Abstract

Expressions are derived for the probability p(n) that n photons are emitted in a given time in the steady state by a two-level atom, when it is placed in a resonant, coherent, exciting field. The distribution p(n) is shown to be narrower than Poissonian. The ratio [〈(Δn)2〉 − 〈n〉]/〈n〉 is negative and has an absolute maximum value of 3/4. The possibility of observing the sub-Poissonian statistics is discussed briefly.

© 1979 Optical Society of America

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  1. H. J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977),Phys. Rev. A 18, 201 (1978); M. Dagenais, L. Mandel, Phys. Rev. A 18, 2217 (1978).
    [CrossRef]
  2. The opposite phenomenon, in which the photoelectric pulses are bunched in time, is usually known as the Hanbury Brown–Twiss effect; R. Hanbury Brown, R. Q. Twiss, Nature (London) 177, 27 (1956).
    [CrossRef]
  3. D. Stoler, Phys. Rev. Lett. 33, 1397 (1974); L. Mista, V. Perinova, J. Perina, Acta Phys. Polon. A 51, 739 (1977); S. Kryszewski, J. Chrostowski, J. Phys. A 10, L261 (1977); L. Mista, J. Perina, Acta Phys. Polon. A 52, 425 (1977).
    [CrossRef]
  4. N. Tornau, A. Bach, Opt. Commun. 11, 46 (1974); I. M. Every, J. Phys. A 8, L69 (1965); H. D. Simaan, R. Loudon, J. Phys A 8, 539 (1975); H. Paul, U. Mohr, W. Brunner, Opt. Commun. 17, 145 (1976); A. Bandilla, H. H. Ritze, Opt. Commun. 19, 169 (1976); M. Kozierowski, R. Tanas, Opt. Commun. 21, 229 (1977); S. Chaturdevi, P. Drummond, D. F. Walls, J. Phys. A 10, L187 (1977); J. Mostowski, K. Rzazewski, Phys. Lett. 66A, 275 (1978); S. Kielich, M. Kozierowski, R. Tanas, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 511.
    [CrossRef]
  5. A simulation of narrowing of the statistical fluctuations was recently reported by J. Wagner, P. Kurowski, W. Martienssen, to be published.
  6. We distinguish those dynamical variables that are Hilbert space operators by the caret.
  7. A formula of this form was first derived by semiclassical arguments by L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958), Proc. Phys. Soc. (London) 74, 233 (1959); L. Mandel, E. C. G. Sudarshan, E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964). The quantum field version of the formula was given by P. L. Kelley, W. H. Kleiner, Phys. Rev. 136, A316 (1964); R. J. Glauber, in Quantum Optics and Electronics, C. deWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), p. 63.
    [CrossRef]
  8. H. J. Carmichael, D. F. Walls, J. Phys. B 9, L43, 1199 (1976).
    [CrossRef]
  9. H. J. Kimble, L. Mandel, Opt. Commun. 14, 167 (1975); Phys. Rev. A 13, 2123 (1976),15, 689 (1977).
    [CrossRef]
  10. C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North-Holland, Amsterdam, 1977).
  11. I. R. Senitzky, Phys. Rev. 123, 1525 (1961); Phys. Rev. A 6, 1171 (1972).
    [CrossRef]
  12. M. Dillard, H. R. Robl, Phys. Rev. 184, 312 (1969).
    [CrossRef]
  13. R. J. Glauber, Phys. Rev. 130, 2529 (1963);
    [CrossRef]
  14. G. S. Agarwal, Phys. Rev. A 15, 814 (1977).
    [CrossRef]
  15. See, for example, J. M. Ziman, Elements of Advanced Quantum Theory (Cambridge U. P., Cambridge, 1969), Sect. 3.6.
  16. See, for example, J. A. Abate, H. J. Kimble, L. Mandel, Phys. Rev. A 14, 788 (1976).
    [CrossRef]

1977 (2)

H. J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977),Phys. Rev. A 18, 201 (1978); M. Dagenais, L. Mandel, Phys. Rev. A 18, 2217 (1978).
[CrossRef]

G. S. Agarwal, Phys. Rev. A 15, 814 (1977).
[CrossRef]

1976 (2)

See, for example, J. A. Abate, H. J. Kimble, L. Mandel, Phys. Rev. A 14, 788 (1976).
[CrossRef]

