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Physical Review A

Dynamic quantum-noise reduction in multilevel-laser systems

Helmut Ritsch, P. Zoller

Published Feb 1, 1992• 54 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.45.1881

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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.
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).
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.
We identify a mechanism of dynamical pump-noise suppression in lasers. It is based on the recycling of the active laser electron from the lower to the upper laser level by a sequence of incoherent step processes. Although each of these steps corresponds to a Poisson process, i.e., is stochastic, the combination of many incoherent steps leads to a regular (deterministic) recycling of the laser electron and, correspondingly, a pump-noise suppression in the laser. The mechanism predicts sub-Poissonian laser output and intensity fluctuations beyond the shot-noise limit for incoherently pumped systems.
We present a general theory that connects the pump process in multilevel lasers with the statistics of the laser field. The key ingredient in our approach is the derivation of an effective master equation that involves only the two laser levels and that contains a non-Markovian pump term. This pump term gives rise to a narrowing of the photon-number distribution in steady state. The qualitative features of this dynamic noise reduction can be inferred from the eigenvalues of the pump term, which are known analytically. As the mechanism that is responsible for the noise reduction, we identify a correlated excitation process in which the effective excitation rate from the lower to the upper laser level depends on the photon number. This correlated excitation process is very different from an excitation at equidistant times, such as in lasers with periodic external injection. We stress that our approach is general and the treatment nonlinear because we do not resort to approximations such as an adiabatic elimination of atomic variables or the linearization of a Fokker-Planck equation. It is therefore particularly relevant for the description of systems where the active medium consists of a few atoms only and where the field losses are not negligible on the time scale of the atomic relaxations. This would be the case in an ion-trap laser. For such a situation our results are substantially different from those obtained by a linearized treatment. \textcopyright{} 1996 The American Physical Society.
We study the statistical properties of a three-level laser where cascade coherent excitation as a two-step process is used as a coherent pump and a laser field is generated on the upper transition. It is shown that the laser intensity fluctuations are suppressed up to 50% below the shot noise limit and that the laser linewidth is reduced below the Schawlow-Townes formula by more than one order of magnitude of the laser intensity. By transforming to the picture dressed by the coherent field on the upper transition, we identify a transfer of the lower coherent excitation to be responsible for the quantum noise reduction. Cascade coherent excitation turns out to be a new way in which one can create a pathway for recycling of active electrons and regularization of laser excitation process. The conditions for the physical realization of the Raman scheme (Ritsch et al 1992 Europhys. Lett. 19 7) as a limiting case is discussed.
Considerable attention has been directed recently to the study of lasing without the need of population inversion. The optical coherence and quantum interference associated with the light amplification may lead to interesting statistical properties in inversionless lasers. Here we analyze a laser model that consists of an ensemble of closed multi-level (three or four levels) atoms driven by a single coherent field. Unlike conventional lasers where population inversion is maintained and kept at the threshold value for arbitrary laser output, the multi-level laser analyzed here can start lasing from population inversion and evolves from population inversion into non-inversion with the increasing atomic pumping. In other words, the system makes transition from light amplification by population inversion into light amplification by coherence. On the other hand, the system can be also made to start lasing without population inversion in any state basis. In both cases, initial inversion and non-inversion, which are controlled by the ratio of spontaneous decay rates, the laser well above threshold is always maintained by lasing without inversion (light amplification by coherence), and the laser intensity fluctuations may drop below the shot noise limit. Two factors contribute to the noise reduction: first, the disappearance of the population inversion leads to the depleted atomic population in the upper lasing state, which reduces the spontaneous emission noises; second, the fast coherent cycling of electrons between excited states leads to the highly regulated absorption and emission processes. We found that the combination of these two mechanisms can lead to the noise reduction in the laser output more than 50% below the shot-noise limit.