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Local oscillator induced degradation of medium-term stability in passive atomic frequency standards

G.J. Dick, John D. Prestage, C. A. Greenhall, Lute Maleki

Published May 1, 1990• 76 citations

Abstract

As the performance of passive atomic frequency standards improves, a new limitation is encountered due to frequency fiuctuations in an ancillary local osciktor (L.O.). The efect is due to time variation in the gain of the feedback which compensates L.O. frequenq fluctuations. The high performance promised by new microwave and optical trapped ion standark may be severely compromised by this effect. We present un anabsk uf this performance limitatwn for the case of sequentially interrogated standards. The time dependence oJ the sensitivity of the interrogution process to L.O. frequenq fluctuations is evalu-ated for singde-pulse and double-puke "Ramvey " RF interrogation and also for amplitude moduluted pukes. The e$ect of these various time dependencies on performance of the standard is calculated for an L.O. with frequency fluctuations showing a typical 1 / f spectral density. A limiting 1 1 6 dependent deviation of fre-quencyjluctuutions k calculated as a function of pulse lengths, dead time, and pulse overlap. We also present conceptual und hardwure+riented solutions to this problem which achieve a much more nearly convtant sensitivity to L.O.fluctuatiofiv. Solutionr involve: we of double-puke interrogation; alternute interrogation of mulliple traps so that the "dead time " of one trap can be covered by operution of the other; and the use of double-pulse interrogation for two traps, so that during the time of the RFpulves, the increasing sensitivity of one trup tends to compensate for the decreasing sensitivity of the otht: A solution making use of amplitude-modulated pulses is aLso presented which shows nominally zero time variation.

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An experimental test of Local-Oscillator (L.O.) degradation of time-sequential passive oscillator performance was accomplished using the TAG-controlled crystal oscillator described in an accompanying paper. For this quantitative demonstration, an electronic frequency-counter discriminator replaced the trapped ion physics package in a closed-loop experiment. With a 10 second cycle time and 5 second dead time, L.O. limited stability was demonstrated in excellent agreement with theoretical calculations. The results are also in agreement with computer simulations and confirm previous noise analysis for the case of a passive standard with short pulse "Ramsey" interrogation.

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We present an analysis of the frequency stability degradation of an optical atomic frequency standard which is operated sequentially, and that is caused by the frequency noise of the laser used to interrogate the clock transition. This is an extension to optical frequencies of the 'Dick effect' already studied in atomic microwave frequency standards. It is shown that the simple concept of laser linewidth is not the relevant parameter to estimate the frequency stability degradation. Rather, the details of the laser frequency noise spectral density must be accounted for together with the parameters of the interrogation method. We compute the optical Dick effect in the case of a four pulse Ramsey–Bordé interferometer. We show that it may be minimized by optimizing the time sequence so that only the Fourier frequencies with minimum laser noise are sampled. Finally, we evaluate the degradation which would be obtained with one of our lasers used as an interrogation oscillator. This laser reaches a white noise floor S ν ( f ) = 10 −2 Hz 2 /Hz at a Fourier frequency of 100 Hz. We show that with this particular laser, a relative frequency stability of the order of 10 −16 in a 1 s averaging time can be reached using realistic experimental parameters.

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Active optical frequency standards provide interesting alternatives to their passive counterparts. Particularly, such a clock alone continuously generates highly stable narrow-line laser radiation. Thus, a local oscillator is not required to keep the optical phase during a dead time between interrogations as in passive clocks, but only to boost the active clock's low output power to practically usable levels with the current state of technology. Here we investigate the spectral properties and the stability of active clocks, including homogeneous and inhomogeneous broadening effects. We find that for short averaging times the stability is limited by photon shot noise from the limited emitted laser power and at long averaging times by phase diffusion of the laser output. Operational parameters for best long-term stability were identified. Using realistic numbers for an active clock with $^{87}\mathrm{Sr}$, we find that optimized stability of ${\ensuremath{\sigma}}_{y}(\ensuremath{\tau})\ensuremath{\approx}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}/\sqrt{\ensuremath{\tau}[\mathrm{s}]}$ is achievable.

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Extending the fundamental limit of atomic clock stability

Ravid Shaniv, Ayush Agrawal, David B. Hume

Optical atomic clocks have been rapidly developing in recent decades, resulting in major improvements in both precision and accuracy. As a result, they have become instrumental in multiple areas of applied and fundamental research. Despite all atomic frequency references having more than two energy-levels, the commonly used model for evaluating their ultimate limits assumes a two-level atom. This leads to frequency interrogation protocols and theoretical stability bounds that are suboptimal for a true multi-level atom. The most fundamental stability bound assumes two noise sources - quantum projection noise and spontaneous decay from the excited state. In this work, we analyze a model that includes these noise types and is generalized beyond the two-level assumption, where spontaneous decay can branch to more than a single ground state. This model allows for detection and exclusion of atomic frequency interrogations in which the atom decayed, leading to a frequency stability improvement of up to $\approx 4.5 \text{ dB}$ compared with the two-level model. Furthermore, we identify an even greater stability enhancement of $\approx 5.4 \text{ dB}$ for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an $^{27}\text{Al}^{+}$ optical ion clock.

