Abstract

By the use of nonclassical depth as a criterion for nonclassicality, we show that states generated by first adding (subtracting) multiphotons to an arbitrary state and then subtracting (adding) multiphotons from the resulting state is certainly nonclassical if the addition number of photons is equal to or larger than the subtraction number. The explicit expressions of nonclassical depth are found for both the single-photon-subtracted Gaussian state and the photon-added-then-subtracted thermal state. Based on the expressions, we show that the nonclassicality of the initial Gaussian state can be enhanced by the photon subtraction process. We also notice that the photon-added-then-subtracted thermal state is nonclassical if the initial thermal mean photon number is not zero, and the nonclassicality increases monotonously as the initial thermal mean photon number increases.

© 2009 Optical Society of America

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  1. M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).
  2. S. L. Braunstein, P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
    [CrossRef]
  3. A. Zavatta, S. Viciani, M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
    [CrossRef] [PubMed]
  4. A. Zavatta, S. Viciani, M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
    [CrossRef]
  5. A. Zavatta, V. Parigi, M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
    [CrossRef]
  6. V. Parigi, A. Zavatta, M. Kim, M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
    [CrossRef] [PubMed]
  7. J. Wenger, R. Tualle-Brouri, P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
    [CrossRef] [PubMed]
  8. A. R. Usha Devi, R. Prabhu, M. S. Uma, “Non-classicality of photon added coherent and thermal radiations,” Eur. Phys. J. D 40, 133–138 (2006).
    [CrossRef]
  9. M. S. Kim, E. Park, P. L. Knight, H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
    [CrossRef]
  10. A. Biswas, G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
    [CrossRef]
  11. S. S. Mizrahi, V. V. Dodonov, “Creating quanta with an ‘annihilation’ operator,” J. Phys. A 35, 8847–8857 (2002).
    [CrossRef]
  12. G. S. Agarwal, K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. W. Vogel, “Nonclassical states: an observable criterion,” Phys. Rev. Lett. 84, 1849–1852 (2000).
    [CrossRef] [PubMed]
  16. Th. Richter, W. Vogel, “Nonclassicality of quantum states: a hierarchy of observable conditions,” Phys. Rev. Lett. 89, 283601 (2002).
    [CrossRef]
  17. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
    [CrossRef]
  18. Arvind, N. Mukunda, R. Simon, “Characterizations of classical and nonclassical states of quantized radiation,” J. Phys. A 31, 565–583 (1998).
    [CrossRef]
  19. G. M. D’Ariano, M. F. Sacchi, “Tomographic measurements of nonclassical radiation states,” Phys. Rev. A 59, 826–830 (1999).
    [CrossRef]
  20. M. S. Kim, W. Son, V. Bužek, P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002).
    [CrossRef]
  21. Wang Xiang-bin, “Theorem for the beam-splitter entangler,” Phys. Rev. A 66, 024303 (2002).
    [CrossRef]
  22. J. K. Asbóth, J. Calsamiglia, H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
    [CrossRef] [PubMed]
  23. G. Vidal, R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
    [CrossRef]
  24. V. Vedral, M. B. Plenio, M. A. Rippin, P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
    [CrossRef]
  25. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [CrossRef]
  26. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
    [CrossRef]
  27. C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991).
    [CrossRef] [PubMed]
  28. C. T. Lee, “Theorem on nonclassical states,” Phys. Rev. A 52, 3374–3376 (1995).
    [CrossRef] [PubMed]
  29. A. Ferraro, S. Olivares, M. G. A. Paris, “Gaussian states in continuous variable quantum information,” quant-ph/0503237, p. 38.
  30. H. R. Li, F. L. Li, Y. Yang, “Entangling two single-mode Gaussian states by use of a beam splitter,” Chin. Phys. 15, 2947–2952 (2006).
    [CrossRef]
  31. E. Shchukin, Th. Richter, W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802(R) (2005).
    [CrossRef]
  32. E. V. Shchukin, W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
    [CrossRef]
  33. T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
    [CrossRef]

