00 ,
| λ , m , n = S 2 ( λ ) ( a cosh λ + b sinh λ ) m ( b cosh λ + a sinh λ ) n | 00 = S 2 ( λ ) sinh n + m λ l = 0 m m ! coth l λ l ! ( m l ) ! b m l a l a n | 00 .
| λ , m , n = S 2 ( λ ) sinh n + m λ l = 0 min ( m , n ) m ! n ! coth l λ l ! ( m l ) ! ( n l ) ! a n l b m l | 00 = sinh ( n + m ) 2 2 λ ( i 2 ) n + m S 2 ( λ ) H m , n ( i tanh λ b , i tanh λ a ) | 00 ,
H m , n ( ε , ɛ ) = k = 0 min ( m , n ) ( 1 ) k m ! n ! ε m k ɛ n k k ! ( m k ) ! ( n k ) ! .
| λ , m , n = S 2 ( λ ) l = 0 min ( m , n ) m ! n ! sinh n + m λ coth l λ l ! ( m l ) ! ( n l ) ! | n l , m l ,
λ , m + s , n + t | λ , m , n = m ! ( n + s ) ! δ s , t sinh 2 n + 2 m + 2 s λ × l = 0 min ( m , n ) ( m + s ) ! n ! coth 2 l + s λ l ! ( m l ) ! ( n l ) ! ( l + s ) ! ,
P m ( α , β ) ( x ) = ( x 1 2 ) m k = 0 m ( m + α k ) ( m + β m k ) ( x + 1 x 1 ) k ,
λ , m + s , n + t | λ , m , n = m ! ( n + s ) ! δ s , t sinh 2 n + s λ cosh s λ P m ( n m , s ) ( cosh 2 λ ) ,
N λ , m , n = λ , m , n | λ , m , n = m ! n ! sinh 2 n λ P m ( n m , 0 ) ( cosh 2 λ ) ,
λ , m , n [ m ! n ! sinh 2 n λ P m ( n m , 0 ) ( cosh 2 λ ) ] 1 2 a m b n S 2 ( λ ) | 00 .
a a = ( m + 1 ) P m + 1 ( n m 1 , 0 ) ( τ ) P m ( n m , 0 ) ( τ ) ,
b b = ( n + 1 ) sinh 2 λ P m ( n m + 1 , 0 ) ( τ ) P m ( n m , 0 ) ( τ ) .
a b a b = ( m + 1 ) ( n + 1 ) sinh 2 λ P m + 1 ( n m , 0 ) ( τ ) P m ( n m , 0 ) ( τ ) .
g 12 ( 2 ) ( λ ) = a b a b a a b b = P m + 1 ( n m , 0 ) ( τ ) P m + 1 ( n m 1 , 0 ) ( τ ) P m ( n m , 0 ) ( τ ) P m ( n m + 1 , 0 ) ( τ ) .
a b = a b = n + 1 2 P m ( n m , 1 ) ( τ ) P m ( n m , 0 ) ( τ ) sinh 2 λ .
( Δ Q ) 2 = 1 2 P m ( n m , 0 ) ( τ ) [ ( m + 1 ) P m + 1 ( n m 1 , 0 ) ( τ ) + ( n + 1 ) P m ( n m + 1 , 0 ) ( τ ) sinh 2 λ + ( n + 1 ) P m ( n m , 1 ) ( τ ) sinh 2 λ + P m ( n m , 0 ) ( τ ) ] ,
( Δ P ) 2 = 1 2 P m ( n m , 0 ) ( τ ) [ ( m + 1 ) P m + 1 ( n m 1 , 0 ) ( τ ) + ( n + 1 ) P m ( n m + 1 , 0 ) ( τ ) sinh 2 λ ( n + 1 ) P m ( n m , 1 ) ( τ ) sinh 2 λ + P m ( n m , 0 ) ( τ ) ] .
n a , n b | λ , m , n = sech λ n a , n b | a m b n e a b tanh λ | 00 = ( m + n a ) ! n a ! n b ! sech λ tanh m + n a λ δ m + n a , n + n b .
P ( n a , n b ) = N λ , m , n 1 | n a , n b | λ , m , n | 2 = [ ( m + n a ) ! sech λ tanh m + n a λ δ m + n a , n + n b ] 2 n a ! n b ! m ! n ! sinh 2 n λ P m ( n m , 0 ) ( cosh 2 λ ) .
P ( n a , n b ) = { sech 2 λ tanh 2 n a λ , n a = n b 0 , n a n b } ,
R a b a 2 a 2 + b 2 b 2 2 a a b b 1 < 0 .
a 2 a 2 = ( m + 1 ) ( m + 2 ) P m + 2 ( n m 2 , 0 ) ( τ ) P m ( n m , 0 ) ( τ ) ,
b 2 b 2 = ( n + 1 ) ( n + 2 ) sinh 4 λ P m ( n m + 2 , 0 ) ( τ ) P m ( n m , 0 ) ( τ ) .
