We demonstrate the operation of a closed-loop fast-light cavity that allows rapid (~10 ms) measurements of the cavity mode frequency and its uncertainty. We vary the scale factor by temperature tuning the atomic density of an intracavity vapor cell. The cavity remains locked even as the system passes through the critical anomalous dispersion where a pole is observed in the scale factor. Positive and negative scale-factor enhancements as large as |S| ≈70 were obtained. To our knowledge, these are the first experiments that demonstrate a scale-factor enhancement in a closed-loop fast-light device by changing the optical path length, laying the groundwork for the improvement of cavity-based metrology instruments such as optical gyroscopes.
A common application for a passive optical cavity is to perform spectrometry, i.e., to characterize the unknown frequency content of a source by changing the optical path length (OPL) of the cavity by a known amount. Conversely, a cavity may be used to characterize an unknown change in OPL generated by some measurand, such as a change in length or rotation rate (via the Sagnac effect), from a known change in the cavity mode frequencies. For the latter set of applications, the source frequency is often locked (via external control loops for a passive cavity, or as a natural consequence of the atom-field interaction for an active laser cavity) to one of the cavity modes so that the change in the mode frequency can be rapidly determined simply by monitoring the frequency change of the source. Recently, a series of studies have predicted that an anomalous dispersion, or fast-light, medium, placed into such a locked cavity, can be used to increase the precision of the measurement of these changes in OPL and the external phenomena of interest that produce them [1–13]. The fast-light enhancement can occur more generally in any system of non-Hermitian coupled oscillators [8,14–16], such as PT-symmetric systems [17,18] near an exceptional point. Coupled oscillators are ubiquitous throughout nature and the physical sciences. Therefore, if it can be confirmed that measurement precision is enhanced, such a finding could have far-reaching consequences, presenting a new strategy for achieving improvements in precision metrology.
There are two factors that determine the measurement precision: (i) the cavity scale factor or sensitivity , i.e., how much the mode frequency shifts for a given change in OPL (where L represents the OPL); and (ii) the uncertainty σ in the measurement of the mode frequency. Thus, the precision is enhanced when the scale factor is increased without a concomitant increase in uncertainty. It then follows that the enhancement in measurement precision can be written as 
In active fast-light cavities, the laser frequency is automatically locked to a cavity resonance. Recent work on active cavities  has estimated a scale-factor enhancement as large as S = 190, but this value is inferred from spectra of the gain profile. Experiments on passive fast-light cavities , on the other hand, have obtained values as large as S = 360 by taking the more direct approach of varying the OPL and monitoring the change in the cavity mode frequencies. Therefore, these passive-cavity experiments have definitively demonstrated the first condition for an enhancement in precision: that the scale factor can be enhanced by several orders of magnitude. In particular, a pole in the scale factor is observed to occur at a critical anomalous dispersion (CAD). In these experiments, however, the cavity is not locked to the laser frequency. Instead, the scale-factor enhancement is determined after the fact, by taking spectra at a variety of OPLs and subsequently finding the extrema. This procedure necessitates taking a large amount of data, is laborious, and is not practical for rapid determination of the scale factor . Moreover, the estimations of the mode frequency and its uncertainty are dependent on the algorithm used to find the mode extremum. Therefore, the values of these quantities obtained using this previous method should not be expected to agree with those that would be obtained from a closed-loop scheme designed for rapid measurements. For this reason, our previous study resorted to inferring the enhancement in precision that would exist in a closed-loop device operating under ideal conditions, i.e., the quantum-noise limit (QNL) . Thus, experiments to date have not directly addressed the second condition: that as the dispersion is increased the uncertainty should not increase as fast as the scale factor.
Another drawback evident in previous studies of fast-light cavities [3,5,7,11,17,18] is the sparse quantity of data taken near the medium resonance peak, which increases the uncertainty in the estimation of S. Data collection in this region is difficult because fast-light cavities display an instability at the CAD condition which results in the mode being forcefully pushed away from the resonance . Small amounts of drift are amplified by the hypersensitivity of the fast-light cavity. Consequently, measurements must be taken rapidly to acquire data in this region and to counteract the effect of drift, which would otherwise degrade the scale-factor measurements.
In this work we demonstrate the operation of a closed-loop passive [21–23] fast-light cavity that enables a rapid  measurement of the cavity mode frequency and mode uncertainty. Our results are the first to demonstrate a scale-factor enhancement in a closed-loop fast-light device by changing the OPL. The active stabilization allows far more data to be recorded near resonance, resulting in reduced uncertainties, and reduces the time for data collection, limiting the effect of drift. The cavity remains locked even as the system passes through the CAD condition, permitting an experimental determination of whether the measurement precision is enhanced near this point. Our experiments do not reveal the predicted increase in measurement precision, but set the stage for future experiments that might, and therefore represent an important proof-of-principle for the application of fast-light to improving devices such as gyroscopes.
