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Physical Review A

Quantum-trajectory description of laser noise with pump depletion

Jorrit Visser, Gerard Nienhuis, S. M. Dutra, Martin P. van Exter, J. P. Woerdman

Published Jun 5, 2002• 1 citation
Publisher

Abstract

The intensity fluctuations of a three-level laser are known to drop below shot noise in the presence of depletion of the ground state. We study the fluctuations of the laser output as a function of the parameter $\ensuremath{\varepsilon},$ defined as the fraction of atoms needed for the laser to operate. For a sufficiently small number of active atoms, the value of $\ensuremath{\varepsilon},$ and thereby the ground-state depletion, can be appreciable even for modest pumping. This suggests that the intensity fluctuations in the laser output would decrease below shot noise as the number of active atoms is reduced. A microscopic approach that uses a quantum-trajectory method and a macroscopic approach using semiclassical rate equations both show, however, that rather than decreasing, the intensity fluctuations actually increase with $\ensuremath{\varepsilon}.$ We find that the fluctuations of the output are determined by the dependence of the cycle time of the atoms on the number of photons in the laser mode.

DOI
10.1103/physreva.65.063809

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The intensity fluctuations of a three-level laser are known to drop below shot noise in the presence of depletion of the ground state. We study the fluctuations of the laser output as a function of the parameter \ensuremathε,\ensuremath{\varepsilon},\ensuremathε, defined as the fraction of atoms needed for the laser to operate. For a sufficiently small number of active atoms, the value of \ensuremathε,\ensuremath{\varepsilon},\ensuremathε, and thereby the ground-state depletion, can be appreciable even for modest pumping. This suggests that the intensity fluctuations in the laser output would decrease below shot noise as the number of active atoms is reduced. A microscopic approach that uses a quantum-trajectory method and a macroscopic approach using semiclassical rate equations both show, however, that rather than decreasing, the intensity fluctuations actually increase with \ensuremathε.\ensuremath{\varepsilon}.\ensuremathε. We find that the fluctuations of the output are determined by the dependence of the cycle time of the atoms on the number of photons in the laser mode.
We show that the nonlinear dynamics of the active atoms in conventionally pumped multilevel laser systems can lead to output intensity fluctuations significantly below the shot noise level1,2 (see also Refs, 3-5). Our calculations are based on the standard model of the laser which assumes a large number of atoms resonantly coupled to a single lasing mode. We identify the multiple recycling of the active electron from the lower lasing level to the upper level through pumping as the key process leading to this dynamic pump noise reduction. It is required that (i) the pump rates be comparable to the decay rates of the atom, and (ii) we have a closed repumping cycle, and a constant number of atoms. We find that the results are closely related to recent calculations based on the assumption of an external regular pump6 (for noise reduction in semiconductor lasers, see Ref. 7).
Based on the standard laser model of a large number N of model atoms resonantly coupled to a single lasing mode, we show that the nonlinear dynamics of the active atoms of the laser can lead to output-intensity fluctuations significantly reduced below the shot-noise level. We identify the multiple recycling of the active electron from the lower lasing level to the upper level through the pumping as the key process leading to this dynamic-pump-noise reduction. This process has been neglected in most of the standard treatments of the laser so far. We find that the results are closely related to recent calculations based on the assumption of an external regular pump. For the widely used four-level model of the active atoms, the intensity noise can be reduced 50% below the shot-noise level. Generalizing the model to an m-level system, we find a quantum-noise reduction by a factor of 1/2m/(m-1)(m\ensuremath{\ge}3), leading to perfect output-intensity noise reduction in the limit of a large number of intermediate steps in the recycling process of the active atoms. Finally, we demonstrate that the bandwidth of the noise reduction can be significantly enhanced using a nonlinear absorber in the cavity.
Using population rate equations, we have calculated the intrinsic intensity fluctuations expected in the output of 3-level and 4-level CW laser oscillators. Such fluctuations stem from the quantum shot noise implicit in a restriction of the atomic and photon population-operator eigenvalues to discrete integral values (in the manner of Shimoda, Takahasi, and Townes). The intrinsic fluctuations below threshold are typical of a Gaussian distribution ofe-mfield amplitudes (Bose-Einstein distribution of photons) and above threshold of an amplitude-stabilizede-mfield (Poisson distribution of photons). The noise properties of 3-level and 4-level lasers are qualitatively similar, although for a given output power the intensity fluctuations of 3-level oscillators are greater than (or equal to) those in 4-level oscillators, principally as a consequence of the increased spontaneous emission associated with the larger upper-state occupation necessary to achieve a net gain in 3-level systems.
We demonstrate that the intensity fluctuations of a laser with a Raman gain medium, described by a large number of independent 3-level atoms, can drop well below the shot noise level. In the limit of a good cavity, where one can adiabatically eliminate the atoms, we find an optimum Mandel Q parameter of Q ≈ −1/2 over a large range of parameters, when the laser is operated well above threshold. This leads to a strong suppression of the low-frequency quantum fluctuations in the laser output. In addition we find that due to suppression of spontaneous emission into the lasing mode, the spectral linewidth of the emitted light can be reduced below the Schawlow-Townes limit by more than one order of magnitude.
We consider the problem of reducing the contributions of pump and spontaneous-emission noise to the output noise of a laser field. We consider two models based on an atomic medium, one analytic, the other involving numerical simulation of a sequence of pump cycles. The predictions of both models are shown to agree. For regular pump excitation the intensity fluctuations of a laser well above threshold may be reduced to zero at the cavity resonance frequency if the ratio of the lifetimes of the upper and lower lasing levels is sufficiently large that spontaneous emission is suppressed. When the atomic lifetimes are comparable, spontaneous emission reduces the amount of noise reduction. For example, for equal atomic lifetimes regular pump excitation will result in a maximum noise reduction in the output field of one-half the shot-noise limit.
Using linearized rate equations, we have calculated the intensity fluctuations expected in the output of four-level cw laser oscillators. Intrinsic quantum effects have been included by restricting the eigenvalues of photon and atomic population operators to discrete integral values and by using equivalent noise sources. The total output fluctuations increase monotonically with output power; the relative fluctuations decrease monotonically, and at high output levels inversely as the output power squared. The low-frequency fluctuations differ in that the relative fluctuations increase with output below threshold (noise amplifier region) but decrease sharply above threshold (dynamic saturation region). Curves typical of gas, solid, and diode lasers are illustrated. The theory agrees well with preliminary experimental results.