Atom interferometry with thousand-fold increase in dynamic range
Dimitry Yankelev, Chen Avinadav, Nir Davidson, Ofer Firstenberg
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y Atom interferometry with thousand-fold increase in dynamic range
Dimitry Yankelev,
1, 2, ∗ Chen Avinadav,
1, 2, ∗ Nir Davidson, and Ofer Firstenberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel Rafael Ltd, Haifa 3102102, Israel
The periodicity inherent to any interferometric signal entails a fundamental trade-off betweensensitivity and dynamic range of interferometry-based sensors. Here we develop a methodology forsignificantly extending the dynamic range of such sensors without compromising their sensitivity,scale-factor, and bandwidth. The scheme is based on operating two simultaneous, nearly-overlappinginterferometers, with full-quadrature phase detection and with different but close scale factors.The two interferometers provide a joint period much larger than π in a moiré-like effect, whilebenefiting from close-to-maximal sensitivity and from suppression of common-mode noise. Themethodology is highly suited to atom interferometers, which offer record sensitivities in measuringgravito-inertial forces but suffer from limited dynamic range. We experimentally demonstrate anatom interferometer with a dynamic-range enhancement of over an order of magnitude in a singleshot and over three orders of magnitude within a few shots, for both static and dynamic signals.This approach can dramatically improve the operation of interferometric sensors in challenging,uncertain, or rapidly varying, conditions. The ambiguity-free dynamic range of interferometricphysical sensors is fundamentally limited to π radians.When the a priori phase uncertainty is larger than a sin-gle fringe, additional information is required to uniquelydetermine the physical quantity measured by the interfer-ometer. If this quantity remains constant over long peri-ods of time, the phase ambiguity may be resolved throughadditional interferometric measurements with differentscale-factors, defined as the ratio between the interferom-eter phase and the magnitude of the physical quantity. Amore challenging scenario arises when the physical quan-tity changes rapidly with time, and measurement withmultiple scale-factors must be realized simultaneously.Overcoming this challenge in cold-atom interferome-ters [1], which have emerged over the past decades asextremely sensitive sensors of gravitational and inertialforces, is an especially ambitious proposition. Applica-tions of atom interferometers vary from fundamental re-search [2–6] and precision measurements [7, 8] to gravitysurveys and inertial navigation [9]. Mobile interferome-ters are being developed by several groups [10–13] withdemonstrations of land-based, marine, and airborne grav-ity surveys [14–16].In these applications, limited dynamic range is espe-cially challenging, as the uncertainty in the accelerationto be measured is potentially very large. Reducing theinterferometer scale-factor or performing multiple mea-surements at each location results in reduced sensitiv-ity or lower temporal bandwidth, respectively. A com-mon solution relies on auxiliary sensors with larger dy-namic range but lower resolution to constrain the in-terferometric measurement to a smaller, non-ambiguous ∗ These authors contributed equally to this [email protected]@weizmann.ac.il range [17, 18]. However, this approach may suffer fromtransfer-function errors, misalignment between the sen-sors, or non-linearities [14]. It is therefore highly de-sirable to have a high-sensitivity, high-bandwidth, atominterferometer with a large dynamic range. While opticalinterferometers may gain such capabilities by employingand detecting multiple wavelengths [19, 20], this feat ismore challenging for matter-wave interferometers.In this work, we achieve a dramatic enhancement ofdynamic range on a single-shot basis by combining twopowerful approaches in atom interferometry: increasingthe dynamic range without sensitivity loss through smallvariations of the interferometer scale factor [21], and ac-quiring multiple phase measurements in a single experi-mental run [22, 23]. First, when the same fundamentalphysical quantity determines two interferometric phaseswith slightly different scale factors, it can be uniquely ex-tracted within an enhanced dynamic range, determinedby a moiré wavelength which is inversely proportionalto the difference between scale-factors [Fig. 1(a)]. Sec-ond, by operating and reading out the two interferome-ters simultaneously within the same experimental shot,major common-mode noises are rejected, increasing thescheme’s robustness to dominant sources of noise. Addi-tionally, such operation maintains the original temporalbandwidth of the measurement. Further exponential in-crease in dynamic range, at the cost of a linear reductionof temporal bandwidth, is achieved by varying the scale-factor ratio between shots.
