Abstract

The nonclassicality of photon-added coherent fields in the thermal channel is investigated by exploring the volume of the negative part of the Wigner function that reduces with the dissipative time. The Wigner functions become positive when the decay time exceeds a threshold value. For the case of the single photon-added coherent state, we derive the exact threshold values of decay time in the thermal channel. For arbitrary partial negative Wigner distribution function, a generic analytical relation between the mean photon number of heat bath and the threshold value of decay time is presented. Finally, the possible application of single photon-added coherent states in quantum computation has been briefly discussed.

© 2008 Optical Society of America

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References

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  1. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).
  2. H. J. Kimble, M. Dagenais, and L. Mandel, "Photon antibunching in resonance fluorescence," Phys. Rev. Lett. 39, 691-695 (1977).
    [CrossRef]
  3. R. Short and L. Mandel, "Observation of sub-Poissonian photon statistics," Phys. Rev. Lett. 51, 384-387 (1983).
    [CrossRef]
  4. V. V. Dodonov, "Nonclassical states in quantum optics: a squeezed review of the first 75 years," J. Opt. B: Quantum Semiclassical Opt. 4, R1-R33 (2002).
    [CrossRef]
  5. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 106, 121-167 (1984).
    [CrossRef]
  6. C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
  7. K. Vogel and H. Risken, "Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase," Phys. Rev. A 40, 2847-2849 (1989).
    [CrossRef] [PubMed]
  8. T. D. Smithey, M. Beck, M. G. Raymer, and A. Faridani, "Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum," Phys. Rev. Lett. 70, 1244-1247 (1993).
    [CrossRef] [PubMed]
  9. D.-G. Welsch, W. Vogel, and T. Opartný, "Homodyne detection and quantum state reconstruction," Prog. Opt. 39, 63-211 (1999).
    [CrossRef]
  10. M. G. Benedict and A. Czirjak, "Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms," Phys. Rev. A 60, 4034-4044 (1999).
    [CrossRef]
  11. V. V. Dodonov and M. A. Andreata, "Shrinking quantum packets in one dimension," Phys. Lett. A 310, 101-109 (2003).
    [CrossRef]
  12. A. Kenfack and K. Zyczkowski, "Negativity of the Wigner function as an indicator of non-classicality," J. Opt. B: Quantum Semiclassical Opt. 6, 396-404 (2004).
    [CrossRef]
  13. 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] [PubMed]
  14. 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]
  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 (2005).
    [CrossRef]
  16. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
    [CrossRef]
  17. S.-B. Li, X.-B. Zou, and G.-C. Guo, "Nonclassicality of quantum excitation of classical coherent field in photon loss channel," Phys. Rev. A 75, 045801 (2007).
    [CrossRef]
  18. S. M. Barnett and P. L. Knight, "Squeezing in correlated quantum systems," J. Mod. Opt. 34, 841-853 (1987).
    [CrossRef]
  19. H. Moya-Cessa and P. L. Knight, "Series representation of quantum-field quasi-probabilities," Phys. Rev. A 48, 2479-2481 (1993).
    [CrossRef] [PubMed]
  20. B.-G. Englert, N. Sterpi, and H. Walther, "Parity states in the one-atom maser," Opt. Commun. 100, 526-535 (1993).
    [CrossRef]
  21. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 1999).
  22. J. K. Asbóth, J. Calsamiglia, and H. Ritsch, "Computable measure of nonclassicality for light," Phys. Rev. Lett. 94, 173602 (2005).
    [CrossRef] [PubMed]
  23. A. Biswas and G. S. Agarwal, "Nonclassicality and decoherence of photon-subtracted squeezed states," Phys. Rev. A 75, 032104 (2007).
    [CrossRef]
  24. T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, "Quantum computation with optical coherent states," Phys. Rev. A 68, 042319 (2003).
    [CrossRef]
  25. T. C. Ralph, W. J. Munro, and G. J. Milburn, "Quantum computation with coherent states, linear interactions and superposed resources," Proc. SPIE 4917, 1 (2002).
    [CrossRef]

2007 (2)

S.-B. Li, X.-B. Zou, and G.-C. Guo, "Nonclassicality of quantum excitation of classical coherent field in photon loss channel," Phys. Rev. A 75, 045801 (2007).
[CrossRef]

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

2005 (2)

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

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 (2005).
[CrossRef]

2004 (2)

A. Kenfack and K. Zyczkowski, "Negativity of the Wigner function as an indicator of non-classicality," J. Opt. B: Quantum Semiclassical Opt. 6, 396-404 (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]

2003 (2)

V. V. Dodonov and M. A. Andreata, "Shrinking quantum packets in one dimension," Phys. Lett. A 310, 101-109 (2003).
[CrossRef]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, "Quantum computation with optical coherent states," Phys. Rev. A 68, 042319 (2003).
[CrossRef]

2002 (2)

T. C. Ralph, W. J. Munro, and G. J. Milburn, "Quantum computation with coherent states, linear interactions and superposed resources," Proc. SPIE 4917, 1 (2002).
[CrossRef]

