## Quantum theory of a spaser-based nanolaser |

Optics Express, Vol. 22, Issue 11, pp. 13671-13679 (2014)

http://dx.doi.org/10.1364/OE.22.013671

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### 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

## 1. Introduction

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 and 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, and 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, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009). [CrossRef] [PubMed]

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

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

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

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

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

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

*Q*-factor resonator with the arbitrary number of quanta both below and above the generation threshold, which interacts with ensemble of

*N*identical active atoms.

*κ*is the fastest rate in the system. Thus, the resonator mode can be adiabatically eliminated [10

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]

*N*identical two-level active atoms. Second, we believe

*N*≫ 1 and thereby the fluctuations of the state of atoms can be considered in the small-noise limit [7, ch.5.1.3]. Note that due to adiabatic mode elimination we can only resolve times

*τ*≫ 1/

*κ*. The smaller times were considered in the paper [11

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

*N*= 1, and below the generation threshold, when mean number of quanta in the resonator is well below unity.

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, and 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, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express **20**, 8100–8116 (2012). [CrossRef] [PubMed]

*κ*. This fact fundamentally distinguishes the behaviour of the bad-cavity nanolasers in comparison with the good-cavity lasers, where the coherence is preserved in a state of photons. The evaluation for the number of plasmons is in accordance with the experimental observations [4

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

*g*

^{(2)}(

*τ*) at

*κτ*≫ 1 and find that above the generation threshold the amplitude fluctuations of polarization of active atoms lead to the damped oscillations in

*g*

^{(2)}(

*τ*). A similar dependence was observed in numerical simulations in the paper [15

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

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

*g*

^{(2)}(

*τ*) can be used as an indicator of the mechanism of the spectral line narrowing. We investigate at what relationship between cavity decay rate

*κ*and homogeneous broadening of active atoms Γ the oscillations occur. Finally, we find attendant peaks in the spectrum

*S*(

*ν*), which are produced by the oscillations in

*g*

^{(2)}(

*τ*). We believe that the obtained results are important for understanding the fundamental principles of operation of spaser-based nanolasers.

## 2. Physical model and methods

*N*≫ 1 identical two-level active atoms with resonant frequency

*ω*coupled to single strongly damped cavity, with a short plasmon lifetime (2

*κ*)

^{−1}centered at the same frequency. The interaction between the atoms and the field is described by the Tavis-Cummings Hamiltonian [7],

*H*=

_{AF}*ih̄g*(

*a*

^{+}

*J*

_{−}−

*aJ*

_{+}), where

*g*is coupling constant identical for all atoms,

*a*

^{+}and

*a*are the creation and annihilation operators of plasmons in the cavity mode [16

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

*σ*,

_{jα}*α*= {

*x*,

*y*,

*z*} – Pauly matrices, and

*σ*= (

_{j±}*σ*±

_{jx}*iσ*)/2. In a bad-cavity limit [10

_{jy}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]

*ρ*= tr

*:*

_{F}ρ_{AF}*γ*

_{↑}and we take into account the spontaneous emission with rate

*γ*

_{↓}. The dephasing processes, which are caused mostly by the interaction with phonons, have rate

*γ*. The last term describes interaction of active atoms through the cavity mode. Adiabatic mode elimination can be performed only if

_{p}*κ*≫

*Ng*

^{2}/

*κ*,

*γ*,

_{p}*γ*

_{↑},

*γ*

_{↓}. Note, that normal-ordered field operator averages can be restored by formal substitutions

*a*

^{+}(

*t*) → (

*g/κ*)

*J*

_{+}(

*t*) and

*a*(

*t*) → (

*g/κ*)

*J*

_{−}(

*t*), [17

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

*N*of active atoms, was considered numerically by V. Temnov and U. Woggon in papers [15

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

18. 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). [CrossRef] [PubMed]

*J*

_{+},

*J*,

_{z}*J*

_{−}. First, we define characteristic function which determines all normal-ordered operator averages in usual way. Next, we introduce the Glauber-Sudarshan

*P*-representation

*P̃*(

*v*,

*v*

^{*},

*m*) as the Fourier transform of

*χ*(

_{N}*ξ*,

*ξ*

^{*},

*η*), which can be interpreted as distribution function and allows to calculate normal-ordered operator averages as in statistical mechanics [7, (6.118a)]. Evolution equation on

*P̃*can be obtained by differentiation Eq. (2) with respect to time and replacement

*ρ*with Eq. (1). The results is where The closed form of the equation confirms the possibility of describing the system in terms of collective atomic operators.