PRINCIPLES OF DUAL- T INTERFEROMETRY
We realize the above concept in a Mach-Zehnder atominterferometer measuring the local acceleration of grav-ity [24]. Such devices use light-pulses as “atom-optics”that split the atomic wavepacket into two ams and later
Figure 1. Concept, scheme and results of dual- T interferometry. (a) Conceptual representation of dynamic-range enhancementby a factor of × , using a pair of simultaneous interferometers with different scale factors. (b) Dual- T atom interferometry. Apair of Raman pulse sequences (red and blue), with different interrogation times T and addressing different velocity classes of theatoms, couple between two atomic states with momentum difference of ~ k eff (bright and dark trajectories). To obtain full phasequadrature information, the Raman retro-reflecting mirror is tilted before the final π/ pulses, generating a transverse phasegradient across the cloud. (c) Top: a single fluorescence image captures the population in one of the atomic states for bothinterferometers. Bottom: measured cross-sections (solid lines) and the fitted fringes (dashed lines) of both interferometers,after vertical integration of the regions indicated by the dashed rectangles and subtraction of the Gaussian envelope. Theinterferometer phases φ , φ are determined by the fringe phase at the center of each cloud. (d) Results of dual- T interferometryfor inertial phase φ a in the range of ± π (color coded), each dot represents a single dual- T measurement. Slope of gray linesis the scale-factor ratio τ = ( T /T ) = 7 / . Shaded region represents the original, ambiguity-free, π dynamic range of asingle interferometer operated at T = T . Full-quadrature phase-detection allows for a unique solution for all phases, comparedto ambiguities generated when detecting only the cosine component (e). (f) Dual- T measurements at constant inertial phase φ a = 0 , demonstrating that the noise in both interferometers is highly correlated with slope ∼ τ . In red, the covariance ellipseat 95% confidence level. recombine them after they traveled on macroscopicallydistinct trajectories. The differential phase accumulatedbetween the arms of the interferometer depends on themotion of the atoms.In our experiment, laser-cooled Rb atoms arelaunched vertically on a free-fall trajectory. Counter-propagating, vertical laser beams at
780 nm drive two-photon Raman transitions between two electronic groundstates while imparting recoil of two photon momenta[25]. The Raman beams are sent from the top and areretro-reflected from a stabilized mirror at the bottom,which defines the reference frame with respect to whichthe motion of the atoms is measured. The interferomet-ric sequence is composed of three Raman pulses, equallyspaced by time T , acting to split the atomic wavepacketinto two components that drift apart, and then to redirectand recombine them, leading to a final atomic populationratio determined by the phase difference between the twoarms.In this configuration, the phase difference is deter-mined by the gravitational acceleration g according to φ a = ( k eff g − α ) T , with ~ k eff the total momentumtransferred by the Raman interaction, and α a chirprate applied to the relative frequency between the Ramanbeams to compensate for the changing Doppler shift ofthe falling atoms. Residual vibrations of the mirror con-tribute noise to the inertial phase φ a .The concept we develop relies on a so-called dual- T operation of the interferometer. Instead of one pulsesequence, two interleaved pulse sequences with slightlydifferent T values are performed [Fig. 1(b)]. By tuningtheir two-photon Doppler-detunings, each set of pulsesaddresses a different vertical velocity class of the atoms.We operate the two interferometers with scale factors dif-fering by the ratio τ . , choosing the interferometricdurations T = T and T = √ τ T , with T = 55 ms (seeMethods).Conventionally, the population ratio between the inter-ferometer states is measured directly, and the cosine ofthe phase is extracted. In our dual- T scheme, we detectthe phases φ , φ of both interferometers by acquiringan image of the atoms in one of the final atomic states. Figure 2. Analysis of dual- T interferometry measurements in single-shot, two-shot, and three-shot operation. (a) Single-shotdual- T operation with D = 8 for φ a in the range of ± π (color coded). Every dot corresponds to a single measurement.Each discrete value of φ diff corresponds to a different sub-range of φ a , and within that sub-range φ sum changes continuouslyand linearly with φ a . (b) Sequential two-shot dual- T operation with D = 7 , for φ a in the range of ± π , presented in the φ ( D =7) diff - φ ( D =8) diff plane. The discrete clusters in this plane correspond to different sub-ranges of φ a . (c) Sequential three-shotdual- T operation with D = 5 , , for φ a in the range of ± π , presented in the φ ( D =5) diff - φ ( D =7) diff - φ ( D =8) diff space. For clarity, onlya subset of the solutions around φ ( D =5 , , diff = 0 is presented, and gray ovals surround the expected solutions. The independent readout of both interferometers is en-abled by the ballistic expansion of the cloud, which mapsthe different velocity classes onto different vertical posi-tions. To obtain the phase, in a manner equivalent to fullquadrature detection where both sine and cosine com-ponents of the phase are measured, we use phase-shearreadout [26]. We tilt the retro-reflecting Raman mirrorby a small angle before the final π/ -pulses to gener-ate a spatial transverse interference pattern across thecloud, as utilized in point-source interferometry [27–29]and shown in Fig. 1(c). The phase offset of this patterncan be directly extracted with constant sensitivity for allinterferometric phases.Figure 1(d) shows single-shot measurements in a dual- T operation with the dynamic range enhanced by a factorof 8. We vary φ a by changing the chirp rate α with re-spect to its nominal value α = k eff g , thereby emulatingchanges in g . We find that φ a is mapped onto a unique setof straight, parallel lines in the plane spanned by φ and φ owing to the quadrature detection capability. Con-versely, conventional detection which resolves only thecosine of the phase, would result in many phase ambigu-ities due to very different values of φ a being mapped tosimilar measured phase components [Fig. 1(e)], severelylimiting the benefits of a dual- T operation. Quadraturedetection, together with the strong suppression of com-mon noise due to operation at very similar scale factors,allows the dual- T scheme to achieve a significantly largerenhancement compared to past implementations of si-multaneous atom interferometers with different scale fac-tors [22]. DUAL- T PHASE ANALYSIS