V. V. Dodonov, "Nonclassical states in quantum optics: a squeezed review of the first 75 years," J. Opt. B: Quantum Semiclassical Opt. 4, R1-R33 (2002).
[CrossRef]

1999 (2)

D.-G. Welsch, W. Vogel, and T. Opartný, "Homodyne detection and quantum state reconstruction," Prog. Opt. 39, 63-211 (1999).
[CrossRef]

M. G. Benedict and A. Czirjak, "Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms," Phys. Rev. A 60, 4034-4044 (1999).
[CrossRef]

1993 (3)

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

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

B.-G. Englert, N. Sterpi, and H. Walther, "Parity states in the one-atom maser," Opt. Commun. 100, 526-535 (1993).
[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] [PubMed]

1989 (1)

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

1987 (1)

S. M. Barnett and P. L. Knight, "Squeezing in correlated quantum systems," J. Mod. Opt. 34, 841-853 (1987).
[CrossRef]

1984 (1)

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

1983 (1)

R. Short and L. Mandel, "Observation of sub-Poissonian photon statistics," Phys. Rev. Lett. 51, 384-387 (1983).
[CrossRef]

1977 (1)

H. J. Kimble, M. Dagenais, and L. Mandel, "Photon antibunching in resonance fluorescence," Phys. Rev. Lett. 39, 691-695 (1977).
[CrossRef]

1963 (1)

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

J. Mod. Opt. (1)

S. M. Barnett and P. L. Knight, "Squeezing in correlated quantum systems," J. Mod. Opt. 34, 841-853 (1987).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (2)

A. Kenfack and K. Zyczkowski, "Negativity of the Wigner function as an indicator of non-classicality," J. Opt. B: Quantum Semiclassical Opt. 6, 396-404 (2004).
[CrossRef]

V. V. Dodonov, "Nonclassical states in quantum optics: a squeezed review of the first 75 years," J. Opt. B: Quantum Semiclassical Opt. 4, R1-R33 (2002).
[CrossRef]

Opt. Commun. (1)

B.-G. Englert, N. Sterpi, and H. Walther, "Parity states in the one-atom maser," Opt. Commun. 100, 526-535 (1993).
[CrossRef]

Phys. Lett. A (1)

V. V. Dodonov and M. A. Andreata, "Shrinking quantum packets in one dimension," Phys. Lett. A 310, 101-109 (2003).
[CrossRef]

Phys. Rep. (1)

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

Phys. Rev. (1)

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

Phys. Rev. A (8)

S.-B. Li, X.-B. Zou, and G.-C. Guo, "Nonclassicality of quantum excitation of classical coherent field in photon loss channel," Phys. Rev. A 75, 045801 (2007).
[CrossRef]

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

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 (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] [PubMed]

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

M. G. Benedict and A. Czirjak, "Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms," Phys. Rev. A 60, 4034-4044 (1999).
[CrossRef]

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

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, "Quantum computation with optical coherent states," Phys. Rev. A 68, 042319 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

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

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

H. J. Kimble, M. Dagenais, and L. Mandel, "Photon antibunching in resonance fluorescence," Phys. Rev. Lett. 39, 691-695 (1977).
[CrossRef]

R. Short and L. Mandel, "Observation of sub-Poissonian photon statistics," Phys. Rev. Lett. 51, 384-387 (1983).
[CrossRef]

Proc. SPIE (1)

T. C. Ralph, W. J. Munro, and G. J. Milburn, "Quantum computation with coherent states, linear interactions and superposed resources," Proc. SPIE 4917, 1 (2002).
[CrossRef]

Prog. Opt. (1)

D.-G. Welsch, W. Vogel, and T. Opartný, "Homodyne detection and quantum state reconstruction," Prog. Opt. 39, 63-211 (1999).
[CrossRef]

Science (1)

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]

Other (3)

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

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

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, 1999).

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Figures (5)

Fig. 1
Fig. 1

Wigner functions of the SPACS with α = 0.5 in a thermal channel with n = 1 are depicted for two different values of decay time γ t .

Fig. 2
Fig. 2

Wigner functions of the TPACS with α = 0.5 in a thermal channel with n = 1 are depicted for two different values of decay time γ t .

Fig. 3
Fig. 3

P N W of the SPACS and TPACS with α = 1.5 in a thermal channel are depicted as the function of γ t for different values of mean thermal photon number n. From top to bottom, n = 0.1 ,0.2,0.3,0.4,0.5,0.6,0.7,0.8, and 0.9.

Fig. 4
Fig. 4

P N W is plotted as the function of the decay time γ t for SPACS and TPACS with different values of α. Solid curve: from top to bottom, SPACS with α = 0.1 ,0.5,1.0, and 1.5. Dashed curve: from top to bottom, TPACS with α = 0.1 ,0.5,1.0, and 1.5. n = 0.5 .

Fig. 5
Fig. 5

Threshold decay time γ t c beyond which P N W = 0 is plotted as the function of the mean thermal photon number n for the case of the SPACS. Solid square: numerical results. Solid curve: γ t c = ln ( 2 + 2 n ) ( 1 + 2 n ) . It is shown that γ t c 0.5 n when n 1 . The numerical calculations are based on the SPACS with α = 0.5 .