It is not straightforward to extend the closed-loop locking scheme, developed previously for passive gyroscopes [21–23], to the case where the cavity contains a resonant medium, because: (i) the simulated-rotation method developed previously did not involve changing the actual resonance frequencies of the cavity and therefore cannot provide a measure of the shift of the cavity mode from the medium resonance frequency; and (ii) it is unclear whether the cavity can remain locked as it passes through the CAD, in particular because the absorption, linewidth, and instability all increase near this point. To address the former, fortunately one does not have to place any additional components inside the cavity. Instead, to shift the cavity mode we can simply vary the laser frequency, and lock the cavity mode to follow it. The laser frequency then provides the detuning of the dispersive cavity, whereas the feedback to the cavity provides a measure of the detuning of the empty cavity. This procedure is analogous to a modified simulated-rotation method where one varies the beat frequency (output) of a gyroscope, and backs out the rotation rate (input) needed to produce it. In other words, we reverse the roles of the input and output.
A narrow linewidth (<1 MHz) tunable external-cavity diode laser (ECDL) at 780 nm was frequency locked to the Doppler-broadened 52S1/2, Fg = 2 → 52P3/2, Fe resonance in 87Rb using a standard first-harmonic demodulation-based peak-locking technique  as shown in Fig. 1. An error signal , produced through phase-sensitive detection of a modulated signal by a lock-in amplifier, is minimized by a proportional-integral (PI) servo controller which provides a feedback signal to the laser controller. The laser is set to a desired detuning and scanned across the peak absorption frequency, while remaining locked, by summing in an input offset signal at A as shown. Subsequently, using the same peak-locking method, one of the modes of a ring cavity containing a second 87Rb vapor cell is locked to the tunable laser. In this case the signal detected is the output (reflection) from the cavity, and the feedback control signal is applied to a piezo-electric transducer (PZT) attached to one of the cavity mirrors. This control signal also serves as the output of the experiment, at B.
Note that the input at A can be thought of as analogous to the output beat frequency from a passive gyroscope [21–23], while the output at B is analogous to the rotation to be measured. In this manner, the configuration shown in Fig. 1 enables a simulated rotation for a unidirectional beam in the cavity. The benefit of common-mode noise rejection that would be obtained from using two propagation directions around the ring cavity is lacking, however. Therefore, in its current form the system cannot distinguish nonreciprocal changes in OPL from reciprocal ones. Consequently, the scale factor is sensitive to cavity drift and jitter. Procedures for minimizing the effect of drift on our measurements are described farther below.
Future embodiments of our scheme can readily use the feedback signal from the rate control loop of a passive gyroscope as the output at B. Nevertheless, because the usual roles of the input and output are reversed in our scheme, such an arrangement would still not allow enhancement of a real rotation. For the enhancement of real changes in rotation rate, acceleration, or any other measurand associated with changes in the cavity resonance frequencies, the laser should be locked to the cavity and the output should be the feedback control signal at the laser control loop, i.e, the input should be at B and the output at A. Closed-loop operation of this more conventional input-output arrangement can be accomplished by using the cavity photodetector as an input to the laser control loop in Fig. 1, and using two separate modulation sources for the control loops, where the frequency difference between the sources is sufficient to avoid coupling between the loops . Such adaptations of our scheme, and the additional experimental complexities associated with them, are not addressed in this paper.
Details of the cavity mirrors, vapor cell, and oven were reported previously in reference . The cavity round-trip length was 40 cm. The quartz cell was 2.5 cm long and was anti-reflection coated. The vapor cell was held in a temperature-controlled oven  and an enclosure was placed around the entire cavity to minimize air currents and temperature variations. To increase the anomalous dispersion, the temperature was raised in steps from 19.3 °C to 45.0 °C and allowed to equilibrate before data collection. The two control loops use the same 2.5 kHz modulation signal, whose amplitude at room temperature is set slightly larger than ideal so that as the temperature and cavity linewidth increase, the modulation approaches the ideal setting. The lock-in amplifier time constant was set to 10 ms, corresponding to integration over 25 modulation cycles. Once established, the gain settings on the laser control loop were not modified, so as not to alter the calibration (described below) as the temperature was increased. In contrast, the gain on the cavity control loop was raised to compensate for the change in mode shape, until the error signal was observed to return to zero, indicating a tight correspondence between the cavity mode and laser frequencies. At each temperature, the laser was scanned multiple times over the approximate center of the Doppler-broadened line by applying a computer-generated sawtooth input signal at A, . The time delay between each data point was set slightly larger than the 10 ms time constant, to allow the loop to respond before moving to the next data point.
