Terahertz Vibrational Molecular Clock with Systematic Uncertainty at the ${10}^{-14}$ Level

Neutral quantum absorbers in optical lattices have emerged as a leading platform for achieving clocks with exquisite spectroscopic resolution. However, the studies of these clocks and their systematic shifts have so far been limited to atoms. Here, we extend this architecture to an ensemble of diatomic molecules and experimentally realize an accurate lattice clock based on pure molecular vibration. We evaluate the leading systematics, including the characterization of nonlinear trap-induced light shifts, achieving a total systematic uncertainty of $4.6\times {10}^{-14}$. The absolute frequency of the vibrational splitting is measured to be 31 825 183 207 592.8(5.1) Hz, enabling the dissociation energy of our molecule to be determined with record accuracy. Our results represent an important milestone in molecular spectroscopy and THz-frequency standards, and may be generalized to other neutral molecular species with applications for fundamental physics, including tests of molecular quantum electrodynamics and the search for new interactions.

Popular Summary

The ability to control the internal states of atoms with electromagnetic radiation has paved the way for quantum devices such as atomic clocks, which have transformed scientific measurements. But extending such control to the various internal states of molecules could usher in a new era of investigations into subtle physical phenomena, such as new particles beyond the paradigm of the standard model, as well as new pathways for terahertz-frequency metrology. Here, we realize an accurate clock based solely on the vibrational transitions of a diatomic molecule—an architecture that combines the best of both worlds in atomic clock making and molecular quantum science—and characterize its performance.

Our clock is based on the transitions between two vibrational states of molecular strontium, ${\mathrm{Sr}}_2$. We assemble the ultracold molecules from laser-cooled atoms and hold them in a standing wave of optical potential. Initially, the molecules are vigorously vibrating and almost dissociating. Then, using a pair of phase-coherent lasers, we bring the internal vibrations of the molecules to a complete halt. This vibrational energy (hence, frequency) that we remove is the basis for our molecular clock. After exploring various systematic effects, we achieve a fractional systematic uncertainty of roughly one part in ${10}^{14}$, matching the performance of analogous atomic clocks just over a decade ago. This latest accuracy is aided by the high precision of the clock, which enables the observation of molecular resonances with high quality factors.

The vibrational molecular clock is an exciting starting point for exploring terahertz metrology, quantum chemistry, and fundamental interactions at atomic length scales.

Figure 1

Vibrational molecular lattice clock. (a) Raman lasers (upleg, red arrow; downleg, orange arrow) detuned from an intermediate state in $(1)0_u^{+}$ probe the two-photon vibrational clock transition between $(v=62,J=0)$ and $(v=0,J=0)$ in the $X^1{\mathrm{\Sigma }}_g^{+}$ ground potential. The optical lattice (brown arrow) off-resonantly addresses an isolated rovibronic state in $(1)1_u$ to induce magic trapping conditions. (b) Experimental setup. The upleg master laser is stabilized to a reference cavity using the Pound-Drever-Hall (PDH) technique, and its phase coherence is transferred to the downleg laser via a frequency comb. The molecules are held in the 1D optical lattice. Copropagating clock lasers are delivered to the molecules via an optical fiber with active fiber noise cancellation (FNC). The spectroscopic signal derives from absorption imaging of $X(62,0)$ photofragments at a slight grazing angle relative to the lattice. A Rb microwave standard acts as a flywheel oscillator linking the molecular clock to GPS time for the absolute frequency measurement. Further information is given in the main text and Appendices pp1 and pp2. (c) Two-photon Rabi oscillations between the clock states driven at the operational probe intensities (filled circles, experimental data averaged over eight consecutive runs, error bars represent $1\sigma$ uncertainties; solid red line, analytical fit to an exponentially decaying sinusoid). We observe lines as narrow as 11(1) Hz (inset, green squares). For clock operation, we perform Rabi spectroscopy with a 30-ms $\pi$-pulse duration (indicated by the black arrow), resolving 30(2)-Hz linewidths consistent with the expected Fourier limit (inset, black open circles). Each point in the inset is a single shot of the experiment, and solid lines are Lorentzian fits.

