Enhancing optical lattice clock coherence times with erasure conversion

TL;DR

This work addresses the decoherence in optical lattice Sr clocks caused by lattice-induced Raman scattering and radiative decay by implementing erasure conversion through hyperfine-resolved readout in a two-ensemble clock. The approach identifies and measures atoms that scatter out of the clock subspace, enabling their erasure from coherent measurements. Experiments demonstrate coherence times exceeding $100~\text{s}$ for Ramsey and $>150~\text{s}$ for spin-echo, with simulations showing consistency when including Raman scattering, lattice Stark shifts, and density effects. Although differential clock stability gains are modest due to imperfect erasure fraction and pulse fidelities, the method reduces the sensitivity of stability to interrogation time and promises enhanced performance for applications requiring long, flexible interrogation windows, including sub-natural linewidth spectroscopy and gravitational-wave sensing.

Abstract

Increasing coherent interrogation times is central to advancing the precision of optical clocks. Synchronous differential optical clock comparisons have now demonstrated atomic coherence times that far exceed the coherence time of the clock laser. While atom coherence times are then primarily limited by errors induced by lattice Raman scattering, excited clock state radiative decay, and broadening from two-body collisions, many of these errors take the atoms out of the clock transition subspace, and can therefore be converted into "erasure" errors if the appropriate readout scheme is employed. Here we experimentally demonstrate a hyperfine-resolved readout technique for ${}^{87}$Sr optical lattice clocks that mitigates decoherence from Raman scattering induced by the lattice as well as radiative decay. By employing hyperfine-resolved readout in synchronous differential comparisons between ${}^{87}$Sr ensembles with both Ramsey and spin echo spectroscopy sequences, we achieve enhanced atomic coherence times exceeding 100 s and 150 s, respectively, enabling longer coherent measurements without a reduction in performance. We anticipate that this hyperfine-resolved readout technique will benefit applications of state-of-the-art optical lattice clock comparisons in which the coherence times are constrained by Raman scattering or radiative decay.

Figures (8)