Equations (41)

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d ρ d t = γ ( n + 1 ) 2 ( 2 a ρ a a a ρ ρ a a ) + γ n 2 ( 2 a ρ a a a ρ ρ a a ) ,
W ( β ) = 2 π Tr [ ( O ̂ e O ̂ o ) D ̂ ( β ) ρ D ̂ ( β ) ] ,
t W ( q , p , t ) = γ 2 ( q q + p p ) W ( q , p , t ) + γ ( 2 n + 1 ) 8 ( 2 q 2 + 2 p 2 ) W ( q , p , t ) ,
W ( q , p , γ t ) = exp ( γ t ) W T ( x , y ) W ( q 1 e γ t x e γ t , p 1 e γ t y e γ t , 0 ) d x d y ,
W T ( x , y ) = 2 π ( 1 + 2 n ) exp ( 2 ( x 2 + y 2 ) 1 + 2 n )
W S ( q , p , 0 ) = 2 L 1 ( 2 q + 2 i p α 2 ) π L 1 ( α 2 ) e 2 q + i p α 2 ,
W T ( q , p , 0 ) = 2 L 2 ( 2 q + 2 i p α 2 ) π L 2 ( α 2 ) e 2 q + i p α 2 ,
W S ( q , p , γ t ) = 2 e γ t [ ( ξ c 2 Re α ) 2 + ( ζ c 2 Im α ) 2 + c 4 1 ] exp [ 2 ( μ 2 + ν 2 ) ( 1 + c 2 ) ] π ( 1 + α 2 ) ( 1 + c 2 ) 3 ,
c = [ ( exp ( γ t ) 1 ) ( 1 + 2 n ) ] 1 2 ,
μ = Re ( α ) q exp ( γ t 2 ) ,
ν = Im ( α ) p exp ( γ t 2 ) ,
ξ = Re ( α ) 2 q exp ( γ t 2 ) ,
ζ = Im ( α ) 2 p exp ( γ t 2 ) .
P N W = Ω W ( q , p ) d q d p ,
γ t c = ln ( 2 + 2 n 1 + 2 n ) ,
γ t c ( n ) = ln e γ t c ( 0 ) + 2 n 1 + 2 n ,
W ( q , p , γ t ) = exp ( γ t ) W 0 ( x , y ) W ( e γ t 2 q e γ t 1 x , e γ t 2 p e γ t 1 y , 0 ) d x d y ,
W 0 ( x , y ) = 2 π exp ( 2 ( x 2 + y 2 ) ) ,
x = x 1 + 2 n ,
y = y 1 + 2 n ,
q = e γ t 2 1 + ( 1 + 2 n ) ( e γ t 1 ) q ,
p = e γ t 2 1 + ( 1 + 2 n ) ( e γ t 1 ) p ,
γ t = ln [ 1 + ( 1 + 2 n ) ( e γ t 1 ) ] .
W ( q , p , γ t ) e γ t = W ( 0 ) ( q , p , γ t ) e γ t ,
P N W ( γ t ) = P N W ( 0 ) ( γ t ) ,
U B S = exp [ i ϕ ( a b + a b ) ] ,
U B S 0 a 0 b = 0 a 0 b ,
U B S 1 N ( α , 1 ) a α a 0 b = 1 N ( α , 1 ) ( cos ( ϕ ) a + i sin ( ϕ ) b ) cos ( ϕ ) α a i sin ( ϕ ) α b ,
U B S 1 N ( α , 1 ) b 0 a α b = 1 N ( α , 1 ) ( cos ( ϕ ) b + i sin ( ϕ ) a ) i sin ( ϕ ) α a cos ( ϕ ) α b ,
U B S 1 N ( α , 1 ) a b α a α b = 1 N ( α , 1 ) ( cos ( ϕ ) a + i sin ( ϕ ) b ) ( cos ( ϕ ) b + i sin ( ϕ ) a ) e i ϕ α a e i ϕ α b ,
0 b a 0 U B S 0 a 0 b = 1 ,
1 N ( α , 1 ) α b a 0 a U B S a α a 0 b = 1 N ( α , 1 ) 0 b a α b U B S b 0 a α b
= cos 2 ( ϕ ) α 2 + cos ϕ N ( α , 1 ) e α 2 ( cos ϕ 1 ) ,
1 N 2 ( α , 1 ) α b a α a b U B S a b α a α b
= α 4 e 4 i ϕ + 2 α 2 e 3 i ϕ + cos ( 2 ϕ ) N 2 ( α , 1 ) e 2 α 2 ( e i ϕ 1 ) .
0 b a 0 U B S 0 a 0 b = 1 ,
1 N ( α , 1 ) α b a 0 a U B S a α a 0 b
= 1 N ( α , 1 ) 0 b a α b U B S b 0 a α b
1 ,
1 N 2 ( α , 1 ) α b a α a b U B S a b α a α b
e 2 i ϕ α 2 .

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