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Journal of the Physical Society of Japan

Validity of Single-Atom Approximation in the Many-Atom Microlaser

Kyungwon An

Published Apr 15, 2003• 9 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.72.811

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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.
We report the observation of multiple laser thresholds in the many-atom cavity QED microlaser. Traveling-wave coupling and a supersonic atom beam are used to create a well-defined atom-cavity interaction. Multiple thresholds are observed as jumps in photon number due to oscillatory gain. Although the number of intracavity atoms is large, up to $N\ensuremath{\sim}{10}^{3}$, the dynamics of the microlaser agree with a single-atom theory. This agreement is supported by quantum trajectory simulations of a many-atom microlaser and a semiclassical microlaser theory. We discuss the relation of the microlaser with the micromaser and conventional lasers.
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.
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.
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.
We consider a single-atom laser (microlaser) operating on three-level atoms interacting with a two-mode cavity. The quantum statistical properties of the cavity field at steady state are investigated by the quantum trajectory method which is a Monte Carlo simulation applied to open quantum systems. It is found that a steady-state solution exists even when the detailed balance condition may not be guaranteed. The differences between a single-mode microlaser and a two-mode microlaser are highlighted. The second-order correlation function ${g}^{(2)}(\ensuremath{\tau})$ of a single mode is studied and special attention is paid to the one-photon trapping state, for which a simple formula is derived for its correlation function. We show the effects of the velocity spread of the atoms used to pump the microlaser cavity on the second-order correlation function, trapping states, and phase transitions of the cavity field.
We have demonstrated laser oscillation with one atom in an optical resonator. In our experiment a beam of 138Ba^{138}\mathrm{Ba}138Ba atoms traverses a single-mode cavity with a finesse of 8 \ifmmode\times\else\texttimes\fi{} 105{10}^{5}105. The atoms are excited by a $\ensuremath{\pi}$ pulse from the 1S0^{1}S_{0}1S0​ ground state to the 3P1(m=0)^{3}P_{1}(m=0)3P1​(m=0) excited state before they enter the cavity. Laser oscillation at 791 nm ($^{3}P_{1}\ensuremath{\rightarrow}^{1}S_{0}$) has been observed, with the mean number of atoms inside the mode ranging from 0.1 to 1.0, resulting in the mean number of photons inside the cavity changing from 0.14 to 11. The results are in good agreement with a fully quantized micromaser theory adapted to our experiment.