### The resonance frequency is repeatedly estimated from sequences of three measurements.

An improved method of making an electronic oscillator track the frequency of an atomic-clock resonance transition is based on fitting a theoretical nonlinear curve to measurements at three oscillator frequencies within the operational frequency band of the transition (in other words, at three points within the resonance peak). In the measurement process, the frequency of a microwave oscillator is repeatedly set at various offsets from the nominal resonance frequency, the oscillator signal is applied in a square pulse of the oscillator signal having a suitable duration (typically, of the order of a second), and, for each pulse at each frequency offset, fluorescence photons of the transition in question are counted. As described below, the counts are used to determine a new nominal resonance frequency. Thereafter, offsets are determined with respect to the new resonance frequency. The process as described thus far is repeated so as to repeatedly adjust the oscillator to track the most recent estimate of the nominal resonance frequency.

The theoretical nonlinear curve is that of the Rabi equation for the shape of the resonance peak

P(y)= | sin^{2}( Π / 2
√
1+ y2) |

1+ y2 |

where the dimensionless variable *y* is related to the duration of the microwave pulse, *T*, and the frequency offset ν – ν_{0 }from the atomic absorption frequency, ν_{0 } , as follows: *y*= 2*T*(ν – ν_{0 }).

Assuming that the signal power has been optimized and that the photon count at a given measurement signal frequency includes a non-resonant background contribution plus a contribution attributable to the resonance, the basic measurement equation for the *i*th measurement is

*C*(*i*) = *B* + *AP*(*y*_{1} – ε)

where *C*(*i*) is the atomic fluorescence photon count, A is atomic fluorescence, and ε is an offset of the nominal resonance frequency from the actual resonance frequency. If measurements are made at three different oscillator frequency offsets (*y*_{1}, *y*_{2}, *y*_{3}), then one has

*C*(*1*) = *B* + *AP*(*y*_{1} – ε)

*C*(*2*) = *B* + *AP*(*y*_{2} – ε)

*C*(*3*) = *B* + *AP*(*y*_{3} – ε)

These Three Plots represent fits of the nonlinear curve to three sets of simulated measurements, each set comprising photon counts at relative frequency offsets (*y* values) of 0 and ±0.761. The parameters of the simulated measurements were *B* = 1.05 × 10^{5} counts, *B* = 3 × 10^{4} counts, and signal-to-noise ratio ≈30." class="caption" align="left">

Repeatedly, for the most recent such set of three measurements (see figure), this set of three equations is inverted to extract *B*, *A*, and ε from the measurement values *C*(1), *C*(2), and *C*(3). Because the solution obtained through inversion of the three equations separates the influences of background light, signal strength, and the offset of the resonance from the nominal resonance frequency, unlike in a prior method, drift in the power of the lamp used to excite the clock atoms to the upper level of the transition does not seem to effect frequency pulling (that is, it does not seem to force a change in the estimate of the resonance frequency).

*This work was done by John D. Prestage, Sang K. Chung, and Meirong Tu of Caltech for NASA’s Jet Propulsion Laboratory. For more information, download the Technical Support Package (free white paper) at www.techbriefs.com/tsp under the Physical Sciences category. NPO-45958*

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