Abstract

We present a quantum theory of a spaser-based nanolaser, under the bad-cavity approximation. We find first- and second-order correlation functions g(1)(τ) and g(2)(τ) below and above the generation threshold, and obtain the average number of plasmons in the cavity. The latter is shown to be of the order of unity near the generation threshold, where the spectral line narrows considerably. In this case the coherence is preserved in a state of active atoms in contradiction to the good-cavity lasers, where the coherence is preserved in a state of photons. The damped oscillations in g(2)(τ) above the generation threshold indicate the unusual character of amplitude fluctuations of polarization and population, which become interconnected in this case. Obtained results allow to understand the fundamental principles of operation of nanolasers.

© 2014 Optical Society of America

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  1. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19, 22029–22106 (2011).
    [CrossRef] [PubMed]
  2. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12, 024004 (2010).
    [CrossRef]
  3. D. Bergman, M. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
    [CrossRef] [PubMed]
  4. M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
    [CrossRef] [PubMed]
  5. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
    [CrossRef] [PubMed]
  6. M. O. Scully, M. Sh. Zubairy, Quantum Optics (Cambridge University Press, 1997).
    [CrossRef]
  7. H.J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, New York, 2010).
  8. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
    [CrossRef] [PubMed]
  9. V. M. Parfenyev, S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86, 043824 (2012).
    [CrossRef]
  10. J. I. Cirac, “Interaction of a two-level atom with a cavity mode in the bad-cavity limit,” Phys. Rev. A 46, 4354–4362 (1992).
    [CrossRef] [PubMed]
  11. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
    [CrossRef]
  12. S. Gnutzmann, “Photon statistics of a bad-cavity laser near threshold,” EPJD 4, 109–123 (1998).
    [CrossRef]
  13. J. Trieschmann, S. Xiao, L. J. Prokopeva, V. P. Drachev, A. V. Kildishev, “Experimental retrieval of the kinetic parameters of a dye in a solid film,” Opt. Express 19, 18253–18259 (2011).
    [CrossRef] [PubMed]
  14. J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20, 8100–8116 (2012).
    [CrossRef] [PubMed]
  15. V. Temnov, U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17, 5774–5782 (2009).
    [CrossRef] [PubMed]
  16. E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
    [CrossRef] [PubMed]
  17. H. J. Carmichael, Statistical Methods in Quantum Optics 2 (Springer, New York, 2008).
    [CrossRef]
  18. V. Temnov, 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).
    [CrossRef] [PubMed]

2013 (2)

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
[CrossRef] [PubMed]

2012 (3)

J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20, 8100–8116 (2012).
[CrossRef] [PubMed]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

V. M. Parfenyev, S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86, 043824 (2012).
[CrossRef]

2011 (2)

2010 (1)

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12, 024004 (2010).
[CrossRef]

2009 (3)

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

V. Temnov, U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17, 5774–5782 (2009).
[CrossRef] [PubMed]

2005 (1)

V. Temnov, 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).
[CrossRef] [PubMed]

2003 (1)

D. Bergman, M. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

1998 (1)

S. Gnutzmann, “Photon statistics of a bad-cavity laser near threshold,” EPJD 4, 109–123 (1998).
[CrossRef]

1992 (1)

J. I. Cirac, “Interaction of a two-level atom with a cavity mode in the bad-cavity limit,” Phys. Rev. A 46, 4354–4362 (1992).
[CrossRef] [PubMed]

Andrianov, E. S.

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
[CrossRef] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

Bakker, R.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Baranov, D. G.

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Belgrave, A.M.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Bergman, D.

D. Bergman, M. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Bohnet, J. G.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Boltasseva, A.

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 2 (Springer, New York, 2008).
[CrossRef]

Carmichael, H.J.

H.J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, New York, 2010).

Chen, Z.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Cirac, J. I.

J. I. Cirac, “Interaction of a two-level atom with a cavity mode in the bad-cavity limit,” Phys. Rev. A 46, 4354–4362 (1992).
[CrossRef] [PubMed]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Dorofeenko, A. V.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
[CrossRef] [PubMed]

Drachev, V. P.

