Abstract

A sufficient condition for nonclassical two-mode states in terms of three probabilities,

(n2+1)p(n1-1, n2+1)2n1p(n1, n2)+(n1+1)p(n1+1, n2-1)2n2p(n1, n2)<1,
is established as a new type II criterion. This criterion is applied to the two-mode coherent state, the two-mode squeezed vacuum state, and the pair coherent state to show how easily it can be used. It is also applied to the photon-added two-mode thermal state and the mixing of two two-photon states by a symmetric beam splitter to show that the type II criterion is superior to its type I counterpart in at least these two examples.

© 1998 Optical Society of America

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References

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  1. L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. 4, 205 (1979).
    [CrossRef] [PubMed]
  2. C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 41, 1721 (1990).
    [CrossRef] [PubMed]
  3. C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A 42, 1608 (1990).
    [CrossRef] [PubMed]
  4. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7 (1996); “The nonclassical light,” Sov. Phys. Usp. 39, 573 (1996) [Usp. Fiz. Nauk. 166, 613 (1966)].
    [CrossRef]
  5. R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84 (1963); E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277 (1963).
    [CrossRef]
  6. R. F. Muirhead, “Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters,” Proc. Edinburgh Math. Soc. 21, 144 (1903).
  7. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications (Academic, New York, 1979).
  8. G. S. Agarwal, “Nonclassical statistics of fields in pair coherent states,” J. Opt. Soc. Am. B 5, 1940 (1988).
    [CrossRef]
  9. M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086 (1985); R. W. Boyd, M. C. Malcuit, D. J. Gauthier, and K. Rzazewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648 (1987).
    [CrossRef] [PubMed]
  10. G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in photon-added thermal state,” Quantum Semiclassic. Opt. 9, 411 (1997).
    [CrossRef]
  11. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
    [CrossRef] [PubMed]
  12. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371 (1989).
    [CrossRef] [PubMed]

1997 (1)

G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in photon-added thermal state,” Quantum Semiclassic. Opt. 9, 411 (1997).
[CrossRef]

1990 (2)

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 41, 1721 (1990).
[CrossRef] [PubMed]

C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A 42, 1608 (1990).
[CrossRef] [PubMed]

1989 (1)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

1979 (1)

Agarwal, G. S.

Campos, R. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371 (1989).
[CrossRef] [PubMed]

Haight, J.

G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in photon-added thermal state,” Quantum Semiclassic. Opt. 9, 411 (1997).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

Jones, G. N.

G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in photon-added thermal state,” Quantum Semiclassic. Opt. 9, 411 (1997).
[CrossRef]

Lee, C. T.

G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in photon-added thermal state,” Quantum Semiclassic. Opt. 9, 411 (1997).
[CrossRef]

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 41, 1721 (1990).
[CrossRef] [PubMed]

C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A 42, 1608 (1990).
[CrossRef] [PubMed]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

L. Mandel, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. 4, 205 (1979).
[CrossRef] [PubMed]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371 (1989).
[CrossRef] [PubMed]

Teich, M. C.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (3)

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 41, 1721 (1990).
[CrossRef] [PubMed]

C. T. Lee, “Nonclassical photon statistics of two-mode squeezed states,” Phys. Rev. A 42, 1608 (1990).
[CrossRef] [PubMed]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt. (1)

G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in photon-added thermal state,” Quantum Semiclassic. Opt. 9, 411 (1997).
[CrossRef]

Other (5)

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213, 7 (1996); “The nonclassical light,” Sov. Phys. Usp. 39, 573 (1996) [Usp. Fiz. Nauk. 166, 613 (1966)].
[CrossRef]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84 (1963); E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

R. F. Muirhead, “Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters,” Proc. Edinburgh Math. Soc. 21, 144 (1903).

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications (Academic, New York, 1979).

