Abstract

We consider the directed superradiant emission from a collection of N two-state atoms with arbitrary spatial locations within the framework of quantum trajectory theory and without a single-mode assumption. The formalism is developed around an unravelling of the master equation in terms of source mode quantum jumps. A modified boson approximation is made to treat the many-atom case, where it is found that strong directional superradiance occurs for a few thousand atoms, even with randomized atomic positions.

© 2008 Optical Society of America

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  1. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99-110 (1954).
    [CrossRef]
  2. M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301-396 (1982).
    [CrossRef]
  3. L. I. Men'shikov, “Superradiance and related phenomena,” Phys. Usp. 42, 107-147 (1999).
    [CrossRef]
  4. J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Collective spontaneous emission from a line of atoms,” Phys. Rev. A 68, 023809 (2003).
    [CrossRef]
  5. J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Shot-to-shot fluctuations in the directed superradiant emission from extended atomic samples,” J. Opt. B: Quantum Semiclassical Opt. 6, S736-S741 (2004).
    [CrossRef]
  6. H. J. Carmichael and K. Kim, “A quantum trajectory unraveling of the superradiance master equation,” Opt. Rev. 179, 417-427 (2000).
  7. J. P. Clemens and H. J. Carmichael, “Stochastic initiation of superradiance in a cavity: an approximation scheme within quantum trajectory theory,” Phys. Rev. A 65, 023815 (2002).
    [CrossRef]
  8. V. Ernst and P. Stehle, “Emission of radiation from a system of many excited atoms,” Phys. Rev. 176, 1456-1479 (1968).
    [CrossRef]
  9. R. Bonifacio and G. Preparata, “Coherent spontaneous emission,” Phys. Rev. A 2, 336-347 (1970).
    [CrossRef]
  10. R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302-313 (1971).
    [CrossRef]
  11. R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: superfluorescence,” Phys. Rev. A 11, 1507-1521 (1975).
    [CrossRef]
  12. R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: superfluorescence. II,” Phys. Rev. A 12, 587-598 (1975).
    [CrossRef]
  13. R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. II,” Phys. Rev. A 4, 854-864 (1971).
    [CrossRef]
  14. V. Degiorgio and F. Ghielmetti, “Approximate solution to the superradiance master equation,” Phys. Rev. A 4, 2415-2418 (1971).
    [CrossRef]
  15. F. Haake and R. J. Glauber, “Quantum statistics of superradiant pulses,” Phys. Rev. A 5, 1457-1466 (1972).
    [CrossRef]
  16. L. M. Narducci, C. A. Coulter, and C. M. Bowden, “Exact diffusion equation for a model for superradiant emission,” Phys. Rev. A 9, 829-845 (1974).
    [CrossRef]
  17. L. Narducci, C. A. Coulter, and C. M. Bowden, “Comments on some recent solutions of the superradiant master equation,” Phys. Rev. A 9, 999-1003 (1974).
    [CrossRef]
  18. R. Glauber and F. Haake, “Superradiant pulses and directed angular momentum states,” Phys. Rev. A 13, 357-366 (1976).
    [CrossRef]
  19. R. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. A 68, 29-32 (1978).
    [CrossRef]
  20. A. A. Belavkin, B. Y. Zeldovich, A. M. Perelomov, and V. S. Popov, “Relaxation of quantum systems with equidistant spectra,” Sov. Phys. JETP 56, 264-274 (1969).
  21. R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883-888 (1970).
    [CrossRef]
  22. R. H. Lehmberg, “Radiation from an N-atom system. II. Spontaneous emission from a pair of atoms,” Phys. Rev. A 2, 889-896 (1970).
    [CrossRef]
  23. G. S. Agarwal, “Master-equation approach to spontaneous emission,” Phys. Rev. A 2, 2038-2046 (1970).
    [CrossRef]
  24. G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches, Vol. 70 of Springer Tracts in Modern Physics (Springer, 1974).
  25. H. J. Carmichael, An Open Systems Approach to Quantum Optics, Vol. m18 of Lecture Notes in Physics, New Series: Monographs (Springer, 1993).
  26. E. Ressayre and A. Tallet, “Basic properties for cooperative emission of radiation,” Phys. Rev. Lett. 37, 424-427 (1976).
    [CrossRef]
  27. E. Ressayre and A. Tallet, “Quantum theory for superradiance,” Phys. Rev. A 15, 2410-2423 (1977).
    [CrossRef]
  28. N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309-312 (1973).
    [CrossRef]
  29. M. Gross, C. Fabre, P. Pillet, and S. Haroche, “Observation of near-infrared Dicke superradiance on cascading transitions in atomic sodium,” Phys. Rev. Lett. 36, 1035-1038 (1976).
    [CrossRef]
  30. Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs, “Quantum beats in superfluorescence in atomic cesium,” Phys. Rev. Lett. 38, 764-767 (1977).
    [CrossRef]
  31. H. M. Gibbs, Q. H. F. Vrehen, and H. M. J. Hikspoors, “Single-pulse superfluorescence in cesium,” Phys. Rev. Lett. 39, 547-550 (1977).
    [CrossRef]
  32. D. Polder, M. F. H. Schuurmans, and Q. H. F. Vrehen, “Superfluorescence: quantum-mechanical derivation of Maxwell-Bloch description with fluctuating field source,” Phys. Rev. A 19, 1192-1203 (1979).
    [CrossRef]

