Abstract

The second-order photon correlation function g (2)(τ) of photons emitted by a continuously pumped ensemble of N two-level systems coupled to a single-mode optical cavity well below the lasing threshold is investigated theoretically. A giant photon bunching is found for N < 10 emitters as the microscopic counterpart of spontaneous emission noise driven quasi-periodic superradiant pulse sequences in macroscopic systems of large numbers of emitters N ≫ 1. The phenomenon of giant photon bunching is preserved even for N = 2 and can be explained by the cooperative evolution via dark and bright two-atom states resulting into emission of superradiant photon pairs. The sensitivity of g (2) to microscopic dephasing processes and resonance frequency detuning opens the door for photon bunching spectroscopy.

© 2009 Optical Society of America

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    [PubMed]
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2007 (1)

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

2006 (1)

J. H. Eberly, "Emission of one photon in an electric dipole transition of one among N atoms," J. Phys. B 39, S599 (2006).

2005 (2)

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, "Quantum correlation between photons from a single quantum dot at room temperature," Nature 406, 968 (2005).

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

2000 (2)

P. Ataloni, E. De Angelis, and F. De Martini, "Bose-Einstein Partition Statistics in Superradiant Spontaneous Emission," Phys. Rev. Lett. 85, 1420 (2000).

G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, "Quantum State Reduction and Conditional Time Evolution of Wave-Particle Correlations in Cavity QED," Phys. Rev. Lett. 85, 3149 (2000).
[PubMed]

1998 (1)

A. A. Belyanin, V. V. Kocharovsky, and V. V. Kocharovsky, "Superradiant generation of femtosecond pulses in quantum-well heterostructures," Quantum Semiclassic. Opt. 10, L13 (1998).

1996 (1)

R. G. DeVoe and R. G. Brewer, "Observation of Superradiant and Subradiant Spontaneous Emission of Two Trapped Ions," Phys. Rev. Lett. 76, 2049 (1996).
[PubMed]

1993 (1)

1990 (1)

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

1989 (2)

J. J. Maki, M. S. Malcuit, M. G. Raymer, R. W. Boyd, and P. D. Drummond, "Influence of collisional dephasing processes on superfluorescence," Phys. Rev. A 40, 5135 (1989).
[PubMed]

G. Benivegna and A. Messina, "Sub-radiant States of a Spatially Extended System of N Two-level Atoms," J. Mod. Opt. 36, 1205 (1989).

1987 (1)

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, "Transition from superfluorescence to amplified spontaneous emission," Phys. Rev. Lett. 59, 1189 (1987).
[PubMed]

1983 (1)

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

1982 (1)

M. Gross and S. Haroche, "Superradiance: An essay on the theory of collective spontaneous emission," Phys. Rep. 93, 301 (1982).

1980 (1)

F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, "Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence," Phys. Rev. Lett. 45, 558 (1980).

1975 (1)

R. Bonifacio and L. A. Lugiato, "Cooperative radiation processes in two-level systems: Superfluorescence," Phys. Rev. A 11, 1507 (1975).

1974 (1)

R. Friedberg and S. R. Hartmann, "Temporal evolution of superradiance in a small sphere," Phys. Rev. A 10, 1728 (1974).

1954 (1)

R. H. Dicke, "Coherence in Spontaneous Radiation Processes," Phys. Rev. 93, 99 (1954).

Ataloni, P.

P. Ataloni, E. De Angelis, and F. De Martini, "Bose-Einstein Partition Statistics in Superradiant Spontaneous Emission," Phys. Rev. Lett. 85, 1420 (2000).

Bacher, G.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

Belyanin, A. A.

A. A. Belyanin, V. V. Kocharovsky, and V. V. Kocharovsky, "Superradiant generation of femtosecond pulses in quantum-well heterostructures," Quantum Semiclassic. Opt. 10, L13 (1998).

Benivegna, G.

G. Benivegna and A. Messina, "Sub-radiant States of a Spatially Extended System of N Two-level Atoms," J. Mod. Opt. 36, 1205 (1989).

Benkert, C.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

Bergou, J.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

Bimbaum, K.

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

Boca, A.

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

Bonifacio, R.

R. Bonifacio and L. A. Lugiato, "Cooperative radiation processes in two-level systems: Superfluorescence," Phys. Rev. A 11, 1507 (1975).

Boozer, A. D.

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

Boyd, R. W.

J. J. Maki, M. S. Malcuit, M. G. Raymer, R. W. Boyd, and P. D. Drummond, "Influence of collisional dephasing processes on superfluorescence," Phys. Rev. A 40, 5135 (1989).
[PubMed]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, "Transition from superfluorescence to amplified spontaneous emission," Phys. Rev. Lett. 59, 1189 (1987).
[PubMed]

Brewer, R. G.

R. G. DeVoe and R. G. Brewer, "Observation of Superradiant and Subradiant Spontaneous Emission of Two Trapped Ions," Phys. Rev. Lett. 76, 2049 (1996).
[PubMed]

Buratto, S. K.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, "Quantum correlation between photons from a single quantum dot at room temperature," Nature 406, 968 (2005).