Figure 2

Clock shifts due to the lattice light. (a) Nonlinear shifts of the molecular clock frequency versus trap depth. For a given lattice frequency (color coded), we make interleaved measurements of clock shifts (open circles) with respect to a reference trap depth (approximately $500E_r$) and fit the data to parabolas (solid lines) with a global quadratic parameter, $-{\beta }^{*}$. (b) Linear light-shift coefficient ${\alpha }^{*}$ versus lattice frequency [color code matches (a)] and the linear fit (black solid line). ${\alpha }^{*}$ is predominantly due to the $E1$ differential polarizability and is nulled at $f_{\mathrm{zero}}$. By tuning ${\alpha }^{*}$, we can find conditions where the sensitivity of $\mathrm{\Delta }{f}_{\text{clock}}$ to fluctuations in $U_0$ is minimal at our operational trap depth of 487(4) $E_r$ (dark green points). Error bars represent $1\sigma$ uncertainties.

Figure 3

Clock shifts at the operational Raman detuning as a function of (a) the upleg laser intensity, and (b) the downleg laser intensity. The horizontal axes are normalized by the respective operational intensities, $I_{\uparrow ,0}$ and $I_{\downarrow ,0}$. Solid lines are linear fits to the data. Residuals are plotted in units of Hz. Error bars represent $1\sigma$ uncertainties.

Figure 4

Density-shift evaluation. (a) Clock shifts due to molecular collisions extrapolated to operating conditions (one molecule per lattice site, averaged over filled sites) plotted versus the change in molecule number per site used for the interleaved measurement. A single constant suffices to fit the data [0.20(10) Hz, ${\chi }_{\mathrm{red}}^2=1.7$]. (b) In the same dataset, the shift between successive resonances taken under identical experimental settings serves as a control experiment to check for technical offsets. As expected, this averages to zero [0.03(20) Hz, ${\chi }_{\mathrm{red}}^2=2.0$]. All statistical errors are scaled up by $\sqrt{{\chi }_{\mathrm{red}}^2}$. Error bars represent $1\sigma$ uncertainties. Both insets show the histogram of normalized residuals, and the solid red lines are Gaussian fits.

Figure 5

(a) Absolute frequency of the clock transition measured over ten trials (filled black circles) with all known frequency offsets corrected, including that of the local Rb time base (see main text for details). Blue error bars are $1\sigma$ statistical uncertainties, dominated by the determination of the comb repetition rate rather than the stability of the scanned molecular clock lines. Red error bars are $1\sigma$ systematic uncertainties due to the molecular clock only (see Table 1). Black error bars are $1\sigma$ total uncertainties, where the uncertainties of the local time-base calibrations are added in quadrature with the statistical and molecular clock systematic uncertainties. The black horizontal line shows the weighted average (${\chi }_{\mathrm{red}}^2=0.5$), and the shaded gray area shows the associated $\pm 1\sigma$ standard error of the mean. (b) Histogram of all clock frequency measurements in the ten trials, relative to the weighted average of $f_{\text{clock}}$. The solid red line is a Gaussian fit to the histogram.

Figure 6

Exemplary spectra of molecular (a) axial and (b) radial sidebands excited using Raman clock transitions, from which the motional frequencies are extracted to calibrate the lattice trap. (c) Measurement of the molecular temperature as a function of the trap depth (black stars) using Raman carrier thermometry. Error bars denote $1\sigma$ uncertainties. A linear fit (solid red line) with the intercept fixed at the origin describes the data well (${\chi }_{\mathrm{red}}^2=0.92$).

Figure 7

The baseline ac polarizability ratio of the clock states in this study (${\alpha }_0/{\alpha }_{62}$) at various trap laser wavelengths. Red circles are experimental measurements. The black dotted line is the theory calculation. $X\to (1)1_u$ resonances are excluded from this plot, but are included in the calculation. Error bars represent $1\sigma$ statistical uncertainties.

Figure 8

Expanded dataset for the absolute clock frequency measurement. For experimental ease, the clock lasers and frequency comb are not actively steered toward the molecular resonance in real time. Instead, each measurement (filled gray circle) is derived from a single scan of the molecular clock transition under operational conditions, and the repetition rate of the frequency comb is counted with a zero dead-time frequency counter (1-s gate time). The shot-to-shot noise in the counted repetition rates is lowered by taking their average over a short window of time centered on the time stamp of the corresponding molecular clock measurement. The data are plotted against the time of measurement (Modified Julian Date, MJD). The frequency values are corrected for all systematic errors in Table 1, the Rb clock calibration, and the gravitational redshift (Appendix pp6). Error bars are not shown for visual clarity. Trial numbers on the top right corner of each plot correspond to those in Fig. 5. The histogram of all measurements is shown in Fig. 5. The axes labels are displayed in the plot for trial 1 (top left), and are the same for the remaining plots.