Measuring spatially extended density profiles using atom-cavity collective strong coupling to higher-order modes

The collective strong coupling of rubidium atoms in a magneto-optical trap (MOT) to the Laguerre-Gaussian (LG) modes of a Fabry-Pérot cavity is investigated. Bright and dark ${}^{85}\mathrm{Rb}$ MOT atoms are prepared at the geometric center of the cavity, and the vacuum Rabi splitting (VRS) of the collectively coupled atom-cavity system is measured for ${\mathrm{LG}}_{l0}$ ($l=0,1,2,3$) modes. The atom number coupled to the cavity mode depends on the overlap of the atomic density distribution and the specific spatial mode function, which is reflected in the measured VRS spectrum. The known mode function and the measured VRS can then be used to test whether the atomic density distribution in the experiment is Gaussian or uniform. A simple theoretical model for this process is described, and the experimental measurements are found to be in close agreement with the model.

Figure 1

Schematic of experiment. MML is a mode-matching lens used for coupling light to the FP cavity, F is an optical bandpass filter to block any stray light of frequencies far from ${\omega }_a$, BS is a beam splitter, PMT1 and PMT2 are photomultiplier tubes, and CCD1 and CCD2 are imaging cameras. The cavity resonant frequency ${\omega }_c$ of a particular transverse mode is matched with ${\omega }_a$ by adjusting the cavity length using the annular PZT, and the cavity probe laser frequency ${\omega }_p$ is scanned around ${\omega }_a$. Inset: The contour plot for the density distribution of the ${}^{85}\mathrm{Rb}$ MOT atoms which fluoresce with natural transition frequency ${\omega }_a$ are imaged by CCD1.

Figure 2

The relevant ${}^{85}\mathrm{Rb}$ energy levels and the transitions used in the experiment are illustrated. Here, CL is the cooling laser, RLB and RLD are the repumping lasers for the bright MOT and the dark MOT, respectively. The probe lasers for the bright-MOT and the dark-MOT VRS, PLB, and PLD, are scanned across the identified transitions.

Figure 3

Top: Empty cavity transmission for the ${\mathrm{LG}}_{00}$ cavity mode. Bottom: Transmission through the cavity with cocentered bright-MOT atoms. The solid blue curve shows VRS for the ${\mathrm{LG}}_{00}$ cavity mode, and the dashed pink curve shows VRS for the ${\mathrm{LG}}_{30}$ cavity mode. The VRS signal for the ${\mathrm{LG}}_{00}$ mode has a larger width and exhibits features of slight nonlinearity due to higher peak intensity as a result of a smaller mode cross section.

Figure 4

Theoretical mode profile functions and images of transverse LG modes of the empty cavity, measured by imaging the probe light transmitted through the cavity. The solid red curve is for the ${\mathrm{LG}}_{00}$ mode, the dashed orange curve is for the ${\mathrm{LG}}_{10}$ mode, the dotted green curve is for the ${\mathrm{LG}}_{20}$ mode, and the dot-dashed blue curve is for ${\mathrm{LG}}_{30}$ mode.

Figure 5

Ratio of atoms coupled to the ${\mathrm{LG}}_{l0}$ cavity mode $N_l$ and the total number of atoms in the MOT $N_{at}$ for different sizes of MOTs cocentered with the cavity. The solid red curve is for the ${\mathrm{LG}}_{00}$ mode, the dashed orange curve is for the ${\mathrm{LG}}_{10}$ mode, the dotted green curve is for the ${\mathrm{LG}}_{20}$ mode, and the dot-dashed blue curve is for the ${\mathrm{LG}}_{30}$ mode. The inset shows a zoomed-in version of the same curves around the value of $\sigma /{\omega }_0$ measured from the CCD1 image of the bright MOT, $2.66\pm 0.06$.

Figure 6

VRS for different LG modes due to bright MOT measured on the $F=3$ to $F^{\prime }=4$ transition. Blue circles are experimental values of measured VRS, and the blue error bars are one standard deviation. The red squares are calculated values of VRS, using $N_{at}$ measured by the PMT1 signal and $\sigma$ obtained from the Gaussian fit to the CCD1 image of the MOT, for different cavity modes. The red error bars include estimated errors in calculated VRS due to statistical and least-count errors in measured $N_{at}$ and $\sigma$.

Figure 7

VRS with a dark MOT for different LG modes measured on the $F=2$ to $F^{\prime }=3$ transition. Blue circles are from experimental measurement of VRS, and the blue error bars are one standard deviation on either side of the mean value of corresponding measurements.