Carmichael, H. J.

G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, "Quantum State Reduction and Conditional Time Evolution of Wave-Particle Correlations in Cavity QED," Phys. Rev. Lett. 85, 3149 (2000).
[PubMed]

Carson, P. J.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, "Quantum correlation between photons from a single quantum dot at room temperature," Nature 406, 968 (2005).

Castin, Y.

Castro-Beltran, H. M.

G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, "Quantum State Reduction and Conditional Time Evolution of Wave-Particle Correlations in Cavity QED," Phys. Rev. Lett. 85, 3149 (2000).
[PubMed]

Claude, Y.

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

Cormier, M.

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

Dalibard, J.

Davidovich, L.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

De Angelis, E.

P. Ataloni, E. De Angelis, and F. De Martini, "Bose-Einstein Partition Statistics in Superradiant Spontaneous Emission," Phys. Rev. Lett. 85, 1420 (2000).

De Martini, F.

P. Ataloni, E. De Angelis, and F. De Martini, "Bose-Einstein Partition Statistics in Superradiant Spontaneous Emission," Phys. Rev. Lett. 85, 1420 (2000).

DeVoe, R. G.

R. G. DeVoe and R. G. Brewer, "Observation of Superradiant and Subradiant Spontaneous Emission of Two Trapped Ions," Phys. Rev. Lett. 76, 2049 (1996).
[PubMed]

Dicke, R. H.

R. H. Dicke, "Coherence in Spontaneous Radiation Processes," Phys. Rev. 93, 99 (1954).

Drummond, P. D.

J. J. Maki, M. S. Malcuit, M. G. Raymer, R. W. Boyd, and P. D. Drummond, "Influence of collisional dephasing processes on superfluorescence," Phys. Rev. A 40, 5135 (1989).
[PubMed]

Eberly, J. H.

J. H. Eberly, "Emission of one photon in an electric dipole transition of one among N atoms," J. Phys. B 39, S599 (2006).

Feld, M. S.

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

Forchel, A.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

Foster, G. T.

G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, "Quantum State Reduction and Conditional Time Evolution of Wave-Particle Correlations in Cavity QED," Phys. Rev. Lett. 85, 3149 (2000).
[PubMed]

Friedberg, R.

R. Friedberg and S. R. Hartmann, "Temporal evolution of superradiance in a small sphere," Phys. Rev. A 10, 1728 (1974).

Gibbs, H. M.

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

Glauber, R.

F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, "Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence," Phys. Rev. Lett. 45, 558 (1980).

Gross, M.

M. Gross and S. Haroche, "Superradiance: An essay on the theory of collective spontaneous emission," Phys. Rep. 93, 301 (1982).

Haake, F.

F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, "Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence," Phys. Rev. Lett. 45, 558 (1980).

Haroche, S.

M. Gross and S. Haroche, "Superradiance: An essay on the theory of collective spontaneous emission," Phys. Rep. 93, 301 (1982).

Hartmann, S. R.

R. Friedberg and S. R. Hartmann, "Temporal evolution of superradiance in a small sphere," Phys. Rev. A 10, 1728 (1974).

Haus, J.

F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, "Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence," Phys. Rev. Lett. 45, 558 (1980).

Hillery, M.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

Hommel, D.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

Imamoglu, A.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, "Quantum correlation between photons from a single quantum dot at room temperature," Nature 406, 968 (2005).

Kimble, H. J.

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

King, H.

F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, "Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence," Phys. Rev. Lett. 45, 558 (1980).

Kocharovsky, V. V.

A. A. Belyanin, V. V. Kocharovsky, and V. V. Kocharovsky, "Superradiant generation of femtosecond pulses in quantum-well heterostructures," Quantum Semiclassic. Opt. 10, L13 (1998).

A. A. Belyanin, V. V. Kocharovsky, and V. V. Kocharovsky, "Superradiant generation of femtosecond pulses in quantum-well heterostructures," Quantum Semiclassic. Opt. 10, L13 (1998).

Lugiato, L. A.

R. Bonifacio and L. A. Lugiato, "Cooperative radiation processes in two-level systems: Superfluorescence," Phys. Rev. A 11, 1507 (1975).

Maki, J. J.

J. J. Maki, M. S. Malcuit, M. G. Raymer, R. W. Boyd, and P. D. Drummond, "Influence of collisional dephasing processes on superfluorescence," Phys. Rev. A 40, 5135 (1989).
[PubMed]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, "Transition from superfluorescence to amplified spontaneous emission," Phys. Rev. Lett. 59, 1189 (1987).
[PubMed]

Malcuit, M. S.

J. J. Maki, M. S. Malcuit, M. G. Raymer, R. W. Boyd, and P. D. Drummond, "Influence of collisional dephasing processes on superfluorescence," Phys. Rev. A 40, 5135 (1989).
[PubMed]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, "Transition from superfluorescence to amplified spontaneous emission," Phys. Rev. Lett. 59, 1189 (1987).
[PubMed]

Mason, M. D.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, "Quantum correlation between photons from a single quantum dot at room temperature," Nature 406, 968 (2005).

