Reducing number fluctuations in an ultracold atomic sample using Faraday rotation and iterative feedback
RReducing number fluctuations in an ultracold atomic sample using Faraday rotationand iterative feedback
R. Thomas , ∗ A. B. Deb , and N. Kjærgaard † Department of Quantum Science, Research School of Physics,The Australian National University, Canberra 2601, Australia and Department of Physics, QSO-Centre for Quantum Science,and Dodd-Walls Centre, University of Otago, Dunedin, New Zealand
We demonstrate a method to reduce number fluctuations in an ultracold atomic sample usingreal-time feedback. By measuring the Faraday rotation of an off-resonant probe laser beam witha pair of avalanche photodetectors in a polarimetric setup we produce a measurement that is aproxy for the number of atoms in the sample. We iteratively remove a fraction of the excess atomsfrom the sample to converge on a target proxy value in a way that is insensitive to environmentalperturbations. We demonstrate a reduction in the number fluctuations from more than to only . for samples at temperatures of 15 µ K and . µ K over the time-scale of several hours. I. INTRODUCTION
The ability to isolate, trap, and cool samples of atomsto ultra-low temperatures ( ∼ µ K ) has allowed for theexploration of many questions in fundamental quantumscience as well as the development of novel sensors withunprecedented precision and accuracy. Ultracold atomicsystems have been used to study soliton dynamics [1, 2],phase separation [3, 4], super and sub-radiance [5, 6],magnetic order [7], quantum phase transitions [8–10], andfundamental chemistry [11–13]. Ultracold atoms are alsothe basis of the next generation of optical clocks [14], in-ertial sensors [15–18], and other metrological devices [19],some of which can take advantage of mesoscopic quantumentanglement to enhance their precision [20, 21].For many of these systems both the number of atomsin an ultracold sample and the sample density are keyparameters that strongly affect the system dynamics. InBose-Einstein condensates the mean-field energy is deter-mined by the total number of atoms in a condensate, andin Fermi gases the Fermi energy is similarly determinedby atom number. In experiments that probe scatteringdynamics, the number of atoms scattered per unit timeis a function of both the density and total number ofatoms [22–24], and in many other studies, ranging fromsub-radiance [5] to quantum droplets [25, 26], the num-ber or density of atoms affects phase transitions. Thus,the precision which can be achieved in such experimentsis highly dependent on the stability of both the atomicdensity and number.Preparing an ultracold atomic cloud with a fixed den-sity of atoms is a task that remains experimentally chal-lenging due to the inherent complexity of the apparatus,which hybridizes several different technologies from opti-cal, microwave, vacuum, electronic, and mechanical engi-neering [27, 28]. The starting point for an ultracold sam-ple is atoms captured in a magneto-optical trap. These ∗ [email protected] † [email protected] atoms are then typically laser cooled in a sub-Dopplerscheme and transferred to a magnetic trap, an opticaldipole trap, or a combination of both. There is an in-evitable stochastic element to this step and when, finally,this trapped sample is evaporatively cooled to the ultra-cold domain, atom number fluctuations on the percentlevel are to be expected even for an optimized experi-mental cycle. Since the trapping geometry and sampletemperature are usually fixed, a stable number of atomsis a sufficient condition for a stable density.In principle all inputs to an experiment can be madearbitrarily stable, but in the final accounting externalmagnetic fields will drift, laser powers will have somenoise, and lab temperatures will change [29]. At somepoint, rather than attempting to stabilise an experimen-tal input (a laser power, for instance) in order to stabi-lize the output (the number of atoms), it becomes morepractical to stabilize the output directly. For this towork we need methods for non-destructively measuringthe number of atoms and removing excess atoms in realtime. An early example was provided by Ref. 30, whoshowed that frequency-modulation spectroscopy of thescalar light shift [31] provided a non-destructive mea-surement that could allow for the selection of experimen-tal cycles where the initial number of atoms was in anacceptable range. Subsequently, Ref. 32 used the vec-tor light shift to provide an impressive demonstration ofthe power of feedback-based number stabilization whereatomic samples with fluctuations below the shot-noiselimit were produced. In a related work, Ref. 33 usedthe same technique to probe number fluctuations in thenumber of Bose-condensed atoms near the critical tem-perature. In these latter two experiments an image of theatomic cloud was taken using dark-field Faraday imaging[34] which allowed the authors to compute a proxy forthe number of atoms in the sample. Excess atoms wereremoved by applying a series of radio-frequency (RF)pulses that each removed a small fraction of the remain-ing atoms. All three examples used a far off-resonantlaser field to minimize inelastically and off-axis scatteredphotons and hence heating of the atoms. