On-line estimation of local oscillator noise and optimisation of servo parameters in atomic clocks

Abstract

For atomic frequency standards in which fluctuations of the local oscillator (LO) frequency are the dominant noise source, we examine the role of the servo algorithm that predicts and corrects these frequency fluctuations. We derive the optimal linear prediction algorithm, showing how to measure the relevant spectral properties of the noise and optimise servo parameters while the standard is running, using only the atomic error signal. We find that, for realistic LO noise spectra, a conventional integrating servo with a properly chosen gain performs nearly as well as the optimal linear predictor. Using simple analytical models and numerical simulations, we establish optimum probe times as a function of clock atom number and of the dominant noise type in the local oscillator. We calculate the resulting LO-dependent scaling of achievable clock stability with atom number for product states as well as for maximally-correlated states.The instability of frequency standards limits the total uncertainty achievable in a measurement of finite duration [1,2]. This limit can be practically relevant even when performing measurements of static frequency ratios, since many-month-long measurement campaigns place stringent demands on the reliability of all components in an experiment. Instability becomes a fundamental concern when attempting to measure time-varying frequency ratios. For instance, in the emerging field of chronometric leveling [3][4][5], direct observation of tidal fluctuations expected in the gravitational red shift [6] requires frequency ratio measurements with a fractional uncertainty at the level of 10 −18 to be completed in a matter of hours. Physics beyond the Standard Model might be detectable in clock frequency ratio measurements as postulated transient shifts associated with dark-matter domain walls [7] or ultralight scalar darkmatter candidates [8,9]. Searches for such signals require the highest possible measurement resolution at timescales where the statistical uncertainty due to instability plays a far greater role than long-term systematic uncertainty.Of the noise processes contributing to the instability of atomic frequency standards, the most fundamental one is quantum projection noise [10], which arises from the discreteness in the measurement results obtainable from a finite number of atoms. For an ensemble of N uncorrelated two-level atoms, this noise imposes a minimum statistical uncertaintyon any measurement of the phase accumulated in an atomic superposition state. For a standard operating at a frequency ω and in the ideal case of Ramsey interrogation without technical noise, this leads to a long-term fractional * Ian.Leroux@nrc-cnrc.gc.ca; Current Address: National Research Council Canada, Ottawa, Ontario, Canada K1A 0R6 instability [11]where T is the duration of a single Ramsey interrogation and T c is the length of the frequency standard's operating cycle, such that τ/T c measurements can be performed in an averaging time τ. This quantum projection noise limit (QPN) 1 for clocks using unco...

Robust optical clock transitions in trapped ions using dynamical decoupling

“…One of the many potential applications of the proposed scheme is a cascaded clock [75][76][77] in which a clock laser is first stabilized to an ensemble of Ca + ions to improve its phase coherence time and thus allow extended probing times [24,25] in a high-accuracy single-ion clock (e.g. Al + or Yb + ).…”

“…It therefore allows the operation of a multi-ion clock [26] using ion species whose clock transitions have a non-vanishing differential electric quadrupole moment. One of the many possible applications of such a multi-ion frequency reference is the phase stabilization of a probe laser for a single ion clock to allow nearlifetime-limited probe times and correspondingly reduced statistical uncertainties [24,25].…”

“…Furthermore, a statistical uncertainty reaching 2.9×10 −16 in 1s is estimated for a cascaded clock scheme in which the dynamically decoupled Coulomb crystal clock stabilizes the interrogation laser for an Al 27 + clock.- [3,22,23]. The statistical uncertainty can be improved by probing for longer times, ultimately limited by the excited clock state lifetime or the laser coherence time [24,25]. Alternatively, the number of probed ions can be increased.Recently, multi-ion clock schemes have been proposed to address this issue [26][27][28][29].…”

“…- [3,22,23]. The statistical uncertainty can be improved by probing for longer times, ultimately limited by the excited clock state lifetime or the laser coherence time [24,25]. Alternatively, the number of probed ions can be increased.…”

Dynamics of bad-cavity-enhanced interaction with cold Sr atoms for laser stabilization

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