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We report on the development of a sub-kelvin, single-crystal silicon Fabry-Perot cavity. Operating such a cavity below 1~K should reduce the thermal noise limit of the cavity, and by this way address the current limitations of ultrastable lasers. To further decrease mechanical losses, mirrors with silicon substrates and crystalline coatings are optically contacted to the spacer, resulting in a room-temperature finesse of 220,000. To operate our cavity at sub-kelvin temperatures, we use a dilution refrigerator able to reach temperatures down to 10 mK. We have designed a mechanical mount to house our cavity in such a cryostat, with optimized heat transfer that will decrease the cooldown time for temperatures below 1~K. The estimated thermal noise is projected to be $\sim 7{\times}10^{-19}$ at 100~mK. However, silicon cavities with crystalline mirror coatings at cryogenic temperatures have shown birefringence correlated frequency fluctuations as well as unknown additional noise mechanisms \cite{yu2023, kedar2023}. We have measured a room-temperature TEM$_{00}$ birefringent mode splitting of about 250 kHz. Understanding and measuring these noise mechanisms will be a key to attaining fractional frequency stabilities beyond state-of-the-art., Comment: 9 pages, 5 figures

Cylindrical WGM resonators machined in high-quality sapphire monocrystal cooled down to liquid helium temperature offer exceptionally-high Q-factors in the microwave frequency domain. Such a resonator constitutes the core of an ultra-stable oscillator featuring fractional frequency stability better than 1e-15 at short integration times. As in any cylindrical resonant structure, the WGM resonator presents a two fold degeneracy. When a defect breaks the cylindrical symmetry of the resonator, the WGMs split and appear as doublets. In the high-quality sapphire resonator, the frequency separation of these twin modes varies from one mode order to another with a maximum value of a few tens of kHz. While the mode splitting for a given mode was considered until now unpredictable and intrinsic to each resonator since resulting a priori from randomly distributed defects. we show here, at the contrary, that the observed mode splitting found on all the sapphire resonators whatever their origin mainly comes from a perfectly determined defect resulting from the manufacturing processes., Comment: 7 pages, 10 figures

We report in this letter the outstanding frequency stability performances of an autonomous cryogenique sapphire oscillator presenting a flicker frequency noise floor below 2e-16 near 1,000 s of integration time and a long term Allan Deviation (ADEV) limited by a random walk process of 1e-18/sqr(tau). The frequency stability qualification at this level called for the implementation of sophisticated instrumentation associated with ultra-stable frequency references and ad hoq averaging and correlation methods., Comment: 4 pages, 2 figures

The cryogenic sapphire oscillator (CSO) is a highly specialized machine, which delivers a reference signal exhibiting the lowest frequency fluctuations. For the best units, the Allan deviation (ADEV) is <1e-15 for integration time between 1 and 10,000 s, with a drift <1e-14 in one day.The oscillator is based on a sapphire monocrystal resonating at 10 GHz in a whispering-gallery mode, cooled at 6 K for highest Q-factor and zero thermal coefficient. We report on the progress accomplished implementing eleven CSOs in about 10 years since the first sample delivered to the ESA station in Argentina. Short-term stability is improved by a factor of 3-10, depending on the integration time, and the refrigerator's electric power is reduced to 3 kW. Frequency stability and overall performances are reproducible, with unattended operation between scheduled maintenance every two years. The CSO is suitable to scientific applications requiring extreme frequency stability with reliable long-term operation. For example, the flywheel for primary frequency standards, the ground segment of GNSS, astrometry, VLBI, and radio astronomy stations., Comment: 7 pages, 12 figures

The Cryogenic Sapphire Oscillator is today recognized for its unprecedented frequency stability, mainly coming from the exceptional physical properties of its resonator made in a high quality sapphire crystal. With these instruments, the fractional frequency measurement resolution, currently of the order of 1e-16, is such that it is possible to detect very small phenomena like residual resonator environmental sensitivities. Thus, we highlighted an unexpected magnetic sensitivity of the Cryogenic Sapphire Oscillator (CSO) at low magnetic field. The fractional frequency sensitivity has been preliminary evaluated to 1e-13/Gauss, making this phenomenon a potential cause of frequency stability limitation. In this paper we report the experimental data related to the magnetic sensitivity of the quasi-transverse magnetic Whispering Gallery (WGH) modes excited in sapphire crystals differing from their paramagnetic contaminants concentration. The magnetic behavior of the WGH modes does not follow the expected theory combining the Curie law and the Zeeman effect affecting the Electron Spin Resonance of the paramagnetic ions present in the crystal., Comment: 7 pages,10 figures, to be submitted to PRB