Phase estimation for single shot dual- T . Themeasured interferometric phases φ , φ are constrainedto the bare dynamic range ± π and can be written as φ = φ a − πn , (1) φ = τ φ a − πn . (2)The integers n and n , which respectively bring φ and φ to the range ± π , are a priori unknown.We define D ≡ (1 − τ ) − , with τ = ( T /T ) the scale-factors ratio. For integer values of D , the dynamic-rangeenhancement is exactly D ; as illustrated in Fig. 1(a), φ and φ have a joint period of Dπ as in a moiré effect,resulting in an extended ambiguity-free dynamic range of ± Dπ (see Methods for discussion on non-integer values).To analyze a dual- T measurement, we define the quan-tities φ diff and φ sum , (cid:18) φ diff φ sum (cid:19) = 11 + τ (cid:18) τ − τ (cid:19) (cid:18) φ φ (cid:19) . (3) φ diff and φ sum act as coarse and fine measurements, re-spectively. As shown in Fig. 2(a), which presents an anal-ysis of D = 8 dual- T measurements, φ diff takes on a dis-crete set of D − values. This constrain uniquely deter-mines the values of n and n and hence the π sub-rangein which φ a lies. Correspondingly, φ sum is a continuousvariable, providing the estimation of the inertial phase φ a within that sub-range (see Methods). Phase estimation for sequential operation.
Wenow turn to discuss further enhancement of dynamicrange obtained by a sequence of several dual- T shots withalternating integer values of D . Here we fix T and al-ternate T between shots. Assuming that changes in φ a Figure 3. Performance analysis of dual- T interferometry. (a) Estimated inertial phase for single-shot measurements (left; insetshown zoom on ± π/ region) and for sequential two-shot (center) and three-shot (right) measurements, with dynamic rangeenhancement factors of , , and , respectively. Outlying measurements appear as data points visibly distant ( > π ) fromtheir expected value. We observe only 10, 24, and 26 such outliers out of 2000 data points in the three data sets, respectively.(b) Residuals of the estimated phases, including only non-outlying measurements. Compared to single-shot measurements,standard deviations of two- and three-shot residuals are smaller by factors of √ and √ , respectively. (c) Outlier probability ǫ as a function of dynamic range enhancement obtained for individual values of D and for various combinations of consecutivecoprime D values. For D ≤ , there were no outliers in the measured data set. Error bars represent 67% confidence intervalsof the estimated value. Solid lines are calculated using Eq. (4) with σ ind = 80 mrad . Dashed lines represent outlier probabilityfor an alternative scheme of averaging two or three sequential shots using a single D value. (d) Estimation error of gravitationphase per shot σ φ, est . The error is dominated by vibration-induced phase noise and is nearly equal for all realizations. In(c),(d), arrows indicate the measurements shown in (a),(b). are small between consecutive shots, the above analysisper shot provides n mod D . Taken together, the full se-quence uniquely determines n within a range defined bythe least common multiple of the employed D values, or,for coprime integers, simply their product (see Methodsand Fig. S1).Analyses of two-shot operation with D = 7 , andthree-shot operation with D = 5 , , are shown inFig. 2(b,c). Each data point is a measurement with arandom value of φ a within the extended dynamic ranges ± π and ± π , respectively. We observe two- andthree-dimensional clustering of the differential phases φ diff , where each cluster corresponds to a unique, non-ambiguous phase range smaller than π . Noise and outlier probability.
By virtue of simul-taneously operating the two interferometers with similarscale factors, vibrations-induced phase noise is highly cor-related between them [Fig. 1(f)] and has negligible contri-bution to φ diff . The dominant noise in φ diff results fromuncorrelated, independent detection noise in φ and φ ,whose standard deviation we denote as σ ind (see Methodsfor a detailed discussion of noise terms).As D is increased, and the discrete values of φ diff be-come denser, the uncorrelated noise may lead to errorsin determining the correct sub-range for φ a , producingan outlier with phase estimation error in multiples of π .The probability ǫ for a measurement to be such an outlieris approximately [see Eq. (10) for exact expression] ǫ ≈ erfc (cid:18) π D · σ ind (cid:19) . (4) Crucially, ǫ depends only on the uncorrelated noise andnot on the vibrations-dominated correlated noise, whichis typically much larger. In the data presented inFig. 2(a), we observe one such outlier out of 5000 mea-surements for D = 8 .For the case of sequential dual- T operation, the totaloutlier probability ǫ seq depends on the outlier probabili-ties in each shot and, in the relevant regime of small errorprobabilities, is given simply by their sum. For any de-sired dynamic range and temporal bandwidth, the outlierprobability is minimized by choosing consecutive coprimevalues of D . EXPERIMENTAL CHARACTERIZATION
Performance analysis.
