Chasing the thermodynamical noise limit in whispering-gallery-mode resonators for ultrastable laser frequency stabilization

Ultrastable high-spectral-purity lasers have served as the cornerstone behind optical atomic clocks, quantum measurements, precision optical microwave generation, high-resolution optical spectroscopy, and sensing. Hertz-level lasers stabilized to high-finesse Fabry-Pérot cavities are typically used for these studies, which are large and fragile and remain laboratory instruments. There is a clear demand for rugged miniaturized lasers with stabilities comparable to those of bulk lasers. Over the past decade, ultrahigh-Q optical whispering-gallery-mode resonators have served as a platform for low-noise microlasers but have not yet reached the stabilities defined by their fundamental noise. Here, we show the noise characteristics of whispering-gallery-mode resonators and demonstrate a resonator-stabilized laser at this limit by compensating the intrinsic thermal expansion, allowing a sub-25 Hz linewidth and a 32 Hz Allan deviation. We also reveal the environmental sensitivities of the resonator at the thermodynamical noise limit and long-term frequency drifts governed by random-walk-noise statistics.


Supplementary Note 1. Device fabrication
To compensate for the extensive thermal expansion of a whispering-gallery-mode (WGM) resonator, the resonator is sandwiched by a nearly zero or slightly negative expansion material (e.g. Zerodur). The compensating layers should be thick enough to hold the resonator's expansion. The ambient temperature variations should be small (< 5 o C) to avoid resonator destruction. The layers are bonded together with a transparent glue and then annealed with slightly varying temperature.
We determined the thickness of the compensating layer on the resonator to fully suppress an overall frequency shift for a given resonator's diameter and thickness via COMSOL simulations (Supplementary Figure 1a). The glue layer has 2 m thickness and is taken into account in the numerical simulation. The colour code indicates the linear expansion of the WGM resonator's equator. The blue colour represents negative expansion dominated by Zerodur components and the red color represents positive expansion where Zerodur components are too weak to hold the resonator's expansion. The green zone shows zero expansion where Zerodur contraction and MgF 2 expansion compensate each other. The thermal-expansion coefficient of the crystalline MgF 2 is approximately 9 ppm K -1 in the direction that is orthogonal to the crystalline/resonator axis. Based on the simulations, we found that the coefficient can be reduced to 0.15 ppm K -1 (60 times) by choosing the proper thickness of the Zerodur layers in the resonator structure described in the inset of Figure 1a unless the stress that propagates to the localized WGM takes into account. We should note here that all other sources of drift such as mechanical creeps or fast gradients of the temperature across the resonator are not eliminated by this technique.
For the compensated resonator, the temperature-dependent resonance frequency shift is introduced by three factors. One of them is the change in the thermorefractive index, which is small (0.6 ppm K -1 ) for the ordinary axis in MgF 2 crystal. The other two factors are the residual thermal expansion and elasto-optical refractive index changes caused by the stress induced in the material when the resonator expansion is prevented by the Zerodur layers. By using the data for Supplementary -2 the thermorefractive effect obtained from the Corning MgF 2 datasheet and data for the elastooptical effect in ref.
[1], we evaluated the stress-induced refractive index change value, averaged over the mode volume and found 0.25-0.5 ppm K -1 (the variation is attributed to the uncontrollable changes of the glue layer), which is smaller than the thermorefractive index change (0.6 ppm K -1 ). The reduction of the impact of the ambient temperature fluctuations on the resonator enables the measurement of the fundamental thermorefractive noise we performed.
The compensation of the free standing resonator may be affected by its mounting technique in a particular resonator's housing. To make sure that our resonator is still compensated after installation, we simulated the expansion of components with an entire mechanical model (Supplementary Figure 1b). Although the complete compensation is lost when the resonator is mounted into the housing, the predicted suppression of the resonator's expansion remains 23×an acceptable value for this experiment. The colour code indicates the distribution of possible device thermal expansion.