The three-cornered hat / Groslambert Covariance methods are widely used to estimate the stability of each individual clock in a set of three, but no method gives reliable confidence intervals for large integration times. We propose a new KLTS (Karhunen-Lo\`eve Tansform using Sufficient statistics) method which uses these estimators to take into account the statistics of all the measurements between the pairs of clocks in a Bayesian way. The resulting Cumulative Density Function (CDF) yields confidence intervals for each clock AVAR. This CDF provides also a stability estimator which is always positive. Checked by massive Monte-Carlo simulations, KLTS proves to be perfectly reliable even for one degree of freedom. An example of experimental measurement is given., Comment: 7 pages, 5 figures

This article shows the first measurement of three 100 MHz signals exhibiting fluctuations from 2E-16 to parts in 1E-15 for integration time tau between 1 s and 1 day. Such stable signals are provided by three Cryogenic Sapphire Oscillators (CSOs) operating at about 10 GHz, also delivering the 100 MHz output via a dedicated synthesizer. The measurement is made possible by a 6-channel Tracking DDS (TDDS) and the two-sample covariance tool, used to estimate the Allan variance. The use of two TDDS channels per CSO enables high rejection of the instrument background noise. The covariance outperforms the Three-Cornered Hat (TCH) method in that the background converges to zero "out of the box," with no need of the hypothesis that the instrument channels are equally noisy, nor of more sophisticated techniques to estimate the background noise of each channel. Thanks to correlation and averaging, the instrument background (AVAR) rolls off with a slope 1/sqrt(m), the number of measurements, down to 1E-18 tau = 1E4 s. For consistency check, we compare the results to the traditional TCH method beating the 10 GHz outputs down to the MHz region. Given the flexibility of the TDDS, our methods find immediate application to the measurement of the 250 MHz output of the FS combs., Comment: 9 pages, 12 figures, two column

Virtually all commercial instruments for the measurement of the oscillator PM noise make use of the Cross Spectrum method (arXiv:1004.5539 [physics.ins-det], 2010). High sensitivity is achieved by correlation and averaging on two equal channels which measure the same input, and reject the background of the instrument. We show that a systematic error is always present if the thermal energy of the input power splitter is not accounted for. Such error can result in noise under estimation up to a few dB in the lowest-noise quartz oscillators, and in a complete nonsense in the case of cryogenic oscillators. As another alarming fact, the presence of metamaterial components in the oscillator results in unpredictable behavior and large errors, even in well controlled experimental conditions. We observed a spread of 40 dB in the phase noise spectra of an oscillator, just replacing the output filter., Comment: 20 pages, 11 figures, 37 bibliography items

We present the design of a compact optical clock based on the $^2S_{1/2} \rightarrow ^2D_{3/2}$ 435.5 nm transition in $^{171}$Yb$^+$. The ion trap will be based on a micro-fabricated circuit, with surface electrodes generating a trapping potential to localize a single Yb ion a few hundred $\mu$m from the electrodes. We present our trap design as well as simulations of the resulting trapping pseudo-potential. We also present a compact, multi-channel wavelength meter that will permit the frequency stabilization of the cooling, repumping and clear-out lasers at 369.5 nm, 935.2 nm and 638.6 nm needed to cool the ion. We use this wavelength meter to characterize and stabilize the frequency of extended cavity diode lasers at 369.5 nm and 638.6 nm., Comment: 7 pages, 5 figures. Proc. of the 8th FSM 2015, Potsdam, Germany. To be published in IOP Journal of Physics: Conference Series

We present the characterization of commercial tunnel diode low-level microwave power detectors at room and cryogenic temperatures. The sensitivity as well as the output voltage noise of the tunnel diodes are measured as functions of the applied microwave power, the signal frequency being 10 GHz. We highlight strong variations of the diode characteristics when the applied microwave power is higher than few microwatt. For a diode operating at ${4}$ K, the differential gain increases from ${1,000}$ V/W to about ${4,500}$ V/W when the power passes from ${-30}$ dBm to ${-20}$ dBm. The diode present a white noise floor equivalent to a NEP of ${0.8}$ pW/ ${\sqrt{\mathrm{Hz}}}$ and ${8}$ pW/${ \sqrt{\mathrm{Hz}}}$ at 4 K and 300 K respectively. Its flicker noise is equivalent to a relative amplitude noise power spectral density ${S_{\alpha}(1~\mathrm{Hz})=-120}$~dB/Hz at ${4}$ K. Flicker noise is 10 dB higher at room temperature., Comment: 8 pages and 16 figures

