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Physical Review A

Reduction of photon-number fluctuations in two-photon lasers

Miguel Orszag, J. C. Retamal

Published Jun 1, 1991• 7 citations
Publisher

Abstract

We analyze the effects of pump statistics on degenerate and nondegenerate two-photon lasers. We find that, although the steady-state mean photon numbers are not affected by pump statistics, there is an interesting reduction in the photon-number fluctuations, both in degenerate and nondegenerate two-photon lasers when the pump statistics is regular (p=1), as compared to the Poissonian case (p=0).

DOI
10.1103/physreva.43.6209

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We analyze the effects of pump statistics on degenerate and nondegenerate two-photon lasers. We find that, although the steady-state mean photon numbers are not affected by pump statistics, there is an interesting reduction in the photon-number fluctuations, both in degenerate and nondegenerate two-photon lasers when the pump statistics is regular (p=1), as compared to the Poissonian case (p=0).
We derive the equation of motion for the field density matrix of the degenerate two-photon laser under conditions of two-photon resonance starting from the full microscopic Hamiltonian. Our results are compared with the corresponding quantities obtained from the standard effective Hamiltonian. As we have shown for the nondegenerate case, the full diagonal density-matrix equations tend to the effective Hamiltonian density-matrix equations in an appropriate limit, but the equations of motion for the off-diagonal elements do not coincide. The equations obtained using a more accurate form of the effective Hamiltonian, in which Stark shifts are included, do agree. In this paper we concentrate on the photon-number distribution and the nature of the phase transition that takes place in the neighborhood of the two-photon lasing transition threshold. We determine the steady-state mean photon numbers and consider the stability of the solutions. A solution is found where the mean photon number is zero, but this is found to be a stable solution only for sufficiently weak pumping, whereas in the standard effective Hamiltonian approach it is always stable. As the pumping strength is increased, a range of detunings is reached in which there are two stable solutions. The amplitude of the zero-photon peak diminishes very rapidly as the pumping strength increases. Finally, for sufficiently large detunings, a single, stable, steady-state solution is obtained. The nature of the phase transition is illustrated by presenting plots of the photon-number distribution against detuning and pumping rate. The photon-number fluctuations about the mean are also discussed.
We show that regularization of pumping leads to large photon-number noise reduction in one- and two-photon micromasers, and is much more important than in macroscopic lasers and masers. Our calculations are based on a step-by-step microscopic approach and are compared to results obtained from a pumping-statistics-dependent master equation, which is shown not to be entirely reliable in the micromaser case. Although noise reduction and the discrepancies betwen the two approaches are more important for monokinetic beams, they are still relevant when a 10% velocity dispersion is allowed.
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.
In laser systems, regular pumping always reduces the photon noise. In micromaser systems, however, regular pumping may lead to photon noise reduction as well as photon noise increase depending on the operation conditions. The effect of higher-order terms in the expansion of ln(1+pu) with respect to the pumping regularity parameter p is discussed. These terms have a significant effect on the photon statistics in micromaser systems. Their effect in laser systems is quite small compared with the second order and can be neglected.
The quantum noise properties of atomic and semiconductor lasers are compared. The intensity fluctuation spectrum of the output field is calculated for the case of regularly, i.e., sub-Poissonian, pumped lasers. The effects of spontaneous emission on the output amplitude fluctuations may be suppressed in regularly pumped atomic lasers by requiring the lifetime of the lower lasing level to be short in comparison with that of the upper laser level. The output amplitude fluctuations then tend to zero at the laser frequency. In regularly pumped semiconductor lasers, net stimulated emission of photons dominates over spontaneous emission because of the combination of pumping and rapid intraband thermalization due to Coulomb scattering.