When the number of points per scan was sufficiently large, drift was observed in as a result of cavity drift. The drift was minimized by the presence of the cavity enclosure and by waiting for the temperature within it to stabilize. In addition, the number of points per cycle was reduced until the output signal was linear for each individual scan, i.e., the timescale of the measurement was reduced to reject the low-frequency drift (we verified the absence of higher-order noise in the individual scans by plotting the Allan deviation). Following the data collection, the effect of the drift was further reduced by smoothing the signal over one scan cycle (200 points) and subtracting it from the original signal. The slope of each scan was then determined by a linear fit, and an average slope was computed. By including scans in both the positive and negative frequency directions in this averaging process, the effect of the drift was substantially reduced. In addition, each of the individual scans were summed together to produce a single averaged data set. The slope of this averaged data set, as determined by a linear fit, was in each case identical to the average of the individual slopes. The standard deviation of the slopes of the individual scans was also recorded to compute the value of and ultimately determine ζ via Eq. (1).
To convert the obtained slopes into scale factor enhancements the input and output voltages at A and B must be converted into frequency detunings. Conversion of is a two-step process. The cavity control loop remains open throughout these steps. First, the laser is scanned across the Doppler-broadened resonance by applying a computer-controlled ramp signal as shown in Fig. 1. The modulation signal is applied but the laser control loop remains open during the scan and the error signal is recorded. The slope of the error signal at the resonance provides a conversion between the error signal and ramp voltages. In the second step, the beam-block is removed, so that saturated absorption (SA) resonances are observed during the laser scan. In this step the loop remains open and the laser is not modulated. The known frequency differences between the SA resonances then provides a conversion of the ramp voltage to frequency. To convert , the laser is locked to the Doppler-broadened resonance as described above. The ramp signal is now summed into the piezo controller to record fringes over several free spectral ranges (FSRs), enabling conversion of the ramp voltage to frequency (assuming the FSR is known). For both conversion processes, the voltage scan was performed rapidly (~1 s) to avoid drift during the measurement, and an average between increasing and decreasing frequency scans was taken to compensate for PZT hysteresis. The PZT response is approximately linear over the relatively small range of the frequency scan employed in the experiment, but one must take care when the range is expanded during these calibration steps.
3. Results and discussion
In Fig. 2, data are shown at three representative temperatures. In Fig. 2(a), raw data of the output signal at B is shown as the input at A was scanned in frequency over a range of 205 MHz in a sawtooth pattern. The reduced range allowed much more data to be taken in the high-dispersion region near zero detuning than in previous works [3,5,7,11,17,18]. At each temperature there were 100 data points per scan and the number of scans was 100 (50 sawtooth cycles), except for the data at 40.7 °C, where the number of scans was increased to 500 to further reduce the measurement uncertainty. Note that as the temperature is increased, the PZT control signal response (output) is weaker because the cavity mode broadens, therefore a larger change in the laser frequency (input) can occur before the loop detects that it is off resonance, i.e., the scale factor increases.
In Fig. 2(b) the average response curves are shown after calibration for these three temperatures. Over a frequency range larger than that shown, the curves in Fig. 2 are in fact nonlinear, corresponding to the anomalous dispersion associated with the atomic resonance, but the scan range was reduced in these experiments to focus on the linear region close to resonance. The detunings along the horizontal and vertical axes are defined as and , respectively, where and are the frequencies of the locked cavity mode for the dispersive and empty cavities, respectively, is the empty-cavity round trip time, and is the frequency where the curves are most linear (near the scale-factor maximum). An offset was applied to to center each scan about , which increased slightly as the critical temperature was approached, but was never more than 90 MHz. The offset arises because (in order of importance): (i) the large modulation amplitude combined with asymmetry in the absorption profile detunes the laser lock point from the absorption maximum; (ii) is shifted from the absorption peak owing to asymmetry in the absorption and mode profiles; and (iii) the two 87Rb cells are not at the same temperature (one is at room temperature) so their absorption maxima occur at slightly different frequencies . The quantities δ and Δ in Fig. 2(b) are in units of the empty-cavity FSR, determined by averaging repeated spectral measurements to be . The scale-factor enhancements are simply the slopes of these curves as obtained from the linear fits, i.e., . Note that, because of the reversed roles of the input and output mentioned earlier, the vertical (horizontal) axis of the uncalibrated data in Fig. 2(a) corresponds to the horizontal (vertical) axis for the calibrated data in Fig. 2(b), i.e., and , respectively. Also note that for the data at 40.7 °C, the signal-to-noise is too low to draw any conclusions from the individual scans in Fig. 2(a), but that the slope is recovered by the averaging process with an uncertainty that is reduced by in Fig. 2(b).
