Comparison of Two Independent Sr Optical Clocks with $1\mathbf{\times }{10}^{-17}$ Stability at ${10}^3\mathbf{s}$

Many-particle optical lattice clocks have the potential for unprecedented measurement precision and stability due to their low quantum projection noise. However, this potential has so far never been realized because clock stability has been limited by frequency noise of optical local oscillators. By synchronously probing two ${}^{87}\mathrm{Sr}$ lattice systems using a laser with a thermal noise floor of $1\times {10}^{-15}$, we remove classically correlated laser noise from the intercomparison, but this does not demonstrate independent clock performance. With an improved optical oscillator that has a $1\times {10}^{-16}$ thermal noise floor, we demonstrate an order of magnitude improvement over the best reported stability of any independent clock, achieving a fractional instability of $1\times {10}^{-17}$ in 1000 s of averaging time for synchronous or asynchronous comparisons. This result is within a factor of 2 of the combined quantum projection noise limit for a 160 ms probe time with $\sim {10}^3$ atoms in each clock. We further demonstrate that even at this high precision, the overall systematic uncertainty of our clock is not limited by atomic interactions. For the second Sr clock, which has a cavity-enhanced lattice, the atomic-density-dependent frequency shift is evaluated to be $-3.11\times {10}^{-17}$ with an uncertainty of $8.2\times {10}^{-19}$.

Figure 1

(a) A cavity-stabilized diode laser is split and sent to each of the lattice clocks. To ensure that both clock laser beams have independent frequency control, Sr1 and Sr2 have separate AOMs. The two clocks have different lattice geometries: Sr1 uses a 1D retroreflected lattice and Sr2 uses a 1D cavity-enhanced lattice. The independent clock laser beams are locked to the ${}^{1}S_0\to {}^{3}P_0$ transition by feeding the measured clock transition frequency back to the rf frequencies of the AOMs. The rf frequencies are recorded to determine the difference between the two clocks. (b) Clock comparisons using our 7 cm vertical cavity-stabilized laser (top) required synchronizing the clock probe pulses to perform correlated spectroscopy. Clock comparisons with our lower noise 40 cm horizontal cavity (bottom) used asynchronous pulses to ensure independent clock operation. Synchronous measurements with the 40 cm cavity (not depicted) were also performed.

Figure 2

(a) The measured thermal noise floor of the two optical reference cavities. The stability of the 7 cm cavity (closed circles) was measured by comparing two cavities of the same design. For the 40 cm cavity (open circles), we determine its frequency stability from a measurement based on the atomic reference. We lock this laser to the ${}^{87}\mathrm{Sr}$ clock transition and subtract off a residual cavity drift of $\sim 1.4\mathrm{mHz}/\mathrm{s}$. These data include contributions from other technical noise and thus represents an upper bound on the thermal noise floor. (b) A scatter plot of the measured excitation fraction when the clock lasers are locked to the two Sr references. Each point represents the measured excitation fraction for Sr1 versus Sr2 for the same duty cycle. The blue points represent data taken under synchronous interrogation using the 7 cm reference cavity, showing a clear correlation arising from common-mode laser noise. The red points represent data taken under asynchronous interrogation with the low-noise 40 cm reference cavity, clearly indicating a lack of classical correlations. Instead, the distribution indicates near-QPN-limited performance for independent Sr1 and Sr2. The inset compares synchronous measurements using the 40 cm cavity (in green) with the asynchronous data using the same cavity. This distribution shows a slight correlation, indicating a small amount of residual laser noise.

Figure 3

(a) The Allan deviation of a synchronous comparison (closed circles) between Sr1 and Sr2 with the low-noise 40 cm cavity. The self-comparison (open circles) is $({\nu }_1-{\nu }_2)/\sqrt{2}$, where ${\nu }_1$ and ${\nu }_2$ are the frequencies to which the two servos are locked. Dividing (${\nu }_1-{\nu }_2$) by $\sqrt{2}$ extrapolates the self-comparison stability to the expected performance of a comparison between the Sr2 system and an identical clock. The dashed line indicates the QPN limit. (b) An asynchronous comparison between the two Sr clocks (also taken with the 40 cm cavity). The Allan deviation of the comparison fits to $4.4\times {10}^{-16}/\sqrt{\tau }$. The estimated Dick effect is roughly equal to the predicted QPN of $2.0\times {10}^{-16}/\sqrt{\tau }$ (dashed line). The inset depicts typical scans of the clock transition (open circles). The red line is a fit to the data using the Rabi model. All stability data shown in this work represent the combined stability of the two systems. To infer a single clock stability, one would need to divide all the data by $\sqrt{2}$.

Figure 4

The measured Sr2 density shift as a function of $\Delta N$. Each point on this plot represents an average over a bin of 30 measurements of $({\nu }_{\mathrm{high}}-{\nu }_{\mathrm{low}})/(N_{\mathrm{high}}-N_{\mathrm{low}})$. Our statistics show that the density shift for our trapping conditions is linear within our quoted uncertainty of $8.2\times {10}^{-19}$. Inset: A single 2000 s long density shift measurement with $\Delta N\simeq 41000$. The shift per atom was measured and then scaled up to 2000 atoms for a typical running condition. This measurement shows that a single density shift evaluation for 1000 s using a large atom number modulation is sufficient for a $1\times {10}^{-18}$ clock.