2008 (2)

T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
[CrossRef]

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

2007 (3)

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

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

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

2006 (2)

A. R. Usha Devi, R. Prabhu, M. S. Uma, “Non-classicality of photon added coherent and thermal radiations,” Eur. Phys. J. D 40, 133–138 (2006).
[CrossRef]

H. R. Li, F. L. Li, Y. Yang, “Entangling two single-mode Gaussian states by use of a beam splitter,” Chin. Phys. 15, 2947–2952 (2006).
[CrossRef]

2005 (6)

E. Shchukin, Th. Richter, W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802(R) (2005).
[CrossRef]

E. V. Shchukin, W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

J. K. Asbóth, J. Calsamiglia, H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef] [PubMed]

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

S. L. Braunstein, P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

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

2004 (2)

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

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

2002 (5)

S. S. Mizrahi, V. V. Dodonov, “Creating quanta with an ‘annihilation’ operator,” J. Phys. A 35, 8847–8857 (2002).
[CrossRef]

G. Vidal, R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Th. Richter, W. Vogel, “Nonclassicality of quantum states: a hierarchy of observable conditions,” Phys. Rev. Lett. 89, 283601 (2002).
[CrossRef]

M. S. Kim, W. Son, V. Bužek, P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002).
[CrossRef]

Wang Xiang-bin, “Theorem for the beam-splitter entangler,” Phys. Rev. A 66, 024303 (2002).
[CrossRef]

2000 (1)

W. Vogel, “Nonclassical states: an observable criterion,” Phys. Rev. Lett. 84, 1849–1852 (2000).
[CrossRef] [PubMed]

1999 (1)

G. M. D’Ariano, M. F. Sacchi, “Tomographic measurements of nonclassical radiation states,” Phys. Rev. A 59, 826–830 (1999).
[CrossRef]

1998 (1)

Arvind, N. Mukunda, R. Simon, “Characterizations of classical and nonclassical states of quantized radiation,” J. Phys. A 31, 565–583 (1998).
[CrossRef]

1997 (1)

V. Vedral, M. B. Plenio, M. A. Rippin, P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

1996 (1)

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
[CrossRef]

1995 (1)

C. T. Lee, “Theorem on nonclassical states,” Phys. Rev. A 52, 3374–3376 (1995).
[CrossRef] [PubMed]

1992 (1)

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

1991 (2)

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991).
[CrossRef] [PubMed]

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

1963 (2)

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

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

Agarwal, G. S.

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

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

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

Arvind,

Arvind, N. Mukunda, R. Simon, “Characterizations of classical and nonclassical states of quantized radiation,” J. Phys. A 31, 565–583 (1998).
[CrossRef]

Asbóth, J. K.

J. K. Asbóth, J. Calsamiglia, H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef] [PubMed]

Bellini, M.

T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
[CrossRef]

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

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

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

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

Biswas, A.

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

Braunstein, S. L.

S. L. Braunstein, P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

Bužek, V.

M. S. Kim, W. Son, V. Bužek, P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002).
[CrossRef]

Calsamiglia, J.

J. K. Asbóth, J. Calsamiglia, H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef] [PubMed]

D’Ariano, G. M.

G. M. D’Ariano, M. F. Sacchi, “Tomographic measurements of nonclassical radiation states,” Phys. Rev. A 59, 826–830 (1999).
[CrossRef]

Dodonov, V. V.

S. S. Mizrahi, V. V. Dodonov, “Creating quanta with an ‘annihilation’ operator,” J. Phys. A 35, 8847–8857 (2002).
[CrossRef]

Fan, H. Y.

Ferraro, A.

A. Ferraro, S. Olivares, M. G. A. Paris, “Gaussian states in continuous variable quantum information,” quant-ph/0503237, p. 38.

Glauber, R. J.

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

Grangier, P.

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

Hu, L. Y.

Jeong, H.

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

Kiesel, T.

T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
[CrossRef]

Kim, M.

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

Kim, M. S.