To quantify the performanceof the dual- T scheme in terms of phase sensitivity andoutlier probability, we extend the phase scan to random,known, values of φ a within the range of ± π , corre-sponding to accelerations of ±
65 mm / s at T = 55 ms .For each phase, we perform measurements with D valuesbetween 5 and 15, and perform dual- T analysis using each D separately, using pairs of consecutive D values, and us-ing triplets of consecutive coprime D values. We analyzeeach measurement within its appropriate extended dy-namic range; data points that are outside the measure-ment’s relevant dynamic range are wrapped back ontoit. We then compare the extracted phase to its expectedvalue, from which we estimate the outlier probability ǫ aswell as the phase residuals of the measurements withoutoutliers.The results, presented in Fig. 3, demonstrate an en-hancement of dynamic range by factors of in a singleshot, ∼ in two shots, and ∼ in three shots, whilemaintaining phase residuals of σ φ, est ∼
160 mrad / shot ( ∼ . µ m / s / shot ), and with outlier probabilities of0.5%, 1.1%, and 1.2%, respectively. In general, we findexcellent agreement with the error model described byEq. (4), with σ ind = 80 mrad estimated from these data.We note that an outlier fraction on the order of 1% isacceptable in most applications, as such outliers can beidentified and removed by comparison to adjacent shotsor using auxiliary measurements. However, even if nearlyzero outlier fraction is required, the dual- T scheme candeliver a significant dynamic range enhancement. Forexample, with the above measured value of σ ind and for D = 6 , we expect ǫ ≈ × − .Furthermore, averaging over N repeated measure-ments with the same D value can decrease the outlierprobability ǫ by effectively reducing σ ind by a factor √ N .However, by employing the same number of sequentialmeasurements with alternating values of D as describedabove, the same value of ǫ may be achieved with signifi-cantly larger dynamic range enhancement, as seen fromcomparing solid and dashed curves in Fig. 3(c). Stability of dual- T interferometry. To demon-strate the long-term stability of dual- T interferometry, wecontinuously measure gravity over 20 hours with D = 10 .As shown in Fig. 4, φ a follows the expected tidal grav-ity variations throughout the measurement period. Itremains stable at time scales of sec , to better than
100 nm / s , showing that the dual- T scheme does not addsignificant drifts to the estimated phase. Conversely, φ diff does exhibit small drifts which we attribute to mutuallight-shift between the two interferometers. However,due to the discrete nature of φ diff , these drifts can beeasily corrected in several ways (see Methods). TRACKING FAST-VARYING SIGNALS
We now turn to discuss dynamic scenarios, such as mo-bile gravity surveys or inertial measurements on a nav-igating platform, where the measured acceleration andthus φ a change dramatically between shots. Dual- T in-terferometry with fixed D can directly track a signalthat randomly varies by up to ± Dπ from shot to shot.Moreover, alternating the value of D between consecu-tive measurements can enable tracking a signal with evenlarger variations; however, the sequential analysis de-scribed above cannot be applied due to the phase chang-ing between shots, and a different analysis method is re-quired.To track such a signal, we employ a particle filter esti-mation protocol [30, 31]. Particle filtering is a powerfuland well-established technique in navigation science, sig- -0.1-0.0500.050.1 -2-1012 a [ r ad ] g e s t - g [ m / s ] Integration time t [sec] -3 -2 -1 -1 [ r ad ] g [ m / s ] (a)(b) Figure 4. Stability of dual- T interferometry. (a) Time seriesof φ sum , with half-hour binning, measured with D = 10 . Theresults follow the expected tidal gravity variation as calcu-lated from solid-earth model (black solid line). (b) Allan de-viation of the residuals of φ sum from the tidal model. Dashedline is a fit to t − / with sensitivity per shot of
155 mrad ( . µ m / s ). -1200-60006001200 -80-4004080 e s t [ r ad ] a e s t [ mm / s ] -0.200.2 -0.0100.010 200 400 600 800 1000 Time step -30030 -202 d / d t [ r ad / s ho t] da / d t [ mm / s / s ho t] (a)(b) Figure 5. Tracking of a time-varying acceleration using dual- T interferometry combined with a particle-filter protocol. Mea-surements are performed with alternating D = 9 , . (a)Acceleration signal extracted from the measurements usingthe particle filter (red), compared to the input signal (black).Bottom panel shows the residuals with standard deviation of