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b While imaging provides a wealth of spatial informa-tion, the addition of a camera adds significant complica-tions to real-time data acquisition and processing. Sincethe total number of atoms in a sample can be relatedto the integrated signal in the dark-field technique, inprinciple one can replace the camera with a sufficientlylow-noise photodetector. This replacement has two mainadvantages. First, readout and processing of a voltagesignal is both simpler and faster than readout and pro-cessing of camera pixel counts. Second, the increasedspeed at which data can be acquired and processed opensup the possibility of stabilizing the number of atoms byan iterative scheme using multiple stages of measurementand feedback rather than the single-stage method demon-strated by Ref. 32. A multi-stage method can be lesssensitive to changes in the environment and the experi-mental parameters, making it a potentially more robuststabilization scheme.In this article, we demonstrate a simplified method forstabilizing the number of atoms in an ultracold atomicsample. By combining the polarimetric signals result-ing from two probe lasers with different frequencies thatinteract with the atoms and impinge on two low-noisephotodetectors, we obtain a measure of the number ofatoms that is insensitive to changes in optical power andphotodetector gain. We use the processed signal to in-form a multi-stage feedback protocol that iteratively con-verges on a target signal and demonstrate a reduction inboth short and long-term fluctuations in the number ofatoms to a relative stability of . . This result is mainlylimited by probe laser beam-pointing instabilities in ourapparatus. II. IMPLEMENTATION
Generally, feedback to control a system parameter re-quires three components: a measurement of the param-eter of interest, an actuator that affects the parameter,and a controller that can compute the actuator from themeasurement at speeds faster than the fluctuations thatone wishes to remove. In the specific case of a sample ofultracold atoms, the measurement must also perturb thesample as little as possible. Our system comprises a sam-ple of Rb atoms in the | F = 2 , m F = 2 (cid:105) state confinedin a Ioffe-Pritchard (IP) magnetic trap with a radial trap-ping frequency of ω r = 2 π × Hz and a
10 : 1 aspectratio. We use Faraday rotation [34–36] of off-resonant,linearly-polarized laser fields propagating along the long-axis of the IP trap to generate a signal that is a proxyfor the number of atoms in our sample at a fixed temper-ature. To describe Faraday rotation, we start by writingthe laser polarization after the field has passed throughthe atoms in terms of the left and right-circularly polar-ized components ˆe ± E = E + e iφ + ˆe + + E − e iφ − ˆe − (1) with amplitudes E ± and phases φ ± . We use a dual-portmeasurement scheme as seen in Fig. 1 for measuring therotation angle which means that we measure the powerin the x (horizontal) and y (vertical) polarization com-ponents P x = P (cid:2) r + 2 r sin ( α + φ a ) (cid:3) (2a) P y = P (cid:2) r − r sin ( α + φ a ) (cid:3) (2b)where P is the total power in the laser field, r = E − /E + is the ratio of amplitudes in the left and right circularlypolarized components, and φ + − φ − = π/ α + φ a with φ a the differential phase shift due solely to the atomicsample and π/ α the differential phase shift from allother sources. For linearly polarized light at ◦ r = 1 and α = 0 . We convert the optical power into voltages us-ing two avalanche photo-detectors (APDs) and then digi-tize the results using the two analog-to-digital convertors(ADCs) on a Red Pitaya development board to producesignals S x and S y which are linearly related to the op-tical powers P x and P y via S x = g x P x and S y = g y P y .The gains g x and g y include both the APD gains and theADC conversion gains.If the gains g x and g y are known then one can formlinear combinations of the x and y signals S ± = g y S x ± g x S y , and the ratio R = S − / S + depends only on thelight polarization and the atomic phase shift R = r sin ( α + φ a ) , (3)where r = 2 r/ (1+ r ) . However, small errors in the gaincoefficients used in cross-multiplication will lead to smalloffsets in R which, if time-dependent, will degrade thefinal stability of the atom number. We use temperature-compensated APDs (Thorlabs APD430A and APD130A)which have gain stabilities on the order of at ambi-ent temperatures. Additionally, our laser fields propa-gate along a 2 m path, and although we use lenses tofocus the laser onto the APDs, beam-pointing instabili-ties still lead to changes in the optical power that reachesthe detectors which mimic changes in the photodetectorgains. Together, these changes can lead to variations onthe order of several percent in our Faraday signal, whichultimately means the atom number stability would belimited to the same level.Fluctuations in the laser power, laser pointing, andphotodetector gains can be circumvented by using twolaser frequencies in a differential measurement. As shownin Fig. 1, we combine two laser fields L and L (cid:48) using afibre beam-splitter and interleave pulses from the twolasers. As long as the time between pulses from L and L (cid:48) is much shorter than the typical time-scale for variationsin the photodetector gain we can assume that the gainsfor each laser field are the same. By choosing the frequen-cies of L and L (cid:48) to be on opposite sides of the resonancethe difference in phase shifts from the two fields leads toan enhanced signal. We combine the signals from lasers P x P y S x S y GTSignal Generator
