We describe the optimization of a Frequency Locked Loop (FLL) in an atomic clock which is based on Coherent Population Trapping (CPT) in 87Rb vapor using the D2 transition. The FLL uses frequency modulation (FM) spectroscopy and we study the effect of FM parameters (modulation frequency and index) on the sensitivity and the signal to noise ratio of the feedback signal in the FLL. The clock which employs a small spherical glass cell containing 87Rb atoms and a buffer gas, exhibits a short term stability of 3×10-11/√τ. The long term relative frequency stability of the 10 MHz output is better than 10-10 with a drift of 10-11 per day.
© 2007 Optical Society of America
Miniature atomic clocks with volumes approaching 1 cm3 and with a power consumption of only few tens of mW have been researched extensively in the past few years . Most small scale clocks employ coherent population trapping (CPT) in either Cesium or Rubidium vapor, and a VCSEL type diode laser [2–4]. The VCSEL is directly modulated at half the hyperfine splitting frequency of the atoms and the first order side bands serve as the two coherent optical fields needed to excite the dark resonance . The small size of the atomic cell, together with the modulation characteristics and various instabilities of the VCSEL  require complex locking schemes  in order to achieve an acceptable clock performance. The locking process entails superimposing low frequency modulation on the microwave drive current to the VCSEL and detection using Lock-in techniques. The interaction of the low frequency components of the optical field with the atomic system has a profound impact on the detected signal. Proper choice of the low frequency modulation parameters increases the signal to noise ratio (SNR) of the demodulated signal which yields optimized clock locking.
This paper describes an experimental optimization of the Frequency-Locked Loop (FLL) in a 87Rb CPT based clock which employs a small spherical glass cell . The FLL is implemented by superimposing low rate frequency modulation (FM) on the microwave drive signal to the diode laser thereby enabling to probe the atomic vapor using the FM spectroscopy scheme [9–11]. Classical FM spectroscopy employs only one FM modulated field. The properties of the demodulated signal for this case have been analyzed [9–11] and measured [10, 11]. However when FM spectroscopy is used in CPT based clocks (where the interacting fields are the side bands of a directly modulated diode laser), each spectral component carries its corresponding FM side bands. The signature of the CPT process on the demodulated signal is consequently different than in the classical case . This difference which may be subtle has nevertheless a profound effect on the sensitivity and SNR of the error signal feeding the FLL. This is demonstrated in the experiments we describe hereon which yield sets of FM parameters (FM modulation frequency and index) that ensure optimum FLL performance.
One set of such optimized FM parameters was used to operate a CPT based Rubidium clock. The clock exhibits a short term stability (in terms of the Allan deviation) of 3×10-11/√τ for time constants of 0.5 sec to 200 sec. The Allan deviation reaches a minimum value of 2.2×10-12 at 500 sec and then increases slightly reaching 2.5×10-12 at 1000 sec. The relative frequency stability, measured over 27 hours was found to be better than 10-10 with a slow drift of only 10-11 per day. These results, obtained for an experimental laboratory set up, resemble state of the art short term stabilities in well engineered small scale atomic clocks [2–4].
2. The system
A schematic of the clock is shown in Fig. 1. A single mode VCSEL emitting at 780.24 nm is driven by a DC bias and a microwave signal at half the 87Rb, D2 transition, hyperfine splitting frequency: 3417.3 MHz. Some CPT based clocks use the D1 (rather than the D2) transition at 794.98 nm [2, 4] since it enables, in principle, a larger contrast of the atomic resonance . The D2 transition was used here since the quality of available 780 nm VCSELs was significantly better than that of the 795 nm VCSELs. Also, the main emphasis of this paper is the optimization of FM spectroscopy and therefore we performed the experiments at 780 nm employing the D2 transition. The VCSEL output is collimated and its polarization is set to be circular. The light impinges on a ball shaped glass cell which contains a mixture of pure 87Rb atoms and a buffer gas. A photograph of such a cell is shown in Fig. 1. The cell used in the present experiments had an outer diameter of 6 mm and an inner diameter of 5 mm which is somewhat larger than those used in chip scale atomic clocks [2–4]. It is constructed using standard glass blowing techniques and is therefore significantly simpler, accessible and more reproducible than cells made by MEMS technologies. The glass cells are somewhat larger than the MEMS fabricated published devices [2–4]. However, a ball of a few mm diameter allows for a very small clock and its slightly larger volume may actually offer somewhat improved performance.
The glass cell, together with a controlled heater and a solenoid are placed in a three layer μ-metal box which attenuates the environmental magnetic field. The cell temperature is stabilized around the optimum temperature of 66 °C to a level of 1 mK. A homogeneous magnetic field of 35 μT pointing in a direction parallel to the optical axis is generated around the vapor cell by the solenoid. This magnetic field lifts the Zeeman degeneracy thereby isolating the 0-0 (clock) transition. The transmitted light is detected by a large area silicon detector and measured by a Lock-in amplifier.
The system uses two control loops: the VCSEL wavelength stabilization servo loop and the FLL or the clock loop. The first loop stabilizes the emitted wavelength by changing the diode laser bias. Various noisy features of the CPT process require detuning the locking point relative to the peak [12, 13]. A red shift of approximately 60 MHz is used in the present system. The clock loop is stabilized on the zero crossing point of the steep FM spectroscopy output signal. In the locked state, the error signal feeds a 10 MHz Oven Controlled Voltage Controlled Crystal Oscillator (OCXO) which serves as the reference for the microwave source driving the diode laser.