R a b = ( m + 1 ) ( m + 2 ) P m + 2 ( n m 2 , 0 ) ( τ ) + ( n + 1 ) ( n + 2 ) sinh 4 λ P m ( n m + 2 , 0 ) ( τ ) 2 ( m + 1 ) ( n + 1 ) sinh 2 λ P m + 1 ( n m , 0 ) ( τ ) 1 .
Δ 1 ( α ) = 1 2 : : δ ( α a ) δ ( α * a ) : : ,
S : : ( ) : : S 1 = : : S ( ) S 1 : : ,
S 2 ( λ ) Δ 1 ( α ) Δ 2 ( β ) S 2 ( λ ) = 1 4 : : δ ( α a cosh λ b sinh λ ) δ ( α * a cosh λ b sinh λ ) × δ ( β b cosh λ a sinh λ ) δ ( β * b cosh λ a sinh λ ) : : = Δ 1 ( α ¯ ) Δ 2 ( β ¯ ) ,
Δ 1 ( α ) = e 2 | α | 2 d 2 z 1 π 2 | z 1 z 1 | e 2 ( z 1 α * α z 1 * ) ,
W ( α , β ) = 1 π 2 sinh n + m 2 λ 2 n + m N λ , m , n e 2 | α ¯ | 2 2 | β ¯ | 2 l = 0 m k = 0 n × [ m ! n ! ] 2 ( tanh λ ) l + k l ! k ! [ ( m l ) ! ( n k ) ! ] 2 | H m l , n k ( B , A ) | 2 ,
W ( α , β ) = ( 1 ) n π 2 e 2 | α ¯ | 2 2 | β ¯ | 2 L n ( 4 | α ¯ | 2 ) ,
d d t ρ ( t ) = ( L 1 + L 2 ) ρ ( t ) ,
L i ρ = κ ( n ¯ + 1 ) ( 2 a i ρ a i a i a i ρ ρ a i a i ) + κ n ¯ ( 2 a i ρ a i a i a i ρ ρ a i a i ) , ( a 1 = a , a 2 = b ) ,
W ( α , β , t ) = 4 ( 2 n ¯ + 1 ) 2 T 2 d 2 ζ d 2 η π 2 W ( ζ , η , 0 ) e 2 | α ζ e κ t | 2 + | β η e κ t | 2 ( 2 n ¯ + 1 ) T .
W ( α , β , t ) = N λ , m , n 1 ( E sinh 2 λ ) m + n π 2 2 n + m ( 2 n ¯ + 1 ) 2 T 2 D e | α β * | 2 e 2 λ 2 κ t + ( 2 n ¯ + 1 ) T | α + β * | 2 e 2 λ 2 κ t + ( 2 n ¯ + 1 ) T × l = 0 n k = 0 m [ m ! n ! ] 2 ( F E tanh λ ) l + k l ! k ! [ ( m k ) ! ( n l ) ! ] 2 | H m k , n l ( G E , K E ) | 2 ,
C = e 2 κ t ( 2 n ¯ + 1 ) T , D = ( 1 + C e 2 λ ) ( 1 + C e 2 λ ) ,
E = e 4 κ t D ( 2 n ¯ T + 1 ) 2 C 2 , F = C 2 1 D ,
G = C e κ t D ( B ¯ + B * C ) ,
B ¯ = i 2 tanh λ ( β * cosh λ + α sinh λ ) ,
K = C e κ t D ( A ¯ + A * C ) ,
A ¯ = i 2 tanh λ ( α * cosh λ + β sinh λ ) .
W ( α , β , ) = 1 π 2 ( 2 n ¯ + 1 ) 2 e 2 2 n ¯ + 1 ( | α | 2 + | β | 2 ) ,
W m = n = 0 ( α , β , t ) = N 1 e E D ( | α | 2 + | β | 2 ) + F D ( α β + α * β * ) ,
κ t > κ t c 1 2 ln 2 n ¯ + 2 2 n ¯ + 1 ,
W ( α , β , t c ) = tanh m + n λ sech 2 λ 4 π 2 N m , n , λ e 4 κ t c e e 2 κ t c [ | α | 2 + | β | 2 ( α * β * + α β ) tanh λ ] × | H m , n ( i tanh λ β * e κ t c , i tanh λ α * e κ t c ) | 2 ,
W ( α , β ) = sinh n + m 2 λ 2 n + m N λ , m , n 00 | H m , n ( i tanh λ b , i tanh λ a ) Δ 1 ( α ¯ ) Δ 2 ( β ¯ ) H m , n ( i tanh λ b , i tanh λ a ) | 00 = sinh n + m 2 λ 2 n + m N λ , m , n e 2 | α ¯ | 2 + 2 | β ¯ | 2 d 2 z 1 d 2 z 2 π 4 e | z 1 | 2 | z 2 | 2 2 ( z 1 α ¯ * α ¯ z 1 * ) 2 ( z 2 β ¯ * β ¯ z 2 * ) × H m , n ( i tanh λ z 2 , i tanh λ z 1 ) H m , n ( i tanh λ z 2 * , i tanh λ z 1 * ) .
