A laser with instability reaching $4 \times 10^{-17}$ based on a 10-cm-long silicon cavity at sub-5-K temperatures

TL;DR

This work demonstrates a 10 cm silicon optical cavity operated at sub-5 K that reaches record-like stability in a closed-cycle cryocooler environment. By combining a cool-quiet quench measurement (CQQM) to bypass cryostat vibrations with a robust horizontal-fastening design and six targeted improvements, the authors isolate the cavity’s intrinsic Brownian-noise limit and achieve a frequency instability of $4.3(2)\times10^{-17}$ for 4–12 s averaging, approaching the calculated thermal-noise floor of $3.3\times10^{-17}$ at 4.9 K. A three-cornered-hat measurement with two room-temperature ULE cavities validates the intrinsic performance, and ultra-narrow linewidths around a few to tens of milliHertz are inferred from PSD-based analysis and direct beat measurements (Si1 beat $\approx 5.7$ mHz). They further reduce operating temperature to $\approx 3.3$ K, attaining stable operation with a long-term performance dominated by cavity-temperature fluctuations and a long-term instability of $\sim 5.5\times10^{-16}$ at $\tau=1000$ s. Overall, this work establishes a practical pathway to low-$10^{-17}$–level laser stabilities using long cryogenic cavities and provides a framework (CQQM and rigid fastening) for exploring even longer cavities and lower temperatures.

Abstract

The realization of ultra-stable lasers with $10^{-17}$-level frequency stability has enabled a wide range of researches on precision metrology and fundamental science, where cryogenic single-crystalline cavities constitute the heart of such ultra-stable lasers. For further improvements in stability, increasing the cavity length at few-kelvin temperatures provides a promising alternative to utilizing relatively short cavities with novel coating, but has yet to be demonstrated with state-of-the-art stability. Here we report on the realization of a relatively long ultra-stable silicon cavity with a length of 10 cm and sub-5-K operating temperatures. We devise a dynamical protocol of cool-quiet quench measurement that reveals the inherent $10^{-17}$-level frequency instability of the silicon cavity despite the substantially larger frequency noise induced by the cryostat vibration. We further develop a method for suppressing the cryostat-vibration-induced frequency noise under continuous cooling, and observe an average frequency instability of $4.3(2) \times 10^{-17}$ for averaging times of 4 to 12 seconds. Using the measured noise power spectral density, we compute a median linewidth of 9.6(3) mHz for the silicon cavity laser at 1397 nm, which is supported by an empirically determined linewidth of 5.7(3) mHz based on direct optical beat measurements. These results establish a new record for optical cavities within a closed-cycle cryocooler at sub-5-K temperatures and provide a prototypical system for using long cryogenic cavities to enhance frequency stabilities to the low-$10^{-17}$ or better level.

Figures (17)