Red PitayaADC1ADC2MWO1O2OpticalPower TimeLL' L L' L L'
SwitchAOM λ /2+ λ /4APDMicrowavesAtoms λ /2 Figure 1. Simplified diagram of the experiment. Two lasers, routed through two acousto-optic modulators (AOMs), arecombined using a fiber beam-combiner/splitter. The polarization of the two lasers is set by a Glan-Thompson prism (GT), andthen rotated to the desired polarization with a half-waveplate. The combined fields are focused onto the atoms in the sciencecell using a lens, and the resulting light is collected by another lens after the light has propagated through approximately 1 mof vacuum system including an aperture between high and ultra-high vacuum sections. A quarter and half-waveplate correctthe polarization and set it to +45 ◦ before it passes through a polarizing beam-splitting cube. The two polarization componentsimpinge on two avalanche photo-detectors (APDs), and the two voltage signals are digitized using a Red Pitaya developmentboard which controls the light pulses and the microwave state. L and L (cid:48) as S ± = S x S (cid:48) y ± S y S (cid:48) x and compute the ratio R = S − /S + which yields R = 2 r sin (cid:0) [ φ a − φ (cid:48) a ] (cid:1) cos (cid:0) α + [ φ a + φ (cid:48) a ] (cid:1) − r sin ( α + φ a ) sin ( α + φ (cid:48) a ) (4)where φ (cid:48) a is the atomic phase difference for laser L (cid:48) , andwe assume that the polarizations of L and L (cid:48) are thesame. R is completely insensitive to changes in the op-tical powers of either L or L (cid:48) and the gains of the pho-todetectors, and, by sharing a common path, R is alsofirst-order insensitive to small changes in the beam align-ment onto the APDs. As long as we operate in a regimewhere the phase shifts are small, corresponding to largelaser detunings and/or low-density atomic samples, R isa linear function of the number of atoms N [34]. Ourtwo laser fields are sourced from our MOT repump laser,which is locked to the Rb F = 1 → F (cid:48) = 2 transition,and a laser that is offset-locked such that its detuning is − . to − GHz from the F = 2 → F (cid:48) = 3 transition.Both fields have optical powers of ≈
600 nW and are fo-cused to beam waists of µ m onto the atomic sample,and this choice of beam waist was made so that the sizeof the beam is close to the transverse size of the cloudat temperatures of interest [37]. Figure 2 shows that themeasured value of R changes linearly with the number ofatoms at a temperature of T = 15 µ K , albeit with a smalloffset. The magnitude of this offset measures the differ-ence in the polarization between the two laser fields, butas long as it does not change with time it has no bearingon the efficacy of the feedback protocol. Figure 2. Measured value of R as a function of the number ofatoms N at a temperature of µ K and a detuning of − . for L and . GHz for L (cid:48) . Blue circles are measuredvalues and the red line is a linear fit. For our actuator, we use microwave pulses resonantwith the | , (cid:105) → | , (cid:105) transition at ∼ . GHz for atomsat the mean energy of the sample. We follow the exampleof Ref. 32 and use many short pulses to remove atomsfrom the sample by transferring small fractions to theanti-trapped | , (cid:105) state where atoms are expelled fromthe trap. We use µ s long pulses which each transferapproximately f ≈ − of the atoms in the | , (cid:105) state tothe | , (cid:105) state, and the microwave pulses are spaced by µ s . For sufficiently small transfer fraction per pulse, wecan approximate the number of pulses needed to removea given fraction of atoms as a linear function of f − . Thischoice of many short pulses, as opposed to a single longerpulse, both simplifies the calculation of the actuator valueand reduces the effect of drifts of the resonance frequencysince the pulse bandwith is large. Additionally, since oneof our laser fields is on resonance with the F = 1 → F (cid:48) =2 transition we need to avoid transferring a significantpopulation of atoms to the | , (cid:105) state in order to ensurethat our laser fields are far off-resonance.To unite our measurement with our actuator and closethe feedback loop, we use programmable logic on boarda 14-bit Red Pitaya (RP) development board. Our feed-back protocol proceeds as follows. First, the RP pulses L and L (cid:48) on for a programmable duration with the L (cid:48) pulse delayed relative to the L pulse, and the resultingAPD voltages are captured by the two ADCs on the RP.The signal values S x,y and S (cid:48) x,y are calculated as the dif-ference between the mean APD voltages when the pulsesare on and when the pulses