We present the characterisation of three Cryogenic Sapphire Oscillators using the three-corner-hat method. Easily implemented with commercial components and instruments, this method reveals itself very useful to analyse the frequency stability limitations of these state-of-the-art ultra-stable oscillators. The best unit presents a fractional frequency stability better than 5e-16 at 1 s and below 2e-16 for integration times less than 5,000s., Comment: 7 pages, 8 figures

We present the characterization of 8-12 GHz whispering gallery mode resonators machined in high-quality sapphire crystals elaborated with different growth techniques. These microwave resonators are intended to constitute the reference frequency of ultra-stable Cryogenic Sapphire Oscillators. We conducted systematic tests near 4 K on these crystals to determine the unloaded Q-factor and the turnover temperature for whispering gallery modes in the 8-12 GHz frequency range. These characterizations show that high quality sapphire crystals elaborated with the Heat Exchange or the Kyropoulos growth technique are both suitable to meet a fractional frequency stability better than 1x10-15 for 1 s to 10.000 s integration times., Comment: 7 figures

Here, we study the paramagnetic ions behavior in presence of a strong microwave electromagnetic field sustained inside a cryogenic sapphire whispering gallery mode resonator. The high frequency measurement resolution that can be now achieved by comparing two CSOs permit for the first time to observe clearly the non-linearity of the resonator power sensitivity. These observations that in turn allow us to optimize the CSO operation, are well explained by the Electron Spin Resonance (ESR) saturation of the paramagnetic impurities contained in the sapphire crystal., Comment: 8 pages, 9 figures

To create a stable signal from a cryogenic sapphire maser frequency standard, the frequency-temperature dependence of the supporting Whispering Gallery mode must be annulled. We report the ability to control this dependence by manipulating the paramagnetic susceptibility of Fe3+ ions in the sapphire lattice. We show that the maser signal depends on other Whispering Gallery modes tuned to the pump signal near 31 GHz, and the annulment point can be controlled to exist between 5 to 10 K depending on the Fe3+ ion concentration and the frequency of the pump. This level of control has not been achieved previously, and will allow improvements in the stability of such devices., Comment: 17 pages, 10 figures

This article reports on the long-term frequency stabilty characterisation of a new type of cryogenic sapphire oscillator using an autonomous pulse-tube cryocooler as its cold source. This new design enables a relative frequency stability of better than 4.5e-15 over one day of integration. This represents to our knowledge the best long-term frequency stability ever obtained with a signal source based on a macroscopic resonator., Comment: 9 pages, 5 figures

The zero-field Fe3+:sapphire whispering-gallery-mode maser oscillator exhibits several alluring features: Its output is many orders of magnitude brighter than that of an active hydrogen maser and thus far less degraded by spontaneous-emission (Schawlow-Townes) and/or receiving-amplifier noise. Its oscillator loop is confined to a piece of mono-crystalline rock bolted into a metal can. Its quiet amplification combined with high resonator Q provide the ingredients for exceptionally low phase noise. We here concentrate on novelties addressing the fundamental conundrums and technical challenges that impede progress. (1) Roasting: The "mase-ability" of sapphire depends significantly on the chemical conditions under which it is grown and heat-treated. We provide some fresh details and nuances here. (2) Simplification: This paper obviates the need for a Ka-band synthesizer: it describes how a 31.3 GHz loop oscillator, operating on the preferred WG pump mode, incorporating Pound locking, was built from low-cost components. (3) "Dark Matter": A Siegman-level analysis of the experimental data determines the substitutional concentration of Fe3+ in HEMEX to be less than a part per billion prior to roasting and up to a few hundred ppb afterwards. Chemical assays, using different techniques (incl. glow discharge mass spectra spectroscopy and neutron activation analysis) consistently indicate, however, that HEMEX contains iron at concentrations of a few parts per million. Drawing from several forgotten-about/under-appreciated papers, this substantial discrepancy is addressed. (4) Excitons: Towards providing a new means of controlling the Fe3+:sapph. system, a cryogenic sapphire ring was illuminated, whilst masing, with UV light at wavelengths corresponding to known electronic and charge-transfer (thus valence-altering) transitions. Preliminary experiments are reported., Comment: pdf only; submitted to the proceedings of the 24th European Frequency and Time Forum, 13-15th April, 2010

