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This paper describes the Light-Shift Laser-Lock (LSLL) technique, a novel method intended for compact atomic clocks that greatly simplifies the laser setup by stabilizing the pumping-laser frequency to the atoms involved in the clock, without the need of an external reference. By alternating two clock sequences with different light shifts, the method estimates and cancels out a controlled amount of induced light shift, acting on the laser frequency. The LSLL technique is compatible with state-of-the-art 3-level clocks and was demonstrated with FPGA-based electronics on a pulsed-optically-pumped (POP) vapor-cell clock developed at INRIM. The results have shown that the LSLL technique operates robustly, having a capture range of gigahertz without significantly compromising clock stability. In our tests, the clock exhibited a white frequency noise of $3.2 \times 10^{-13}$ $\tau^{-1/2}$ for averaging time up to 4000 s, reaching a floor below $1 \times 10^{-14}$ up to 100 000 s. These performance levels meet the requirements of future Global Navigation Satellite Systems (GNSS) on-board clocks, and offer the added benefits of a reduced clock footprint, as well as increased reliability and robustness.

We present the frequency stability performances of a vapor cell Rb clock based on the pulsed optically pumping (POP) technique. The clock has been developed in the frame of a collaboration between INRIM and Leonardo SpA, aiming to realize a space-qualified POP frequency standard. The results here reported were obtained with an engineered physics package, specifically designed for space applications, joint to laboratory-grade optics and electronics. The measured frequency stability expressed in terms of Allan deviation is $1.2 \times 10^{-13}$ at 1 s and achieves the value of $6 \times 10^{-16}$ for integration times of 40000 s (drift removed). This is, to our knowledge, a record result for a vapor-cell frequency standard. In the paper, we show that in order to get this result, a careful stabilization of microwave and laser pulses is required.

This work focuses on the generation of a composite clock for time scales and shows the advantages of a more recent digital approach with respect to a traditional analog one. A digital approach directly processes the information contained into the clock sinusoids, instead of the sinusoids themselves and leads to significant advantages in terms of reliability, performance, complexity, flexibility, size, power consumption and cost. A practical example based on a new digital instrument is provided to show how it is possible to combine state-of-the-art clocks from the Oscillator Imp platform at FEMTO-ST and FEMTO Engineering (Besan\c{c}on, France)., Comment: Proceeding of the IFCS-EFTF 2023, held in Toyama

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 knowledge of the frequency noise spectrum of a diode laser is of interest in several high resolution experiments. Specifically, in laser-pumped vapor cell clocks, it is well established that the laser frequency noise plays a role in affecting the clock performances. It is then important to characterize the frequency noise of a diode laser, especially since such measurements are rarely found in the literature and hardly ever provided by vendors. In this paper, we describe a technique based on a frequency-to-voltage converter that transforms the laser frequency fluctuations into voltage fluctuations. In this way, it is possible to characterize the laser frequency noise power spectral density in a wide range of Fourier frequencies, as required in cell clock applications., Comment: accepted for publication on IEEE Transactions on Instrumentation and Measurement (Oct 2022)

Vapor-cell devices based on microwave interrogation provide a stable frequency reference with a compact and robust setup. Further miniaturization must focus on optimizing the physics package, containing the microwave cavity and atomic reservoir. In this paper we present a compact cavity-cell assembly based on a dielectric-loaded cylindrical resonator. The structure accommodates a clock cell with $0.9 \, \mathrm{cm^3}$ inner volume and has an outer volume of only $35 \, \mathrm{cm^3}$. The proposed design aims at strongly reducing the core of the atomic clock, maintaining at the same time high-performing short-term stability ($\sigma_y(\tau) \leq 5\times 10^{-13} \,\tau^{-1/2}$ standard Allan deviation). The proposed structure is characterized in terms of magnetic field uniformity and atom-field coupling with the aid of finite-elements calculations. The thermal sensitivity is also analyzed and experimentally characterized. We present preliminary spectroscopy results by integrating the compact cavity within a rubidium clock setup based on the pulsed optically pumping technique. The obtained clock signals are compatible with the targeted performances. The loaded-cavity approach is thus a viable design option for miniaturized microwave clocks., Comment: Submitted to IEEE-UFFC