Supplementary Note 2. Thermodynamical noise limits of the cylindrical WGM resonator
The Supplementary Figure 2a illustrates the WGM resonator used in this experiment and its ring-down time (4 s) shows the loaded Q of 2.4×10 9 . The WGM resonator has the radius (r) of 3.45 mm, the rim thickness (L) of 0.025 mm. The thermodynamical noise limit of the compensated MgF 2 WGM resonator is calculated by our theoretical model [2,3]. Here, we consider the two major thermal noise sources that are the thermal-expansion noise and the thermorefractive noise. We numerically quantify the frequency noise power spectral density (FNPSD) imposed by thermal expansion noise for comparing it with experimental measurements where l is the thermal expansion coefficient and V c is the volume of the WGM resonator. The thermorefractive noise limit is also evaluated by Eq. 2, which is derived for the cylindrical WGM resonator possessing r >> L.
where  0 is the carrier frequency, k B is Boltzmann constant,  n is the thermorefractive coefficient of the material,  is the material density, C is the specific heat capacity, V m is the mode volume, D is the heat diffusion coefficient, and m is the mode order defined by m=2rn -1 .
Supplementary Figure 4a shows the FNPSD curves for two different ambient pressure levels.
We observe many noise spikes in the acoustic frequency regime originating from the laboratory environment when the WGM resonator is just sealed without evacuating the vacuum chamber (black curve). These spikes are suppressed by increasing the vacuum level to 1.33 mPa (red curve). The remaining spikes mostly come from the 60 Hz harmonics of the electrical power-line.
In this measurement, we observed that the evacuated chamber substantially reduces acoustic noise peaks and provides further noise reduction near the carrier frequency.  Figure 4b), which allows us to investigate the thermodynamical noise limit of the thermal-compensation WGM resonator more clearly.

Supplementary Note 4. Why thermorefractive noise of a WGM resonator is pink
In our work we reached the fundamental thermodynamical noise floor of the resonator and found out that the power spectrum of the noise scales as f -1. 5 . This is so called pink or flicker noise which is general for various physical processes, ranging from traffic flow to DNA sequence structure [4]. Flicker noise is the major physical mechanism that limits spectral purity of lasers stabilized to super-cavities [5]. The origin of this noise is still not completely understood and is usually studied case-specifically [4][5][6]. It was theoretically predicted that the fundamental thermorefractive frequency noise of a WGM resonator has distinct pink noise dependence [7]. In this work, we provide a careful experimental study of this noise.
where f is the spectral frequency, k B is the Boltzmann constant, T is the ambient temperature,  is the density of the resonator host material, C is the heat capacity of the material, From this equation, we find that spectral density of the fundamental thermorefractive noise has Brownian noise frequency dependence ( -2 ) when we only consider one thermal mode of the WGM resonator. However, summing over many modes results in the pink frequency noise ( -1.5 ). Our experimental data confirms this theoretical prediction. This is a fundamentally important observation for a physical structure with limited dimensionality. It is worth noting that similar results were predicted for optical mirror coatings which is important in classical and quantum metrology [8,9].

Supplementary Note 5. Sub-100 Hz linewidth WGM resonator-stabilized laser
Characterizing the spectral linewidth of the cavity-stabilized laser from the FNPSD is not trivial due to the non-unified definition of linewidth of a flickering or drifting laser frequency and therefore we estimate spectral linewidth with two different methods. The integral linewidth is first calculated from the power spectral density with the phase noise method [10]. An effective linewidth of 119  2 Hz is deduced from the raw FNPSD measurement from the olive curve in Figure 3a. We note that most of the residual noise and linewidth contribution arises from the 60 Hz and 120 Hz electrical power line frequency noise. When these two dominant noise peaks are excluded, an upper bound 25  0.3 Hz is estimated. By fitting the frequency noise dependence using frequency decomposition method that removes the rest of the spurious frequencies, we estimate the fundamental noise limited spectral purity and its resultant linewidth is determined to be 8.7 Hz. The spectral linewidth is double-checked by the -separation line method [11]. This approach implements a simple geometric line based on the low-pass filtered white noise to determine the laser linewidth for an arbitrary frequency noise spectrum. The -separation line is plotted in Figure 3a (dashed gray line). By integrating the FNPSD up to 16 Hz offset frequency which is the region above the -separation line, we determine the linewidth of 30.6  0.1 Hz.

Supplementary Note 6. Random walk noise distribution
We calculated the probability distribution [12] for N=100, where N is the number of samples.
The probability distributions for the larger N are also similarly calculated. We took 30 sets of