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arXiv:quant-ph/0210039 · Physical Review A

Optimal sizes of dielectric microspheres for cavity QED with strong coupling

J. R. Buck, Harry J. Kimble

Published Mar 21, 2003• 241 citations
arXivHTMLPDF

Abstract

The whispering gallery modes (WGMs) of quartz microspheres are investigated for the purpose of strong coupling between single photons and atoms in cavity quantum electrodynamics (cavity QED). Within our current understanding of the loss mechanisms of the WGMs, the saturation photon number ${n}_{0}$ and critical atom number ${N}_{0}$ cannot be minimized simultaneously, so that an ``optimal'' sphere size is taken to be the radius for which the geometric mean $\sqrt{{n}_{0}{N}_{0}},$ is minimized. While a general treatment is given for the dimensionless parameters used to characterize the atom-cavity system, detailed consideration is given to the ${D}_{2}$ transition in atomic cesium at ${\ensuremath{\lambda}}_{0}=852\mathrm{nm}$ using fused-silica microspheres, for which the maximum coupling coefficient ${g}_{a}/(2\ensuremath{\pi})\ensuremath{\approx}750\mathrm{MHz}$ occurs for a sphere radius $a=3.63\ensuremath{\mu}\mathrm{m}$ corresponding to the minimum for ${n}_{0}\ensuremath{\approx}6.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.$ By contrast, the minimum for ${N}_{0}\ensuremath{\approx}9.00\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ occurs for a sphere radius of $a=8.12\ensuremath{\mu}\mathrm{m},$ while the optimal sphere size for which $\sqrt{{n}_{0}{N}_{0}}$ is minimized occurs at $a=7.83\ensuremath{\mu}\mathrm{m}.$ On an experimental front, we have fabricated fused-silica microspheres with radii $a\ensuremath{\sim}10\ensuremath{\mu}\mathrm{m}$ and consistently observed quality factors $Q>~0.8\ifmmode\times\else\texttimes\fi{}{10}^{7}.$ These results for the WGMs are compared with corresponding parameters achieved in Fabry-Perot cavities to demonstrate the significant potential of microspheres as a tool for cavity QED with strong coupling.