Mattar, E. P.

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

McCall, S. L.

E. A. Watson, H. M. Gibbs, E. P. Mattar, M. Cormier, Y. Claude, S. L. McCall, and M. S. Feld, "Quantum fluctuations and transverse effects in superfluorescence," Phys. Rev. A 27, 1427 (1983).

Messina, A.

G. Benivegna and A. Messina, "Sub-radiant States of a Spatially Extended System of N Two-level Atoms," J. Mod. Opt. 36, 1205 (1989).

Michler, P.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, "Quantum correlation between photons from a single quantum dot at room temperature," Nature 406, 968 (2005).

Miller, R.

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

Mølmer, K.

Northup, T. E.

K. Bimbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87 (2005).

Orozco, L. A.

G. T. Foster, L. A. Orozco, H. M. Castro-Beltran, and H. J. Carmichael, "Quantum State Reduction and Conditional Time Evolution of Wave-Particle Correlations in Cavity QED," Phys. Rev. Lett. 85, 3149 (2000).
[PubMed]

Orszag, M.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

Passow, T.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

Raymer, M. G.

J. J. Maki, M. S. Malcuit, M. G. Raymer, R. W. Boyd, and P. D. Drummond, "Influence of collisional dephasing processes on superfluorescence," Phys. Rev. A 40, 5135 (1989).
[PubMed]

Scheibner, M.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

Schmidt, T.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, "Superradiance of quantum dots," Nat. Physics 3, 106 (2007).

Schröder, G.

F. Haake, J. Haus, H. King, G. Schröder, and R. Glauber, "Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence," Phys. Rev. Lett. 45, 558 (1980).

Scully, M. O.

C. Benkert, M. O. Scully, J. Bergou, L. Davidovich, M. Hillery, and M. Orszag, "Role of pumping statistics in laser dynamics: Quantum Langevin approach," Phys. Rev. A 41, 2756 (1990).
[PubMed]

Simkin, D. J.

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, "Transition from superfluorescence to amplified spontaneous emission," Phys. Rev. Lett. 59, 1189 (1987).
[PubMed]

Strouse, G. F.

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Figures (5)

Fig. 1.
Fig. 1.

Emission intensity I(t) of superradiant pulse trains (a) for different numbers N ≫ 1 of incoherently pumped two-level systems and (b) corresponding trajectories in the population inversion vs. polarization diagram, dashed circles show the direction of motion (see text).

Fig. 2.
Fig. 2.

Second-order photon correlation function g (2)(τ) obtained from semiclassical intensity traces I(t) for different N. A curve for N=10 reveals a strong bunching maximum.

Fig. 3.
Fig. 3.

Second-order photon correlation function g (2)(τ) for N incoherently pumped two-level systems (γ pump = 0.01) for different values of dephasing rate γ deph. For N=1 anti-bunching is observed, which does not depend on dephasing. For N > 1 a pronounced bunching is observed, which is extremely sensitive to γ deph. Inset shows the dependence of bunching amplitude g (2)(0) on N. All rates are in units of g 2/κ.

Fig. 4.
Fig. 4.

Level structure for two two-level systems. The bunching pathway after simultaneous excitation of both atoms via a subradiant (dark) state leads to emission of photon pairs via a superradiant (bright) state. See text for details.

Fig. 5.
Fig. 5.

(a) Dependence of bunching amplitude on the detuning δω 1: (a) N = 2, (b) N = 4; the other three frequencies are kept constant: δω 2 = -0.5g 2/κ, δω 3 = δω 4 = 0.5g 2/κ. The full width at half maximum of individual atomic resonances is 2g 2/κ.

Equations (10)

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dP dt = 2 g 2 κ PZ + N γ pump F p ( t )
dZ dt = 2 g 2 κ P 2 + N γ pump F z ( t )
g ( 2 ) ( τ ) = I ( t ) I ( t + τ ) dt I 2 ( t ) dt
dt = L SR ρ + L pump ρ + L deph ρ + L sp ρ
L SR ρ = g 2 κ [ 2 J ( t ) ρ J + ( t ) J + ( t ) J ( t ) ρ ρ J ( t ) J + ( t ) ]
L pump ρ = γ pump 2 j = 1 N ( 2 σ + ( j ) ρ σ ( j ) σ ( j ) σ + ( j ) ρ ρ σ ( j ) σ + ( j ) )
L sp ρ = γ sp 2 j = 1 N ( 2 σ ( j ) ρ σ + ( j ) σ + ( j ) σ ( j ) ρ ρ σ + ( j ) σ ( j ) )
L deph ρ = γ deph j = 1 N ( P e ( j ) ρ P g ( j ) + P g ( j ) ρ P e ( j ) )
g ( 2 ) ( 0 ) = g ( 2 ) / ( 2 κ ( γ pump + γ deph + γ sp ) ) .
Δω = 2 g ( 2 ) γ pump / κ

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