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Dual-axis, high data-rate atom interferometer via cold ensemble exchange

Akash V. Rakholia,1, 2 Hayden J. McGuinness,1 and Grant W. Biedermann1, 2, ∗

1Sandia National Laboratories, Albuquerque, NM 87185, USA2Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy,

University of New Mexico, Albuquerque, New Mexico, 87131, USA(Dated: July 18, 2018)

We demonstrate a dual-axis accelerometer and gyroscope atom interferometer, which forms thebuilding blocks of a six-axis inertial measurement unit. By recapturing the atoms after the interfer-ometer sequence, we maintain a large atom number at high data-rates of 50 to 100 measurementsper second. Two cold ensembles are formed in trap zones located a few centimeters apart, and arelaunched toward one-another. During their ballistic trajectory, they are interrogated with a stimu-lated Raman sequence, detected, and recaptured in the opposing trap zone. We achieve sensitivitiesat µg/

√Hz and µrad/s/

√Hz levels, making this a compelling prospect for expanding the use of

atom interferometer inertial sensors beyond benign laboratory environments.

I. INTRODUCTION

Light-pulse atoms interferometers (LPAIs) [1–3] havedemonstrated remarkably precise and stable measure-ments of acceleration and rotation. Numerous applica-tions have been suggested, including gravimetry [4], seis-mology, inertial navigation [5–7], and near-surface forcecharacterization [8]. However, such systems are typicallydesigned for laboratory environments in pursuit of max-imum sensitivity, and thus are bulky and fragile, operat-ing at the few Hertz rate. Lately, there has been greatinterest in adapting such systems to field use [9, 10].Adapting LPAI technology to this application space re-quires a compact sensor explicitly tailored for dynamicenvironments. Furthermore, inertial navigation requiresa sensor capable of measuring both accelerations and ro-tations. Building on our previously reported high data-rate accelerometer [11], we report a compact LPAI thatis capable of simultaneous acceleration and rotation mea-surements, and suitable for dynamic environments.Our LPAI uses stimulated Raman transitions between

hyperfine levels in a π/2 − π − π/2 pulse sequence [12]to coherently separate, redirect, and recombine atomicwavepackets from a cold atomic sample. The spatially-dependent phase of the light field is imprinted on theatoms during each pulse. To first order, the resultingphase difference between the hyperfine levels is given by∆φ = ke · (a − 2v ×Ω)T 2, where ke is the effective Ra-man wavevector, T is the delay between pulses, and v,a, and Ω are the velocity, acceleration, and rotation ofthe atoms relative to the platform, respectively. Sinceke is typically locked to an atomic transition, a precisedetermination of ∆φ results in a precise measurement ofacceleration and rotation.The phase shift ∆φ includes both acceleration and ro-

tation contributions. If we consider a system of two in-dependent interferometers [13] with phase shifts φa andφb, and opposing velocities va = −vb, the sum and dif-

∗ Corresponding author: gbieder@sandia.gov

ference of the phases result in a term proportional toacceleration and rotation, respectively.In a dynamic environment, a free-fall measurement

such as this requires a short interrogation time [14] tocurtail otherwise significant and uncontrolled excursionsof the ensemble. Due to the quadratic dependence of ∆φon T , a significant loss of sensitivity ensues. This canbe compensated by using a large atom number and mini-mizing the dead time between measurements. We achieveboth through recapture, recycling as many atoms as pos-sible between shots [11]. Given the constraint of a smallT , it is preferable to maximize velocity to maximize therotation rate sensitivity. One method to achieve a largevelocity while maintaining a high data-rate is through theuse of our ensemble exchange technique, reported herein.