174 mrad ( . µ m / s ). (b) Temporal derivative (shot-to-shotvariation) of the measured acceleration signal (red) comparedto the input (black). nal processing and machine learning, among other fields.It is a sequential, Monte-Carlo estimation approach basedon a large number of particles which represent possi-ble hypotheses of the system’s current state, e.g. , theinertial phase measured by the sensor. These hypothe-ses are weighted through Bayesian estimation after everymeasurement, converging on a solution that is consistentwith the sensor readings over time. In our context, un-der some model assumptions on the signal dynamics, useof particle filter enables full recovery of the single-shotbandwidth [29] while maintaining the large increase indynamic range rendered by the sequential operation.An experimental realization of tracking a dynamic sig-nal is presented in Fig. 5. We change the chirp rate α between shots to simulate a band-limited random walkof a and perform dual- T measurements with alternating D = 9 , . The sequence of measured phases is thenanalyzed with a particle filter protocol using a second-order derivative model (see Methods), to extract bestestimate for the time-series of φ a . Following a brief con-vergence period (Supplementary Fig. S4), we successfullytrack this time-varying signal which spans over π and changes by up to π between shots, with sensitivityper shot similar to measurements of static signals undersimilar conditions, and with no outliers. We note thatwhile the analysis was carried out in post-process, it is inprinciple compatible with implementation as a real-timeprotocol. DISCUSSION
In conclusion, we present a novel approach to atominterferometry for significant enhancement of dynamicrange without a reduction in sensitivity and with highmeasurement bandwidth. In applications where tradi-tional atom interferometers must be operated at reducedsensitivity due to the expected dynamic range of the mea-sured signal, our approach enables measurements with asubstantial increase in sensitivity while maintaining thenecessary dynamic range.Taking advantage of full-quadrature phase detectionand common-noise rejection, we experimentally demon-strate an increase of dynamic range by more than anorder of magnitude in a single shot. Incorporating datafrom several consecutive shots, the dynamic range fur-ther increases in exponential fashion, allowing us to reachthree orders of magnitude gain using only three measure-ments. Finally, we demonstrate tracking of a dynami-cal signal with tens of radians shot-to-shot variation bycombining the dual- T measurement with a particle-filterprotocol, representing a major improvement compared torecent works [21].This approach can dramatically enhance performanceof sensors, and in particular inertial-sensing atom in-terferometers, under challenging conditions, by enablingnon-ambiguous operation without sacrificing either sen-sitivity or bandwidth. Such conditions are encounteredin field operation of such sensors, for example in mobilegravity surveys or when used for inertial navigation ona moving platform. By extending the sensor real-time dynamic range, the requirements on vibration isolationor corrections based on auxiliary measurements can berelaxed, or equivalently, existing sensors can be operatedin more demanding environments.Dual- T measurements can be realized by multiplemeans, based on known atom interferometry tools, e.g. ,dual-species interferometry [22] or momentum-state mul-tiplexing [23], in addition to phase-shear readout [26]used in this work. It is also compatible with importantatom-interferometry practices, such as k -reversal [32, 33]and zero-dead-time operation [34]. Further improvementof the scheme is possible by incorporating more than twointerferometric sequences within the same experimentalshot, enabling the gain demonstrated here for sequentialoperation within a single shot. Extension of the approachto other atom interferometry configurations is also pos-sible. ACKNOWLEDGMENTS
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Experimental sequence.
We load a cloud of Rbatoms in a magneto-optical trap (MOT) and launch itupwards at . / s with moving optical molasses, whichalso cools the cloud to µ K . Atoms initially populateequally all m F sub-levels in the F = 2 hyperfine mani-fold. We select atoms in two distinct velocity classes andin the m F = 0 state using two counter-propagating Ra-man π -pulses, with µ s duration and a relative Dopplerdetuning of
80 kHz . Two interferometric sequences of π/ - π - π/ pulses, with durations of , , and µ s re-spectively, address each of the velocity classes as shown inFig. 1. The timing of the π pulses of the two interferome-ters is set to
22 ms and . after the apex of the tra-jectories. The precise ratio of T /T contains empirically-calibrated corrections on the order of − with respectto the naive √ τ value, attributed mainly to finite Ra-man pulse durations [35]. Before the final π/ -pulses,the Raman mirror is tilted by µ rad . With the MOTbeams tuned on resonance with the F = 2 → F = 3 cycling transition, a fluorescence image of atoms in the F = 2 level is taken on a CCD camera oriented perpen-dicularly to the Raman mirror tilt axis. The experimentis repeated every 2 to 3 seconds. Extraction of the measured phases φ , φ . Wefirst integrate the image horizontally to find the verticalGaussian envelopes of the fringe patterns, which are usedto define the analysis region-of-interest for each interfer-ometer (Supplementary Fig. S2). We then vertically in-tegrate the image over those regions and fit the resultingprofile to Gaussian envelopes with sinusoidal modulation.The phases of the measurement are taken as the phasesof the fitted fringes at the horizontal center of the cloud.Finally, we calculate and correct the vibration-inducedphase based on the auxiliary accelerometer signal, tak-ing into account the different interrogation times of eachinterferometer.