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Gnutzmann, S.

S. Gnutzmann, “Photon statistics of a bad-cavity laser near threshold,” EPJD 4, 109–123 (1998).
[CrossRef]

Herz, E.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Holland, M. J.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Jacob, Z.

Kildishev, A. V.

Kim, J.

Lisyansky, A. A.

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
[CrossRef] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

Ma, R.-M.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Meiser, D.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Naik, G. V.

Narimanov, E. E.

Narimanov, E.E.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Noginov, M.A.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Parfenyev, V. M.

V. M. Parfenyev, S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86, 043824 (2012).
[CrossRef]

Prokopeva, L. J.

Pukhov, A. A.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
[CrossRef] [PubMed]

Scully, M. O.

M. O. Scully, M. Sh. Zubairy, Quantum Optics (Cambridge University Press, 1997).
[CrossRef]

Shalaev, V. M.

Shalaev, V.M.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Stockman, M.

D. Bergman, M. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19, 22029–22106 (2011).
[CrossRef] [PubMed]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12, 024004 (2010).
[CrossRef]

Stout, S.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Suteewong, T.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Temnov, V.

V. Temnov, U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17, 5774–5782 (2009).
[CrossRef] [PubMed]

V. Temnov, 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).
[CrossRef] [PubMed]

Thompson, J. K.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Trieschmann, J.

Vergeles, S. S.

V. M. Parfenyev, S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86, 043824 (2012).
[CrossRef]

Vinogradov, A. P.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express 21, 13467–13478 (2013).
[CrossRef] [PubMed]

Weiner, J. M.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Wiesner, U.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Woggon, U.

V. Temnov, U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express 17, 5774–5782 (2009).
[CrossRef] [PubMed]

V. Temnov, 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).
[CrossRef] [PubMed]

Xiao, S.

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Zhang, X.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Zhu, G.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Zubairy, M. Sh.

M. O. Scully, M. Sh. Zubairy, Quantum Optics (Cambridge University Press, 1997).
[CrossRef]

EPJD (1)

S. Gnutzmann, “Photon statistics of a bad-cavity laser near threshold,” EPJD 4, 109–123 (1998).
[CrossRef]

J. Opt. (1)

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12, 024004 (2010).
[CrossRef]

JETP (1)

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP 117, 205–213 (2013).
[CrossRef]

Nature (3)

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Opt. Express (5)

Phys. Rev. A (2)

V. M. Parfenyev, S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A 86, 043824 (2012).
[CrossRef]

J. I. Cirac, “Interaction of a two-level atom with a cavity mode in the bad-cavity limit,” Phys. Rev. A 46, 4354–4362 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

D. Bergman, M. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

V. Temnov, 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).
[CrossRef] [PubMed]

Other (3)

H. J. Carmichael, Statistical Methods in Quantum Optics 2 (Springer, New York, 2008).
[CrossRef]

M. O. Scully, M. Sh. Zubairy, Quantum Optics (Cambridge University Press, 1997).
[CrossRef]

H.J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, New York, 2010).

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Figures (3)

Fig. 1
Fig. 1

(a) The dependence of the average number of plasmons in the cavity on the pump-parameter , for the following parameters: κ = 2 · 1015 s−1, Γ = 5 · 1012 s−1, g = 1011 s−1, γ = 9 · 1010 s−1, γ = 1010 s−1. The dashed line corresponds to the mean-field theory, the solid line takes into account quantum fluctuations. (b) Normalized spectrum with corrections arising from the amplitude fluctuations.

Fig. 2
Fig. 2

(a) The second-order correlation function above the generation threshold. The parameters are as for the Fig. 1. (b) The vector field obtained from the right parts of the macroscopic equations with = 1.1, ns = 0.8 and ΓT1 = 50. The spiral movement to the steady-state leads to the damped oscillations in g > ( 2 ) ( τ ) .

