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Physical Review A

Pump-operator treatment of the ion-trap laser

Georg M. Meyer, Hans‐Jürgen Briegel

Published Oct 1, 1998• 25 citations
Publisher

Abstract

We present a detailed treatment of a laser process that has been predicted for an experiment with a single trapped ${\mathrm{Ca}}^{+}$ ion. It is demonstrated how the complex multilevel system can be understood as a coherently pumped three-level scheme with time-delayed pumping out of the metastable lower laser level. This is described by a pump operator, which simplifies the numerical analysis substantially and allows for a treatment of more complicated systems.

DOI
10.1103/physreva.58.3210

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We present a detailed treatment of a laser process that has been predicted for an experiment with a single trapped Ca+{\mathrm{Ca}}^{+}Ca+ ion. It is demonstrated how the complex multilevel system can be understood as a coherently pumped three-level scheme with time-delayed pumping out of the metastable lower laser level. This is described by a pump operator, which simplifies the numerical analysis substantially and allows for a treatment of more complicated systems.
Laser operation with a single trapped Ca + ion in a high-finesse optical cavity is predicted at a wavelength of 866 nm. Different from a conventional laser, the formation of two thresholds can be observed. For both small and large pump strengths the field is quasi-thermal, in an intermediate region laser light with Poissonian and even sub-Poissonian statistics is produced. Atomic coherence effects play an important role. The multi-level system is treated by generalizing the pump-operator approach ( Europhys. Lett. , 33 (1996) 515).
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 present theoretical and experimental results of coherent population trapping spectra on a multilevel $\mathrm{\ensuremath{\Lambda}}$-type configuration, adding a third beam to the standard two-laser $\mathrm{\ensuremath{\Lambda}}$ system to avoid undesired optical pumping. We show that the extra laser can preserve the nature of the dark resonances or introduce decoherence depending on its power. Experiments are carried out using a single trapped 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ ion in the ${S}_{1/2}\text{\ensuremath{-}}{P}_{1/2}\text{\ensuremath{-}}{D}_{3/2}$ manifold. Theoretically, the problem is solved with a Floquet-type expansion of the Liouvillian that correctly predicts all of the measured spectra without the need of full time integration. As a first application of the multilaser technique, we show that the richer spectra obtained can be used as a vectorial polarimeter of one of the beams, allowing one to measure the electrical field at the ion's position in any spatial direction. We also explain how our setup could realize a thermometer with tunable sensitivity and no laser-linewidth dependence.
Laser cooling of trapped multilevel ions is studied numerically. The ions are assumed to be localized to spatial dimensions smaller than the optical wavelength (Lamb-Dicke regime). A master equation for the center-of-mass motion is used to numerically evaluate cooling rates and final temperatures for arbitrary light field configurations. The results show both well-known cooling mechanisms (Doppler cooling, sideband cooling) and effects introduced by the presence of multiple atomic levels. Quantitative results are given for a trapped Ba+{\mathrm{Ba}}^{+}Ba+ ion. The numerical procedure can easily be adapted for all ions used in today's trapping experiments. \textcopyright{} 1996 The American Physical Society.
In Chapter 1, we introduced the standard rate equations (SRE) for a laser containing two-level atoms between which stimulated emission is possible. But real lasers exhibit much more complicated energy level schemes. Throughout this chapter, we consider several models in which the light–matter interaction is described by population equations for all the levels involved in the laser operation. Although they are mathematically more complicated, we shall investigate these equations in the same way as in Chapter 1, i.e. by formulating dimensionless equations and by analyzing the stability properties of the steady states. The basic ingredients of the SRE model used up to here, namely pumping and relaxation processes, play key roles in the efficiency of lasers. But already during the pioneering days of the laser, it appeared important to investigate three- and four-level models in order to obtain more reliable information on quantities such as the power conversion efficiency or the response time. The common extensions of the two-level SRE typically consider an open two-level system, or three- or four-level systems, depending on the nature of the active medium. Restriction to as few as four levels is again a crude simplification of the complex population dynamics occurring in most lasers. But it is rather surprising to see how good these simple kinetic models are. As we demonstrate by studying specific examples, the solution of the three- or four-level rate equations differs only slightly from the solution of the SRE.
A numerical analysis is presented of several potential pulsed-laser applications of adiabatic population inversion in atoms or molecules. A simple numerical method is demonstrated for evaluating adiabatic processes in multilevel systems for both swept-frequency and resonant, time-delayed pulses. Adiabatic inversion with time-delayed pulses is generalized to systems with any number of levels. The effect of Doppler broadening on time-delayed adiabatic pumping of a three-level system is investigated.