Single-shot dual- T analysis. For each dual- T shot,we rotate the measured φ , φ according to Eq. (3), φ diff = 2 π τ (cid:16) n D − ∆ n (cid:17) , (5) φ sum = φ a − π ( n + τ n )1 + τ . (6)Within the extended dynamic range of ± Dπ for φ a , theinteger n takes values within ± ⌊ D/ ⌋ , and ∆ n = n − n takes either , ± . From φ diff we uniquely determine ∆ n , ∆ n = ( | φ diff | < π τ (cid:0) − D (cid:1) − sgn ( φ diff ) | φ diff | > π τ (cid:0) − D (cid:1) , (7)and n follows as the round value of D (cid:2)(cid:0) τ (cid:1) φ diff / (2 π ) + ∆ n (cid:3) . Finally, we estimate φ a by substituting n and n = n − ∆ n back into φ sum . We focused the discussion on integer D . Rational D yields joint phase periodicity according to the lowest termnumerator of D , but with less efficient common-modenoise rejection. For irrational D , there is no well-definedperiodicity and hence no discrete set of allowed φ diff solu-tions. While in both cases dynamic range enhancementis attained, optimal results are achieved for integer D . Sequential dual- T analysis. From a sequenceof N ( N = 2 , in this work) shots with alternat-ing D ( i ) , where i = 1 , . . . , N , we retrieve N pairs ofphases [ φ ( i )1 , φ ( i )2 ] . Analyzing each shot separately asdescribed above, we extract from them a set of val-ues ˜ n ( i )1 , each within ± (cid:4) D ( i ) / (cid:5) . Joint analysis of thesequential measurements in principle amounts to find-ing the integer n that satisfies the set of equations ˜ n ( i )1 = n mod D ( i ) . The solution is unique within therange ± LCM (cid:0) D (1) , . . . , D ( N ) (cid:1) , LCM denoting the leastcommon multiple. This analysis assumes that n ( i )1 = n (1)1 for all i , as the first interferometer always measures φ a with the same interrogation time T . However, for valuesof φ a close to odd multiples of π , phase noise may causevariations of up to ± in n ( i )1 . We calculate the varia-tions ∆ n ( i )1 = n (1)1 − n ( i )1 for i > as the round valueof ( φ (1)1 − φ ( i )1 ) / (2 π ) and take them into account whensolving the set of equations described above for n . InFig. 2(b), only measurements with ∆ n (2)1 = 0 are shownfor clarity; the full range of results is shown in Supple-mentary Fig. S3. Experimental noise parametrization.
Extendingon Eqs. (1-2), we write the phases φ , φ as φ = ( φ a + δφ corr ) + δφ ind , − πn , (8) φ = τ ( φ a + δφ corr ) + δφ ind , − πn . (9)Here δφ corr ∼ N (cid:0) , σ corr (cid:1) is the noise term on the in-ertial signal common to both interferometers, whereas δφ ind , , δφ ind , ∼ N (cid:0) , σ ind (cid:1) are independent noiseterms, e.g. , due to detection noise of each interferom-eter. While the methodology and data processing willwork well for any noise covariance, this parametrizationis natural to operating the two interferometers simultane-ously and with similar scale factors, such that the inertialnoise is highly correlated as demonstrated in Fig. 1(f).With this parametrization and based on Eq. (3), φ diff and φ sum are characterized by random noisewith standard deviations σ ind / √ τ and σ φ, est = p σ corr + σ ind / (1 + τ ) , respectively. An outlier mea-surement occurs when the random deviation of φ diff fromits theoretical value is larger than half the difference be-tween its discrete solutions, which is π/ (cid:2) D (cid:0) τ (cid:1)(cid:3) .The probability of such an event is given by ǫ = erfc (cid:18) π √ D √ τ σ ind (cid:19) , (10)and approximated by Eq. (4) for τ ≈ . See Supplemen-tary Information for experimental noise characterization. Systematic phase shifts.
Dual- T measurementshave several systematic phase shifts which are commonalso to conventional atom interferometers, due to factorssuch as one-photon light shifts, two-photon light shifts[36], and offset of the Raman frequency from Doppler res-onance [37]. Typically, these effects are either estimatedand accounted for theoretically, or they are eliminatedthrough wave-vector reversal ( k -reversal) [32, 33].Nevertheless, some of these shifts may be complicatedor modified by the existence of two simultaneous interfer-ometer pulse sequences, while new sources of systematicshifts may arise, such as due to an estimation error of thecloud center position when using phase shear readout. Asdemonstrated in Fig. 4, these effects do not contribute tobias instability in the measured phase up to few mrad , al-though they may introduce a constant bias which can bedetermined and calibrated in advance by comparison ofdual- T measurements with standard interferometry. Wecorrect this bias by performing 15 to 50 initial calibrationmeasurements for different D values and k eff signs, wherewe assume prior knowledge of φ a . For the time-varyingexperiment in Fig. 5, these calibration measurements arenot included in the particle filter analysis. Correction of drifts in the differential phase.