are off to account for a pos-sibly changing offset voltage. The ratio R j − is thencalculated from the integrated signals as in Eq. (4) to 16bits of precision. R j − is compared with a target value, R target , and we calculate the number of microwave pulses n j to apply for the current feedback round j as n j = (cid:22) bf − (cid:18) R j − − R target R j − (cid:19)(cid:23) (5)with b the fraction of excess atoms to remove in a givenround j . After j rounds of feedback, the measured ratioshould be R j = R target + ( R − R target )(1 − b ) j . (6)The feedback protocol described by Ref. 32 effectivelyimplements Eq. (6) with b = 1 where all the excess atomsare removed after a single measurement. If, however,the measurements R j have some noise or the fraction ofatoms removed with each microwave pulse changes or isnot well-known, then a single round of feedback will notnecessarily produce a sample with the target number ofatoms. Instead, we use b = 0 . to b = 0 . , and weterminate feedback when R j < (1 + tol ) R target where tolis a fractional tolerance value that is typically × − .The number of rounds of feedback required to reach thetolerance is then j = (cid:24) log − b (cid:20) tol × R target R − R target (cid:21)(cid:25) . (7)For a removal fraction b = 0 . and a starting number ofatoms R = 2 R target we require 8 rounds of feedback toreach a value of R that is less than . × R target . III. RESULTS
Figure 3 shows the effect of our iterative feedbackscheme on a sample of Rb atoms held in our IP trap at atemperature of µ K . We measure the number of atomsand the temperature using standard absorption imagingon the F = 2 → F (cid:48) = 3 transition along one of the radialaxes. We perform feedback at a temperature of µ K asthis is approximately where our value of R is maximizeddue to the interplay between the atomic sample size and Figure 3. Effect of feedback on the number of atoms in a µ K cloud. a Number of atoms in the sample as measuredby absorption imaging. Red squares show the experimentalcycles where feedback was disabled. Blue circles show cycleswhere feedback was enabled but R < R target so feedbackdid not engage. Green circles show cycles where feedbackwas enabled and R > R target so that feedback was actuallyapplied. b Measured ratio R for those cycles where feedbackwas enabled. The target value of R was set to . , so whenthe initial value was less than the target no feedback wasapplied. the laser beam waists [37]. Each round of feedback com-prises a measurement of R using pulses from L and L (cid:48) that are µ s long with the pulses from L (cid:48) delayed by µ s . L is detuned from the F = 2 → F (cid:48) = 3 transitionby − , and L (cid:48) is locked to the F = 1 → F (cid:48) = 2 transition. Each pair of pulses, which define the start ofa feedback round, are separated in time by 5 ms whichis chosen to allow enough time for all possible microwavepulses to be applied. The number of microwave pulsesis calculated by the Red Pitaya using Eq. (5) with theproduct bf − = 250 which defines the maximum num-ber of microwave pulses that can be applied. The exactvalue for the maximum number of microwave pulses isunimportant, except that a larger value leads to fewerfeedback rounds at the cost of less precision in reachingthe target value.We run our experimental cycle times with a randomselection of runs with feedback disabled and the other runs with feedback enabled; each run takes approxi-mately one minute. To demonstrate the efficacy of ourapproach we deliberately vary the number of atoms inthe sample by randomly changing the MOT cooling laserpower with each run, and this produces a relative varia-tion in number of . . when feedback is disabled asseen in Fig. 3a. With feedback enabled and R target = 0 . we find a relative variation of . . We can improveon this result by noting that feedback can only be appliedwhen R > R target since the actuator can only removeatoms, not add them. If, while running the experiment,we keep track of R , as can be seen in Fig. 3b, then wecan post-select cycles where R > R target ; in this case,the relative fluctuations as seen by the green circles inFig. 3a are only . . This is an 8 dB reduction inthe bare fluctuations, and it is an approximately 5.9 dBreduction in the fluctuations if one post-selected on R but did not perform feedback. Post-selection allows oneto maintain a high mean number of atoms at the costof increasing the effective experimental cycle time: inour case, where R target is approximately the mean value R over all cycles, we discard cycles in Fig. 3 andexpect, on average, to discard approximately half of allcycles. However, the mean number of atoms for the post-selected, feedback-enabled cycles is only lower thanfor those with feedback disabled.If an increase in the effective cycle time cannot be tol-erated, or in the case of significant drift in the uncon-trolled number of atoms, then one can instead set R target to be well below the lowest number of atoms expectedin the experiment. In Fig. 4 we show the effect on theatom number of setting R target to be lower than theinitial value R , which corresponds to setting the targetnumber of atoms 7 standard deviations below the initial,fluctuating number of atoms. For this measurement wereduce the temperature of the sample to approximately . µ K . We run the experimental cycle 400 times with arandomly selected 200 runs having feedback enabled. InFig. 