2004 (1)

J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Shot-to-shot fluctuations in the directed superradiant emission from extended atomic samples,” J. Opt. B: Quantum Semiclassical Opt. 6, S736-S741 (2004).
[CrossRef]

2003 (1)

J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Collective spontaneous emission from a line of atoms,” Phys. Rev. A 68, 023809 (2003).
[CrossRef]

2002 (1)

J. P. Clemens and H. J. Carmichael, “Stochastic initiation of superradiance in a cavity: an approximation scheme within quantum trajectory theory,” Phys. Rev. A 65, 023815 (2002).
[CrossRef]

2000 (1)

H. J. Carmichael and K. Kim, “A quantum trajectory unraveling of the superradiance master equation,” Opt. Rev. 179, 417-427 (2000).

1999 (1)

L. I. Men'shikov, “Superradiance and related phenomena,” Phys. Usp. 42, 107-147 (1999).
[CrossRef]

1982 (1)

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301-396 (1982).
[CrossRef]

1979 (1)

D. Polder, M. F. H. Schuurmans, and Q. H. F. Vrehen, “Superfluorescence: quantum-mechanical derivation of Maxwell-Bloch description with fluctuating field source,” Phys. Rev. A 19, 1192-1203 (1979).
[CrossRef]

1978 (1)

R. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. A 68, 29-32 (1978).
[CrossRef]

1977 (3)

E. Ressayre and A. Tallet, “Quantum theory for superradiance,” Phys. Rev. A 15, 2410-2423 (1977).
[CrossRef]

Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs, “Quantum beats in superfluorescence in atomic cesium,” Phys. Rev. Lett. 38, 764-767 (1977).
[CrossRef]

H. M. Gibbs, Q. H. F. Vrehen, and H. M. J. Hikspoors, “Single-pulse superfluorescence in cesium,” Phys. Rev. Lett. 39, 547-550 (1977).
[CrossRef]

1976 (3)

E. Ressayre and A. Tallet, “Basic properties for cooperative emission of radiation,” Phys. Rev. Lett. 37, 424-427 (1976).
[CrossRef]

M. Gross, C. Fabre, P. Pillet, and S. Haroche, “Observation of near-infrared Dicke superradiance on cascading transitions in atomic sodium,” Phys. Rev. Lett. 36, 1035-1038 (1976).
[CrossRef]

R. Glauber and F. Haake, “Superradiant pulses and directed angular momentum states,” Phys. Rev. A 13, 357-366 (1976).
[CrossRef]

1975 (2)

R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: superfluorescence,” Phys. Rev. A 11, 1507-1521 (1975).
[CrossRef]

