Experimental and Theoretical Study of Dynamic Polarizabilities in the $5S_{1/2}$–$5D_{5/2}$ Clock Transition in Rubidium-87 and Determination of Electric Dipole Matrix Elements

The interaction between light and an atom causes perturbations in the atom’s energy levels, known as the light shift. These light shifts are a key source of inaccuracy in atomic clocks, and can also deteriorate their precision. We present a study of light shifts and associated dynamic polarizabilities for a two-photon atomic clock based on the $5S_{1/2}$–$5D_{5/2}$ transition in rubidium-87 over the range 770–800 nm. We determine experimental and theoretical values for a magic wavelength in this range and the electric dipole (E1) matrix element for the $5P_{3/2}$–$5D_{5/2}$ transition. We find a magic wavelength of 776.179(5) nm (experimental) and 776.21 nm (theoretical) in the vicinity of the $5P_{3/2}$–$5D_{5/2}$ resonance, and the corresponding reduced E1 matrix element $1.80(6)e{a}_0$ (experimental) and $1.96(15)e{a}_0$ (theoretical). These values resolve a previous discrepancy between theory and experiment.

Figure 1

Relevant energy levels in rubidium-87, and experimental schematic of the clock modified to measure light shift. AOM: acousto-optic modulator, SHG: second harmonic generator, WDM: wavelength division multiplexer, EDFA: erbium doped fiber amplifier, SFG: sum frequency generator, EOM: electro-optic modulator, PD: photodiode, PBS: polarization beam splitter, QWP: quarter wave plate, PMT: photo multiplier tube, Pol: linear polarizer.

Figure 2

Modeled fluorescence spectrum as a function of 776-nm laser detuning, ${\mathrm{\Delta }}_{776}$. The dashed line is an example of an averaged spectrum over the laser beam from which the value of light shift $\overline{\mathrm{\Delta }\nu }(\omega )$ is extracted. Solid lines are fluorescence spectra at different radial positions within the laser beam of beam waist $w$ ($1/e^2$ intensity diameter).

Figure 3

Measured values for differential polarizability $\mathrm{\Delta }\alpha (\lambda )$ [with $m=1/2$; see Eq. (6)] for the $5S_{1/2}$–$5D_{5/2}$ transition of $\mathrm{Rb}$ (purple markers), overlaid with theoretical prediction (solid blue line). The results of calculations for the individual $5S$ and $5D_{5/2}$ polarizabilities are also shown (dotted and dashed lines, respectively). The inset shows the same, zoomed into the region of the magic wavelength around 776 nm; the purple and blue circles show the experimental and theoretical values for the magic wavelength, respectively.