Realization of a pulsed optically pumped Rb clock with a frequency stability below [Formula: see text].

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 [Formula: see text] at 1s and achieves the value of [Formula: see text] 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.

Kr-Based Buffer Gas for Rb Vapor-Cell Clocks.

Optically pumped Rb vapor cell clocks are by far the most used devices for timekeeping in all ground and space applications. The compactness and the robustness of this technology make Rb clocks extremely well fit to a large number of applications, including GNSS, telecommunication, and network synchronization. Many efforts are devoted to improve the stability of Rb clocks and reduce their environmental sensitivity. In this article, we investigate the use of a novel mixture of buffer gas based on Kr and N2, capable of reducing by more than one order of magnitude the barometric and temperature sensitivities of the clock, with possible improvement of their long-term stability.

Loaded Microwave Cavity for Compact Vapor-Cell Clocks.

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 article, we present a compact cavity-cell assembly based on a dielectric-loaded cylindrical resonator. The loaded cavity resonating at 6.83 GHz has an external volume of only 35 cm3 and accommodates a vapor cell with 0.9-cm3 inner volume. The proposed design aims at strongly reducing the core of the atomic clock, maintaining, at the same time, high-performing short-term stability ( σy(τ) ≤ 5×10-13 τ-1/2 standard Allan deviation). The proposed structure is characterized in terms of microwave field uniformity and atom-field coupling with the aid of finite-element 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.

Phase-Noise and Amplitude-Noise Measurement of DACs and DDSs.

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 (DACs) and direct digital synthesizers (DDSs) based on the 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: 1) the noise specs of the PN analyzer are relaxed by a factor equal to the carrier suppression ratio and 2) 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 an instrument is extremely simple and easy to implement with a power-detector diode followed by an FFT analyzer. Unlike the classical bridge (interferometric) method, there is no need for external line stretcher and variable attenuators because phase and amplitude controls are 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 ±1 dB over 7.5 decades. Due to the simplicity, reliability, and low background noise, this method has the potential to become the standard method for the AN and PN measurements of DACs and DDSs.

Intensity Detection Noise in Pulsed Vapor-Cell Frequency Standards.

Laser intensity noise is currently recognized as one of the main factors limiting the short-term stability of vapor-cell clocks. In this article, we propose a signal theory approach to estimate the contribution of the laser intensity fluctuations to the short-term stability of vapor-cell clocks working in a pulsed regime. Specifically, given the laser intensity noise spectrum, an analytical expression is derived to evaluate its impact on the clock Allan deviation (ADEV). The theory has been tested for two intensity noise spectra of interest in clock applications: white frequency noise and flicker noise. The predicted results turn out to be in good agreement with experiments performed with a prototype of pulsed optically pumped Rb-cell clock, and can be extended to other compact clocks.

KLTS: A Rigorous Method to Compute the Confidence Intervals for the Three-Cornered Hat and for Groslambert Covariance.

The three-cornered hat/Groslambert Covariance (GCov) 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-Loève Tansform using Sufficient statistics) method which uses these estimators to consider 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 Allan variance (AVAR). This CDF provides also a stability estimator that 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.

Frequency Stability Measurement of Cryogenic Sapphire Oscillators With a Multichannel Tracking DDS and the Two-Sample Covariance.

This paper shows the first measurement of three 100-MHz signals exhibiting fluctuations from 2×10-16 to parts in 10-15 for an integration time τ 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 six-channel tracking direct digital synthesizer (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/√m , the number of measurements, down to 10-18 at τ = 104 s. For consistency check, we compare the results to the traditional TCH method beating the 10-GHz outputs down to the megahertz region. Given the flexibility of the TDDS, our methods find an immediate application to the measurement of the 250-MHz output of the femtosecond combs.

Phase Noise and Frequency Stability of the Red-Pitaya Internal PLL.

In field-programmable gate array platforms, the main clock is generally a low-cost quartz oscillator whose stability is of the order of 10-9 to 10-10 in the short term and 10-7 to 10-8 in the medium term, with the uncertainty of tens of ppm. Better stability is achieved by feeding an external reference into the internal phase-locked loop (PLL). We report the noise characterization of the internal PLL of Red-Pitaya platform, an open-source embedded system architected around the Zynq 7010 System on Chip, with analog-to-digital and digital-to-analog converters. Our experiments show that, providing an external 10-MHz reference, the PLL exhibits a residual frequency stability of 1.2×10-12 at 1 s and 1.3×10-15 at 4000 s, Allan deviation in 5-Hz bandwidth. These results help to predict the PLL stability as a function of frequency and power of the external reference, and provide guidelines for the design of precision instrumentation, chiefly intended for time and frequency metrology.

A scalable hardware and software control apparatus for experiments with hybrid quantum systems.

Modern experiments with fundamental quantum systems - like ultracold atoms, trapped ions, and 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 time scale. 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 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. 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.

Reducing Cavity-Pulling Shift in Ramsey-Operated Compact Clocks.

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. The stabilization allows us to reduce the microwave field-amplitude fluctuations, which in turn impact the clock frequency through cavity pulling. The proposed technique has shown to be effective to improve our clock frequency stability 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.

Avoiding Aliasing in Allan Variance: An Application to Fiber Link Data Analysis.

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 (AVAR) 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.

Active disturbance rejection control of temperature for ultrastable optical cavities.