• Figure 1: Lattice photon-induced Raman scattering and its effect on hyperfine state populations. (a) Two ensembles of $^{87}$Sr atoms, initially prepared in the clock subspace, are simultaneously probed by the 698 nm clock laser under a magnetic bias field. (b) The relevant energy level diagram for a $^{87}$Sr optical lattice clock. Atoms are initialized in the clock subspace, consisting of $\ket{^1\text{S}_0, m_F=5/2}$ (blue) and $\ket{^3\text{P}_0, m_F=3/2}$ (yellow). Raman scattering from the lattice light redistributes the population from $\ket{^3\text{P}_0, m_F=3/2}$ to the $^3$P$_J\,$ manifolds, while radiative decay directly returns excited-state atoms to ${}^1$S$_0\,$. Radiative decay from $^3$P$_1\,$ further redistributes the population among the ${}^1$S$_0\,$$m_F$ states. Detection is performed via fluorescence imaging on the ${}^1$S$_0\,$-${}^1$P$_1\,$ transition. (c) The excitation fractions of ensembles 1 and 2 from synchronous Ramsey measurement are plotted parametrically, resulting in ellipses determined by the relative detuning between the clock transitions of the atoms in the two ensembles. Over time, Raman scattering and radiative decay transfer atoms out of the clock subspace, leading to the shrinking size of the ellipses, which corresponds to decoherence.
• Figure 2: Hyperfine state population ratio measurement of erasures. The top panel of (a) illustrates the sequence used to measure the excited-state population distribution. The system is first initialized in $\ket{e, 3/2}$, followed by a clean-up pulse to remove the ground-state population. After holding the atoms in the lattice for 10 s, another clean-up pulse eliminates remaining ground-state atoms. A specific clock transition $\ket{g, m_F} \leftrightarrow \ket{e, m_F'}$ is then driven for varying pulse durations to map out the $\ket{g, m_F}$ state population. The bottom panel displays the resulting excited-state population distribution, with an inset that zooms into the probability below 0.02 in order to show the population in $m_F$-states other than $\ket{e, 3/2}$ more clearly. Similarly, the top panel of (b) depicts the sequence used to measure the ground-state population. The bottom panel presents the corresponding population distribution. Dark purple bars indicate measured normalized $m_F$-state populations in $g$ (lower panel) and $e$ (upper panel), while light purple bars indicate the theoretically predicted populations. Insets: Representative Rabi oscillation plots of the $\ket{g,-5/2}\leftrightarrow\ket{e,-7/2}$ transition (left panel) and the $\ket{g,5/2}\leftrightarrow\ket{e,7/2}$ transition (right panel), respectively.
• Figure 3: Illustration of pulse sequence used for hyperfine state resolved readout, starting with a quantum superposition between $\ket{g,5/2}$ and $\ket{e,3/2}$. The $\ket{g,5/2}$ population is shelved in the clock state by a first $\pi$-pulse on resonant with the $\ket{g,5/2} \rightarrow \ket{e,7/2}$ transition; The remaining populations in $g$ is measured ($N_{g,res}$) using a global imaging pulse ($\sim25~\mu$s) on the ${}^1\text{S}_0 \rightarrow {}^1\text{P}_1$ transition and at the same time erased from the remaining operations; A second clock $\pi$-pulse resonant with the $\ket{e,7/2} \rightarrow \ket{g,5/2}$ transition is used to coherently unshelve the population back to the ground state; The $\ket{g,5/2}$ population ($N_{g,5/2}$) is measured using a second imaging pulse; The $\ket{e,3/2}$ population is transferred to $g$ via a third clock pulse on the $\ket{e,3/2} \rightarrow \ket{g,5/2}$ transition; The $\ket{e,3/2}$ population ($N_{e,3/2}$) is measured using a third imaging pulse; The remaining populations in ${}^3\text{P}_0$ and ${}^3\text{P}_2$ are transferred back to $g$ using standard global repump pulses on the ${}^3\text{P}_0\rightarrow{}^3\text{S}_1$ and ${}^3\text{P}_2\rightarrow{}^3\text{S}_1$ transitions; A final imaging pulse is applied to measure the residual excited state population ($N_{e,res}$).
• Figure 4: Comparison of coherence times using standard readout (blue), hyperfine-resolved readout (cyan), and reconstructed standard readout (orange) methods. Representative plots of coherence decay are shown in (a) for the Ramsey sequence at 75 $E_{\rm rec}$ (the highest trap depth measured in this work) and in (b) for the spin echo sequence at 15 $E_{\rm rec}$ (the lowest depth), with the corresponding comparison ellipses shown in panels (i) and (ii), respectively. Maximum likelihood estimation (MLE) is used to extract the atomic coherence. The solid curves are calculated through numerical simulation based on the known experimental parameters. (c) shows coherence time versus trap depth for the Ramsey sequence, while (d) presents the coherence time for the spin-echo sequence. Numerically simulated coherence times are indicated by star-shaped markers with matching color coding. (e) compares the enhancement ratio of the measured coherence time using erasure conversion for the Ramsey sequence (solid circles) and spin-echo sequence (hollow circles) as a function of depth. The measured and simulated coherence decay for the other trap depths are presented in Appendix \ref{['fig8_no_labels']}.
• Figure 5: (a)&(b) show the extracted Allan deviations at 1 second for the differential clock comparisons for Ramsey (filled circles) and spin echo (empty circles), as well as using standard readout (blue), hyperfine-resolved readout (cyan), and reconstructed standard readout (orange), all taken at 15 $E_{\rm rec}$ depth. The error bar of each data point is on the order of $1\times10^{-18}$, and is thus not visible in the plot. The dashed curve in (a) represents the idealized Allan deviation, based only on the simulated coherence and the remaining number of atoms as a function of time, and does not include the relative phase difference between the two ensembles. In (b), the solid lines show the modeled Allan deviations for standard (blue) and hyperfine-resolved (cyan) readout, accounting for the accumulated phase between the two ensembles. Please see Appendix \ref{['Appendix: Numerical simulation']} for details.