We report observations of the gyrotropic change in magnetic susceptibility of the Fe$^{3+}$ electron paramagnetic resonance at 12.037GHz (between spin states $|1/2>$ and $|3/2>$) in sapphire with respect to applied magnetic field. Measurements were made by observing the response of the high-Q Whispering Gallery doublet (WGH$_{\pm17,0,0}$) in a Hemex sapphire resonator cooled to 5 K. The doublets initially existed as standing waves at zero field and were transformed to traveling waves due to the gyrotropic response., Comment: Accepted for publication in Phys. Rev. B

As a consequence of a general trend in the physics of oscillators and clocks towards optics, phase and frequency metrology is rapidly moving to optics too. Yet, optics is not replacing the traditional radio-frequency (RF) and microwave domains. Instead, it adds tough challenges. Precision frequency-stability measurements are chiefly based on the measurement of phase noise, which is the main focus of this article. Major progress has been achieved in two main areas. The first is the extreme low-noise measurements, based on the bridge (interferometric) method in real time or with sophisticated correlation and averaging techniques. The second is the emerging field of microwave photonics, which combines optics and RF/microwaves. This includes the femtosecond laser, the two-way fiber links, the noise measurement systems based on the fiber and the photonic oscillator. Besides, the phenomenology of flicker (1/f) noise is better understood, though the ultimate reasons are still elusive., Comment: Invited. 7th Symposium on Frequency Standards and Metrology, 5-11 October 2008, Pacific Grove, California, USA. 10 pages, 10 figures, 20 references

We report observations of the Schawlow-Townes noise limit in a cryogenic sapphire secondary frequency standard. The effect causes a fundamental limit to the frequency stability, and was measured through the novel excitation of a bimodal maser oscillation of a Whispering Gallery doublet at $12.04 GHz$. The beat frequency of $10 kHz$ between the oscillations enabled a sensitive probe for this measurement of fractional frequency instability of $10^{-14}\tau^{-1/2}$ with only 0.5 $pW$ of output power., Comment: Published in PRL 100, 233901 (2008)

The frequency flicker of an oscillator, which appears as a 1/f^3 line in the phase noise spectral density, and as a floor on the Allan variance plot, originates from two basic phenomena, namely: (1) the 1/f phase noise turned into 1/f frequency noise via the Leeson effect, and (2) the 1/f fluctuation of the resonator natural frequency. The discussion on which is the dominant effect, thus on how to improve the stability of the oscillator, has been going on for years without giving a clear answer. This article tackles the question by analyzing the phase noise spectrum of several commercial oscillators and laboratory prototypes, and demonstrates that the fluctuation of the resonator natural frequency is the dominant effect. The investigation method starts from reverse engineering the oscillator phase noise in order to show that if the Leeson effect was dominant, the resonator merit factor Q would be too low as compared to the available technology., Comment: 20 pages, list of symbols, 1 table, 6 figures, 43 references

The measurement of the close-to-the-carrier noise of rf and microwave devices is a relevant issue in time and frequency metrology and in some fields of electronics, physics and optics. While phase noise is the main concern, amplitude noise is often of interest. The highest sensitivity is achieved with the bridge (interferometric) method, which consists of the amplification and synchronous detection of the noise sidebands after suppressing the carrier by vector subtraction of an equal signal. A substantial progress in understanding the flicker noise mechanism of the interferometer results in new schemes that improve by 20--30 dB the sensitivity at low Fourier frequencies. The article provides the complete theory and detailed design criteria, and reports on the implementation of a prototype. In real-time measurements, a background noise of -175 -180 dBrad^2/Hz has been obtained at f=1 Hz off the 100 MHz carrier. Exploiting correlation and averaging in similar conditions, the sensitivity exceeds -185 dBrad^2/Hz at f=1 Hz. A residual noise of -203 dBrad^2/Hz at f=250 Hz off the carrier has been obtained, while the ultimate noise floor is still limited by the averaging capability of the correlator. This is equivalent to a S/N ratio of 2E20 with a frequency spacing of 2.5E-6. Applications include the measurement of the properties of materials and the observation of weak flicker-type physical phenomena, out of reach for other instruments. We measured the flicker noise of a by-step attenuator (-171 dB[rad^2]/Hz at f=1 Hz) and of the ferrite noise of a reactive power divider (-173.7 dB[rad^2]/Hz at f=1 Hz) without need of correlation., Comment: 30 pages, 15 figures, 35 references