*L*, which describe transitions in active atoms. Moreover, the solution cannot be found in analytical form. However, we can obtain an approximate nonsingular distribution, replacing Eq. (3) by a Fokker-Planck equation. The key element to such replacement is a system size expansion procedure [7, ch. 5.1.3]. The large system size parameter in our case is the number

*N*≫ 1 of active atoms. Following the system size expansion method we will obtain an adequate treatment of quantum fluctuations in the first order in 1/

*N*.

*N*, we move into rotating frame and introduce dimensionless polarization

*σ*= tr[

*ρJ*

_{−}

*e*]/

^{iωt}*N*and inverse population

*n*= tr[

*ρJ*]/

_{z}*N*per one atom. Next, we separate the mean values and fluctuations in the phase-space variables and introduce a distribution function

*P*(

*ν*,

*ν*

^{*},

*μ*,

*t*) ≡

*N*

^{3/2}

*P̃*(

*v*(

*ν*,

*t*),

*v*

^{*}(

*ν*,

^{*}*t*),

*m*(

*μ*,

*t*),

*t*), which depends on variables, corresponding to fluctuations. Using the Eq. (3) and neglecting terms ∼

*O*(

*N*

^{−1/2}), we obtain the equation for a scaled distribution function

*P*. More accurately, we obtain two sets of equations: the first set describes dynamics of macroscopic variables, and one more equation characterizes fluctuations.

## 3. Macroscopic equations and generation threshold

*T*

_{1}= 1/(

*γ*

_{↑}+

*γ*

_{↓}), homogeneous broadening Γ =

*γ*+ 1/(2

_{p}*T*

_{1}) and equilibrium inverse population

*n*= (

_{s}*γ*

_{↑}−

*γ*

_{↓})/(

*γ*

_{↑}+

*γ*

_{↓}). The system has two different stable steady-states solutions, depending on

*pump*-parameter

*℘*=

*℘*

_{0}

*n*, where

_{s}*℘*

_{0}=

*Ng*

^{2}/(

*κ*Γ).

*℘*< 1, one obtains the solution

*n*=

*n*,

_{s}*σ*= 0. Thus, there is no macroscopic polarization and this situation corresponds to the nanolaser operating below generation threshold. In the opposite case

*℘*> 1, the solution takes a form

*n*= 1/

*℘*

_{0},

*℘*= 1 and this point corresponds to the spaser generation threshold, which was obtained in earlier semiclassical papers, e.g. [2

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

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

## 4. Quantum fluctuations below threshold

*℘*< 1, we obtain The equation can be solved by separation of variables, and we calculate the steady-state correlation functions as in statistical mechanics

*κ*, since we adiabatically eliminate the cavity mode, rather than the polarization of active atoms [7, ch. 8.1.4]. Note, that due to adiabatic mode elimination we can only resolve times

*τ*≫ 1/

*κ*. The smaller times were resolved in paper [11

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

*℘*= 1 the drift term in the Eq. (7) vanishes and there is no restoring force to prevent the fluctuations from growing without bound. Thus, the average number of plasmons in the cavity mode (8) diverges at the point

*℘*= 1. Thereby, the Eq. (7) cannot correctly describe the behaviour of system at the generation threshold. Note, that the operation of a bad-cavity laser at the threshold was discussed in the paper [12

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

## 5. Quantum fluctuations above threshold

*℘*> 1, the phase of polarization is undetermined. Thus, in place of the first equation in (4), we write where the variable

*ν*represents real amplitude fluctuations now, which must fall within the range −

*N*

^{1/2}|

*σ*| ≤

*ν*≤ ∞, and the variable

*ψ*represents phase fluctuations. The distribution function in scaled variables, normalized with respect to the integration measure

*dνdψdμ*, is defined by

*P*(

*ν*,

*ψ*,

*μ*,

*t*) ≡

*N*

^{3/2}(|

*σ*| +

*N*

^{−1/2}

*ν*)

*P̃*(

*v*(

*ν*,

*ψ*,

*t*),

*v*

^{*}(

*ν*,

*ψ*,

*t*),

*m*(

*μ*,

*t*),

*t*).