H. J. Carmichael, D. F. Walls, J. Phys. B 9, L43, 1199 (1976).
[CrossRef]

1975 (1)

H. J. Kimble, L. Mandel, Opt. Commun. 14, 167 (1975); Phys. Rev. A 13, 2123 (1976),15, 689 (1977).
[CrossRef]

1974 (2)

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974); L. Mista, V. Perinova, J. Perina, Acta Phys. Polon. A 51, 739 (1977); S. Kryszewski, J. Chrostowski, J. Phys. A 10, L261 (1977); L. Mista, J. Perina, Acta Phys. Polon. A 52, 425 (1977).
[CrossRef]

N. Tornau, A. Bach, Opt. Commun. 11, 46 (1974); I. M. Every, J. Phys. A 8, L69 (1965); H. D. Simaan, R. Loudon, J. Phys A 8, 539 (1975); H. Paul, U. Mohr, W. Brunner, Opt. Commun. 17, 145 (1976); A. Bandilla, H. H. Ritze, Opt. Commun. 19, 169 (1976); M. Kozierowski, R. Tanas, Opt. Commun. 21, 229 (1977); S. Chaturdevi, P. Drummond, D. F. Walls, J. Phys. A 10, L187 (1977); J. Mostowski, K. Rzazewski, Phys. Lett. 66A, 275 (1978); S. Kielich, M. Kozierowski, R. Tanas, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 511.
[CrossRef]

1969 (1)

M. Dillard, H. R. Robl, Phys. Rev. 184, 312 (1969).
[CrossRef]

1963 (1)

R. J. Glauber, Phys. Rev. 130, 2529 (1963);
[CrossRef]

1961 (1)

I. R. Senitzky, Phys. Rev. 123, 1525 (1961); Phys. Rev. A 6, 1171 (1972).
[CrossRef]

1958 (1)

A formula of this form was first derived by semiclassical arguments by L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958), Proc. Phys. Soc. (London) 74, 233 (1959); L. Mandel, E. C. G. Sudarshan, E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964). The quantum field version of the formula was given by P. L. Kelley, W. H. Kleiner, Phys. Rev. 136, A316 (1964); R. J. Glauber, in Quantum Optics and Electronics, C. deWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), p. 63.
[CrossRef]

1956 (1)

The opposite phenomenon, in which the photoelectric pulses are bunched in time, is usually known as the Hanbury Brown–Twiss effect; R. Hanbury Brown, R. Q. Twiss, Nature (London) 177, 27 (1956).
[CrossRef]

Abate, J. A.

See, for example, J. A. Abate, H. J. Kimble, L. Mandel, Phys. Rev. A 14, 788 (1976).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. A 15, 814 (1977).
[CrossRef]

Bach, A.

N. Tornau, A. Bach, Opt. Commun. 11, 46 (1974); I. M. Every, J. Phys. A 8, L69 (1965); H. D. Simaan, R. Loudon, J. Phys A 8, 539 (1975); H. Paul, U. Mohr, W. Brunner, Opt. Commun. 17, 145 (1976); A. Bandilla, H. H. Ritze, Opt. Commun. 19, 169 (1976); M. Kozierowski, R. Tanas, Opt. Commun. 21, 229 (1977); S. Chaturdevi, P. Drummond, D. F. Walls, J. Phys. A 10, L187 (1977); J. Mostowski, K. Rzazewski, Phys. Lett. 66A, 275 (1978); S. Kielich, M. Kozierowski, R. Tanas, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 511.
[CrossRef]

Carmichael, H. J.

H. J. Carmichael, D. F. Walls, J. Phys. B 9, L43, 1199 (1976).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North-Holland, Amsterdam, 1977).

Dagenais, M.

H. J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977),Phys. Rev. A 18, 201 (1978); M. Dagenais, L. Mandel, Phys. Rev. A 18, 2217 (1978).
[CrossRef]

Dillard, M.

M. Dillard, H. R. Robl, Phys. Rev. 184, 312 (1969).
[CrossRef]

Glauber, R. J.

R. J. Glauber, Phys. Rev. 130, 2529 (1963);
[CrossRef]

Hanbury Brown, R.

The opposite phenomenon, in which the photoelectric pulses are bunched in time, is usually known as the Hanbury Brown–Twiss effect; R. Hanbury Brown, R. Q. Twiss, Nature (London) 177, 27 (1956).
[CrossRef]

Kimble, H. J.