R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: superfluorescence. II,” Phys. Rev. A 12, 587-598 (1975).
[CrossRef]

1974 (2)

L. M. Narducci, C. A. Coulter, and C. M. Bowden, “Exact diffusion equation for a model for superradiant emission,” Phys. Rev. A 9, 829-845 (1974).
[CrossRef]

L. Narducci, C. A. Coulter, and C. M. Bowden, “Comments on some recent solutions of the superradiant master equation,” Phys. Rev. A 9, 999-1003 (1974).
[CrossRef]

1973 (1)

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309-312 (1973).
[CrossRef]

1972 (1)

F. Haake and R. J. Glauber, “Quantum statistics of superradiant pulses,” Phys. Rev. A 5, 1457-1466 (1972).
[CrossRef]

1971 (3)

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302-313 (1971).
[CrossRef]

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. II,” Phys. Rev. A 4, 854-864 (1971).
[CrossRef]

V. Degiorgio and F. Ghielmetti, “Approximate solution to the superradiance master equation,” Phys. Rev. A 4, 2415-2418 (1971).
[CrossRef]

1970 (4)

R. Bonifacio and G. Preparata, “Coherent spontaneous emission,” Phys. Rev. A 2, 336-347 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883-888 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. II. Spontaneous emission from a pair of atoms,” Phys. Rev. A 2, 889-896 (1970).
[CrossRef]

G. S. Agarwal, “Master-equation approach to spontaneous emission,” Phys. Rev. A 2, 2038-2046 (1970).
[CrossRef]

1969 (1)

A. A. Belavkin, B. Y. Zeldovich, A. M. Perelomov, and V. S. Popov, “Relaxation of quantum systems with equidistant spectra,” Sov. Phys. JETP 56, 264-274 (1969).

1968 (1)

V. Ernst and P. Stehle, “Emission of radiation from a system of many excited atoms,” Phys. Rev. 176, 1456-1479 (1968).
[CrossRef]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99-110 (1954).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Shot-to-shot fluctuations in the directed superradiant emission from extended atomic samples,” J. Opt. B: Quantum Semiclassical Opt. 6, S736-S741 (2004).
[CrossRef]

Opt. Rev. (1)

H. J. Carmichael and K. Kim, “A quantum trajectory unraveling of the superradiance master equation,” Opt. Rev. 179, 417-427 (2000).

Phys. Lett. A (1)

R. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. A 68, 29-32 (1978).
[CrossRef]

Phys. Rep. (1)

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301-396 (1982).
[CrossRef]

Phys. Rev. (2)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99-110 (1954).
[CrossRef]

V. Ernst and P. Stehle, “Emission of radiation from a system of many excited atoms,” Phys. Rev. 176, 1456-1479 (1968).
[CrossRef]

Phys. Rev. A (17)

R. Bonifacio and G. Preparata, “Coherent spontaneous emission,” Phys. Rev. A 2, 336-347 (1970).
[CrossRef]

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302-313 (1971).
[CrossRef]

R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: superfluorescence,” Phys. Rev. A 11, 1507-1521 (1975).
[CrossRef]

R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: superfluorescence. II,” Phys. Rev. A 12, 587-598 (1975).
[CrossRef]

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. II,” Phys. Rev. A 4, 854-864 (1971).
[CrossRef]

V. Degiorgio and F. Ghielmetti, “Approximate solution to the superradiance master equation,” Phys. Rev. A 4, 2415-2418 (1971).
[CrossRef]

F. Haake and R. J. Glauber, “Quantum statistics of superradiant pulses,” Phys. Rev. A 5, 1457-1466 (1972).
[CrossRef]

L. M. Narducci, C. A. Coulter, and C. M. Bowden, “Exact diffusion equation for a model for superradiant emission,” Phys. Rev. A 9, 829-845 (1974).
[CrossRef]

L. Narducci, C. A. Coulter, and C. M. Bowden, “Comments on some recent solutions of the superradiant master equation,” Phys. Rev. A 9, 999-1003 (1974).
[CrossRef]