• Figure 1: Experimental setup and safe cooling down of a silicon cavity to few-kelvin temperatures with intact optical performance. (a) Silicon cavity (Si1) and its vacuum chamber for cryogenic cooling. In the insets, red arrows indicate a vertical support under the Si1 central ring at three points marked by centers of red circles, and blue arrows indicate a three-point fastening of the Si1 horizontal position (not shown in the side view). (b) Diagram of three-cornered-hat (TCH) frequency stability measurements based on the Si1 cavity and two room-temperature ultra-low-expansion (ULE) glass cavities, ULE2 and ULE3. DM: dichroic mirror; PD: photo-detector; FNC: fiber noise cancellation; PPLN: periodically poled lithium niobate waveguide for frequency doubling. (c) The Si1 cavity finesse (solid circles) remains stable and slightly increases during the cooling. Inset: a ringdown measurement for determining the finesse at a platform temperature of about 3 K. (d) Step-function response of the Si1 resonance frequency for in situ determination of cavity temperature, with improvements (1) to (4) implemented. The blue line is the platform temperature. Gray circles denote the beat frequency between Laser3 (based on the ULE3 cavity) and Laser1 (Si1). The red line shows a fit using a third-order low-pass filter model. Error bars represent $1\sigma$ statistical uncertainties.
• Figure 2: Dynamically revealing the $10^{-17}$-level frequency stability of a cryostat-vibration-free Si1 cavity by cool-quiet quench measurement (CQQM). (a) Diagram of the CQQM. Here, $\Delta f_{\mathrm{Si1-ULE3}}$ denotes an optical beat frequency between Laser1 and Laser3 (mixed down to near d.c.), and res $\Delta f_{\mathrm{Si1-ULE3}}$ in the inset further denotes the residual frequency noise after subtracting an offset and an overall linear drift from the $\Delta f_{\mathrm{Si1-ULE3}}$ data. (b) Frequency stabilities of the Si1, ULE2, and ULE3 cavities. The vertical axis corresponds to the modified Allan deviation. A dashed line shows a typical Brownian thermal noise floor of the Si1 cavity at 4.9 K. Magenta empty circles denote the frequency instability of the Si1 cavity measured by CQQM (illustrated by the 50-second-long data set in the inset of (a)), which bypasses the cryostat vibration and reveals a sub-total effect mainly contributed by the optical performance of the Si1 cavity and electro-optically induced technical noise sources. Here, improvements (1) to (4) are implemented, while (5) and (6) are not (see Table \ref{['table:SiCavityConfig']}). Olive squares and blue diamonds denote the corresponding frequency stabilities of the ULE2 and ULE3 cavities respectively. For comparison, gray solid circles show the substantially larger frequency instability of the Si1 cavity under a room temperature of 292 K. Error bars represent $1\sigma$ statistical uncertainties.
• Figure 3: Overcoming two major obstacles to realizing $10^{-17}$-level frequency stability under continuous cooling. (a) Modified Allan deviations. With technical improvements (1) to (4) implemented, the frequency stability of the silicon cavity ("Si1a" configuration, green empty circles) has significantly improved with respect to that of the earlier Si0 cavity with mirror contamination (orange solid squares). (b) Power spectral densities (PSDs) for measurements in (a). (c) Modified Allan deviations. With technical improvements (1) to (5) implemented, the frequency stability of the silicon cavity ("Si1b" configuration, blue empty circles) further shows stability enhancement over Si1a (by a factor of more than three at $\tau = 1$ s). (d) PSDs for measurements in (c). To better show the Si1b PSD, the Si1a PSD is only plotted in part in (d). In (a) to (d), the Brownian thermal noise floor is shown as red dashed lines, and the CQQM results, as a reference for comparison, are shown as magenta empty circles (modified Allan deviations) and magenta lines (PSDs). Error bars represent $1\sigma$ statistical uncertainties.
• Figure 4: Optimized frequency instability approaching the Si1 thermal noise floor at 4.9 K under continuous cooling. (a) Modified Allan deviations. Measured frequency instabilities are shown for the silicon cavity with improvements (1) to (6) implemented ("Si1c", red circles) as well as for ULE2 (olive squares) and ULE3 (blue diamonds) cavities. Red, blue, gray, and brown dashed lines show the computed Brownian thermal noise floor at 4.9 K (as also shown in Figs. \ref{['fig:cqqm']} and \ref{['fig:twomajorimprovements']}), the estimated instability induced by seismic vibrations, the measured instability due to residual amplitude modulation (RAM), and the projected instability due to temperature fluctuation. The inset shows the Si1 frequency instability (purple hexagons) in a configuration (denoted as Si1c$'$) that is almost the same as Si1c except for a reduced platform temperature of 3.3 K. (b) PSDs. The Si1c spectrum (solid line) is compared with that of Si1b (short dotted line) and the Brownian thermal noise floor. (c) Based on the Si1c PSDs (measured for (b)) and a systematic scheme in Ref. [Bishof13prl], the Si1c linewidths are computed and the histogram constructed, with a median full width at half-maximum (FWHM) of 9.6(3) mHz (shown in the inset of (c)). (d) A sample 100-s-long beat signal (mixed down to near d.c. and measured with a digital oscilloscope) between the Si1 and ULE3 cavities. (e) Average line profile (green circles) for 12 data sets similar to the beat signal measured in (d), analyzed via normalized fast Fourier transform using a 10-mHz resolution bandwidth (RBW) and Hanning window. The profile is fitted to a Lorentzian form (red line). Error bars represent $1\sigma$ statistical uncertainties.
• Figure S1: Illustration of the horizontal fastening for Si1 cavity and strain relief based on indium metal film. This supplemental figure provides an enlarged and marked view of the upper inset of Fig. 1a of the main text. A monolithic PEEK structure (beige structure marked by red letters) surrounds the Si1 cavity and "touches" the cavity via three small horizontal cylinders (marked by the three blue arrows), which thus fasten the position of the silicon cavity in the horizontal direction. The PEEK structure itself is fixed onto a stainless steel supporting structure under it, while the stainless steel structure is in turn fixed onto a base made of oxygen-free high-conductivity copper (with goldish yellow color). To provide strain relief and further protect the cavity, we add a single layer of relatively soft indium metal film with 10-$\mu$m thickness onto the surface of each small cylinder.