H. J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977),Phys. Rev. A 18, 201 (1978); M. Dagenais, L. Mandel, Phys. Rev. A 18, 2217 (1978).
[CrossRef]

See, for example, J. A. Abate, H. J. Kimble, L. Mandel, Phys. Rev. A 14, 788 (1976).
[CrossRef]

H. J. Kimble, L. Mandel, Opt. Commun. 14, 167 (1975); Phys. Rev. A 13, 2123 (1976),15, 689 (1977).
[CrossRef]

Kurowski, P.

A simulation of narrowing of the statistical fluctuations was recently reported by J. Wagner, P. Kurowski, W. Martienssen, to be published.

Mandel, L.

H. J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977),Phys. Rev. A 18, 201 (1978); M. Dagenais, L. Mandel, Phys. Rev. A 18, 2217 (1978).
[CrossRef]

See, for example, J. A. Abate, H. J. Kimble, L. Mandel, Phys. Rev. A 14, 788 (1976).
[CrossRef]

H. J. Kimble, L. Mandel, Opt. Commun. 14, 167 (1975); Phys. Rev. A 13, 2123 (1976),15, 689 (1977).
[CrossRef]

A formula of this form was first derived by semiclassical arguments by L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958), Proc. Phys. Soc. (London) 74, 233 (1959); L. Mandel, E. C. G. Sudarshan, E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964). The quantum field version of the formula was given by P. L. Kelley, W. H. Kleiner, Phys. Rev. 136, A316 (1964); R. J. Glauber, in Quantum Optics and Electronics, C. deWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), p. 63.
[CrossRef]

Martienssen, W.

A simulation of narrowing of the statistical fluctuations was recently reported by J. Wagner, P. Kurowski, W. Martienssen, to be published.

Robl, H. R.

M. Dillard, H. R. Robl, Phys. Rev. 184, 312 (1969).
[CrossRef]

Senitzky, I. R.

I. R. Senitzky, Phys. Rev. 123, 1525 (1961); Phys. Rev. A 6, 1171 (1972).
[CrossRef]

Stoler, D.

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974); L. Mista, V. Perinova, J. Perina, Acta Phys. Polon. A 51, 739 (1977); S. Kryszewski, J. Chrostowski, J. Phys. A 10, L261 (1977); L. Mista, J. Perina, Acta Phys. Polon. A 52, 425 (1977).
[CrossRef]

Tornau, N.

N. Tornau, A. Bach, Opt. Commun. 11, 46 (1974); I. M. Every, J. Phys. A 8, L69 (1965); H. D. Simaan, R. Loudon, J. Phys A 8, 539 (1975); H. Paul, U. Mohr, W. Brunner, Opt. Commun. 17, 145 (1976); A. Bandilla, H. H. Ritze, Opt. Commun. 19, 169 (1976); M. Kozierowski, R. Tanas, Opt. Commun. 21, 229 (1977); S. Chaturdevi, P. Drummond, D. F. Walls, J. Phys. A 10, L187 (1977); J. Mostowski, K. Rzazewski, Phys. Lett. 66A, 275 (1978); S. Kielich, M. Kozierowski, R. Tanas, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 511.
[CrossRef]

Twiss, R. Q.

The opposite phenomenon, in which the photoelectric pulses are bunched in time, is usually known as the Hanbury Brown–Twiss effect; R. Hanbury Brown, R. Q. Twiss, Nature (London) 177, 27 (1956).
[CrossRef]

Wagner, J.

A simulation of narrowing of the statistical fluctuations was recently reported by J. Wagner, P. Kurowski, W. Martienssen, to be published.

Walls, D. F.

H. J. Carmichael, D. F. Walls, J. Phys. B 9, L43, 1199 (1976).
[CrossRef]

Ziman, J. M.

See, for example, J. M. Ziman, Elements of Advanced Quantum Theory (Cambridge U. P., Cambridge, 1969), Sect. 3.6.