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

M. S. Kim, W. Son, V. Bužek, P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002).
[CrossRef]

Klyshko, D. N.

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7–15 (1996).
[CrossRef]

Knight, P. L.

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

M. S. Kim, W. Son, V. Bužek, P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002).
[CrossRef]

V. Vedral, M. B. Plenio, M. A. Rippin, P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Lee, C. T.

C. T. Lee, “Theorem on nonclassical states,” Phys. Rev. A 52, 3374–3376 (1995).
[CrossRef] [PubMed]

C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991).
[CrossRef] [PubMed]

Li, F. L.

H. R. Li, F. L. Li, Y. Yang, “Entangling two single-mode Gaussian states by use of a beam splitter,” Chin. Phys. 15, 2947–2952 (2006).
[CrossRef]

Li, H. R.

H. R. Li, F. L. Li, Y. Yang, “Entangling two single-mode Gaussian states by use of a beam splitter,” Chin. Phys. 15, 2947–2952 (2006).
[CrossRef]

Mizrahi, S. S.

S. S. Mizrahi, V. V. Dodonov, “Creating quanta with an ‘annihilation’ operator,” J. Phys. A 35, 8847–8857 (2002).
[CrossRef]

Mukunda, N.

Arvind, N. Mukunda, R. Simon, “Characterizations of classical and nonclassical states of quantized radiation,” J. Phys. A 31, 565–583 (1998).
[CrossRef]

Olivares, S.

A. Ferraro, S. Olivares, M. G. A. Paris, “Gaussian states in continuous variable quantum information,” quant-ph/0503237, p. 38.

Parigi, V.

T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
[CrossRef]

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

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

Paris, M. G. A.

A. Ferraro, S. Olivares, M. G. A. Paris, “Gaussian states in continuous variable quantum information,” quant-ph/0503237, p. 38.

Park, E.

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

Plenio, M. B.

V. Vedral, M. B. Plenio, M. A. Rippin, P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Prabhu, R.

A. R. Usha Devi, R. Prabhu, M. S. Uma, “Non-classicality of photon added coherent and thermal radiations,” Eur. Phys. J. D 40, 133–138 (2006).
[CrossRef]

Richter, Th.

E. Shchukin, Th. Richter, W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802(R) (2005).
[CrossRef]

Th. Richter, W. Vogel, “Nonclassicality of quantum states: a hierarchy of observable conditions,” Phys. Rev. Lett. 89, 283601 (2002).
[CrossRef]

Rippin, M. A.

V. Vedral, M. B. Plenio, M. A. Rippin, P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Ritsch, H.

J. K. Asbóth, J. Calsamiglia, H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev. Lett. 94, 173602 (2005).
[CrossRef] [PubMed]

Sacchi, M. F.

G. M. D’Ariano, M. F. Sacchi, “Tomographic measurements of nonclassical radiation states,” Phys. Rev. A 59, 826–830 (1999).
[CrossRef]

Scully, M. O.

M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Shchukin, E.

E. Shchukin, Th. Richter, W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802(R) (2005).
[CrossRef]

Shchukin, E. V.

E. V. Shchukin, W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

Simon, R.

Arvind, N. Mukunda, R. Simon, “Characterizations of classical and nonclassical states of quantized radiation,” J. Phys. A 31, 565–583 (1998).
[CrossRef]

Son, W.

M. S. Kim, W. Son, V. Bužek, P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002).
[CrossRef]

Sudarshan, E. C. G.

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

Tara, K.

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

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

Tualle-Brouri, R.

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

Uma, M. S.

A. R. Usha Devi, R. Prabhu, M. S. Uma, “Non-classicality of photon added coherent and thermal radiations,” Eur. Phys. J. D 40, 133–138 (2006).
[CrossRef]

Usha Devi, A. R.

A. R. Usha Devi, R. Prabhu, M. S. Uma, “Non-classicality of photon added coherent and thermal radiations,” Eur. Phys. J. D 40, 133–138 (2006).
[CrossRef]

van Loock, P.