In Fig. 3(a) the scale factor enhancements, S, are plotted vs. temperature for all the temperatures included in the experiment and compared with a semi-empirical model [11,29] that takes into account all the Doppler-broadened Fg = 2 → Fe hyperfine transitions of the 87Rb D2 transition. Cavity parameters used as inputs to the model, such as mirror reflection coefficients and round-trip losses, are described in detail in reference . The model is valid provided the levels are Zeeman-degenerate, the intensity is sufficiently weak that the atoms remain in thermal equilibrium (saturation, power broadening, and optical pumping are negligible), and the polarization is linear. These conditions were satisfied throughout the experiments. The intensity of the beam entering the cavity was held at 1.6 mW/cm2, corresponding to intracavity intensities less than 10 μW/cm2, far below the lowest saturation intensity for the hyperfine transitions with π-polarized light (3.6 mW/cm2 for the Fg = 2 to Fe = 3 transition ). The usual buildup of intensity within the cavity is limited when the mode is on-resonance with the atoms owing to the strong absorption. The use of a ring rather than a linear cavity also prevents the appearance of additional narrow peaks in the absorption spectrum that might occur at higher intensities due to counter-propagating cavity beams. It is, therefore, possible to use higher intensities in these experiments without added complication. In this work, however, we restricted ourselves to these lower intensities for comparison with the theoretical model. As predicted by the theory, a pole was observed in the scale factor data around 40.8 °C, corresponding to the CAD. In this case CAD conditions occurred at an effective group index of (or a material group index of ), which is significantly different than the value predicted by models that treat the dispersion as linear about the resonance . As we have discussed previously [3,11], such linear-dispersion models are only applicable when the linewidth of the cavity is much smaller than that of the dispersive atomic resonance so that the entire mode (not just the peak) falls within the linear region, which was not the case here owing to the low cavity finesse (F < 10). For temperatures larger than the critical temperature, the scale factor was negative, corresponding to modes that split . The largest positive and negative values recorded were S = 66.6 ± 3.8 at 40.7 °C and S = −71.7 ± 4.9 at 41.0 °C, respectively.
In Fig. 3(b) the enhancement in measurement precision ζ is plotted. In the QNL, ε is approximately the ratio of the cavity linewidth to the signal-to-noise, W/M, where W and M are each normalized to its corresponding empty-cavity value . Thus, the enhancement in precision in this limit is . The solid curve in Fig. 3(b) is the predicted QNL enhancement in precision, , determined from theoretical calculation of S, W, and M, as previously reported . The data points, on the other hand, are obtained from Eq. (1). At each temperature, the value of ε was computed by taking the standard deviation of the slopes obtained in Fig. 2(a), converting into a standard deviation in S by error propagation, and normalizing to the result obtained at room temperature. The data has been further scaled for comparison so that the points lie slightly below the QNL curve. We did not make an absolute measurement of ζ in this work. A more careful procedure would involve an independent measurement of the standard deviation in S for the empty cavity, , at each temperature, under conditions identical to those used with the atomic vapor. Nevertheless, our relative measurement is more than sufficient to conclude that the predicted enhancement in precision is not observed in these experiments. The data deviates prominently in functional form from the QNL prediction near the critical temperature, where ζ is reduced by a factor of ≈100 from its room-temperature value.
To investigate the physical origin of this deviation, we performed Monte-Carlo simulations of the sort reported in reference  and found that a systematic underestimation of S can occur as a result of mode skew. The cavity mode becomes skewed when slightly detuned from the medium resonance, especially near the critical temperature, as a result of mode reshaping from the absorption profile . In the high-finesse linear-dispersion approximation the mode reshaping is negligible and the skew is not present, but we do not deal with that case in this work because the measurement precision is not predicted to be enhanced within that approximation . The results of our simulations show that the underestimation in S grows when the signal-to-noise is reduced or the cavity jitter is increased, demonstrating that quantum or classical noise, respectively, can reduce the observed enhancement. The origin of this problem lies in how the mode frequency is determined. In the theory plots in Fig. 3, the mode frequency is the frequency corresponding to an extremum in the cavity spectrum. In the simulations, on the other hand, the mode frequency is computed from peak-finding algorithms applied to statistical ensembles of photons. The results of the simulations show that algorithms that more accurately determine the extremum result in larger values of S. Therefore, experiments employing better peak-finding methods are expected to hew closer to the theory. When the signal-to-noise is sufficiently high, however, the discrepancies between algorithms is minimized, and S converges toward the theoretical ideal. Thus, we should only expect the experimental value of to approach the theoretical prediction shown in Fig. 3(b) at high signal-to-noise.