This article proposes a method for the measurement of Phase Noise (PN, or PM noise) and Amplitude Noise (AN, or AM noise) of Digital-to-Analog Converters (DAC) and Direct Digital Synthesizers (DDS) based on modulation-index amplification. The carrier is first reduced by a controlled amount (30-40 dB) by adding a reference signal of nearly equal amplitude and opposite in phase. Then, residual carrier and noise sidebands are amplified and sent to a conventional PN analyzer. The main virtues of our method are: (i) the noise specs of the PN analyzer are relaxed by a factor equal to the carrier suppression ratio; and, (ii) the capability to measure the AN using a PN analyzer, with no need for the analyzer to feature AN measurement. An obvious variant enables AN and PN measurements using an AN analyzer with no PN measurement capability. Such instrument is extremely simple and easy to implement with a power-detector diode followed by a FFT analyzer. Unlike the classical bridge (interferometric) method, there is no need for external line stretcher and variable attenuators because phase and amplitude control is implemented in the device under test. In one case (AD9144), we could measure the noise over 10 decades of frequency. The flicker noise matches the exact $1/f$ law with a maximum discrepancy of $\pm1$ dB over 7.5 decades. Thanks to simplicity, reliability, and low background noise, this method has the potential to become the standard method for the AN and PN measurement of DACs and DDSs., Comment: Extended version, after a talk given at the IEEE IFCS, Orlando, FL, USA, April 2019. 22 pages, 10 figures, 48 references

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

Modern experiments with fundamental quantum systems - like ultracold atoms, trapped ions, single photons - are managed by a control system formed by a number of input/output electronic channels governed by a computer. In hybrid quantum systems, where two or more quantum systems are combined and made to interact, establishing an efficient control system is particularly challenging due to the higher complexity, especially when each single quantum system is characterized by a different timescale. Here we present a new control apparatus specifically designed to efficiently manage hybrid quantum systems. The apparatus is formed by a network of fast communicating Field Programmable Gate Arrays (FPGAs), the action of which is administrated by a software. Both hardware and software share the same tree-like structure, which ensures a full scalability of the control apparatus. In the hardware, a master board acts on a number of slave boards, each of which is equipped with an FPGA that locally drives analog and digital input/output channels and radiofrequency (RF) outputs up to 400 MHz. The software is designed to be a general platform for managing both commercial and home-made instruments in a user-friendly and intuitive Graphical User Interface (GUI). The architecture ensures that complex control protocols can be carried out, such as performing of concurrent commands loops by acting on different channels, the generation of multi-variable error functions and the implementation of self-optimization procedures. Although designed for managing experiments with hybrid quantum systems, in particular with atom-ion mixtures, this control apparatus can in principle be used in any experiment in atomic, molecular, and optical physics., Comment: 10 pages, 12 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

We describe a method to stabilize the amplitude of the interrogating microwave field in compact atomic clocks working in a Ramsey approach. In this technique, we take advantage of the pulsed regime to use the atoms themselves as microwave amplitude discriminators. Specifically, in addition to the dependence on the microwave detuning, the atomic signal after the Ramsey interrogation acquires a dependence on the microwave pulse area (amplitude times duration) that can be exploited to implement an active stabilization of the microwave field amplitude, in a similar way in which the Ramsey clock signal is used to lock the local oscillator frequency to the atomic reference. This stabilization results in a reduced sensitivity of the clock frequency to microwave amplitude fluctuations that are transferred to the atoms through the cavity-pulling effect. The proposed technique is then effective to improve the clock stability and drift on medium and long term. We demonstrate the method for a vapor-cell clock working with a hot sample of atoms but it can be extended to cold-atom compact clocks., Comment: Accepted for publication by IEEE UFFC on April 16th 2018

This article explains phase noise, jitter, and some slower phenomena in digital integrated circuits, focusing on high-demanding, noise-critical applications. We introduce the concept of phase type and time type phase noise. The rules for scaling the noise with frequency are chiefly determined by the spectral properties of these two basic types, by the aliasing phenomenon, and by the input and output circuits. Then, we discuss the parameter extraction from experimental data and we report on the measured phase noise in some selected devices of different node size and complexity. We observed flicker noise between -80 and -130 dBrad^2/Hz at 1 Hz offset, and white noise down to -165 dBrad^2/Hz in some fortunate cases and using the appropriate tricks. It turns out that flicker noise is proportional to the reciprocal of the volume of the transistor. This unpleasant conclusion is supported by a gedanken experiment. Further experiments provide understanding on: (i) the interplay between noise sources in the internal PLL, often present in FPGAs; (ii) the chattering phenomenon, which consists in multiple bouncing at transitions; and (iii) thermal time constants, and their effect on phase wander and on the Allan variance., Comment: 25 pages, 16 figures, 41 references

