Light shifts in a pulsed cold-atom coherent-population-trapping clock

Field-grade atomic clocks capable of primary standard performance in compact physics packages would be of significant value in applications ranging from network synchronization to inertial navigation. A coherent-population-trapping clock featuring laser-cooled ${}^{87}\mathrm{Rb}$ atoms and pulsed Ramsey interrogation is a strong candidate for this technology if the frequency biases can be minimized and controlled. Here we characterize the light shift in a cold-atom coherent-population-trapping clock, explaining observed shifts in terms of phase shifts that arise during the formation of dark-state coherences combined with optical-pumping effects caused by unwanted incoherent light in the interrogation spectrum. Measurements are compared with existing and new theoretical treatments, and a laser configuration is identified that would reduce clock frequency uncertainty from light shifts to a fractional frequency level of $\Delta \nu /\nu =4\times {10}^{-14}$ per 100 kHz of laser frequency uncertainty.

Figure 1

(a) Energy levels in the $\mathrm{lin}\parallel \mathrm{lin}$ CPT configuration for ${}^{87}\mathrm{Rb}$ (not to scale). Two independent $\Lambda$ systems are shown (red and blue arrows). Labels in parentheses identify states of the three-level system used in the theory below. The Rabi frequencies of the light connecting $|F=1,2\rangle \to |F=1^{\prime }\rangle$ are ${\Omega }_1$ and ${\Omega }_2$, respectively. (b) A typical single Ramsey sequence. The widths of the magneto-optical trap (MOT), pump, Ramsey, and probe pulses are to scale with typical operating parameters.

Figure 2

Measured clock frequency shift from the ${}^{87}\mathrm{Rb}$ ground-state hyperfine splitting versus the Raman saturation parameter ${\Omega }^{2}S{\tau }_1$ for three values of ${\tau }_1$ and the theoretically predicted shift (line) [20]. The motivation for expressing the intensity in terms of ${\Omega }^{2}S{\tau }_1$ is clear since data collected with three different values of ${\tau }_1$ fall roughly on the same curve. Here, ${\delta }_{\mathrm{opt}}=2\mathrm{MHz},T_R=4\mathrm{ms}$, and $F_0=1$. A Zeeman shift of $1.3\times {10}^{-10}$ was subtracted from the data to account for the magnetic bias field. The theoretical model used here does not include scattering from incoherent light.

Figure 3

Measured clock frequency difference from the ${}^{87}\mathrm{Rb}$ ground-state hyperfine splitting versus ${\Omega }^{2}S{\tau }_1$ for varied , and with ${\delta }_{\mathrm{opt}}=2\mathrm{MHz}$ and ${\tau }_1=400\mu \mathrm{s}$. The theory curves (lines), calculated from Eqs. (4) and (5), incorporate incoherent pumping. A Zeeman shift of $1.3\times {10}^{-10}$ was subtracted from the data to account for the magnetic bias field. Observed light shifts are equal and opposite for negative detunings.

Figure 4

Comparison of coherent light shifts measured for different $I_{\mathrm{CPT}}$ values (points) to the full theoretical model (lines) [20] for $T_R=4\mathrm{ms}$ and ${\tau }_1=400\mu \mathrm{s}$.

Figure 5

(a) Pumping-induced light shift versus $P_{\mathrm{carrier}}$ for $T_R=4\mathrm{ms}.P_{\mathrm{carrier}}$ was adjusted by deoptimizing the PLL settings. Given the fitting error, the linear fit is consistent with zero for 100% carrier power as would be predicted by Eq. (4). (b) Beat note between master and slave lasers with $P_{\mathrm{carrier}}=0.73$. (Resolution bandwidth = 100 kHz.)

Figure 6

Light shifts for three different values of the with $T_R=4\mathrm{ms}$ and $F_0=1$. Data are shown with the laser system in (a) Configuration 1 and (b) Configuration 2 with the corresponding laser configurations shown to the right of the plots.