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arXiv:quant-ph/0410218 · Physical Review A

Ultrahigh- Q toroidal microresonators for cavity quantum electrodynamics

S. M. Spillane, Tobias Jan Kippenberg, Kerry J. Vahala, Kok Win Goh, E. Wilcut, Harry J. Kimble

Published Jan 26, 2005• 569 citations
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Abstract

We investigate the suitability of toroidal microcavities for strong-coupling cavity quantum electrodynamics (QED). Numerical modeling of the optical modes demonstrate a significant reduction of the modal volume with respect to the whispering gallery modes of dielectric spheres, while retaining the high-quality factors representative of spherical cavities. The extra degree of freedom of toroid microcavities can be used to achieve improved cavity QED characteristics. Numerical results for atom-cavity coupling strength $g$, critical atom number ${N}_{0}$, and critical photon number ${n}_{0}$ for cesium are calculated and shown to exceed values currently possible using Fabry-Perot cavities. Modeling predicts coupling rates $g∕2\ensuremath{\pi}$ exceeding $700\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$ and critical atom numbers approaching ${10}^{\ensuremath{-}7}$ in optimized structures. Furthermore, preliminary experimental measurements of toroidal cavities at a wavelength of $852\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ indicate that quality factors in excess of ${10}^{8}$ can be obtained in a $50\text{\penalty1000-\hskip0pt}\ensuremath{\mu}\mathrm{m}$ principal diameter cavity, which would result in strong-coupling values of $\mathbf{(}g∕(2\ensuremath{\pi}),{n}_{0},{N}_{0}\mathbf{)}=(86\phantom{\rule{0.3em}{0ex}}\mathrm{MHz},4.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4},1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3})$.

DOI
10.1103/physreva.71.013817

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We investigate the suitability of toroidal microcavities for strong-coupling cavity quantum electrodynamics (QED). Numerical modeling of the optical modes demonstrate a significant reduction of the modal volume with respect to the whispering gallery modes of dielectric spheres, while retaining the high-quality factors representative of spherical cavities. The extra degree of freedom of toroid microcavities can be used to achieve improved cavity QED characteristics. Numerical results for atom-cavity coupling strength ggg, critical atom number N0{N}_{0}N0​, and critical photon number n0{n}_{0}n0​ for cesium are calculated and shown to exceed values currently possible using Fabry-Perot cavities. Modeling predicts coupling rates g∕2\ensuremathπg∕2\ensuremath{\pi}g∕2\ensuremathπ exceeding 700MHz700\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}700MHz and critical atom numbers approaching 10\ensuremath−7{10}^{\ensuremath{-}7}10\ensuremath−7 in optimized structures. Furthermore, preliminary experimental measurements of toroidal cavities at a wavelength of 852nm852\phantom{\rule{0.3em}{0ex}}\mathrm{nm}852nm indicate that quality factors in excess of 108{10}^{8}108 can be obtained in a 50\penalty1000-\ensuremathμm50\text{\penalty1000-\hskip0pt}\ensuremath{\mu}\mathrm{m}50\penalty1000-\ensuremathμm principal diameter cavity, which would result in strong-coupling values of (g∕(2\ensuremathπ),n0,N0)=(86MHz,4.6\ifmmode×\else\texttimes\fi10\ensuremath−4,1.0\ifmmode×\else\texttimes\fi10\ensuremath−3)\mathbf{(}g∕(2\ensuremath{\pi}),{n}_{0},{N}_{0}\mathbf{)}=(86\phantom{\rule{0.3em}{0ex}}\mathrm{MHz},4.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4},1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3})(g∕(2\ensuremathπ),n0​,N0​)=(86MHz,4.6\ifmmode×\else\texttimes\fi10\ensuremath−4,1.0\ifmmode×\else\texttimes\fi10\ensuremath−3).
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 ${\ensuremath{\lambda}}_{0}=852\mathrm{nm}$ 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 n0N0\sqrt{{n}_{0}{N}_{0}}n0​N0​​ 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.
We report the demonstration of strong coupling between single Cesium atoms and a high-Q chip-based microresonator. Our toroidal microresonators are compact, Si chip-based whispering gallery mode resonators that confine light to small volumes with extremely low losses, and are manufactured in large numbers by standard lithographic techniques. Combined with the capability to couple efficiently light to and from these microresonators by a tapered optical fiber, toroidal microresonators offer a promising avenue towards scalable quantum networks. Experimentally, laser cooled Cs atoms are dropped onto a toroidal microresonator while a probe beam is critically coupled to the cavity mode. When an atom interacts with the cavity, it modifies the resonance spectrum of the cavity, leading to rejection of some of the probe light from the cavity, and thus to an increase in the output power. By observing such transit events while systematically detuning the cavity from the atomic resonance, we determine the maximal accessible single-photon Rabi frequency of Ω0/2π ≈ (100 ± 24) MHz. This value puts our system in the regime of strong coupling, being significantly larger than the dissipation rates in our system.
Of the many applications of whispering-gallery mode (WGM) microresonators, Single-atom cavity-QED poses the most extreme demands on mode-volume and quality factor. Here we present a model-based procedure for the fabrication of small mode-volume ultrahigh Q silica WGM microresonators of varying geometries, from toroidal to micro-spheres. We experimentally demonstrate WGM resonators with ultra-high qualities as high as 1.7×1081.7\times 10^81.7×108 at 780nm. We present a theoretical model that allows tailoring the recipe to attain the desired geometry of the fabricated WGM microresonators.
Of the many applications of whispering-gallery-mode (WGM) micro-resonators, single-atom cavity quantum electrodynamics (cavity-QED) poses the most extreme demands on mode volume, dimensions, and quality factor ( Q ). Here we present the procedure for the fabrication of chip-based, small mode-volume, ultrahigh- Q silica WGM micro-resonators, designed for single-emitter cavity-QED. We demonstrate micro-resonators at varying geometries, from toroidal to micro-spheres, yielding ultrahigh-qualities as high as 1.7 × 10 8 at 780nm. We present a comprehensive theoretical model that allows tailoring the fabrication process to attain the desired micro-resonator geometry.
We theoretically study the properties of highly prolate-shaped dielectric microresonators. Such resonators sustain whispering-gallery modes that exhibit two spatially well-separated regions with enhanced field strength. The field per photon on the resonator surface is significantly higher than, e.g., for equatorial whispering-gallery modes in microsphere resonators with a comparable mode volume. At the same time, the frequency spacing of these modes is much more favorable, so that a tuning range of several free spectral ranges should be attainable. We discuss the possible application of such resonators for cavity quantum electrodynamics experiments with neutral atoms and reveal distinct advantages with respect to existing concepts.
Open Fabry-Perot microcavities represent a promising route for achieving a quantum electrodynamics (cavity-QED) platform with diamond-based emitters. In particular, they offer the opportunity to introduce high-purity, minimally fabricated material into a tunable, high quality factor optical resonator. Here, we demonstrate a fiber-based microcavity incorporating a thick ($>10\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\text{m}$) diamond membrane with a finesse of 17 000, corresponding to a quality factor $Q\ensuremath{\sim}{10}^{6}$. Such minimally fabricated thick samples can contain optically stable emitters similar to those found in bulk diamond. We observe modified microcavity spectra in the presence of the membrane, and we develop analytic and numerical models to describe the effect of the membrane on cavity modes, including loss and coupling to higher-order transverse modes. We estimate that a Purcell enhancement of approximately 20 should be possible for emitters within the diamond in this device, and we provide evidence that better diamond surface treatments and mirror coatings could increase this value to 200 in a realistic system.