We demonstrate a vapor cell atomic clock prototype based on continuous-wave (CW) interrogation and double-modulation coherent population trapping (DM-CPT) technique. The DM-CPT technique uses a synchronous modulation of polarization and relative phase of a bi-chromatic laser beam in order to increase the number of atoms trapped in a dark state, i.e. a non-absorbing state. The narrow resonance, observed in transmission of a Cs vapor cell, is used as a narrow frequency discriminator in an atomic clock. A detailed characterization of the CPT resonance versus numerous parameters is reported. A short-term frequency stability of $3.2 \times 10^{-13} \tau^{-1/2}$ up to 100 s averaging time is measured. These performances are more than one order of magnitude better than industrial Rb clocks and comparable to those of best laboratory-prototype vapor cell clocks. The noise budget analysis shows that the short and mid-term frequency stability is mainly limited by the power fluctuations of the microwave used to generate the bi-chromatic laser. These preliminary results demonstrate that the DM-CPT technique is well-suited for the development of a high-performance atomic clock, with potential compact and robust setup due to its linear architecture. This clock could find future applications in industry, telecommunications, instrumentation or global navigation satellite systems., Comment: 14 pages, 25 figures, 2 tables

Optical fiber links are known as the most performing tools to transfer ultrastable frequency reference signals. However, these signals are affected by phase noise up to bandwidths of several kilohertz and a careful data processing strategy is required to properly estimate the uncertainty. This aspect is often overlooked and a number of approaches have been proposed to implicitly deal with it. Here, we face this issue in terms of aliasing and show how typical tools of signal analysis can be adapted to the evaluation of optical fiber links performance. In this way, it is possible to use the Allan variance as estimator of stability and there is no need to introduce other estimators. The general rules we derive can be extended to all optical links. As an example, we apply this method to the experimental data we obtained on a 1284 km coherent optical link for frequency dissemination, which we realized in Italy.

In this paper we propose a simple method to reject the high-frequency noise in the evaluation of statistical uncertainty of coherent optical fiber links. Specifically, we propose a preliminary data filtering, separated from the frequency stability computation. In this way, it is possible to use the Allan deviation as estimator of stability, to get unbiased data, which are representative of the noise process affecting the delivered signal. Our approach is alternative to the use of the modified Allan deviation, which is largely adopted in this field. We apply this processing to the experimental data we obtained on a 1284 km coherent optical link for frequency dissemination, which we realized in Italy. We also show how the so-called Lambda-type commercial phase/frequency counters can be used to this purpose.

We performed a two-way remote optical phase comparison on optical fiber. Two optical frequency signals were launched in opposite directions in an optical fiber and their phases were simultaneously measured at the other end. In this technique, the fiber noise was passively cancelled, and we compared two optical frequencies at the ultimate 1E-21 stability level. The experiment was performed on a 47 km fiber that is part of the metropolitan network for Internet traffic. The technique relies on the synchronous measurement of the optical phases at the two ends of the link, that is made possible by the use of digital electronics. This scheme offers several advantages with respect to active noise cancellation, and can be upgraded to perform more complex tasks.

We report on the implementation and the metrological characterization of a vapor-cell Rb frequency standard working in pulsed regime. The three main parts that compose the clock, physics package, optics and electronics, are described in detail in the paper. The prototype is designed and optimized to detect the clock transition in the optical domain. Specifically, the reference atomic transition, excited with a Ramsey scheme, is detected by observing the interference pattern on a laser absorption signal. \ The metrological analysis includes the observation and characterization of the clock signal and the measurement of frequency stability and drift. In terms of Allan deviation, the measured frequency stability results as low as $1.7\times 10^{-13} \ \tau^{-1/2}$, $\tau$ being the averaging time, and reaches the value of few units of $10^{-15}$ for $\tau=10^{4}$ s, an unprecedent achievement for a vapor cell clock. We discuss in the paper the physical effects leading to this result with particular care to laser and microwave noises transferred to the clock signal. The frequency drift, probably related to the temperature, stays below $10^{-14}$ per day, and no evidence of flicker floor is observed. \ We also mention some possible improvements that in principle would lead to a clock stability below the $10^{-13}$ level at 1 s and to a drift of few units of $10^{-15}$ per day.

We stabilized the frequency of a compact planar-waveguide external cavity laser (ECL) on a Fabry-P\'erot cavity (FPC) through a Pound-Drever-Hall scheme. The residual frequency stability of the ECL is 1E-14, comparable to the stability achievable with a fiber laser (FL) locked to a FPC through the same scheme. We set up an optical link of 100 km, based on fiber spools, that reaches 1E-19 relative stability, and we show that its performances using the ECL or FL are comparable. Thus ECLs could serve as an excellent replacement for FLs in optical links where cost-effectiveness and robustness are important considerations.