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Physical Review A

Dynamic noise reduction in multilevel lasers: Nonlinear theory and the pump-operator approach

Hans‐Jürgen Briegel, Georg M. Meyer, Berthold‐Georg Englert

Published Feb 1, 1996• 32 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.53.1143

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arXiv:physics/9911034•Physical Review A•Dec 1, 1998•7 citations
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.
In a multi-level laser the pump process usually involves several intermediate levels as well as spontaneous relaxations and an external pump field. These details of the atomic excitation are often irrelevant and can be summarized by an effective heat-bath description and a two-level- laser model instead. Under certain circumstances, however, the heat-bath approximation will be too crude and it cannot account for phenomena such as intensity-noise squeezing and sub- Poissonian field statistics. We show how in the general situation the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation. In steady state the atomic excitation is then described by a rate matrix rather than a single rate. The matrix or operator nature of the pump process gives rise to a correlation of the excitation rates with the statistical moments of the laser field. This correlation represents a correction to the standard laser model. In particular, it can be identified as the physical origin of the phenomenon of dynamic noise reduction in multi-level lasers.
In a multi-level laser the pump process involves several intermediate levels. The details of the atomic excitation are often irrelevant so that an effective heat bath description and a two-level–laser model are applicable. The heat bath approximation may be too crude, however; it does not predict phenomena such as intensity-noise squeezing and sub-Poissonian field statistics. We show how, in the general situation, the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation where a rate matrix, rather than a single rate, accounts for the excitation process. The resulting dependence of the excitation rates on the state of the laser field constitutes a correction to the standard laser model. We hold it responsible for the dynamic noise reduction in multi-level lasers.
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.
A quantum theory of a two-level single-mode laser with injected atomic coherence is developed by generalizing the Scully-Lamb laser theory to a form appropriate for the analysis of a coherently pumped laser. We assume that the active atoms are prepared initially in a coherent superposition of the upper and lower levels, and we derive the master equation for the field density operator by treating the interaction of the laser field with many active atoms simultaneously. It is shown that the photon-number distribution can be exactly Poissonian. The laser operation is analyzed in terms of the Fokker-Planck equation for the laser field. Both the intensity and phase diffusion coefficients are phase sensitive and, for stable laser operation, become much smaller than those of an ordinary laser. Consequently, the injected atomic coherence reduces both the photon-number noise and phase noise simultaneously. The intensity diffusion coefficient can vanish exactly, and at the same time the phase diffusion coefficient can become very small. This leads to spontaneous-emission noise quenching in the photon-number distribution, and the laser field can become very close to a coherent state. A scheme to generate the proper form of the initial atomic coherence necessary for the quantum noise quenching is proposed and analyzed.
The nonlinear dynamics of dissipative quantum systems in incoherent laser fields is studied in the framework of a master equation with the random telegraph model describing the laser noise and the Markovian approximation dealing with the system-bath couplings. Floquet theory and time-dependent perturbation methods are used to facilitate both analytical and numerical solutions. We develop a theoretical formalism that provides a powerful tool for the detailed analysis of the dissipative quantum dynamics of multilevel systems driven by intense stochastic laser fields. It is found that the system relaxes to a steady state from the effect of the laser phase and frequency noise and the kinetics of this relaxation increases with the addition of dissipative terms, introduced by the coupling to the reservoir. Amplitude fluctuations show a different behavior. Other results concerning the destruction of quantum coherence and the dynamical localization will be established and further relaxation mechanisms such as spontaneous emission and the ionization process will also be considered.