R. Glauber and F. Haake, “Superradiant pulses and directed angular momentum states,” Phys. Rev. A 13, 357-366 (1976).
[CrossRef]

J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Collective spontaneous emission from a line of atoms,” Phys. Rev. A 68, 023809 (2003).
[CrossRef]

J. P. Clemens and H. J. Carmichael, “Stochastic initiation of superradiance in a cavity: an approximation scheme within quantum trajectory theory,” Phys. Rev. A 65, 023815 (2002).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883-888 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. II. Spontaneous emission from a pair of atoms,” Phys. Rev. A 2, 889-896 (1970).
[CrossRef]

G. S. Agarwal, “Master-equation approach to spontaneous emission,” Phys. Rev. A 2, 2038-2046 (1970).
[CrossRef]

E. Ressayre and A. Tallet, “Quantum theory for superradiance,” Phys. Rev. A 15, 2410-2423 (1977).
[CrossRef]

D. Polder, M. F. H. Schuurmans, and Q. H. F. Vrehen, “Superfluorescence: quantum-mechanical derivation of Maxwell-Bloch description with fluctuating field source,” Phys. Rev. A 19, 1192-1203 (1979).
[CrossRef]

Phys. Rev. Lett. (5)

E. Ressayre and A. Tallet, “Basic properties for cooperative emission of radiation,” Phys. Rev. Lett. 37, 424-427 (1976).
[CrossRef]

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309-312 (1973).
[CrossRef]

M. Gross, C. Fabre, P. Pillet, and S. Haroche, “Observation of near-infrared Dicke superradiance on cascading transitions in atomic sodium,” Phys. Rev. Lett. 36, 1035-1038 (1976).
[CrossRef]

Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs, “Quantum beats in superfluorescence in atomic cesium,” Phys. Rev. Lett. 38, 764-767 (1977).
[CrossRef]

H. M. Gibbs, Q. H. F. Vrehen, and H. M. J. Hikspoors, “Single-pulse superfluorescence in cesium,” Phys. Rev. Lett. 39, 547-550 (1977).
[CrossRef]

Phys. Usp. (1)

L. I. Men'shikov, “Superradiance and related phenomena,” Phys. Usp. 42, 107-147 (1999).
[CrossRef]

Sov. Phys. JETP (1)

A. A. Belavkin, B. Y. Zeldovich, A. M. Perelomov, and V. S. Popov, “Relaxation of quantum systems with equidistant spectra,” Sov. Phys. JETP 56, 264-274 (1969).

Other (2)

G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches, Vol. 70 of Springer Tracts in Modern Physics (Springer, 1974).

H. J. Carmichael, An Open Systems Approach to Quantum Optics, Vol. m18 of Lecture Notes in Physics, New Series: Monographs (Springer, 1993).

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Figures (4)

Fig. 1
Fig. 1

Superradiant emission from a regular linear array of 1000 atoms positioned on the z axis with a spacing of λ 0 40 and the dipoles aligned in the x ̂ direction. We plot the (a) intensity, (b) spatial emission pattern, (c) angular distribution, and (d) eigenvalues of ( γ i j ) .

Fig. 2
Fig. 2

Superradiant emission from a regular three-dimensional array of dimensions λ 0 ( 3 × 3 × 18 ) and spacing of λ 0 3 . We plot the (a) intensity, (b) spatial emission pattern, (c) angular distribution, and the (d) eigenvalues of ( γ i j ) .

Fig. 3
Fig. 3

Superradiant emission from randomized linear samples of 1000 atoms. We plot the (a) intensity and (b) angular distribution for a regular array (solid curve), randomized arrays with Δ s = 2.5 λ 0 (long-dashed curve) and 25 λ 0 (short-dashed curve), and a uniform distribution of length 25 λ 0 (dotted curve) averaged over 2400 random distributions.