J. Phys. B (1)

H. J. Carmichael, D. F. Walls, J. Phys. B 9, L43, 1199 (1976).
[CrossRef]

Nature (London) (1)

The opposite phenomenon, in which the photoelectric pulses are bunched in time, is usually known as the Hanbury Brown–Twiss effect; R. Hanbury Brown, R. Q. Twiss, Nature (London) 177, 27 (1956).
[CrossRef]

Opt. Commun. (2)

N. Tornau, A. Bach, Opt. Commun. 11, 46 (1974); I. M. Every, J. Phys. A 8, L69 (1965); H. D. Simaan, R. Loudon, J. Phys A 8, 539 (1975); H. Paul, U. Mohr, W. Brunner, Opt. Commun. 17, 145 (1976); A. Bandilla, H. H. Ritze, Opt. Commun. 19, 169 (1976); M. Kozierowski, R. Tanas, Opt. Commun. 21, 229 (1977); S. Chaturdevi, P. Drummond, D. F. Walls, J. Phys. A 10, L187 (1977); J. Mostowski, K. Rzazewski, Phys. Lett. 66A, 275 (1978); S. Kielich, M. Kozierowski, R. Tanas, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), p. 511.
[CrossRef]

H. J. Kimble, L. Mandel, Opt. Commun. 14, 167 (1975); Phys. Rev. A 13, 2123 (1976),15, 689 (1977).
[CrossRef]

Phys. Rev. (3)

I. R. Senitzky, Phys. Rev. 123, 1525 (1961); Phys. Rev. A 6, 1171 (1972).
[CrossRef]

M. Dillard, H. R. Robl, Phys. Rev. 184, 312 (1969).
[CrossRef]

R. J. Glauber, Phys. Rev. 130, 2529 (1963);
[CrossRef]

Phys. Rev. A (2)

G. S. Agarwal, Phys. Rev. A 15, 814 (1977).
[CrossRef]

See, for example, J. A. Abate, H. J. Kimble, L. Mandel, Phys. Rev. A 14, 788 (1976).
[CrossRef]

Phys. Rev. Lett. (2)

H. J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977),Phys. Rev. A 18, 201 (1978); M. Dagenais, L. Mandel, Phys. Rev. A 18, 2217 (1978).
[CrossRef]

D. Stoler, Phys. Rev. Lett. 33, 1397 (1974); L. Mista, V. Perinova, J. Perina, Acta Phys. Polon. A 51, 739 (1977); S. Kryszewski, J. Chrostowski, J. Phys. A 10, L261 (1977); L. Mista, J. Perina, Acta Phys. Polon. A 52, 425 (1977).
[CrossRef]

Proc. Phys. Soc. (London) (1)

A formula of this form was first derived by semiclassical arguments by L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958), Proc. Phys. Soc. (London) 74, 233 (1959); L. Mandel, E. C. G. Sudarshan, E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964). The quantum field version of the formula was given by P. L. Kelley, W. H. Kleiner, Phys. Rev. 136, A316 (1964); R. J. Glauber, in Quantum Optics and Electronics, C. deWitt, A. Blandin, C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965), p. 63.
[CrossRef]

Other (4)

C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North-Holland, Amsterdam, 1977).

A simulation of narrowing of the statistical fluctuations was recently reported by J. Wagner, P. Kurowski, W. Martienssen, to be published.

We distinguish those dynamical variables that are Hilbert space operators by the caret.

See, for example, J. M. Ziman, Elements of Advanced Quantum Theory (Cambridge U. P., Cambridge, 1969), Sect. 3.6.

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Figures (2)

Fig. 1
Fig. 1

The variation of the normalized second factorial moment Q with T for various values of Ω/β.

Fig. 2
Fig. 2

Comparison of the probability distribution p(n) for photon emission, with βT = 10, Ω = β (full curve) with a Poisson distribution having the same mean (broken curve). Although the probabilities are defined only for integer values of n, they are shown as continuous curves for convenience.

Equations (20)

