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Physical Review A

Semiclassical four-level single-atom laser

Kyungwon An, Michael S. Feld

Published Aug 1, 1997• 24 citations
Publisher

Abstract

The recent single-atom microlaser experiment [K. An et al., Phys. Rev. Lett. 73, 3375 (1994)], in which a stream of inverted two-level atoms is injected into an ultrahigh $Q$ cavity, can be understood in the context of a semiclassical four-level laser model. Transit time broadening due to short atom-cavity interaction time effectively introduces nonradiative decay of the two levels in the model. The steady-state solution of semiclassical photon and atom rate equations for the intracavity mean photon number versus the intracavity atom number provides a good fit to the experimental data.

DOI
10.1103/physreva.56.1662

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The recent single-atom microlaser experiment [K. An et al., Phys. Rev. Lett. 73, 3375 (1994)], in which a stream of inverted two-level atoms is injected into an ultrahigh QQQ cavity, can be understood in the context of a semiclassical four-level laser model. Transit time broadening due to short atom-cavity interaction time effectively introduces nonradiative decay of the two levels in the model. The steady-state solution of semiclassical photon and atom rate equations for the intracavity mean photon number versus the intracavity atom number provides a good fit to the experimental data.
The single-atom microlaser is one and the only type of single atom lasers realized experimentally so far. The gain medium is a beam of excited, non-interacting two-level atoms (/spl lambda/=791 nm, /sup 3/P/sup 1//spl rarr//sup 1/S/sub 0/ transition of /sup 138/Ba) entering an ultra-low loss resonator one at a time. This laser device utilizes the coherent atom-cavity interaction, often referred to as quantized Rabi oscillation, as the light amplification method, With the mean atom number about unity, tens of photons could be stored inside the cavity. The laser oscillation persisted with the mean number of atoms in the cavity well below unity.
We have achieved the first realization of a single atom microlaser, a laser oscillator with only one atom in an optical resonator. In the experiment a beam of two-level barium atoms traverses an ultrahigh Q single-mode cavity. The atoms are inverted by a (pi) -pulse field before they enter the cavity. Laser oscillation ((lambda) equals 791 nm) has been observed, with the mean number of atoms inside the mode <N> as small as 0.1, resulting in the mean number of photons in the mode <N> or 0.14. To understand the results quantitatively we used a fully-quantized one-atom microlaser theory adapted from its counterpart micromaser theory. The present theory was found to be in good agreement with the data for small <N> and <n>. Discrepancy between experiment and theory was observed for <n> very much greater than 1 and <N> approximately equal to 1. This discrepancy may be explained by the standing-wave nature of the cavity mode, in combination with the saturation effect occurring at large cavity photon number, as well as by the breakdown of single-atom interaction assumption in the theory for <N> greater than or equal to 1.
The quantum mechanical master equation for a single two-level atom in a single-mode optical cavity is numerically solved in both the quantum and the semiclassical limits. The quantum limit of few cavity photons shows semiclassically forbidden behaviour such as steady state two-level population inversion. Qualitatively new fluorescent spectra, having sidebands broadened by the cavity interaction, also occur. The quantum theory of the single-atom laser with injected signal is presented. At the interface between its quantum and semiclassical dynamics we elucidate the signature of semiclassical limit cycles.
In a recent microlaser experiment [K. An et al., Phys. Rev. Lett. 73, 3375 (1994)], a thresholdlike transition of intracavity mean photon number as a function of intracavity mean atom number has been observed. In this paper the behavior is explored with quantum trajectory simulations. It is shown that the transition is caused by enhanced atom-cavity Rabi interaction due to the increase of the intracavity photon number as the intracavity atom number is increased. The transition is further accentuated by the position-dependent variation of the coupling constant in the Fabry-P\'erot cavity. In addition, it is demonstrated that multiatom collective effects are negligible in the microlaser under consideration, in which atoms are injected into the cavity at random times and the product of the coupling constant and atom-cavity interaction time is much less than \ensuremath{\pi}. In this case the analytic theory of the one-atom micromaser [P. Filipowicz et al., Phys. Rev. A 34, 3077 (1986)] can be extrapolated into the multiatom region, assuming uniform atom-cavity coupling throughout the cavity and monovelocity atomic injection. Finally, simulations are performed which account for spatial variation of coupling constant, velocity distribution of injected atoms, and spontaneous atomic decay in the actual experiment. The results are in good agreement with experiment.
This thesis describes a study of a cavity QED microlaser in which many atoms are present simultaneously and atom-cavity interaction is well-defined. The microlaser is found to display multiple thresholds analogous to first-order phase transitions of the cavity field. Hysteresis is observed as a function of atom-cavity detuning and number of atoms. Data is compared with a rate equation model and fully quantized treatment based on micromaser theory. Good agreement between theory and experiment is found when the cavity is resonant with atoms of the most probable velocity, but long lifetimes of metastable states preclude the observation of true steady-state transition points. For nonzero atom-cavity detuning the microlaser displays broadenings and shifts which are not yet well-understood. Quantum trajectory simulations are performed to investigate many-atom and finite transit time effects in the microlaser. We show that over a wide range of parameters the many-atom microlaser scales with the single-atom theory, with a perturbation in the photon statistics due to cavity decay during the atom transit time.
A criterion for the validity of the single-atom approximation in the many-atom microlaser was proposed and its adequateness has been tested in semiclassical, quantum mechanical and quantum trajectory simulations. Through these studies we conclude that the single-atom approximation, employing the same emission probability formula for individual atoms as that of the single-atom micromaser theory is valid as long as the product of the atom-field coupling constant and the atom-field interaction time is much smaller than the square root of the mean cavity photon number.