II. APPARATUS

Our ensemble exchange apparatus swaps cold atomicensembles between two regions via launch and recapture.Two ensembles are simultaneously loaded in magneto-optical traps (MOTs) located a few centimeters apart.Through the use of conventional fountain launch tech-niques, the ensembles are launched towards each otherat a velocity of a few meters per second. The interferom-eter takes place over several milliseconds during the bal-listic transit between the two loading regions. At the endof the experiment, atoms are recaptured in the oppositetrap from where they were launched. After a few mil-liseconds of recapture/loading, the cycle repeats. Thus,the atoms are “exchanged” between the trapping regions.Typically, the background-limited time it takes to loada MOT from vapor is on the order of a second. Withrecapture, the loading time is determined by the trap’stime-constant, τMOT, which is typically on the order ofseveral milliseconds [15].We demonstrate this concept with a custom-built sen-

sor head. The sensor head is designed around a rect-angular quartz vacuum cell with inner dimensions of20 × 30 × 60 mm3 and 3 mm thick walls (Fig. 1). Theuse of high vacuum, as opposed to ultra-high vacuum,

2

xy

z

^

^

^

(a)

(b)To Vacuum

System

FIG. 1. (Color online) Diagram of the apparatus implement-ing the cold ensemble exchange and dual-axis, high data-rateatom interferometer. (a) Front view: Two MOTs are loaded36 mm apart. Cooling beams are shown in blue, probe beamsin pink, and Raman beams in yellow. The trap is turnedoff, and the outer and inner cooling beams are blue and reddetuned, respectively, which launches the ensembles towardseach other. After the experiment, atoms are recaptured in theopposite trap to facilitate loading. (b) Side view: The designallows for four planes of optical access, enabling a compactapparatus. The vector g shows the direction of gravity, whilea and Ω are the directions of acceleration and rotation mea-surement, respectively.

significantly relaxes the vacuum requirements for the ap-paratus. Two magneto-optical traps of 87Rb are loaded36 mm apart at opposite ends of the cell from a back-ground pressure of 2 × 10−7 Torr. The high vapor pres-sure is generated from a temperature-controlled sourcecontaining Rubidium from a crushed ampoule. The highbackground vapor density allows for a fast short-termloading rate of 1× 105 atoms/ms. This limits our coher-ence time to less than 100 ms, which is sufficiently highfor our purposes. This vapor pressure represents a nearlyoptimal trade-off between coherence time and MOT load-ing rate, and occurs conveniently near the room temper-ature vapor pressure of Rb.

The loading of two MOTs in close proximity is facili-tated by small (approximately 16 mm length) quadrupole

Tim

e (

ms)

Launch

Inte

rfero

me

ter

Re

ca

ptu

re

0.0

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FIG. 2. (Color online) Frames from a movie demonstratingthe ensemble exchange with a cycle time of 20 ms. The ensem-bles are launched towards each other over 2 ms to a velocityof 2.5 m/s, with simultaneous sub-Doppler cooling. The in-terferometer takes place during their ballistic trajectory withan interrogation time T ≈ 4 ms. Imaging beams are off dur-ing the interferometer to reduce photon scatter which wouldotherwise cause decoherence. Dashed circles represent the ap-proximate positions of the ensembles during this stage. Atomsare then recaptured in the opposite trapping region, with ad-ditional atoms loaded from vapor for 7 ms. This cycle repre-sents a high atom number, low bandwidth cycle to optimizethe image contrast. Images were taken with a Lumenera CCDcamera (Lm075) using a 0.2 ms exposure.

coils located 36 mm apart, and a custom-designedopto-mechanical frame to support the necessary optics.The sensor head encloses a volume of approximately5000 cm3, excluding components of the high vacuumsystem. Due to the close proximity of the MOTs, themagnetic field geometry changes based on the relativealignment of the quadrupole field polarities. Aligningthe polarities in the same direction causes the magneticfield gradient to vanish at points between the two traps,roughly one quarter the distance between them. We alignthe polarities in a complementary, opposing geometry toextend the high radial field gradient of 9 G/cm by a fac-tor of 1.5. Furthermore, the opto-mechanical frame ismachined out of the insulator G10 to diminish eddy cur-rents and allow for fast magnetic field switching, whilemaintaining a coefficient of thermal expansion similar tothat of aluminum. The time-constants for magnetic fieldswitching are 14 µs and 37 µs for bias and quadrupolecoils, respectively. This ensures a sufficiently stable mag-netic field during the interrogation time of our high band-width experiment.The apparatus uses optical access along four indepen-