Fig. 3
Fig. 3

The “phase diagram” contained information about oscillations in g > ( 2 ) ( τ ) , for different pump-parameters . The area below and to the right to the corresponding curves responds to the non-oscillating regime. The painted area above the dotted line corresponds to the bad-cavity lasers.

Equations (17)

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ρ ˙ = i 1 2 ω [ J z , ρ ] + γ 2 ( j = 1 N 2 σ j + ρ σ j + 1 2 J z ρ + 1 2 ρ J z N ρ ) + + γ 2 ( j = 1 N 2 σ j ρ σ j + 1 2 J z ρ 1 2 ρ J z N ρ ) + γ p 2 ( j = 1 N σ j z ρ σ j z N ρ ) + + g 2 κ ( 2 J ρ J + J + J ρ ρ J + J ) ,
χ N ( ξ , ξ * , η ) tr ( ρ e i ξ * J + e i η J z e i ξ J ) ,
P ˜ t = L ( v , v * , m , v , v * , m ) P ˜ ,
L = γ 2 [ ( e 2 m 1 ) ( N m ) + 4 v 2 v * 2 e 2 m ( N + m ) + 2 N 2 v v * ] + + γ 2 ( 2 e 2 m 1 + 2 2 v v * ) ( v v + v * v * ) + + γ 2 [ ( e 2 m 1 ) ( N + m ) + v v + v * v * ] + + γ p [ v v + v * v * + 2 v v * e 2 m ( N + m ) ] + i ω [ v v v * v * ] + + g 2 κ [ 2 ( 2 e 2 m ) v v * ( v v m + v * v * m ) + 2 v 2 v 2 + 2 v * 2 v * 2 ] .
v e i ω t / N = σ + N 1 / 2 ν , m / N = n + N 1 / 2 μ ,
d ( σ / n s ) Γ d t = ( 1 n n s ) σ / n s ,
d ( n / n s ) Γ d t = n / n s 1 Γ T 1 4 | σ / n s | 2 ,
P t = Γ ( 1 ) [ ν ν + ν * ν * ] P + 1 T 1 μ μ P + 2 γ ( Γ + 2 γ ) ( γ + γ ) 2 P ν ν * + 4 γ γ γ + γ 2 P μ 2 .
a + a s s , < = g 2 κ 2 J + J s s , < = N g 2 κ 2 ( 1 ) γ ( Γ + 2 γ ) Γ ( γ + γ ) ,
g < 1 ( τ ) = lim t a + ( t ) a ( t + τ ) < a + a s s , < = e Γ ( 1 ) τ e i ω τ , τ 1 / κ ,
g < ( 2 ) ( τ ) = lim t a + ( t ) a + ( t + τ ) a ( t + τ ) a ( t ) < a + a s s , < 2 = 1 + e 2 Γ ( 1 ) τ , τ 1 / κ .
v e i ω t / N = e i N 1 / 2 ψ ( | σ | + N 1 / 2 ν ) ,
A t = Γ ( 1 ) 4 T 1 [ 8 μ ν ν μ ] A + 1 T 1 μ μ A + γ p 4 ( 1 + 1 0 ) 2 ν 2 A ,
Φ t = γ p Γ T 1 ( 0 + 1 ) 0 ( 1 ) 2 ψ 2 Φ .
a + a s s , > Γ ( 1 ) 4 T 1 g 2 = γ p ( 0 + 1 ) 8 κ [ 2 Γ T 1 + 1 / ( 1 ) ] .
g > ( 1 ) ( τ ) = e ( i ω + D ) τ [ 1 D T 1 + D T 1 e τ / 2 T 1 cos ( 2 Γ T 1 ( 1 ) τ / T 1 ) ] , D = γ p Γ T 1 N 0 ( 0 + 1 ) ( 1 ) = γ p Γ 4 h ¯ ω ( 0 + 1 ) P > 1 / T 1 ,
g > ( 2 ) ( τ ) = 1 + 4 D T 1 e τ / 2 T 1 cos ( 2 Γ T 1 ( 1 ) τ / T 1 ) .

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