*P*(

*ν*,

*ψ*,

*μ*,

*t*) =

*A*(

*ν*,

*μ*,

*t*)Φ(

*ψ*,

*t*) in the limit of small amplitude fluctuations above the generation threshold, |

*σ*| ≫

*N*

^{−1/2}

*ν*. Moreover, in accordance with experimental papers [13

13. J. Trieschmann, S. Xiao, L. J. Prokopeva, V. P. Drachev, and 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, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express **20**, 8100–8116 (2012). [CrossRef] [PubMed]

*T*

_{1}≫ 1. Under the assumptions, the evolution of the distribution functions

*A*and Φ is governed by equations:

*T*

_{1}(

*℘*− 1) ≫ 1, this leads to the restriction

*℘*− 1 ≫ (

*γ*)(

_{p}/κ*gT*

_{1})

^{2}(

*℘*

_{0}+ 1). Thus, our theory is self-consistent if (

*γ*)(

_{p}/κ*gT*

_{1})

^{2}≪ 1.

*τ*≫ 1/

*κ*)

*D*defines the width of spectral line above the generation threshold, and we rewrite it in term of the output power

*P*

_{>}=

*κh̄ω*〈

*a*

^{+}

*a*〉

_{ss,>}. Summands proportional to

*DT*

_{1}≪ 1 correspond to amplitude fluctuations. Qualitatively, the result is similar to the case of good-cavity lasers, compare with [7, (8.138)]. However, the mechanism which leads to the narrowing of the spectral line is quite different. We will discuss it in detail in the next section.

## 6. Numerical parameters and discussion

^{12}

*s*

^{−1},

*γ*

_{↓}= 10

^{10}

*s*

^{−1},

*γ*

_{↑}= 9 · 10

^{10}

*s*

^{−1}for active atoms, based on the paper [13

13. J. Trieschmann, S. Xiao, L. J. Prokopeva, V. P. Drachev, and A. V. Kildishev, “Experimental retrieval of the kinetic parameters of a dye in a solid film,” Opt. Express **19**, 18253–18259 (2011). [CrossRef] [PubMed]

*κ*= 2 · 10

^{15}

*s*

^{−1}and

*g*= 10

^{11}

*s*

^{−1}. In experiments and theoretical papers, e.g. [4

**460**, 1110–1112 (2009). [CrossRef] [PubMed]

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

*κ*is usually less and the coupling constant

*g*is usually larger than ours. Thus, proposed numerical parameters are easily achievable in experiment and reasonable. To change pump-parameter

*℘*we variate the number

*N*of active atoms.

*℘*below and above the generation threshold. The dashed line corresponds to the semiclassical mean-field theory, see Eqs. (5)–(6), and the solid line takes into account quantum fluctuations, see Eqs. (8), (14). Emphasize, that near the threshold

*℘*∼ 1 the average number of plasmons in the cavity mode 〈

*a*

^{+}

*a*〉 < 1. Despite this fact, a linewidth of the order of Γ well below threshold is changed into a considerably narrower line of the order of

*D*above threshold. When

*℘*− 1 = 0.1, we find

*D*/Γ ∼ 1/850. Note, that amplitude fluctuations slightly changes the shape of the spectral curve above generation threshold, see Fig. 1(b), which was obtained as the Fourier transform of the first-order correlation function (15). The height of the side peaks is small compared with the height of the central peak as (

*DT*

_{1})

^{2}≪ 1.

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

*κ*, but the coherence is still alive in active atoms, which relax slowly, 1/Γ ≫ 1/

*κ*. The next plasmon generated by such atoms can be coherent to the previous one. This mechanism of the spectral line narrowing was demonstrated in experiment with the laser, which deals with photons, in the paper [8

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

*T*

_{1}≫ 1 is not fulfilled. Another experimental realization of the lasing regime was presented in the paper [4