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p ( n ) = T : 1 n ! [ t t + T d t I ^ ( t ) ] n × exp [ - t t + T d t I ( t ) ] : .
n ( r ) = n = 0 n ( n - 1 ) ( n - r + 1 ) p ( n ) = t t + T d t r t t + T d t r - 1 × t t + T d t 1 T : I ^ ( t 1 ) I ^ ( t 2 ) I ^ ( t r ) : ,
p ( m ) = 1 m ! r = 0 ( - 1 ) r r ! n ( m + r ) ,
T : I ^ ( t ) I ^ ( t + τ ) : = I ^ I ^ ( τ ) G ,             τ 0 ,
I ^ ( τ ) G = I ^ [ 1 + λ ( τ ) ] ,
I ^ = ( ½ Ω 2 / β ) / ( ½ Ω 2 / β 2 + 1 ) ,
1 + λ ( τ ) = 1 - e - 3 β τ / 2 [ cos Ω β τ + ( 3 / 2 Ω ) sin Ω β τ ] , Ω ( Ω 2 / β 2 - ¼ ) 1 / 2 .
T : I ^ ( t 1 ) I ( t 2 ) I ^ ( t 3 ) : = T : I ^ ( t 2 ) I ^ ( t 3 ) : T : I ^ ( t 1 ) I ^ ( t 2 ) : / I ^ = I ^ 3 [ 1 + λ ( t 3 - t 2 ) ] [ 1 + λ ( t 2 - t 1 ) ] ,
T : I ^ ( t 1 ) I ^ ( t 2 ) I ^ ( t r ) : = I ^ r [ 1 + λ ( t r - t r - 1 ) ] [ 1 + λ ( t 2 - t 1 ) ] ,             r = 2 , 3 , , t 1 t 2 t r .
n ( r ) = I ^ r r ! 0 T d t r 0 t 2 d t 1 [ 1 + λ ( t r - t r - 1 ) ] [ 1 + λ ( t 2 - t 1 ) ] ,             r = 2 , 3 , , n ( 1 ) = I ^ T ,             n ( 0 ) = 1.
Q n ( 2 ) - n 2 n = ( Δ n ) 2 - n n = 2 I ^ T 0 T d t 2 0 t 2 d t 1 [ 1 + λ ( t 1 ) ] - I ^ T ,
Q = - ½ Ω 2 / β 2 ( ½ Ω 2 / β 2 + 1 ) 2 1 2 β T ( 6 β T + 1 ½ Ω 2 / β 2 + 1 × { - 7 + Ω 2 β 2 + e - 3 β T / 2 [ ( 7 - Ω 2 β 2 ) cos Ω β T + 9 2 Ω ( 1 - Ω 2 β 2 ) sin Ω β T ] } ) .
Q - [ 3 Ω 2 / β 2 / 2 ( ½ Ω 2 / β 2 + 1 ) 2 ]
1 + λ ( τ ) ½ τ 2 ( Ω 2 + 2 β 2 ) .
0 t r + 1 d t r 0 t r d t r - 1 0 t 2 d t 1 [ 1 + λ ( t r + 1 - t r ) ] [ 1 + λ ( t 2 - t 1 ) ] = ( Ω 2 + 2 β 2 ) r t r + 1 3 r ( 3 r ) ! ,
n ( r ) = ( ½ Ω 2 / β 2 ½ Ω 2 / β 2 + 1 ) β T ( β Ω 2 T 3 ) r - 1 r ! ( 3 r - 2 ) !
p ( n ) = δ n 0 + ( ½ Ω 2 / β 2 ½ Ω 2 / β 2 + 1 ) β T × [ ( 1 - δ n 0 ) ( β Ω 2 T 3 ) n - 1 ( 3 n - 2 ) ! + 1 n ! × r = 1 ( - 1 ) r r ! ( β Ω 2 T 3 ) n + r - 1 ( n + r ) ! ( 3 n + 3 r - 2 ) ! ] .
p ( n ) δ n 0 + ( ½ Ω 2 / β 2 ½ Ω 2 / β 2 + 1 ) × β T { ( 1 - δ n 0 ) ( β Ω 2 T 3 ) ( 3 n - 2 ) ! - ( n + 1 ) ( β Ω 2 T 3 ) n ( 3 n + 1 ) ! × [ 1 - n + 2 ( 3 n + 2 ) ( 3 n + 3 ) ( 3 n + 4 ) β Ω 2 T 3 ] } .
0 t 2 d t 1 [ 1 + λ ( t 2 - t 1 ) ] t 2 + 0 λ ( τ ) d τ = t 2 - 3 β Ω 2 + 2 β 2 .
n ( r ) = ( I ^ T ) r r ! m = 1 r ( r - 1 m - 1 ) 1 m ! ( - 3 β / T Ω 2 + 2 β 2 ) r - m = ( ½ Ω 2 / β 2 ½ Ω 2 / β 2 + 1 ) β T ( - ¾ Ω 2 / β 2 ½ Ω 2 / β 2 + 1 ) r - 1 ( r - 1 ) ! × L 1 r - 1 [ 2 3 β T ( 1 2 Ω 2 β 2 + 1 ) ] ,

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