Another significant factor contributing to the deviation in Fig. 3(b) is a byproduct of the reversal of the roles of the input and output, which effectively scales the data by a factor of ≈S. To understand why this is so, consider two complementary experimental arrangements: (I) the input is at the B and the output is at A; and (II) the input is at A and the output is at B. Case I is the conventional input-output arrangement, whereas case II is the reversed arrangement used in our experiments. For case I, the total noise in the variable at the output A is , where and represent the noise present in the laser and cavity control loops, respectively, when they are separated and with no medium present. Thus, the last term on the RHS is the amplification by S of the noise in the cavity control loop. For case II, on the other hand, the total noise in at the output B is , where the first term on the RHS is the deamplification by S of the noise in the laser control loop. We must now convert the error in at B into one for at A. It is a simple matter to show that the error propagates as , so that , and therefore . Substituting this result into Eq. (1), we obtain , independent of the relative strengths of the noise sources, i.e., for any ratio of . Consequently, the reversed input-output arrangement is more susceptible to noise at the output. This is not surprising considering that given a constant background, amplifying a signal above the background is better than deamplifying the same signal below the background, if one wishes to remain limited by the noise of the signal rather than the background. Adjustment of the data by a factor of S results in the open circles shown in Fig. 3(b). The predicted enhancement in ζ is still not observed, suggesting that the geometry of the experiment is not the only factor in the divergence of the data from the prediction, and that the experiment does not approximate the ideal, high signal-to-noise, QNL conditions .
At temperatures above 42.0°C the cavity lock could not be maintained. Importantly, however, the cavity did remain locked at all other temperatures, even as the cavity passed through the CAD condition. If the cavity linewidth broadened to the same degree that the scale factor increased, as previously suggested , then locking should not be possible at the pole in S. Locking is possible, though, because the broadening of the cavity mode under CAD conditions is limited by the finite width of the Doppler-broadened line. Locking the laser to this same broad feature (rather than to a much narrower saturated-absorption hyperfine resonance) ensures that the modulation amplitude is sufficiently large to encompass the cavity mode, even when it’s broadened to its maximum extent by the dispersion. The relatively low finesse of the cavity also minimizes the change in the cavity linewidth as the dispersion increases. This is important for another reason: achieving the enhancement in precision in a passive cavity requires comparable cavity and medium linewidths .
In summary, we have demonstrated the closed-loop operation of a fast-light cavity, which enables rapid measurements of both the mode frequency and the mode uncertainty, and found that for this particular cavity and experimental arrangement, the uncertainty increases faster than the scale factor. The use of the reversed input-output geometry simplifies the experiment because it enables use of a common modulation source, and the laser control loop parameters, once set, do not need to be adjusted, but also results in a decrease in the measured values for ζ by a factor of ≈S. Even after taking into account this scaling, we do not observe the predicted increase in measurement precision. This is not surprising considering the strong influence of noise on our cavity, which was constructed using only standard off-the-shelf components. Future experiments can benefit from increased signal-to-noise and by more closely approaching the QNL, through better mechanical and thermal stabilization, and through the use of two counterpropagating optical paths, i.e., a passive gyroscope configuration, for the cancellation of common-mode noise. This would have the added benefit of eliminating the averaging required, resulting in faster measurements. In addition, a modification to our scheme  that permits a more conventional input-output arrangement could substantially increase the observed value of ζ near the critical temperature, as well as permitting direct monitoring of scale-factor enhancements for real changes in rotation rate, acceleration, or any other measurand associated with changes in the cavity resonance frequencies.
NASA Game Changing Development Program; U.S. Army AMRDEC Missile S & T Program.
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32. We also considered the possibility that the process of increasing the feedback gain might increase the noise in loop B, which would decrease the measured value of ζ in comparison with the theoretical prediction. However, inspection of Fig. 2(a) reveals that this is likely not a factor preventing observation of the enhancement in precision. The proportional gain is increased by a factor of 6.6 in this figure, yet after removing the slopes the noise that remains is roughly the same in the three sets of data. Thus, does not change much with temperature. This is because the noise on loop B is dominated by OPL fluctuations in the cavity, and is largely unaffected by noise introduced by the servo controller gain. Furthermore, the gain was increased more rapidly at lower temperatures, but then was held almost constant around the critical temperature, quite opposite the trend predicted for .
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