Asshown in Fig. S4(a), φ diff exhibits small drifts over timefrom its expected discrete value. While these drifts donot directly enter into the estimation of φ a , they mayhave a large impact on outlier probability ǫ . By per-forming k -reversal, we observe that the drift in φ diff isanti-symmetric with respect to k eff . We therefore at-tribute the observed phase drifts to differential light-shiftbetween the two interferometric states of the Ramanpulses. As the temporal response function to externalphase-shifts is anti-symmetric with respect to the cen-tral π -pulse, normally the effect of light shifts due to theinterferometer pulses cancels up to changes in the lightshift during the interferometer due to laser intensity fluc-tuations [38]. In our dual- T realization, the light shiftinduced by the π/ -pulses of the shorter interferometeron the longer one still cancel as before, but each inter-ferometer experiences an uncompensated light shift dueto the π -pulse of its counterpart. A realization of dual- T with simultaneous π -pulses for both interferometers willcircumvent this effect [22].These mutual light shifts will be of approximatelyequal amplitude but opposite signs, therefore they aresuppressed in φ sum by a factor (1 − τ ) / (cid:0) τ (cid:1) but am-plified in φ diff by a factor (1 + τ ) / (cid:0) τ (cid:1) . These ef-fects of light shifts are entirely canceled by performing k -reversal, and indeed, as we observe in Fig. S4(b), theaverage value of φ diff over ± k eff remains stable at timescales of sec to better than . In the particle fil-ter demonstration, we used both k eff signs to correct suchdrifts, demonstrating the compatibility of the k -reversal technique with the dual- T approach.Additionally, due to the discrete nature of φ diff , theobserved drifts can also be deterministically correctedwithout requiring k -reversal, and thus with practically noimpact on the interferometer performance or bandwidth.For the data presented in Figs. 2 and 3, we continuouslycorrect drifts in φ diff , without assuming prior knowledgeof φ a , by tracking the difference between the measured φ diff values from the nearest discrete values and subtract-ing their long-term, moving average. Particle filter implementation.
We choose as statevariables the instantaneous value of the inertial phase φ a and its first- and second-order time derivatives, denoting x ( m,i ) = [ φ ( m,i ) a ˙ φ ( m,i ) a ¨ φ ( m,i ) a ] T for the m -th particleat the i -th time step. As observables, we choose the twointerferometer phases φ , . The initial value and deriva-tives of the input φ a signal are approximately − π , . π/dt , and . π/dt , respectively. We represent a sce-nario where some imperfect knowledge about the startingconditions exists by drawing the initial values of the par-ticles from normal distributions characterized by µ (0) φ = − π σ (0) φ = 50 π,µ (0)˙ φ = 0 σ (0)˙ φ = 10 π/dt,µ (0)¨ φ = 0 σ (0)¨ φ = 8 π/dt . (11)At each time step of the filter, we first propagate theparticles’ state according to x ( m,i +1) = F · x ( m,i ) + w ( m,i ) ,with F being the state propagation matrix and w ( m,i ) a random process noise with zero mean and covariancematrix Q . For our model, we have F = dt dt dt , Q = dt q a , (12)where dt is the time interval between consecutive mea-surements. Following state propagation, we calculatethe expected interferometer signals for each particle as φ ( m,i )1 = φ ( m,i ) a and φ ( m,i )2 = τ ( i ) φ ( m,i ) a , where τ ( i ) is thescale-factor ratio in the i -th measurement. We calculatetheir residuals from the actual measurements modulo π and weigh each particle according to the likelihood thatthese residuals are consistent with the independent mea-surement noise σ ind , which we take as
73 mrad accord-ing to the spread of δφ ind,diff . State variables estimationis achieved by using a ridge-detection algorithm (MAT-LAB tfridge function) on the time-dependent particlehistogram to estimate φ ( i ) est , as demonstrated in Supple-mentary Fig. S5. For q ¨ a , we took a value of π/dt , as itminimizes the mean error of the estimated φ a from themeasured φ , .0 SUPPLEMENTARY FIGURES
Measured phases ProbedquantityProbedquantity
Figure S1. Principle of dynamic-range enhancement with sequential operation. In this example, combining dual- T interferom-eters with enhancement factors of × (top) and × (bottom) provides overall enhancement of × in two shots.Figure S2. Dual- T image analysis. Center panel shows raw fluorescence image and, on the right, the horizontal integrationwith fit to two Gaussian envelopes. From the fit, we determine the analysis region-of-interest for each interferometer (dashedrectangles), over which we integrate the image vertically and fit to a Gaussian envelope with sinusoidal modulation (top andbottom panels). - - /20 /2 d i ff D = [ r ad ] - - /2 0 /2 d i ff D = [ r ad ] - - /2 0 /2 diff(D=8) [rad]-- /20/2 s u m ( D = ) [ r ad ] diff(D=7) diff(D=8) [rad] diff(D=7) diff(D=8) [rad] diff(D=7) diff(D=8) [rad] diff(D=7) -56-40-24-88244056 n = 1 n = 0 n = -1 (a) (b) a Figure S3. Extended display of the sequential (two-shot) measurements with D = 7 , shown in Fig. 2(b). (a) Results in the φ ( D =7) diff - φ ( D =8) diff plane, including data points with ∆ n = 0 which result from measurements where φ a is near odd multiples of π .Note that ∆ n is extracted from the measurements as well, and therefore all these data points are valid sensor measurements.(b) Results in the φ ( D =8) diff - φ ( D =8) sum plane, for data points with ∆ n = 0 and specific values of φ ( D =7) diff as specified in the title andindicated by red rectangles in (a). -0.06-0.0300.030.06 d i ff [ r ad ] +k -k avg. Integration time t [sec] -4 -3 -2 -1 d i ff [ r ad ] -0.4-0.200.20.4 (a)(b) Figure S4. Analysis of drifts in φ diff based on the data collected in the experiment presented in Fig. 4, with D = 10 . (a) Timeseries of φ diff , with half-hour binning. Systematic phase shifts, which drift by tens of mrad , are evident for both signs of k eff but are strongly suppressed after averaging. The observed drifts are much smaller that the spacing between the discrete valuesof φ diff ( ∼ ± π/ ∼ = ± .