4a we clearly see that feedback results in the elim-ination of the long-term drift in the number of atoms Figure 4. Feedback performance with target value 7 standarddeviations below initial fluctuations. a Number of atoms at . µ K with feedback disabled (red squares) and enabled (bluecircles). b Relative Allan deviation with feedback disabled(red squares) and enabled (blue circles). Filled markers arefor the number of atoms, and empty markers are for the in-trap density. which is due to a slow drift in the IP trap bottom thatcauses the fixed radio-frequency evaporative cooling tocut deeper into the sample and reduce the temperatureand number of atoms. Fig. 4b shows the relative Allandeviation for both the number of atoms in the sampleand the calculated in-trap density. While feedback re-duces the short-term number fluctuations and eliminatesthe long-term number drift, represented by the Allan de-viations for small and long cycles respectively, it doesnot affect the overall drift in the temperature, which canbe seen as an increase in the density fluctuations after10 runs as seen in Fig. 4b. Nevertheless, feedback sig-nificantly reduces relative fluctuations in the density atall times compared to the situation without feedback,and a different trap design, such as a Ioffe-Pritchard trapwith smaller electric currents or an all-optical trap, wouldshow significantly improved results.Our measurements conducted at sample temperaturesof 15 µ K and 2.8 µ K both show a lower bound to therelative cycle-to-cycle number fluctuations of about . regardless of the starting fluctuations. A lower bound onthe fluctuations in the parameter of interest is expectedwhen using feedback, as one cannot control a parame-ter to a higher precision than one can measure it or itsproxy. Based on the applied laser power ( ≈
600 nW ), theAPD noise-equivalent powers, the laser pulse durations,and the RP input noise, we expect that the contribu-tion of shot-noise and electronic noise to the stabilizedatom number from the Faraday detection system is ap-proximately . . Absorption imaging, with an inten-sity that is of the saturation intensity, contributesanother . to the residual atom number fluctuationswith the majority of that arising from technical noise(such as fringes) in the images. We believe that the re-maining atom number fluctuations come from a combi-nation of beam-pointing noise caused by a long opticalpath between the atoms and the APDs, lensing from theatomic sample, and an aperture in our vacuum system.The atoms act as a lens [37] that focuses the left andright circularly polarized components differently, whichmeans that the vacuum aperture affects the two polariza-tion components differently, and therefore small changesin the beam alignment onto the atoms can cause changesin the optical power in the circularly polarized compo-nents when they reach the final set of waveplates andPBS. Effectively, this leads to cycle-to-cycle changes in r = E − /E + and hence R . While we are restricted tousing this particular beam path for our setup, other ap-paratuses need not suffer the same problem and may beable to obtain number stabilization to the level of theatomic shot-noise. IV. CONCLUSION
In this article we have demonstrated a simple methodfor reducing the fluctuations in the number of atoms inan ultracold sample. We use a differential measurementof off-resonant Faraday rotation of two laser fields as theypropagate through the sample to compute a proxy for thenumber of atoms that is independent of changes in laserpower and photodetector gain and is only weakly sensi-tive to alignment. For sufficiently large detunings, thisproxy value is linear in the number of atoms for a fixedtemperature. By using an iterative feedback scheme, wereduce the need for pre-calibration of our control systemwhich makes it more robust to environmental perturba-tions. We demonstrate a reduction in the cycle-to-cyclenumber fluctuations down to approximately . anda relative Allan deviation of . when averaged overabout . hours.While our method does not obtain the same limitingperformance as Ref. 32, which would correspond to rela-tive number fluctuations of . × − , our control systemhas a number of advantages. It is both simpler in termsof equipment and data processing, which may be usefulin situations where size, weight, and power are impor-tant considerations. Our method is also faster, whichallows for an iterative scheme to be used. Finally, byusing microwave pulses instead of radio-frequency pulsesour method is easily adapted to feedback control of thenumber of atoms in dual species experiments.Upon finalizing this article, we became aware of recentrelated work using an optical cavity for dispersive realtime tracking of evaporative cooling with cavity-assistedfeedback stabilization of atom number as a future goal[38]. [1] S. Burger, K. Bongs, S. 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