Active compensation of the AC Stark shift in a two-photon rubidium optical frequency reference using power modulation

TL;DR

The paper addresses the AC Stark shift in a two-photon rubidium optical frequency reference and demonstrates a two-loop auto-compensated shift (ACS) method that uses power modulation and a secondary feedback loop to cancel the shift. The approach relies on the relations $\nu_{int} = \nu_{LO} + \xi P(t)$ and $P(t) = P_0[1 + A\sin(2\pi f_{PM} t)]$, with the secondary loop tuning $\xi$ based on a lock-in tone at $f_{PM}$; the primary loop keeps $\nu_{LO}$ locked to the atomic transition. Results show a ~1000-fold suppression of AC Stark sensitivity, achieving short-term instabilities of $3\times10^{-14}$ at 1 s and long-term instabilities of $2\times10^{-14}$ at $10^4$ s, albeit with a LO-noise–limited floor described by the derived stability limit ${\sigma_y}(\tau> T)$. The work highlights practical prospects and limitations for ACS in compact, field-deployable cw optical references, and suggests pathways to further mitigate LO noise and implement two-loop schemes without an AOM.

Abstract

We implement a feedback protocol to suppress the AC Stark shift in a two-photon rubidium optical frequency reference, reducing its sensitivity to optical power variations by a factor of 1000. This method alleviates the tradeoff between short-term and long-term stability imposed by the AC Stark shift, enabling us to simultaneously achieve instabilities of $3\times10^{-14}$ at 1 s and $2\times10^{-14}$ at $10^4$ s. We also quantitatively describe, and experimentally explore, a stability limit imposed on clocks using this method by frequency noise on the local oscillator.

Figures (5)

• Figure 1: A high-level block diagram of the ACS method for suppressing the AC Stark shift in a generic CW frequency reference. PID = proportional/ integral/ derivative controller.
• Figure 2: A block diagram representing our experimental implementation of ACS in a two-photon rubidium OFR. For simplicity, this diagram does not show the RAM suppression signal pathway or the optical frequency measurement setup used to characterize our OFR stability. The waveform generator and the multiplier connected to it are shown as separate instruments for clarity, but they are implemented in a single FPGA device. PD = photodiode; PMT = photomultiplier tube; PID = proportional/ integral/ derivative controller; AOM = acousto-optic modulator; PLL = phase locked loop; LP = lowpass.
• Figure 3: The response of our OFR to intentional optical power variations, caused by alternatingly raising and lowering a piece of glass in front of the power measurement photodiode once per hour. ACS suppresses the OFR response by a factor of about 1000; the residual shift on each step is near or below the 7.4 Hz uncertainty (Allan deviation at $\tau = 30 \text{ min}$). For the ACS data, $T = 170$ s.
• Figure 4: Allan deviations of the OFR with and without ACS. Shaded regions denote standard error. For the ACS data, $A = 0.19$ and $T \approx 200 \text{ s}$. Our previously reported two-photon rubidium performance, Ref. newman_high-performance_2021, is shown in gray for context. (Note that for the data shown here, a nanoparticle polarizer was used before the power monitor pickoff, while in Figs. \ref{['fig_step_test']} and \ref{['fig_adev_transitions']}, a polarizing beamsplitter cube was used. We believe this to be responsible for the slight difference in the shapes of the Allan deviations.)
• Figure 5: Allan deviations of our OFR for different choices of the ACS parameters $A$ and $T$. Shaded regions denote standard error. For integration times $\tau \gg T$, the Allan deviations show reasonable agreement with the stability limits predicted by Eq. (\ref{['eq_stability_limit']}), shown in dashes. For these measurements, we chose time constants considerably faster than the optimal value of about 200 s to ensure that the OFR stability is governed by this limit, without contributions from systematic instabilities that ultimately limit performance at the longest integration times.