High Cooperativity Using a Confocal-Cavity–QED Microscope

Cavity quantum electrodynamics (QED) with cooperativity far greater than unity enables high-fidelity quantum sensing and information processing. The high-cooperativity regime is often reached through the use of short single-mode resonators. More complicated multimode resonators, such as the near-confocal optical Fabry-Prot cavity, can provide intracavity atomic imaging in addition to high cooperativity. This capability has recently proved important for exploring quantum many-body physics in the driven-dissipative setting. In this work, we show that a confocal-cavity–QED microscope can realize cooperativity in excess of 110. This cooperativity is on par with the very best single-mode cavities (which are far shorter) and 21 times greater than single-mode resonators of similar length and mirror radii. The 1.7-$\text{μ}\mathrm{m}$ imaging resolution is naturally identical to the photon-mediated interaction range. We measure these quantities by determining the threshold of cavity superradiance when small optically tweezed Bose-Einstein condensates are pumped at various intracavity locations. Transmission measurements of an ex situ cavity corroborate these results. We provide a theoretical description that shows how cooperativity enhancement arises from the dispersive coupling to the atoms of many near-degenerate modes.

Popular Summary

Optical-cavity quantum electrodynamics (QED) provides the means to generate strong coupling through the interaction of the electric dipole of an atom with the electromagnetic field confined as a cavity mode. The larger the field, the stronger is the coupling. Single-mode resonators are typically employed to enhance the field at the atom. However, multimode degenerate cavities are of increasing interest. They accommodate many modes at (nearly) the same resonant frequency and thereby enable light-matter coupling involving the participation of spatially distinct modes.

We demonstrate the capabilities of a particular type of multimode resonator called a confocal cavity. We show that it simultaneously allows the strong interaction of light and matter as well as the detection of that matter through the high-resolution imaging of the light. The relative strength of light-matter interactions is captured by the cooperativity $C$. This figure of merit compares the single-atom single-photon interaction strength to dissipative rates. We measure one of the highest cooperativities ever achieved in an optical cavity, $C=112$. Moreover, we show that the system provides images of the atoms trapped within at the micron scale, which is naturally matched to the length scale of photon-mediated interactions among these atoms.

The large cooperativity of this confocal-cavity–QED microscope allows strong light-matter interactions to occur before decoherence from dissipation sets in. This is a key ingredient in many quantum information and sensing platforms and may render observable exotic nonequilibrium many-body phenomena. The new confocal-cavity–QED system benchmarked here opens new directions in quantum simulation and device physics.

Figure 1

(a) An illustration of the experiment. An ${}^{87}$$\mathrm{Rb}$ Bose-Einstein condensate (BEC) (orange) is positioned using an optical-tweezer dipole trap (blue) inside a confocal cavity. It is pumped by a transverse running-wave laser (red), which induces a superradiant self-organization phase transition above a critical pump power. This scatters light into a synthetic mode—i.e., a near-degenerate mode superposition—of the cavity (green), which can be imaged onto a detector (gray box). The single synthetic mode cavity field appears as two green beams or spots because the crossed “nonlocal” portion connecting them is far less intense than the straight “local” portions shown in the figure [17]. The inset image shows an example of the synthetic mode emission; the two spots correspond to the image and mirror components of the local field and demonstrate the ability of the cavity to image the BEC. The waist of the fundamental mode of the cavity is $w_0=35\text{μ}\mathrm{m}$. (b) The transmission spectrum of the employed confocal cavity taken by using a probe beam focused at the cavity center with a waist of $25\text{μ}\mathrm{m}$ (at the cavity midplane). The red line indicates a typical transverse pump detuning from a reference point within the mode spread ${\omega }_B$, here chosen to be the first local maximum in the transmission. (c) An example of the cavity emission (solid blue) detected by a single-photon counter versus time. The pump power is shown as a dashed black line. The threshold power is indicated by the red dotted line. Superradiant emission is observed once the threshold is reached and persists for $\lesssim 1\mathrm{ms}$, limited by the excitation of atomic motion by the running-wave pump.

Figure 2

Threshold measurements. (a) The normalized threshold pump strength $\sqrt{N}{\mathrm{\Omega }}_c$ for a BEC of three different shapes, each located at the cavity center, versus $-{\mathrm{\Delta }}_C/2\pi$. The pump strength is normalized by the typical BEC number, $N_0=3\times {10}^5$, and is plotted in units of $2\pi \sqrt{N_0}\mathrm{MHz}$. The BEC radii are listed as $(R_x,R_y,R_z)$. The required pump power increases with the pump-cavity detuning ${\mathrm{\Delta }}_C$ because the photon-mediated interaction becomes weaker. However, the increase is less than in a single-mode cavity due to the participation of high-order modes. The single-mode ${\mathrm{\Delta }}_C^{1/2}$ scaling is shown as a black dashed line. The error bars represent the standard error for five shots per point. (b) The normalized threshold strength as a function of the BEC position. Data are taken at ${\mathrm{\Delta }}_C/2\pi =-120\mathrm{MHz}$, with $N=0.76N_0$. The dip at the cavity center is due to the interaction enhancement from the overlap of the local and mirror fields [9]. The minimum width of this feature is set by the size of the BEC in $\hat{x}$; the BEC probe used here has dimensions $[R_x,R_y,R_z]=[3.1,7.6,5.3]\text{μ}\mathrm{m}$, as indicated by the orange dashed line. The fact that the width of the dip exceeds that of the BEC indicates that the synthetic mode has a finite waist. The blue star in each panel corresponds to data taken at the same position, detuning, and trap frequencies.

Figure 3

Cooperativity enhancement and the microscope resolution or interaction range. (a) The data in Fig. 2 presented in terms of multimode cooperativity enhancement. The enhancement increases with detuning and for narrower BECs. All solid lines are derived from the parameters $ϵ$ and $\alpha$ obtained from a single global fit to all measurements via the threshold expression given in Eq. (7). The black curve indicates the enhancement that a point particle would experience at the cavity center. The gray band denotes uncertainty in $ϵ$ and $\alpha$. The enhancement would reach 96(49) at a detuning of 300 MHz. The maximum directly measured enhancement is 21.5(3). The error bars represent the standard error for five shots per point. (b) The resolution of the multimode cavity, at the cavity center, measured by the HWHM of the synthetic mode profile; see Fig. 2 for an example of a profile (convolved with the BEC probe width). This resolution is equivalent to the photon-mediated interaction range [9]. The data points are derived from single scans of the BEC versus position, such as that displayed in Fig. 2 (see text for explanation). The microscope reaches a minimum measured resolution of $1.7(2)\text{μ}\mathrm{m}$. As in (a), the solid black line and its error band are derived from the global fit to all measurements. The blue triangle corresponds to the data presented in Fig. 2.

Figure 4

The all-optical characterization of confocal-cavity resolution. (a) The estimated widths of the cavity Green’s function along the major and minor astigmatic axes versus the cavity detuning $-{\mathrm{\Delta }}_C/2\pi$. (b) An example of a typical Gaussian-like intensity profile of the cavity transmission projected onto the minor axis when the probe beam is detuned by approximately $155\mathrm{MHz}$. The horizontal axis is the distance in the object plane. The inset shows the full image, with the minor axis indicated by the red dashed line. Together, these clearly demonstrate the ability to image micron-sized features via the cavity.