Probing beyond the laser coherence time in optical clock comparisons

We develop differential measurement protocols that circumvent the laser noise limit in the stability of optical clock comparisons by synchronous probing of two clocks using phase-locked local oscillators. This allows for probe times longer than the laser coherence time, avoids the Dick effect, and supports Heisenberg-limited measurement precision. We present protocols for such frequency comparisons and develop numerical simulations of the protocols with realistic noise sources. These methods provide a route to reduce frequency ratio measurement durations by more than an order of magnitude.

Figure 1

Optical clock comparison with phase-locked LOs. (a) A cavity-stabilized laser simultaneously probes clocks at two different frequencies, which are phase locked via a frequency comb and active path-length stabilization. (b) Timing diagram of a near-synchronous Ramsey experiment. The phase ${\varphi }_1^{\mathrm{est}}$ measured at clock 1 is used to correct the laser phase before the final pulse of the second clock. (c) Transition probabilities as a function of ${\varphi }_1$ for clock 1 (red) and clock 2 before and after applying the feed-forward phase (blue dotted and blue solid lines, respectively). The size of the projection noise for the two clocks is denoted by the thicker lines, and the distribution of laser phase noise is depicted as the gray region.

Figure 2

Noise reduction for a frequency ratio measurement of a many-atom clock with a single-atom clock. Precise laser phase measurements on clock 1 allow the unambiguous determination of the clock 2 phase for probe durations longer than the laser coherence limit, giving a reduction of measurement noise compared to the projection noise limit for asynchronous clock comparisons with otherwise identical noise. Simulation results (points) reproduce the analytical estimates (dash-dotted lines) up to the point that the projection noise for the two clocks is comparable. Inset: Minimum relative variance $R_{\beta ,\min }$ plotted vs $N_1$, showing that higher atom numbers support greater suppression of the noise.

Figure 3

Simulated stability of a comparison between a ytterbium optical lattice clock and an aluminum ion clock operating with five ions in a GHZ state. The fractional clock 1 frequency stability (blue points) and the fractional ratio measurement stability (black points) are shown, along with the common-mode, unstabilized, laser frequency noise (red points) and the differential laser frequency noise (green points). Clock 1 reaches the Dick effect stability limit (blue dashed line), while the ratio stability exceeds that, reaching the calculated projection noise limit for the aluminum ion clock (black dashed line).

Figure 4

Illustration of the maximum-likelihood phase estimation algorithm. (a) The transition probabilities of clock 1 (red line) and clock 2 (blue line) are plotted as a function of the phase of clock 1 for zero frequency ratio error. Example measurement outcomes for the two clocks are indicated by the horizontal dashed lines, with thick line segments indicating possible phase-inversion outcomes. The prior distribution of laser phase values (gray) and clock 1 phase values (red) and clock 2 phase values (blue) based on the measurements are shown as shaded areas, with the product of the three shaded green. (b) Simulated frequency ratio measurement stability of a strontium optical lattice clock with a ytterbium optical lattice clock using the maximum-likelihood protocol. The projection noise limit is shown by dashed lines, and the simulation results are shown by solid lines.