The demodulated FM spectroscopy signal yields two outputs from the Lock-in amplifier: the “in-phase” and ‘quadrature’ signals. For each set of FM modulation parameters, we find a linear superposition of the two components (which results from a simple rotational transformation) that maximizes the signal to noise ratio of the feedback signal of the FLL and leads to optimum clock performance.
Typical output signals of the Lock-in amplifier are shown in Fig. 2(a). The blue dashed line represents the “in-phase” signal while the red solid line represents the ‘quadrature’. These signals result from a CPT resonance whose width was measured to be less than 190 Hz around the RF modulation frequency: 3.4173 GHz. Figure 2(b) shows a direct measurement of the narrow CPT the resonance obtained with no FM modulation.
The details of the CPT resonance and hence the efficiency and stability of the FLL depend on the FM modulation parameters . Maximization of the slope and the SNR of the feedback signal requires an optimization of the FM parameters and the phase of the reference to the Lock-in amplifier. The slope of the “in-phase” signal near its zero crossing point was optimized in a two step procedure. First, the FM parameters were scanned over a wide range. For each of the 85 points we examined, the two Lock-in amplifier outputs were recorded. Keeping the phase reference constant yields a clear optimum modulation frequency and index. Each two components were rotated in the second stage with respect to each other in order to maximize the slope. This rotation is equivalent to changing the phase of the reference signal by the opposite angle.
Figure 3(a) shows in a contour plot the slope of the “in-phase” signal for different FM parameters. Each point in the plot was obtained with its corresponding optimized reference phase. The maximum slope is achieved along a continuous curve in the FM parameters space. Approximately the same maximum value can be achieved for each modulation frequency by choosing a proper modulation index and a particular reference phase. Higher modulation frequencies require lower modulation indexes with the index converging asymptotically towards a value of about 0.5. The reasoning behind the observed behavior is related to the fact that the CPT resonance results from a combined interaction of several pairs of spectral lines whose complex amplitude distributions change with the FM modulation frequency and index.
For each FM parameter set, we also measured the accompanying noise by using the noise measurement feature of the Lock-in amplifier. The noise in the “in-phase” component, shown in Fig. 3(b) in contour form, is low and essentially constant in the entire space of FM parameters which was examined. The noise of the feedback signal has therefore a negligible effect on the choice of the optimum FM parameters.
One set of optimized FM parameters (a modulation frequency of 870 Hz and an index of 0.55) was used to operate the CPT based atomic clock. While a feedback signal with a large SNR can be obtained for many other FM parameters, we chose to operate at a relatively high frequency because it ensures a fast but still accurate response of the system.
The short term clock performance is described in Fig. 4(a). Shown is the measured Allan deviation which is linear in the range of time constants between 0.5 sec to 200 sec where the short term stability is 3 × 10-11/√τ. The Allan deviation reaches a minimum value of 2.2×10-12 at about 500 sec and then increases gradually to a level of 2.5×10-12 at 1000 sec. These results resemble state of the art performance of small scale atomic clocks [2–4].
Long term frequency stability was examined by sampling the 10 MHz output every few seconds over a period of 27 hours. The results shown in Fig. 4(b) demonstrate that the frequency deviates by less than ±1 mHz with a slow drift of approximately 0.1 mHz per day. The slight drift observed in Fig. 4(b) should be noticeable in the Allan deviation for time constants much longer than the maximum 1000 sec value we have used. Allan deviation measurements with extremely long time constants are not possible in the present experimental laboratory system and would anyways not add much basic information about the optimization of the FM spectroscopy scheme itself which is the main objective of this study.
The large slope and low noise level of the feedback signal can be obtained under a range of FM parameters (Fig. 3). It is therefore expected that the clock performance can also be maintained for many FM parameter sets. Since detailed clock characterization is a cumbersome process and since the experimental results of Fig. 4 are consistent with the expected Allan deviation values, based on Fig. 3 and Ref. 1 and 13, we predict that the clock performance will stay unchanged for any of the optimized FM parameter sets.
To conclude, we have described an optimization of a FLL in 87Rb CPT based atomic clock which employs FM spectroscopy. The optimization amounts to determining the optimum FM parameters that yield the largest signal to noise ratio in the FLL feedback signal. Optimum operation was found to be possible within a wide range of the FM parameters space. Using one set of optimized parameters, we operated a clock that uses a small spherical glass vapor cell and obtained a short term stability of 3×10-11/√τ for time constants of 0.5 sec to 200 sec, a minimum Allan deviation of 2.2×10-12 at 500 sec and a long term frequency stability better than 10-10 with a slow relative drift of 10-11 per day. The clock performance, which is similar to that of state of the art small atomic clocks, is limited by VCSEL noise; mainly frequency-and polarization-noise, and by environmental disturbances of the large laboratory set up which is hard to thermally stabilize over long time spans.
The authors thank Dr. A. Stern and Mr. B. Levi of AccuBeat Ltd. and Professor M. Rosenbluh of Bar Ilan University.
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