DOI
10.1103/physreva.67.033806

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The whispering gallery modes (WGMs) of quartz microspheres are investigated for the purpose of strong coupling between single photons and atoms in cavity quantum electrodynamics (cavity QED). Within our current understanding of the loss mechanisms of the WGMs, the saturation photon number n0{n}_{0}n0​ and critical atom number N0{N}_{0}N0​ cannot be minimized simultaneously, so that an ``optimal'' sphere size is taken to be the radius for which the geometric mean n0N0,\sqrt{{n}_{0}{N}_{0}},n0​N0​​, is minimized. While a general treatment is given for the dimensionless parameters used to characterize the atom-cavity system, detailed consideration is given to the D2{D}_{2}D2​ transition in atomic cesium at \ensuremathλ0=852nm{\ensuremath{\lambda}}_{0}=852\mathrm{nm}\ensuremathλ0​=852nm using fused-silica microspheres, for which the maximum coupling coefficient ga/(2\ensuremathπ)\ensuremath≈750MHz{g}_{a}/(2\ensuremath{\pi})\ensuremath{\approx}750\mathrm{MHz}ga​/(2\ensuremathπ)\ensuremath≈750MHz occurs for a sphere radius a=3.63\ensuremathμma=3.63\ensuremath{\mu}\mathrm{m}a=3.63\ensuremathμm corresponding to the minimum for n0\ensuremath≈6.06\ifmmode×\else\texttimes\fi10\ensuremath−6.{n}_{0}\ensuremath{\approx}6.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.n0​\ensuremath≈6.06\ifmmode×\else\texttimes\fi10\ensuremath−6. By contrast, the minimum for N0\ensuremath≈9.00\ifmmode×\else\texttimes\fi10\ensuremath−6{N}_{0}\ensuremath{\approx}9.00\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}N0​\ensuremath≈9.00\ifmmode×\else\texttimes\fi10\ensuremath−6 occurs for a sphere radius of a=8.12\ensuremathμm,a=8.12\ensuremath{\mu}\mathrm{m},a=8.12\ensuremathμm, while the optimal sphere size for which n0N0\sqrt{{n}_{0}{N}_{0}}n0​N0​​ is minimized occurs at a=7.83\ensuremathμm.a=7.83\ensuremath{\mu}\mathrm{m}.a=7.83\ensuremathμm. On an experimental front, we have fabricated fused-silica microspheres with radii a\ensuremath∼10\ensuremathμma\ensuremath{\sim}10\ensuremath{\mu}\mathrm{m}a\ensuremath∼10\ensuremathμm and consistently observed quality factors Q> 0.8\ifmmode×\else\texttimes\fi107.Q>~0.8\ifmmode\times\else\texttimes\fi{}{10}^{7}.Q> 0.8\ifmmode×\else\texttimes\fi107. These results for the WGMs are compared with corresponding parameters achieved in Fabry-Perot cavities to demonstrate the significant potential of microspheres as a tool for cavity QED with strong coupling.
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We report measurements of cavity-QED effects for the radiative coupling of atoms in a dilute vapor to the external evanescent field of a whispering-gallery mode (WGM) in a fused silica microsphere. The high QQQ $(5\ifmmode\times\else\texttimes\fi{}{10}^{7})$, small mode volume (10${}^{\ensuremath{-}8}$ cm3){}^{3})3), and unusual symmetry of the microcavity evanescent field enable velocity-selective interactions between fields with photon number of order unity in the WGM and ${N}_{T}\ensuremath{\sim}1$ atoms in the surrounding vapor.
Summary form only given, as follows: The high Q whispering gallery modes of silica microspheres are promising for quantum optics and nonlinear optics experiments involving atoms and photons stored in very small volumes. We have developed the technology of these resonators, learning how to couple light in and out through optical fibers, how to obtain and measure record Q factors, and how to probe, by near-field optical microscopy, the electric field distribution immediately outside the sphere. Whispering gallery mode lasers have also been obtained with microspheres made of silica doped with neodymium ions. We describe these experiments and discuss projects involving the coupling of whispering gallery modes to single atoms or ions to perform cavity QED experiments in the optical domain.
Remarkable demonstrations of a cavity-induced giant reduction of the Raman lasing threshold in silica microspheres have been reported in Spillane et al. [Nature (London) 415, 621 (2002)], yet a complete quantum electrodynamics (QED) treatment of this process including the rigorous expressions for the electromagnetic modes of the sphere is lacking. In this paper, we employ a cavity QED approach to describe Raman lasing in microscopic spherical dielectric resonators with third-order nonlinearity to derive explicit expressions for the Raman gain and lasing threshold in terms of mode-overlapping coefficients determined from full analytic expressions of the electromagnetic eigenmodes of a dielectric sphere. We present dependencies of the Raman lasing threshold on the the order nnn of overlapping whispering gallery modes for microspheres with diameter in the range 20--35 $\ensuremath{\mu}$m, demonstrating ultralow thresholds of between 70 and 90 $\ensuremath{\mu}$W, consistent with recent experimental results. We explain the reduction of the lasing threshold as due to an increase in the overlapping coefficients as the mode order is increased toward the regime where the modal energy is mostly confined to the surface region of the resonator. The presented theory can be easily generalized to any microcavity of regular morphology and for any combination of interacting whispering gallery modes.
Silica microspheres have been demonstrated to provide a permanent resonator which combines small mode volumes with the highest observed quality factors (Q>2.10/sup 9/) corresponding to photon storage times of the order of one microsecond. These properties have allowed us to make a very low-threshold laser with Nd-doped silica microspheres (200 nW absorbed pump power, i.e. 10/sup 6/ excited ions/mode). These microlasers are now operated at 2 K to benefit from a smaller homogeneous linewidth. We then expect to reach the "thresholdless" regime with only a few excited ions. These experiments can also be considered as a first step toward Cavity QED experiments in the strong coupling regime with single atoms or molecules coupled to a whispering gallery mode.
We have tuned the whispering-gallery modes of a fused-silica microresonator over nearly 1 nm at 800 nm, i.e., over half a free spectral range, or 10(6) linewidths of the resonator. This result has been achieved by use of a new method based on the stretching of a two-stem microsphere. We describe devices that will permit new cavity QED experiments with this high- Q optical resonator when it is desirable to optimize its coupling to emitters with given transition frequencies. The demonstrated tuning capability is compatible with both UHV and low-temperature operation, which should be useful for future experiments with laser-cooled atoms or single quantum dots.