**460**, 1110–1112 (2009). [CrossRef] [PubMed]

*a*

^{+}

*a*〉 ∼ 0.2. Indeed, the total pumping energy absorbed per one nanolaser is

*P*∼ 〈

_{W}*a*

^{+}

*a*〉

*h̄ω*

^{2}

*τ*, where

_{p}/Q*τ*= 5

_{p}*ns*is the duration of the pumping pulse, the measured value

*P*= 10

_{W}^{−13}

*J*near the generation threshold,

*Q*= 13.2, and

*h̄ω*= 2.3

*eV*. Note, that numerical parameters in the Fig. 1 are slightly different from those from the paper [4

**460**, 1110–1112 (2009). [CrossRef] [PubMed]

*τ*≫ 1/

*κ*, because it was obtained under the bad-cavity approximation. In order to explain the nature of the damped oscillations, we plot the vector field on the

*n*-|

*σ*|-plane, see Fig. 2(b), which corresponds to the right parts of the mean-field Eqs. (5)–(6). The red point represents the steady-state solution of these equations. Fluctuations move the system from its equilibrium state and then it relaxes to the steady-state. The spiral movement corresponds to the damped oscillations in polarization amplitude |

*σ*| and inverse population

*n*, and as a consequence in the second-order correlation function.

*Q*-factor, but we should go beyond the bad-cavity approximation. Instead of macroscopic Eqs. (5)–(6), we need to consider a full system of three Maxwell-Bloch equations [2

**12**, 024004 (2010). [CrossRef]

*κT*

_{1}, Γ

*T*

_{1}and

*℘*. In the Fig. 3 we plot a ”phase diagram” in logarithmic coordinates for different pump-parameters

*℘*> 1. The area to the right and below to the corresponding curves responds to the non-oscillating regime. The parameters from the painted area above the dotted line always correspond to the oscillations in

*℘*≫ 1) of the dotted line was obtained numerically and it corresponds to the dependencies

*κT*

_{1}≈ 0.16

*℘*and Γ

*T*

_{1}≈ 2.5

*℘*. In the area below the dotted line both the oscillating and non-oscillating behaviour of

*℘*. Thus, the shape of the second-order correlation function provides insufficient information to obtain a mechanism of spectral line narrowing. The answer on this question is contained in the ”phase diagram” in the Fig. 3.

## 7. Conclusion

**460**, 1110–1112 (2009). [CrossRef] [PubMed]

*g*

^{(2)}(

*τ*) above the generation threshold. It is unusual behaviour for the good-cavity lasers, and we investigated in detail what relationship between cavity decay rate

*κ*and homogeneous broadening of active atoms Γ corresponds to the bad-cavity damped oscillations and non-oscillating regime.

## Acknowledgments

## References and links

1. | M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express |

2. | M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. |

3. | D. Bergman and M. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. |

4. | M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E.E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature |

5. | R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature |

6. | M. O. Scully and M. Sh. Zubairy, |

7. | H.J. Carmichael, |

8. | J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature |

9. | V. M. Parfenyev and S. S. Vergeles, “Intensity-dependent frequency shift in surface plasmon amplification by stimulated emission of radiation,” Phys. Rev. A |

10. | J. I. Cirac, “Interaction of a two-level atom with a cavity mode in the bad-cavity limit,” Phys. Rev. A |

11. | E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Spectrum of surface plasmons excited by spontaneous quantum dot transitions,” JETP |

12. | S. Gnutzmann, “Photon statistics of a bad-cavity laser near threshold,” EPJD |

13. | J. Trieschmann, S. Xiao, L. J. Prokopeva, V. P. Drachev, and A. V. Kildishev, “Experimental retrieval of the kinetic parameters of a dye in a solid film,” Opt. Express |

14. | J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express |

15. | V. Temnov and U. Woggon, “Photon statistics in the cooperative spontaneous emission,” Opt. Express |

16. | E. S. Andrianov, D. G. Baranov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Loss compensation by spasers in plasmonic systems,” Opt. Express |

17. | H. J. Carmichael, |

18. | 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. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(270.3430) Quantum optics : Laser theory

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 17, 2014

Revised Manuscript: May 16, 2014

Manuscript Accepted: May 19, 2014

Published: May 29, 2014

**Citation**

Vladimir M. Parfenyev and Sergey S. Vergeles, "Quantum theory of a spaser-based nanolaser," Opt. Express **22**, 13671-13679 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13671

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### References

- 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]
- 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]
- 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]
- 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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