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arXiv:quant-ph/0309133 · Physical Review A

Comparison of theory and experiment for a one-atom laser in a regime of strong coupling

A. David. Boozer, Andreea Boca, J. R. Buck, J. McKeever, Harry J. Kimble

Published Aug 25, 2004• 38 citations
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Abstract

Our recent paper reports the experimental realization of a one-atom laser in a regime of strong coupling [J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, Nature (London) 425, 268 (2003)]. Here we provide the supporting theoretical analysis relevant to the operating regime of our experiment. By way of a simplified four-state model, we investigate the passage from the domain of conventional laser theory into the regime of strong coupling for a single intracavity atom pumped by coherent external fields. The four-state model is also employed to exhibit the vacuum-Rabi splitting and to calculate the optical spectrum. We next extend this model to incorporate the relevant Zeeman hyperfine states as well as a simple description of the pumping processes in the presence of polarization gradients and atomic motion. This extended model is employed to make quantitative comparisons with our earlier measurements for the intracavity photon number versus pump strength and for the photon statistics as expressed by the intensity correlation function ${g}^{(2)}(\ensuremath{\tau})$.

DOI
10.1103/physreva.70.023814

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Our recent paper reports the experimental realization of a one-atom laser in a regime of strong coupling [J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, Nature (London) 425, 268 (2003)]. Here we provide the supporting theoretical analysis relevant to the operating regime of our experiment. By way of a simplified four-state model, we investigate the passage from the domain of conventional laser theory into the regime of strong coupling for a single intracavity atom pumped by coherent external fields. The four-state model is also employed to exhibit the vacuum-Rabi splitting and to calculate the optical spectrum. We next extend this model to incorporate the relevant Zeeman hyperfine states as well as a simple description of the pumping processes in the presence of polarization gradients and atomic motion. This extended model is employed to make quantitative comparisons with our earlier measurements for the intracavity photon number versus pump strength and for the photon statistics as expressed by the intensity correlation function g(2)(\ensuremathτ){g}^{(2)}(\ensuremath{\tau})g(2)(\ensuremathτ).
Although conventional lasers operate with a large number of intracavity atoms, the lasing properties of a single atom in a resonant cavity have been theoretically investigated for more than a decade. Here we report the experimental realization of such a one-atom laser operated in a regime of strong coupling. Our experiment exploits recent advances in cavity quantum electrodynamics that allow one atom to be isolated in an optical cavity in a regime for which one photon is sufficient to saturate the atomic transition. In this regime the observed characteristics of the atom-cavity system are qualitatively different from those of the familiar many atom case. Specifically, we present measurements of intracavity photon number versus pump intensity that exhibit "thresholdless" behavior, and infer that the output flux from the cavity mode exceeds that from atomic fluorescence by more than tenfold. Observations of the second-order intensity correlation function demonstrate that our one-atom laser generates manifestly nonclassical light that exhibits both photon antibunching and sub-Poissonian photon statistics.
We consider a laser composed of a single atom in a microcavity, with a coherent or incoherent pump. We consider both three- and four-level gain schemes, and examine the output spectrum of such lasers. We find that the linewidth generally scales as the inverse of the photon number. For large atom-field coupling, a vacuum-Rabi doublet structure is obtained. In the three-level case, this vacuum-Rabi splitting is apparent only for small intracavity photon numbers, and vanishes for large pumps. In the four-level scheme, the vacuum-Rabi structure appears at a nonzero pump level, and is maintained for large pumps, even when the intracavity photon number is larger than unity. This behavior is explained utilizing the quantum trajectory approach.
We investigate the transient and the steady-state behaviour of the photon statistics of a single-atom laser, consisting of a three-level Λ system, driven by an incoherent external pump field and strongly coupled to a single mode of an optical cavity. For several limiting cases we are able to obtain analytical results. A comparison with standard multi-atom laser theory reveals a surprisingly good agreement down to very small intracavity photon numbers. Under appropriate operating conditions a calculation of the second-order intensity correlation function yields sub-Poissonian statistics as well as strong antibunching induced by the nonlinear dynamics of the atom.
The experiments discussed in this thesis focus on the interaction of a single trapped atom with the single mode of a high-finesse optical cavity, in the regime of strong coupling. Chapter 1 gives a brief introduction, after which Chapter 2 describes our recent measurements of the transmission spectrum of the atom-cavity system. The spectrum exhibits a clearly resolved vacuum-Rabi splitting, in good quantitative agreement with theoretical predictions. A new Raman scheme for cooling atomic motion along the cavity axis enables a complete spectrum to be recorded for an individual atom trapped within the cavity mode, in contrast to all previous measurements of this type that have required averaging over 10^3-10^5 atoms. Chapter 3 discusses our observations of photon blockade for the transmitted light in the presence of one trapped atom. Excitation of the atom-cavity system by a first photon blocks the transmission of a second one, thereby converting an incident Poissonian stream of photons into a sub-Poissonian, anti-bunched stream, as confirmed by measurements of the photon statistics of the transmitted field. The intensity correlations of the cavity transmission also reveal the energy distribution for oscillatory motion of the trapped atom. Chapter 4 details a set of simple but necessary measurements of relevant experimental parameters such as cavity geometry, linewidth, mirror properties, birefringence, and detection efficiency. The thesis concludes with Appendix A, describing the efficient laser setup we use for our magneto-optical traps.
The limits of the well-established single-atom model for describing photon-pair generation via four-wave mixing in a diamond configuration in atomic ensembles are experimentally tested. Using a cold-atom source, biphotons are generated and detected through polarization analyzers that resolve the emitted light into horizontal and vertical components in the laboratory frame. The pump lasers driving the first and second excitation transitions are horizontally and vertically polarized, respectively. To predict single-photon counts and coincidence rates in all polarization channels, the first- and second-order correlation functions are calculated directly using a comprehensive model that accounts for all Zeeman sublevels of the relevant hyperfine states. Furthermore, the population dynamics with and without the re-pump laser of the magneto-optical trap are compared, revealing substantial differences in the populations of the Zeeman sublevels. This motivates the inclusion of two additional hyperfine levels and their corresponding Zeeman sublevels in the final model. The resulting density matrix is used to calculate expectation values of the far-field electric-field operators and compare them with experimental measurements over a range of pump-laser powers. Excellent agreement with the measured photon counts is obtained over most of this range. For the coincidence measurements, good agreement is found when the polarization of the photon generated by the first (second) decay is parallel to that of the first (second) pump beam. In contrast, the model consistently underestimates the experimental coincidence rates for the opposite polarization configuration. These findings indicate that effects beyond the internal level dynamics of individual atoms, most notably collective phenomena, are required to fully account for photon coincidences generated by four-wave mixing in atomic ensembles.
We calculate the output spectrum of a single-atom laser in a microcavity across a wide range of operating conditions. We considered both three-level and four-level atomic level structures. We used a numerical routine to calculate spectra that is more efficient than others used previously. We found that the linewidth of a single-atom laser generally scales as the inverse of the photon number and that there is no pump value at which an abrupt change occurs that might locate a lasing threshold. For a three-level gain atom we found vacuum–Rabi splitting similar to that found by Loffler et al. [Phys. Rev. A 55, 3923 (1997)] and used quantum trajectory theory to obtain a new interpretation of the results. For a four-level gain atom the vacuum–Rabi structure can appear at a small nonzero pump level and is maintained for large pumps, even when the intracavity photon number is larger than unity and the laser is on. We use the quantum trajectory approach to explain these results.