35 rad for D = 10 , see inset). (b) Allan deviation of φ diff , after averaging ± k eff . The uncertainty pershot is
39 mrad , corresponding to σ ind = 53 mrad . Figure S5. Detailed presentation of the first ten time-steps of the particle filter analysis for the dynamic signal presented inFig. 5. For each time step, we present a histogram of the φ a values of the particles, weighted based on their likelihood ofdescribing the actual measurement (red curves). Dashed curve in the first time step is the initial distribution of particles,representing uncertainty in the initial φ a value. For reference, we present also the input signal (black curve) and the φ a valuesestimated from ridge analysis of the particles data (blue markers). In the first time steps, multiple solutions exist (spaced bymultiples of π and π apart, for odd and even shots with D = 9 and , respectively), and the ridge detection algorithmdoes not necessarily select the correct one. After several time steps, most of the particles (and eventually all) converge to followthe input signal. Inset: zoom-in on the particle distribution in the 7th time step. SUPPLEMENTARY INFORMATION
Raman beams generation and optics.
The Raman beams are derived from a single distributed Bragg reflectorlaser diode tuned . below the F = 2 → F ′ = 1 of theD transition. The Raman laser is phase modulatedwith an electro-optic modulator (EOM) at ∼ .
834 MHz , followed by dual-stage optical amplification. The relativelysmall one-photon detuning causes significant spontaneous scattering, and in fact limits our ability to implement morethan two interferometric sequences in a single shot due to loss of contrast. The maximal in-fiber Raman beamsintensity, including all modulation sidebands, is . For dual- T operation, we use only . to reduce the velocityacceptance range of the Raman pulses. The beams are collimated to a
40 mm diameter and have a circular polarization.After traversing the vacuum chamber, they are retro-reflected from a mirror mounted on a piezo tip-tilt stage anda passive vibration-isolation platform. Residual vibrations are measured with a sensitive classical accelerometer andthe associated phase noise is subtracted in post-processing [17, 18].The Raman EOM is driven by a fixed . microwave signal mixed with a ∼
34 MHz signal from an agile directdigital synthesizer (DDS). For dual- T operation, each interferometer employs one of two phase-coherent channels ofthe DDS, which are electronically switched prior to each Raman pulse. The two signals are step-wise chirped at anequal rate α with a relative offset of
80 kHz to address the two velocity classes. We set α = α − k eff δg , where α = k eff g ( g is the local gravity acceleration), and δg is the simulated change in gravity. k -reversal is achieved by flipping thesign of α . Noise characterization.
We estimate σ φ, est directly as the standard deviation of φ est − φ in . Generally, weobserve σ φ, est in the range of −
180 mrad ( . − . µ m / s ) per shot, with the dominant contribution attributed toresidual vibration noise. As such, it varies slightly between measurements done at different times, depending on theenvironmental noise. We also observe some dependence on D , with larger noise at lower D values [Fig. 3(d)], whichis in part attributed to weaker common-noise rejection within each dual- T shot.The estimate of σ ind is based on the standard deviation of φ diff − φ diff, , with φ diff, calculated from φ a accordingto Eq. (3). We note that for low D values with negligible outlier probabilities, σ ind can also be estimated accordingto the standard deviation of δφ ind,diff , which is the residual between measured φ diff and its nearest allowed discretevalue, as defined in Eq. (5), without using any prior information on φ a ; however, when the outlier fraction becomessignificant, this method would result in under-estimation of σ ind . We observe σ ind in the range of −
80 mrad80 mrad