This paper describes the application of a novel active disturbance rejection control (ADRC) to the stabilization of the temperature of two ultra-stable Fabry-Perot cavities. The cavities are 10 cm long and entirely made of ultralow- expansion glass. The control is based on a linear extended state observer that estimates and compensates the disturbance in the system in real time. The resulting control is inherently robust and easy to tune. A digital implementation of ADRC gives a temperature instability of 200 µK at one day of integration time.

Enhanced temperature sensitivity in vapor-cell frequency standards.

We report on the measurement of an anomalously large temperature sensitivity of the clock frequency in a Rb cell with buffer gas. The effect is observed in a prototype of pulsed optically pumped frequency standard which allows high resolution measurements because of its frequency stability at the level 1.7 × 10(-13) for 1 s of measurement time. We attribute this phenomenon to the geometry of the interaction and to the presence in the cell of temperature inhomogeneities that may enhance the temperature sensitivity of the clock frequency via the buffer gas pressure coefficient. We also propose some solutions to reduce this unwanted effect that may limit the medium-long-term performances of high-frequency-stability vapor-cell clocks.

Pulsed optically pumped rubidium clock with high frequency-stability performance.

In this paper, we present the performance of a vapor-cell rubidium frequency standard working in the pulsed regime, in which the clock signal is represented by a Ramsey pattern observed on an optically detected laser absorption signal. The main experimental results agree with previously reported theoretical predictions. In particular, we measured a relative frequency stability of σy(τ) - 1.6 × 10(-13)τ-1/2 for integration times, τ, up to 200 s, which represents a record in short-term stability for a vapor-cell clock. We also discuss the most important physical phenomena that contribute to this result.

Microwave cavities for vapor cell frequency standards.

In this paper, we report an analysis of the design criteria of microwave cavities for vapor cell frequency standards. Two main geometries exploited in those devices are considered: the cylindrical cavity, used, for example, in the coherent population trapping maser and in the pulsed optically pumped (POP) clock, and the spherical cavity used in the isotropically laser cooled clock. The cavity behavior is described through a lumped equivalent circuit in which the input coupling loop, the dielectric cell containing the atoms and the diodes for frequency tuning or Q control are taken into account. In particular, the effect of the cell on the cavity resonance frequency is analytically evaluated via a first-order perturbation approach. The theory is found in good agreement with the experiments performed with two different cylindrical cavities used for the POP clock; the model here developed can then be helpful in the design of the cavity system. The general principles here reported can be adapted to other standards, such as atomic fountains and hydrogen masers, and to other modes and/or geometries.

Planar-waveguide external cavity laser stabilization for an optical link with 10(-19) frequency stability.

We stabilized the frequency of a compact planar-waveguide external cavity laser (ECL) on a Fabry-Perot cavity (FPC) through a Pound-Drever-Hall scheme. The residual frequency stability of the ECL is 10(-14), comparable to the stability achievable with a fiber laser (FL) locked to an FPC through the same scheme. We set up an optical link of 100 km, based on fiber spools, that reaches 10(-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.

Medium-long term frequency stability of pulsed vapor cell clocks.

In this paper we report an analysis of the physical phenomena that can affect the frequency stability of optically pumped vapor cell clocks working in pulsed regime. It is well known that the pulsed approach allows a strong reduction of the light shift that is one of the main sources of frequency instability. However, other instability sources can degrade clock performance by limiting both short- and medium-term frequency stability. After recognizing the different noise sources and realizing how they are transferred to the clock transition, we propose some technical solutions to limit their effects, extending the region of white frequency noise up to integration times tau of the order of 10(4) s.

Cryogenic fountain development at NIST and INRIM: preliminary characterization.

This paper describes the new twin laser-cooled Cs fountain primary frequency standards NIST-F2 and ITCsF2, and presents some of their design features. Most significant is a cryogenic microwave interrogation region which dramatically reduces the blackbody radiation shift. We also present a preliminary accuracy evaluation of IT-CsF2.

Electronics for the pulsed rubidium clock: design and characterization.

Pulsing the different operation phases of a vapor-cell clock (optical pumping, interrogation, and detection) has been recognized as one of the most effective techniques to reduce light shift and then to improve the stability perspectives of vapor cell clocks. However, in order to take full advantage of the pulsed scheme, a fast-gated electronics is required, the times involved being of the order of milliseconds. In this paper we describe the design and the implementation of the electronics that synchronizes the different phases of the clock operation, as well as of the electronics that is mainly devoted to the thermal stabilization of the clock physics package. We also report some characterization measurements, including a measurement of the clock frequency stability. In particular, in terms of Allan deviation, we measured a frequency stability of 1.2 x 10(-12) tao(-1/2) for averaging times up to tao = 10(5) s, a very interesting result by itself and also for a possible space application of such a clock.

The pulsed rubidium clock.

In this paper, we describe a laboratory prototype of pulsed optically pumped clock based on a rubidium vapor cell with buffer gas. The measured frequency stability (overlapping Allan deviation) is sigma(y)(tau) = 3 x 10(-12)tau(-1/2) and the level of 4 x 10(-14) is reached for averaging time of r = 3 x 10(14) s. For the same set of data, the statistical tool Theol predicts a frequency stability of 2 x 10(-14) for tau = 10(5) s. This result confirms the theoretical predictions regarding this kind of frequency standard and makes it very attractive for satellite navigation and space applications in which a simple and reliable implementation is required, and the short and medium term stability (till one day) is the main concern.