S. L. Braunstein, P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[CrossRef]

Vedral, V.

V. Vedral, M. B. Plenio, M. A. Rippin, P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Viciani, S.

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

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

Vidal, G.

G. Vidal, R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Vogel, W.

T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
[CrossRef]

E. V. Shchukin, W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

E. Shchukin, Th. Richter, W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802(R) (2005).
[CrossRef]

Th. Richter, W. Vogel, “Nonclassicality of quantum states: a hierarchy of observable conditions,” Phys. Rev. Lett. 89, 283601 (2002).
[CrossRef]

W. Vogel, “Nonclassical states: an observable criterion,” Phys. Rev. Lett. 84, 1849–1852 (2000).
[CrossRef] [PubMed]

Wenger, J.

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

Werner, R. F.

G. Vidal, R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Xiang-bin, Wang

Wang Xiang-bin, “Theorem for the beam-splitter entangler,” Phys. Rev. A 66, 024303 (2002).
[CrossRef]

Yang, Y.

H. R. Li, F. L. Li, Y. Yang, “Entangling two single-mode Gaussian states by use of a beam splitter,” Chin. Phys. 15, 2947–2952 (2006).
[CrossRef]

Zavatta, A.

T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, M. Bellini, “Experimental determination of a nonclassical Glauber–Sudarshan P function,” Phys. Rev. A 78, 021804(R) (2008).
[CrossRef]

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

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

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

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Figures (1)

Fig. 1
Fig. 1

Nonclassical depth of the photon-added-then-subtracted Gaussian state τ m ( s a ) (solid) in comparison with that of the photon-subtracted Gaussian state τ m ( s ) (dashed) and that of the original Gaussian state τ m (dotted) as a function of m with (a) n = 1 and (b) n = 2 .

Equations (30)

