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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References

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  1. R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93, 99 (1954).
  2. R. Friedberg and S. R. Hartmann, “Temporal evolution of superradiance in a small sphere,” Phys. Rev. A 10, 1728 (1974).
  3. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301 (1982).
  4. 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).
  5. 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]
  6. 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).
  7. J. H. Eberly, “Emission of one photon in an electric dipole transition of one among N atoms,” J. Phys. B 39, S599 (2006).
  8. C. F. Lee and N. F. Johnson, “First-Order Superradiant Phase Transitions in a Multiqubit Cavity System,” Phys. Rev. Lett. 93, 083001 (2004).
  9. V. V. Temnov and U. Woggon, “Superradiance and Subradiance in an Inhomogeneously Broadened Ensemble of Two-Level Systems Coupled to a Low-Q Cavity,” Phys. Rev. Lett. 95, 243602 (2005).
  10. M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).
  11. T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).
  12. 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]
  13. 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]
  14. 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]
  15. L. Schneebeli, M. Kira, and S. W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. 101, 097401 (2008).
  16. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).
  17. 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).
  18. 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).
  19. M. Hennrich, A. Kuhn, and G. Rempe, “Transition from Antibunching to Bunching in Cavity QED,” Phys. Rev. Lett. 94, 053604 (2005).
  20. S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).
  21. P. Ataloni, E. De Angelis, and F. De Martini, “Bose-Einstein Partition Statistics in Superradiant Spontaneous Emission,” Phys. Rev. Lett. 85, 1420 (2000).
  22. 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]
  23. V. V. Temnov, M. I. Kolobov, and F. Haake, “Superradiant and Subradiant Behavior of the Overdamped Many-Atom Micromaser,” in Directions in Quantum Optics, H. J. Carmichael, R. Glauber, and M. Scully, eds. , no. 561 in Lecture Notes in Physics (Springer, Berlin, 2001), p. 261.
  24. 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).
  25. R. Bonifacio and L. A. Lugiato, “Cooperative radiation processes in two-level systems: Superfluorescence,” Phys. Rev. A 11, 1507 (1975).
  26. K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” J. Opt. Soc. Am. B 10, 524 (1993).
  27. G. Benivegna and A. Messina, “Sub-radiant States of a Spatially Extended System of N Two-level Atoms,” J. Mod. Opt. 36, 1205 (1989).

2008 (2)

L. Schneebeli, M. Kira, and S. W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. 101, 097401 (2008).

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

2007 (3)

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (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 (4)

V. V. Temnov and U. Woggon, “Superradiance and Subradiance in an Inhomogeneously Broadened Ensemble of Two-Level Systems Coupled to a Low-Q Cavity,” Phys. Rev. Lett. 95, 243602 (2005).

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).

M. Hennrich, A. Kuhn, and G. Rempe, “Transition from Antibunching to Bunching in Cavity QED,” Phys. Rev. Lett. 94, 053604 (2005).

2004 (1)

C. F. Lee and N. F. Johnson, “First-Order Superradiant Phase Transitions in a Multiqubit Cavity System,” Phys. Rev. Lett. 93, 083001 (2004).

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)

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

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]

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).

Ates, S.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

Auer, T.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

Bacher, G.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

Bayer, M.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (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]

Berstermann, T.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

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,” Nature Phys. 3, 106 (2007).

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (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).

Gies, C.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

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).

V. V. Temnov, M. I. Kolobov, and F. Haake, “Superradiant and Subradiant Behavior of the Overdamped Many-Atom Micromaser,” in Directions in Quantum Optics, H. J. Carmichael, R. Glauber, and M. Scully, eds. , no. 561 in Lecture Notes in Physics (Springer, Berlin, 2001), p. 261.

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).

Hennrich, M.

M. Hennrich, A. Kuhn, and G. Rempe, “Transition from Antibunching to Bunching in Cavity QED,” Phys. Rev. Lett. 94, 053604 (2005).

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]

Hofmann, C.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

Hommel, D.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 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).

Jahnke, F.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

Johnson, N. F.

C. F. Lee and N. F. Johnson, “First-Order Superradiant Phase Transitions in a Multiqubit Cavity System,” Phys. Rev. Lett. 93, 083001 (2004).

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).

Kira, M.

L. Schneebeli, M. Kira, and S. W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. 101, 097401 (2008).

Koch, M.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

Koch, S. W.

L. Schneebeli, M. Kira, and S. W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. 101, 097401 (2008).

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).

Kolobov, M. I.