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As the performance of passive atomic frequency standards improves, a new limitation is encountered due to frequency fiuctuations in an ancillary local osciktor (L.O.). The efect is due to time variation in the gain of the feedback which compensates L.O. frequenq fluctuations. The high performance promised by new microwave and optical trapped ion standark may be severely compromised by this effect. We present un anabsk uf this performance limitatwn for the case of sequentially interrogated standards. The time dependence oJ the sensitivity of the interrogution process to L.O. frequenq fluctuations is evalu-ated for singde-pulse and double-puke "Ramvey " RF interrogation and also for amplitude moduluted pukes. The e$ect of these various time dependencies on performance of the standard is calculated for an L.O. with frequency fluctuations showing a typical 1 / f spectral density. A limiting 1 1 6 dependent deviation of fre-quencyjluctuutions k calculated as a function of pulse lengths, dead time, and pulse overlap. We also present conceptual und hardwure+riented solutions to this problem which achieve a much more nearly convtant sensitivity to L.O.fluctuatiofiv. Solutionr involve: we of double-puke interrogation; alternute interrogation of mulliple traps so that the "dead time " of one trap can be covered by operution of the other; and the use of double-pulse interrogation for two traps, so that during the time of the RFpulves, the increasing sensitivity of one trup tends to compensate for the decreasing sensitivity of the otht: A solution making use of amplitude-modulated pulses is aLso presented which shows nominally zero time variation.

An experimental test of Local-Oscillator (L.O.) degradation of time-sequential passive oscillator performance was accomplished using the TAG-controlled crystal oscillator described in an accompanying paper. For this quantitative demonstration, an electronic frequency-counter discriminator replaced the trapped ion physics package in a closed-loop experiment. With a 10 second cycle time and 5 second dead time, L.O. limited stability was demonstrated in excellent agreement with theoretical calculations. The results are also in agreement with computer simulations and confirm previous noise analysis for the case of a passive standard with short pulse "Ramsey" interrogation.

We present an analysis of the frequency stability degradation of an optical atomic frequency standard which is operated sequentially, and that is caused by the frequency noise of the laser used to interrogate the clock transition. This is an extension to optical frequencies of the 'Dick effect' already studied in atomic microwave frequency standards. It is shown that the simple concept of laser linewidth is not the relevant parameter to estimate the frequency stability degradation. Rather, the details of the laser frequency noise spectral density must be accounted for together with the parameters of the interrogation method. We compute the optical Dick effect in the case of a four pulse Ramsey–Bordé interferometer. We show that it may be minimized by optimizing the time sequence so that only the Fourier frequencies with minimum laser noise are sampled. Finally, we evaluate the degradation which would be obtained with one of our lasers used as an interrogation oscillator. This laser reaches a white noise floor S ν ( f ) = 10 −2 Hz 2 /Hz at a Fourier frequency of 100 Hz. We show that with this particular laser, a relative frequency stability of the order of 10 −16 in a 1 s averaging time can be reached using realistic experimental parameters.

Active optical frequency standards provide interesting alternatives to their passive counterparts. Particularly, such a clock alone continuously generates highly stable narrow-line laser radiation. Thus, a local oscillator is not required to keep the optical phase during a dead time between interrogations as in passive clocks, but only to boost the active clock's low output power to practically usable levels with the current state of technology. Here we investigate the spectral properties and the stability of active clocks, including homogeneous and inhomogeneous broadening effects. We find that for short averaging times the stability is limited by photon shot noise from the limited emitted laser power and at long averaging times by phase diffusion of the laser output. Operational parameters for best long-term stability were identified. Using realistic numbers for an active clock with

^{87}\mathrm{Sr}

, we find that optimized stability of ${\ensuremath{\sigma}}_{y}(\ensuremath{\tau})\ensuremath{\approx}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}/\sqrt{\ensuremath{\tau}[\mathrm{s}]}$ is achievable.

Optical atomic clocks have been rapidly developing in recent decades, resulting in major improvements in both precision and accuracy. As a result, they have become instrumental in multiple areas of applied and fundamental research. Despite all atomic frequency references having more than two energy-levels, the commonly used model for evaluating their ultimate limits assumes a two-level atom. This leads to frequency interrogation protocols and theoretical stability bounds that are suboptimal for a true multi-level atom. The most fundamental stability bound assumes two noise sources - quantum projection noise and spontaneous decay from the excited state. In this work, we analyze a model that includes these noise types and is generalized beyond the two-level assumption, where spontaneous decay can branch to more than a single ground state. This model allows for detection and exclusion of atomic frequency interrogations in which the atom decayed, leading to a frequency stability improvement of up to

\approx 4.5 \text{ dB}

compared with the two-level model. Furthermore, we identify an even greater stability enhancement of

\approx 5.4 \text{ dB}

for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an

^{27}\text{Al}^{+}

optical ion clock.