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Communications in Theoretical Physics

The Photon-Number Distribution of the Lasers with Binomial Pump Model

Ming Yang, Cao Li, Wu Da-Jin, Zaiguang Li

Published Jun 15, 1998• 0 citations
Publisher

Abstract

A method on how to take into account pump statistics and spontaneous atomic decays when calculating the photon-number distribution of lasers in photon-number representation is presented. The intracavity phonon-number and extracavity photocount distributions with binomial pump and spontaneous atomic decays are calculated. In two limit cases, the results of calculations transit respectively to the photon-number distributions of systems with common Poissonian pump and regular pump, and both with the spontaneous atomic decays being suppressed.

DOI
10.1088/0253-6102/29/4/539

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A method on how to take into account pump statistics and spontaneous atomic decays when calculating the photon-number distribution of lasers in photon-number representation is presented. The intracavity phonon-number and extracavity photocount distributions with binomial pump and spontaneous atomic decays are calculated. In two limit cases, the results of calculations transit respectively to the photon-number distributions of systems with common Poissonian pump and regular pump, and both with the spontaneous atomic decays being suppressed.
The photon-number distribution of lasers generated via pump-noise suppression is derived in the photon-number representation. The derivation shows that the negative probability emerging in the calculation of the photon-number distribution in the photon-number representation is not caused by the representation itself but by the truncation of the master equation. Here without truncating the master equation we derive the characteristic function and the distribution function of the photon-number distribution of the laser system with regular pump and with spontaneous atomic decays being eliminated. It is convenient to use the characteristic function to calculate any order of the moment of the photon-number distribution. Such a method can be used to some more general cases, in which the laser system has imperfectly regular pump and the spontaneous atomic decays are also considered.
We calculate intracavity photon-number and extracavity photocount distributions for a pump-noise-suppressed laser, using both the Lamb-Scully laser model in the number-state representation and the Haken-Risken laser model in the coherent representation. When pump fluctuations and spontaneous decay are suppressed, a pump-noise-suppressed laser has intracavity photon-number fluctuations squeezed by a factor up to 1/2. The extracavity photocount fluctuations are at the Poisson level for counting times much shorter than the cavity lifetime, whereas the variance for photocounts can be reduced to the mean of the intracavity photon number, which is markedly sub-Poissonian, under ideal conditions when the counting time is much longer than the cavity lifetime.
A laser with a Poissonian random pump, operated far above threshold, has Poissonian statistics both for the intracavity photon number and for extracavity photocounts. We analyze a pump-noise-suppressed laser using the positive P representation. When the pump noise is suppressed below a critical level, the intracavity photon-number distribution becomes sub-Poissonian.1,2 When the spontaneous decay of the upper lasing level is also eliminated, intracavity photon-number fluctuations can be diminished by a factor up to half. For counting times much shorter than the cavity decay time, the extracavity photocount statistics are essentially Poissonian, since cavity decay dominates. For counting times comparable with or larger than the cavity lifetime, the extracavity photocount statistics become sub-Poissonian. In the limit that intracavity photon-number fluctuations are diminished by one half, the extracavity photocount noise can be completely suppressed for counting times much longer than the cavity lifetime. The cavity mode occupies an incoherent superposition of two states 180° out of phase, both of which diffuse in phase. In the microlaser regime, quantum diffusion between these two states can result in 180° phase jumps, which are in addition to ordinary phase diffusion.
The effect of the pump-noise on the statistical properties of the output photon-number is discussed. The master equation for the radiation field valid for any pump fluctuation with steady atomic injection is presented. Far above threshold, the characteristic function for the photon-number distribution is derived without truncating the master equation, the mean value and fluctuation of the output photon-number are also calculated. Thereby the condition of generating sub-Poissonian light is derived and the effect of the pump-noise on the higher-order moment of the output photon-number distribution is analyzed.
It is shown that, introducing quantum-mechanical degrees of freedom for the pump field, one can model optical-pumping processes of a laser with sub-Poissonian statistics. For a class of pump models containing regular pumping at one end of the range and Poissonian pumping at the other, the exact stationary moments for the pump-field-averaged laser field are calculated in the strong-saturation limit of the lasing transition. It is demonstrated that, in a photodetection experiment of the laser output, complete noise suppression in the photocurrent fluctuation spectrum is, at least in principle, achievable. Finally, an approximate Fokker-Planck equation for photon distribution of the laser field with explicit appearance of the pump-light intensity correlation function in the diffusion term is derived and solved in the stationary limit.
In the present work the model of a single-atom laser generating in the regime when incoherent pumping rate coincides with the cavity decay rate is theoretically investigated. Using the stationary equation for phase averaged Glauber P function the analytical expressions for mean number of photons and its dispersion are obtained. In the limiting case of the strong-coupling regime the exact expression for the photon number distribution function is derived. Obtained results describe the sub-Poissonian photon statistics and show an increase in mean number of photons to non-zero value when the pumping tends to zero.