V. V. Temnov, M. I. Kolobov, and F. Haake, “Superradiant and Subradiant Behavior of the Overdamped Many-Atom Micromaser,” in Directions in Quantum Optics, H. J. Carmichael, R. Glauber, and M. Scully, eds. , no. 561 in Lecture Notes in Physics (Springer, Berlin, 2001), p. 261.

Kubanek, A.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

Kuhn, A.

M. Hennrich, A. Kuhn, and G. Rempe, “Transition from Antibunching to Bunching in Cavity QED,” Phys. Rev. Lett. 94, 053604 (2005).

Kurtze, H.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

Lee, C. F.

C. F. Lee and N. F. Johnson, “First-Order Superradiant Phase Transitions in a Multiqubit Cavity System,” Phys. Rev. Lett. 93, 083001 (2004).

Löffler, A.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

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.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

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.

Murr, K.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

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]

Ourjoumtsev, A.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

Passow, T.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

Pinkse, P. W. H.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

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]

Reitzenstein, S.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

Rempe, G.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

M. Hennrich, A. Kuhn, and G. Rempe, “Transition from Antibunching to Bunching in Cavity QED,” Phys. Rev. Lett. 94, 053604 (2005).

Reuter, D.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

Scheibner, M.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

Schmidt, T.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

Schneebeli, L.

L. Schneebeli, M. Kira, and S. W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. 101, 097401 (2008).

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).

Schuster, I.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

Schwab, M.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

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.

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).

Temnov, V. V.

V. V. Temnov and U. Woggon, “Superradiance and Subradiance in an Inhomogeneously Broadened Ensemble of Two-Level Systems Coupled to a Low-Q Cavity,” Phys. Rev. Lett. 95, 243602 (2005).

V. V. Temnov, M. I. Kolobov, and F. Haake, “Superradiant and Subradiant Behavior of the Overdamped Many-Atom Micromaser,” in Directions in Quantum Optics, H. J. Carmichael, R. Glauber, and M. Scully, eds. , no. 561 in Lecture Notes in Physics (Springer, Berlin, 2001), p. 261.

Ulrich, S. M.

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

Watson, E. A.

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).

Wieck, A. D.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

Wiersig, J.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

Woggon, U.

V. V. Temnov and U. Woggon, “Superradiance and Subradiance in an Inhomogeneously Broadened Ensemble of Two-Level Systems Coupled to a Low-Q Cavity,” Phys. Rev. Lett. 95, 243602 (2005).

Worschech, L.

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

Yakovlev, D. R.

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

J. Mod. Opt. (1)

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

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

J. Phys. B (1)

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

Nature (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).

Nature Phys. (1)

M. Scheibner, T. Schmidt, L. Worschech, A. Forchel, G. Bacher, T. Passow, and D. Hommel, “Superradiance of quantum dots,” Nature Phys. 3, 106 (2007).

Phys. Rep. (1)

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

Phys. Rev. (1)

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

Phys. Rev. A (5)

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

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]

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).

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]

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

Phys. Rev. B (1)

T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. D. Wieck, “Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots,” Phys. Rev. B 76, 165318 (2007).

Phys. Rev. Lett. (11)

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]

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]

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]

L. Schneebeli, M. Kira, and S. W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. 101, 097401 (2008).

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-Photon Gateway in One-Atom Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008).

M. Hennrich, A. Kuhn, and G. Rempe, “Transition from Antibunching to Bunching in Cavity QED,” Phys. Rev. Lett. 94, 053604 (2005).

S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon Statistics of Semiconductor Microcavity Lasers,” Phys. Rev. Lett. 98, 043906 (2007).

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

C. F. Lee and N. F. Johnson, “First-Order Superradiant Phase Transitions in a Multiqubit Cavity System,” Phys. Rev. Lett. 93, 083001 (2004).

V. V. Temnov and U. Woggon, “Superradiance and Subradiance in an Inhomogeneously Broadened Ensemble of Two-Level Systems Coupled to a Low-Q Cavity,” Phys. Rev. Lett. 95, 243602 (2005).

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).

Quantum Semiclassic. Opt. (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).

Other (1)

V. V. Temnov, M. I. Kolobov, and F. Haake, “Superradiant and Subradiant Behavior of the Overdamped Many-Atom Micromaser,” in Directions in Quantum Optics, H. J. Carmichael, R. Glauber, and M. Scully, eds. , no. 561 in Lecture Notes in Physics (Springer, Berlin, 2001), p. 261.

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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)

Equations on this page are rendered with MathJax. Learn more.

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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