Fig. 4
Fig. 4

Superradiant emission from randomized three-dimensional samples of 4374 atoms. We plot the (a) intensity and (b) angular distribution for a regular array (solid curve), randomized samples with Δ i s i = λ 0 (long-dashed curve) and 10 λ 0 (short-dashed curve), and a uniform distribution of dimensions λ 0 ( 3 × 3 × 18 ) (dotted curve) averaged over 240 random distributions.

Equations (29)

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ρ ̇ = 1 2 i , j = 1 N γ i j ( 2 σ ̂ j ρ σ ̂ i + σ ̂ i + σ ̂ j ρ ρ σ ̂ i + σ ̂ j ) ,
γ i j = γ 3 2 { [ 1 ( d ̂ r ̂ i j ) 2 ] sin ξ i j ξ i j + [ 1 3 ( d ̂ r ̂ i j ) 2 ] ( cos ξ i j ξ i j 2 sin ξ i j ξ i j 3 ) } ,
ρ ̇ = i [ H ̂ , ρ ] + i ( 2 O ̂ i ρ O ̂ i O ̂ i O ̂ i ρ ρ O ̂ i O ̂ i ) ,
S ̂ ( θ , ϕ ) = γ D ( θ , ϕ ) d Ω j = 1 N e i k 0 R ̂ ( θ , ϕ ) r j σ ̂ j ,
D ( θ , ϕ ) = 3 8 π [ 1 ( d ̂ R ̂ ( θ , ϕ ) ) 2 ] ,
( γ i j ) = B T Λ B ,
ρ ̇ = 1 2 l = 1 N ( 2 J ̂ l ρ J ̂ l J ̂ l J ̂ l ρ ρ J ̂ l J ̂ l ) ,
J ̂ l = λ l i = 1 N b l i σ ̂ i .
S ̂ ( θ , ϕ ) = γ D ( θ , ϕ ) d Ω l = 1 N λ l 1 2 ξ l ( θ , ϕ ) J l ,
ξ l ( θ , ϕ ) = { } S ̂ ( θ , ϕ ) l γ D ( θ , ϕ ) d Ω ,
Q l ( θ , ϕ ) d Ω = l S ̂ ( θ , ϕ ) S ̂ ( θ , ϕ ) l = ( γ λ l ) D ( θ , ϕ ) ξ l ( θ , ϕ ) 2 d Ω .
l = 1 N λ l Q l ( θ , ϕ ) = γ D ( θ , ϕ ) l = 1 N ξ l ( θ , ϕ ) 2 = N γ D ( θ , ϕ ) .
( λ l λ l ) 1 2 [ J ̂ l , J ̂ l ] = i = 1 N b l i b l i σ ̂ i z ,
( λ l λ l ) 1 2 [ J ̂ l , J ̂ l ] { ± } = ± i = 1 N b l i b l i { ± } , = ± δ l l { ± } .
J ̂ l λ l a ̂ l , with [ a ̂ l , a ̂ l ] = δ l , l .
J ̂ l λ l b ̂ l , with [ b ̂ l , b ̂ l ] = δ l , l ,
J ̂ l λ l b ̂ a ̂ l ,
R k + 1 l 1 , , l k ( θ , ϕ ) = ( 1 k N ) l = 1 N λ l Q l ( θ , ϕ ) ( n l + 1 ) ,
I k + 1 l 1 , , l k = ( 1 k N ) l = 1 N λ l ( n l + 1 ) .
x i = y i = 0 , z i = N s 2 + i s ,
x i = L x 2 + i s x , i = 1 , , L x s x ,
y j = L y 2 + j s y , j = 1 , , L y s y ,
z k = L z 2 + k s z , k = 1 , , L z s z ,
x i = y i = 0 , z i = N s 2 + i s + Δ ζ i ,
x i = y i = 0 , z i = L z ζ i ,
x i = L x 2 + i s x + Δ x ζ x i j k , i = 1 , , L x s x ,
y j = L y 2 + j s y + Δ y ζ y i j k , j = 1 , , L y s y ,
z k = L z 2 + k s z + Δ z ζ z i j k , k = 1 , , L z s z ,
x i = L x ζ x i , y i = L y ζ y i , z i = L z ζ z i ,

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