Doppler Shift Mitigation in a Chip-Scale Atomic Beam Clock

TL;DR

This work addresses laser-frequency sensitivity in a chip-scale atomic beam clock (CSABC) based on Ramsey-CPT in a thermal $^{87}$Rb beam. The authors reveal a competition between Doppler shifts and resonant light shifts stemming from asymmetric CPT decay, and demonstrate an operational point where the clock’s detuning sensitivity $ abla \\Delta/\\nabla \\delta_L$ is effectively nulled by tuning the excited-state pathways and optical pumping. They identify a zero-crossing for $F'=2$ near $\\Omega_2^2/\\Omega_1^2 \approx 1.1$ and $P_{\rm res} \approx 113\ \mu$W, achieving $\\chi \approx 0$ and enabling long-term stability with clock performance reaching the $10^{-11}$–$10^{-12}$ level over 1000 s. This Doppler-cancellation scheme, together with a compact chip-scale design, points toward week-long timing holdover with low power and is a step toward robust, low-drift chip-scale clocks.

Abstract

Chip-scale microwave atomic systems based on thermal atomic beams offer a promising approach to realize low-power and low-drift clocks for timing holdover applications. Miniature beam clocks are expected to suppress many of the shifts that commonly limit existing chip-scale atomic clocks based on coherent population trapping, including collisional shifts and some light shifts. However, the beam geometry can amplify some challenges such as Doppler shifts, which generate a strong sensitivity to laser frequency variation. Using a cm-scale 87Rb atom beam clock, we identify a surprisingly strong competition between Doppler shifts and resonant light shifts arising from asymmetric decay in the clock spectroscopy Λ-system. Leveraging this competition between Doppler and resonant light shifts, we demonstrate clock operation at specific, convenient experimental parameters consistent with zero sensitivity to laser frequency variation and white-noise-limited clock frequency averaging for 1000 s of integration.

Figures (5)

• Figure 1: Ramsey CPT spectroscopy using a chip-scale atomic beam. (a) Image of the atomic beam device showing internal components and the location of the atomic beam (cartoon arrow). (b) Energy level diagram of the $^{87}$Rb, D1 lines indicating the relevant atomic levels, CPT Rabi rates (red), and decay rates (blue). (c) Schematic of light and optics shows how the atomic beams are probed. (d) Typical Ramsey fringe measured relative to the Raman detuning $\Delta$.
• Figure 2: Clock sensitivity to laser detuning. (a) Clock frequency is measured vs. laser detuning for $F^{\prime} = 1$ (purple) and $F^{\prime} = 2$ (green) using $P_{\rm res} \approx 113 \ \mu{\rm W}$ and $\Omega_{2}^{2}/\Omega_{1}^{2} \approx 1$. Solid lines are linear least squares fits used to determine $\chi$. (b) Measured $\chi$ vs. laser power for $F^{\prime} =$ 1 (purple), $F^{\prime} = 2$ (green) and $\Omega_{2}^{2}/\Omega_{1}^{2} \approx$ 0.33 (triangles), 1 (circles), and 3 (squares). Dashed lines are guides to the eye. The Raman damping parameter $\Omega^{2}S\tau$ is calculated for $F' = 2$, $\tau = 2 \ \mu {\rm s}$, and $\delta = 0$ (values are three times lower for $F' = 1$). Laser detuning sensitivity adds (subtracts) from $\chi_D$ for $F^{\prime}$ = 1(2) due to light shifts, and the sensitivity is shown to cross zero for $F^{\prime} = 2$ at certain values of $P_{\rm res}$. Error bars indicate the standard error of the mean; some are smaller than their marker.
• Figure 3: Resonant light shifts and laser detuning sensitivity. (a) The resonant light shift is plotted relative to common-mode laser detuning in the limiting cases of $r = 1/4$, $\Delta\rho_0 = 0$ (red) and $r = 0$, $\Delta\rho_0 = 1 \ \%$ (blue) for $P_{\rm res} = 100 \ \mu{\rm W}$. The Doppler shift (dashed line) is $\approx 18.1 \ {\rm Hz/MHz}$, and the relative line pulling weight for each detuning class in the measured geometry is indicated by the shaded area. The clock sensitivity to laser detuning using the experimental parameters is plotted for these cases considering the behavior near $\delta = 0$ (b) and the sensitivity averaged over the line pulling weights (c) using Eq. (\ref{['eq:<chi>']}).
• Figure 4: Laser detuning sensitivity $\chi$ vs. $\Omega_{2}^{2}/\Omega_{1}^{2}$ for $F^{\prime} = 1$ (open circles) and $F^{\prime} = 2$ (solid circles) with $P_{\rm res} \approx 113 \ \mu{\rm W}$. Inset shows the nulled clock frequency dependence on common-mode laser detuning near $\Omega_{2}^{2}/\Omega_{1}^{2} = 1.1$.
• Figure 5: Clock stability comparison at optimal operation point for $F^{\prime} = 2$. The clock stability (modified Allan deviation, ${\rm Mod} \ \sigma_{y}$) is measured vs. averaging time using $\Omega_2^2/\Omega_1^2 = 1.1$ and $P_{\rm res} = 113 \ \mu{\rm W}$ to compare $F^{\prime} = 2$ (solid circles, $\chi \approx 0$) and $F' = 1$ (open circles, $\chi \approx 38 \ {\rm Hz/MHz}$). The $F^{\prime} = 1$ laser-detuning-limited stability (shaded band) is plotted using the measured detuning sensitivity and laser frequency stability, explaining the drift of this data set. The $F^{\prime} = 2$ data shows better short-term stability and stable averaging over 1000 s.