M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086 (1985); R. W. Boyd, M. C. Malcuit, D. J. Gauthier, and K. Rzazewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648 (1987).
[CrossRef] [PubMed]

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Equations (51)

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(n2+1)p(n1-1, n2+1)2n1p(n1, n2)+(n1+1)p(n1+1, n2-1)2n2p(n1, n2)<1,
Qn(2)-n2n<0,
n(l+1)n(m-1)-n(l)n(m)<0,lm.
n1(2)+n2(2)-2(n1n2)<0,
(n+1)p(n-1)p(n+1)n[p(n)]2<1,n=1,2,3, ,
R(n1, n2)(n2+1)p(n1-1, n2+1)2n1p(n1, n2)+(n1+1)p(n1+1, n2-1)2n2p(n1, n2)<1,
n1, n2=1,2,3, ,
ρˆ=1π2 d2α1d2α2P(α1, α2)|α2|α1α1|α2|,
n1(k1)n2(k2)=1π2 d2α1d2α2P(α1, α2)×|α1|2k1|α2|2k2,
p(n1, n2)=1π2n1!n2! d2α1d2α2P(α1, α2)×exp(-|α1|2-|α2|2)|α1|2n1|α2|2n2.
Ps(α1, α2)P(α1, α2) exp[-s(|α1|2+|α2|2)],
μs(k1, k2)1π2 d2α1d2α2Ps(α1, α2)|α1|2k1|α2|2k2,
μ0(k1, k2)=n1(k1)n2(k2),
μ1(n1, n2)=n1!n2!p(n1, n2)
Ms(k1, k2)μs(k1-1, k2+1)+μs(k1+1,k2-1)-2μs(k1, k2)=1π2 d2α1d2α2Ps(α1, α2)|α1|2k1-2×|α2|2k2-2(|α1|2-|α2|2)2
Ms(k1, k2)0forallk11,k21;
Ms(k1, k2)<0forsomek11,k21.
12(|α1|2+|α2|2)|α1||α2|.
p(n1, n2)=exp(-n1-n2) n1n1n2n2n1!n2!.
R(n1, n2)=n2(n2+1)n22+n1(n1+1)n122n1n2n1n2
>n22n22+n12n122n1n2n1n21,
Sˆ(ζ)exp[ζa^1a^2-ζ*a^1a^2]=sech r exp[a^1a^2 exp(iθ) tanh r]×exp[(a^1a^1+a^2a^2) ln(sech r)]×exp[-a^1a^2 exp(-iθ) tanh r]
|ζSˆ(ζ)|0, 0=sech r exp[a^1a^2 exp(iθ) tanh r]|0, 0=sech rn=0[exp(iθ) tanh r]n|n, n,
p(n1, n2)=sech2 r(tanh r)2n1δn1, n2.
R(n, n)=0<1,
a^1a^2|ζ, q=ζ|ζ, q,
(a^1a^1-a^2a^2)|ζ, q=q|ζ, q,
|ζ, q=Nqn=0 ζn[n!(n+q)!]1/2 |n+q, n,
Nqn=0 r2n(n+q)!n!-1/2=[rq/Iq(2r)]1/2
p(n1, n2)=Nq2 r2n2n1!n2!ifn1-n2=q0ifn1-n2q.
R(n+q, n)=0<1,
ρ^s(t)n=1m=1 sin(n|g|t) sin(m|g|t)×ρ0(n-1, m-1)|nm|,
n=1m=1ρ0(n, m)|nm|,
p(n1, n2)=(1-x1)(1-x2)x1n1x2n2,
xi=exp(-ωi/kT),i=1, 2.
ps(n1, n2)0ifn2=0x1n1x2n2-1 sin2(n2|g|t)ifn21,
R(n1, n2)=(n2+1)x2 sin2(n2+1|g|t)2n1x1 sin2(n2|g|t)+(n1+1)x1 sin2(n2-1|g|t)2n2x2 sin2(n2|g|t).
R(n1, 1)=x2 sin2(n2+1|g|t)n1x1 sin2(n2|g|t).
n1>x2 sin2(n2+1|g|t)x1 sin2(n2|g|t).
|ψ=(R-T)|11|12+i2RT|21|02+i2RT|01|22,
p(1, 1)=δ,
p(2, 0)=p(0, 2)=(1-δ)/2,
p(n1, n2)=0otherwise,
n1(2)+n2(2)-2n1n2=2(1-δ)-2δ<0.
δ>12.
R(1, 1)=2p(0, 2)+2p(2, 0)2p(1, 1)=2(1-δ)2δ<1,
p(2, 2)=0.25,
p(4, 0)=p(0, 4)=0.375,
p(n1, n2)=0otherwise.
n1(2)+n2(2)-2n1n2=5.0+5.0-2<0.
R(2, 2)3p(1, 3)4p(2, 2)+3p(3, 1)4p(2, 2)=01<1,

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