H m , n ( ε , ɛ ) = m + n t m t n | exp [ t t + ε t + ɛ t ] | t = t = 0 .
W ( α , β ) = sinh n + m 2 λ 2 n + m N λ , m , n e 2 | α ¯ | 2 2 | β ¯ | 2 m + n t m τ n m + n t m τ n × | e t τ t τ + A * τ + B * t + A τ + B t ( t t + τ τ ) tanh λ | t = τ = t = τ = 0 ,
d 2 z π e ζ | z | 2 + ξ z + η z * = 1 ζ e ξ η ζ , Re ( ζ ) < 0 .
W ( α , β ) = sinh n + m 2 λ 2 n + m N λ , m , n e 2 | α ¯ | 2 2 | β ¯ | 2 l = 0 k = 0 ( tanh λ ) l + k l ! k ! × l + k B l A k l + k B * l A * k H m , n ( B , A ) H m , n ( B * , A * ) .
l + k ε l ɛ k H m , n ( ε , ɛ ) = m ! n ! H m l , n k ( ε , ɛ ) ( m l ) ! ( n k ) ! ,
| η a = exp [ 1 2 | η a | 2 + η a a η a * a ̃ + a a ̃ ] | 0 0 ̃ ,
| η b = exp [ 1 2 | η b | 2 + η b b η b * b ̃ + b b ̃ ] | 0 0 ̃ ,
( a a ̃ ) | η a = η a | η a , ( a a ̃ ) | η a = η a * | η a ,
η a | ( a a ̃ ) = η a * η a | , η a | ( a a ̃ ) = η a η a | ,
a | I a , I b = a ̃ | I a , I b , a | I a , I b = a ̃ | I a , I b ,
b | I a , I b = b ̃ | I a , I b , b | I a , I b = b ̃ | I a , I b ,
d d t | ρ ( t ) = [ κ ( n ¯ + 1 ) ( 2 a a ̃ a a a ̃ a ̃ ) + κ n ¯ ( 2 a a ̃ a a a ̃ a ̃ ) + κ ( n ¯ + 1 ) ( 2 b b ̃ b b b ̃ b ̃ ) + κ n ¯ ( 2 b b ̃ b b b ̃ b ̃ ) ] | ρ ( t ) .
| ρ ( t ) = exp [ κ t ( n ¯ + 1 ) ( 2 a a ̃ a a a ̃ a ̃ ) + κ t n ¯ ( 2 a a ̃ a a a ̃ a ̃ ) + κ t ( n ¯ + 1 ) ( 2 b b ̃ b b b ̃ b ̃ ) + κ t n ¯ ( 2 b b ̃ b b b ̃ b ̃ ) ] | ρ 0 ,
| ρ ( t ) = exp [ ( a a ̃ a ̃ a + 1 ) κ t ] × exp [ 2 n ¯ + 1 2 ( 1 e 2 κ t ) ( a a ̃ ) ( a a ̃ ) ] × exp [ ( b b ̃ b ̃ b + 1 ) κ t ] × exp [ 2 n ¯ + 1 2 ( 1 e 2 κ t ) ( b b ̃ ) ( b b ̃ ) ] | ρ 0 ,
η a , η b | ρ ( t ) = exp [ 2 n ¯ + 1 2 T ( | η a | 2 + | η b | 2 ) ] η a e κ t , η b e κ t | ρ 0 ,
| ρ ( t ) = 1 ( n ¯ T + 1 ) 2 exp [ T 1 ( a a ̃ + b b ̃ ) ] × exp [ ( a a + b b + a ̃ a ̃ + b ̃ b ̃ ) ln T 2 ] × exp [ T 3 ( a a ̃ + b b ̃ ) ] ρ 0 | I a , I b .
W ( α ) = m , n n , n ̃ | Δ ( α ) ρ | m , m ̃ = 1 π ξ = 2 α | ρ ,
W ( α , β ) = Tr [ Δ a ( α ) Δ b ( β ) ρ ] = 1 π 2 ξ a = 2 α , ξ b = 2 β ρ .
W ( α , β , t ) = d 2 η a d 2 η b 4 π 4 e α * η a α η a * + β * η b β η b * η a , η b | ρ ( t ) .
W ( α , β , t ) = d 2 η a d 2 η b 4 π 4 e 2 n ¯ + 1 2 T | ( | η a | 2 + | η b | 2 ) × e α * η a α η a * + β * η b β η b * η a e κ t , η b e κ t | ρ 0 = d 2 ξ a d 2 ξ b π 2 W ( ζ , η , 0 ) d 2 η a d 2 η b 4 π 2 e 2 n ¯ + 1 2 T | ( | η a | 2 + | η b | 2 ) × e α * η a α η a * + β * η b β η b * η a e κ t , η b e κ t | ξ a = 2 ζ , ξ b = 2 η .
W ( α , β , t ) = 4 e 4 κ t d 2 ζ d 2 η W a th ( ζ ) W b th ( η ) × W { e κ t ( α T ζ ) , e κ t ( β T η ) , 0 } ,

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