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Q Function for a Single-Atom Laser Operating in the “Classical” Regime

  • ATOMS, MOLECULES, OPTICS
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Abstract

A model of single-atom laser with incoherent pumping is investigated theoretically. In the stationary case, a linear homogeneous differential equation for the phase-averaged Husimi Q function is derived from the equation for the density operator of the system. In the regime in which the coupling of the cavity mode with an atom is much stronger than the coupling of the mode with the reservoir ensuring its damping, the asymptotic solution is obtained to this equation. This solution makes it possible to describe some statistical features of the single-atom laser (in particular, the weak sub-Poissonian photon statistics).

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Funding

This work was supported by the State assignment for fundamental research works (theme code FSEG-2020-0024).

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Authors and Affiliations

  1. St. Petersburg State Marine Technical University, 190121, St. Petersburg, Russia

    N. V. Larionov

  2. Peter the Great St. Petersburg Polytechnic University, 195251, St. Petersburg, Russia

    N. V. Larionov

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  1. N. V. Larionov
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Correspondence to N. V. Larionov.

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Translated by N. Wadhwa

APPENDIX

APPENDIX

Coefficients bik from Eq. (9) are given by

$$\begin{gathered} {{b}_{{02}}} = - \frac{1}{2}[1 + {{I}_{s}}(r + 1)],\quad {{b}_{{03}}} = 1; \\ {{b}_{{11}}} = - 3[1 + {{I}_{s}}(r + 1)], \\ {{b}_{{12}}} = \frac{1}{2}[7 - 3{{I}_{s}}(r + 1)],\quad {{b}_{{13}}} = 3; \\ {{b}_{{20}}} = - 3[1 + {{I}_{s}}(r + 1)], \\ {{b}_{{21}}} = \frac{1}{4}[ - 21 - 26{{I}_{s}}(r + 1) + 3I_{s}^{2}{{(r + 1)}^{2}}], \\ \end{gathered} $$
$$\begin{gathered} {{b}_{{22}}} = - 3[ - 4 + {{I}_{s}}(r + 1)],\quad {{b}_{{23}}} = 3; \\ {{b}_{{30}}} = \frac{1}{2}[ - 15 - 8{{I}_{s}}(r + 1) + 3I_{s}^{2}{{(r + 1)}^{2}}], \\ {{b}_{{31}}} = - \frac{1}{2}{{I}_{s}}[c(1 + {{I}_{s}}(r + 1)) - (r + 1)( - 13 + 3{{I}_{s}}(r + 1))], \\ {{b}_{{32}}} = \frac{1}{2}[23 + {{I}_{s}}(2c - 7(r + 1))],\quad {{b}_{{33}}} = 1; \\ {{b}_{{40}}} = \frac{1}{4}[ - 24 + {{I}_{s}}(r + 1 - 3c) - I_{s}^{3}(r + 1)(c + {{(r + 1)}^{2}}) \\ \, + I_{s}^{2}(8{{(r + 1)}^{2}} - c(3r + 4))], \\ \end{gathered} $$
(22)
$$\begin{gathered} {{b}_{{41}}} = \frac{1}{4}[15 - 2{{I}_{s}}(c + 10(r + 1)) \\ \, + I_{s}^{2}(5{{(r + 1)}^{2}} - 2c(2r + 1))], \\ {{b}_{{42}}} = \frac{1}{2}[7 + {{I}_{s}}(4c - 3(r + 1))]; \\ {{b}_{{50}}} = \frac{1}{4}[ - 6 + {{I}_{s}}(5(r + 1) - 3c) + I_{s}^{3}(r + 1)(c(r - 1) \\ \, - {{(r + 1)}^{2}}) + I_{s}^{2}(2{{(r + 1)}^{2}} - 4c)], \\ {{b}_{{51}}} = \frac{1}{2}[3 - 4{{I}_{s}}(r + 1) + I_{s}^{2}({{(r + 1)}^{2}} - 2cr)],\quad {{b}_{{52}}}\, = \,c{{I}_{s}}. \\ \end{gathered} $$

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Larionov, N.V. Q Function for a Single-Atom Laser Operating in the “Classical” Regime. J. Exp. Theor. Phys. 134, 135–143 (2022). https://doi.org/10.1134/S1063776122020078

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