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R ( z , τ ) = 1 π τ d 2 α exp ( z α 2 τ ) P ( α ) .
ρ = n = 0 m = 0 ρ ( n , m ) n m .
ρ s a l k = N s a l k ( a ) l ( a ) k ρ ( a ) k ( a ) l ,
ρ s a l k = N s a l k n m ρ ( n , m ) ( n + k ) ! n ! ( n + k l ) ! n + k l m + k l ( m + k ) ! m ! ( m + k l ) ! .
ρ a s k l = N a s k l ( a ) k ( a ) l ρ ( a ) l ( a ) k = N a s k l n m ρ ( n , m ) n ! ( n + k l ) ! ( n l ) ! n + k l m + k l m ! ( m + k l ) ! ( m l ) ! ,
C ( n ) ( λ , λ * ) = tr ( e i λ a e i λ * a ρ ) = e n λ 2 1 2 m * λ 2 1 2 m λ * 2 ,
τ m = max { 0 , m n } .
ρ s = N s a ρ a ,
C s ( n ) ( λ , λ * ) = N s [ n ( n 2 + m 2 ) λ 2 n m * λ 2 n m λ * 2 ] e n λ 2 1 2 m * λ 2 1 2 m λ * 2 .
R s ( z , τ ) = 1 π τ d 2 α e z a 2 τ 1 π 2 d 2 λ e i λ α * i λ * α C s ( n ) ( λ , λ * ) = 1 π 2 d 2 λ e τ λ 2 i λ z * i λ * z C s ( n ) ( λ , λ * ) = N s π 2 d 2 λ [ n ( n 2 + m 2 ) λ 2 n m * λ 2 n m λ * 2 ] × e ( n + τ ) λ 2 1 2 m * λ 2 1 2 m λ * 2 i λ z * i λ * z .
n + τ m
R s ( z , τ ) = N s π [ ( n + τ ) 2 m 2 ] 5 2 exp [ ( n + τ ) z 2 + 1 2 m * z 2 + 1 2 m z * 2 ( n + τ ) 2 m 2 ] { 2 N z 2 + M * z 2 + M z * 2 + τ [ ( n + τ ) 2 m 2 ] [ n ( n + τ ) m 2 ] } ,
N = 1 2 { [ n ( n + τ ) m 2 ] 2 + m 2 τ 2 } 0 ,
M = m τ n ( n + τ ) m 2 .
N M = 1 2 [ n ( n + τ ) m 2 m τ ] 2 0 ,
n ( n + τ ) m 2 0 ,
τ m 2 n 2 n .
τ m ( s ) = max { 0 , m 2 n 2 n } .
τ m ( s ) = m + n n ( m n ) τ m .
ρ s a = N s a a a ρ a a ,
R s a ( z , τ ) = N s a π [ ( n + τ ) 2 m 2 ] 9 2 exp [ ( n + τ ) z 2 + 1 2 m * z 2 + 1 2 m z * 2 ( n + τ ) 2 m 2 ] R 0 ,
R 0 = [ ( n 2 + m 2 ) B 2 + n ( m * B 2 + m B * 2 ) ] × [ ( ( n + 1 ) 2 + m 2 ) B 2 + ( n + 1 ) ( m * B 2 + m B * 2 ) ] + [ ( n 2 + m 2 ) ( ( n + 1 ) 2 + m 2 ) + 2 n ( n + 1 ) m 2 ] { 6 ( n + τ ) A B 2 + A 2 [ 2 ( n + τ ) z 2 m * z 2 m z * 2 ] + A 2 [ 2 ( n + τ ) 2 + m 2 ] } + [ ( n + 1 ) ( n 2 + m 2 ) + n ( ( n + 1 ) 2 + m 2 ) ] { 5 ( n + τ ) A ( m * B 2 + m B * 2 ) + 2 m 2 A B 2 + 2 A 2 [ ( n + τ ) ( m * z 2 + m z * 2 ) 2 m 2 z 2 ] 6 ( n + τ ) m 2 A 2 } + n ( n + 1 ) 6 m 2 A ( m * B 2 + m B * 2 + m 2 A ) + ( 4 n 3 + 7 n 2 + 3 n + 8 n m 2 + 5 m 2 ) A 2 [ B 2 ( n + τ ) A ] + ( 5 n 2 + 6 n + 1 + m 2 ) A 2 ( m * B 2 + m B * 2 + 2 m 2 A ) + ( 2 n 2 + 3 n + 1 + m 2 ) A 4 ,
R s a ( z , τ ) = N s a ( n + 1 ) π ( n + τ ) 5 exp ( z 2 n + τ ) { ( n + 1 ) n 2 z 4 + n ( n + τ ) [ ( 4 n + 3 ) τ n ] z 2 + ( n + τ ) 2 τ [ ( 2 n + 1 ) τ n ] } .
τ m ( s a ) = n 2 n + 1 .
ρ η = tr b [ B ( ρ 0 b 0 ) B ] ,
C η ( n ) ( λ , λ * ) = tr a { e i λ a e i λ * a tr b [ B ( ρ 0 b 0 ) B ] } = tr a , b ( B e i λ a e i λ * a B ρ 0 b 0 ) = tr a , b ( e i λ ( a cos θ b sin θ ) e i λ * ( a cos θ b sin θ ) ρ 0 b 0 ) = tr a ( e i λ cos θ a e i λ * cos θ a ρ ) tr b ( e i λ sin θ b e i λ * sin θ b 0 b 0 ) = C ( n ) ( λ cos θ , λ * cos θ ) .
R s a , λ ( z , τ ) = N s a , η ( n + 1 ) π ( n η + τ ) 5 exp ( z 2 n η + τ ) { ( n + 1 ) n 2 η 2 z 4 + n η ( n η + τ ) [ ( 4 n + 3 ) τ n η ] z 2 + ( n η + τ ) 2 τ [ ( 2 n + 1 ) τ n η ] } .
τ m , η ( s a ) = η n 2 n + 1 = η τ m ( s a ) ,
τ m , η ( a ) = η = η τ m ( a ) ,
τ m , η ( s ) = max { 0 , η m 2 n 2 n } = η τ m ( s ) .

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