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tom

Num

ber

(×10

6)

Time (ms)

Recaptu

reV

apor

Ns

FIG. 3. (Color online) Recapture process during steady-stateensemble exchange. Black dots show the atom number calcu-lated from an avalanche photodiode (APD) signal as atomsare being recaptured and loaded into the trap. The red dashedline indicates the total steady-state atom number Ns, whilethe red dotted line is the vapor loading rate contribution.Many atoms are loaded via recapture over a few milliseconds,followed by vapor loading to replenish the lost atoms. By sub-tracting out the vapor loading contribution, we determine thenet recapture efficiency to be r = 85%. This includes lossesdue to background collisions during the cycle.

dent geometric planes to densely pack the required 19 op-tical beams and achieve a compact, modular design. Wefind that a useful and compact simplification is to em-ploy uncollimated cooling beams that diverge freely fromthe fiber at a half-angle of 5 degrees [5]. For each MOT,we use 20 mW of cooling light locked 12 MHz red of the|F = 2〉 → |F ′ = 3〉 cycling transition, correspondingto a total saturation parameter s0 = 15. Additionally,1 mW of repump light resonant with |F = 1〉 → |F ′ = 2〉prevents accumulation of population in the dark state.

Fig. 2 illustrates the launch and recapture process.Prior to the experiment, the quadrupole field is extin-guished, and the inner and outer cooling beams are se-lectively detuned 2.2 MHz red and blue of the cooling fre-quency, respectively. This results in optical molasses ina moving frame at 2.5 m/s towards the center of the cellfor both ensembles. The interferometer takes place as theatoms travel ballistically to the other side of the cell over14 ms. To facilitate loading for the next shot, at the endof the experiment the traps are turned on and atoms arerecaptured in the opposite trap. The characteristic cap-ture time is given by τMOT = 2~k/(µ′A), where A is themagnetic-field gradient and µ′ is the effective magneticmoment of the transition used [15]. With our typical ra-dial field gradient of 9 G/cm, we have τMOT = 2.0 ms.Following recapture, atoms are launched again for thenext shot of the experiment.

III. RECAPTURE

We extend the well-known MOT loading equation [16]by including recapture between discrete shots of the ex-periment. We model the atom number for each shot nas a constant recapture fraction r0, with a loss rate βand linear loading rate α. Furthermore, we assume acycle time Tc, and a fraction of the cycle η reserved forrecapture. The atom number may then be modeled asthe sequential sum of atoms loaded from vapor, atomsrecaptured, and atoms lost to background collisions. Ifwe assume no density-induced losses, the atom numberis given by the geometric sequence

Nn+1 = αηTc + (r0 − βTc)Nn, (1)

with solution,

Nn =αηTc

1− r(1− rn) , (2)

where r = r0 − βTc is the net recapture efficiency. Thisassumes that the recapture time ηTc is sufficiently largerthan the MOT relaxation time, in order to recapturemost of the available atoms. When the number of atomswhich were not recaptured equals the number of atomsloaded from vapor, we achieve a steady-state atom num-ber Ns, given by

Ns = limn→∞

Nn =αηTc

1− r=

αηTc

βTc + (1− r0). (3)

For r0 < 1 under low vapor pressure (βTc ≪ 1− r0), theloss rate is dominated by imperfect recapture. Thus, thesteady-state atom number Ns = αηTc/(1− r0) grows lin-early in α. Under higher vapor pressure (βTc ≫ 1− r0),the loss rate is dominated by background vapor collisions,resulting in a steady-state atom number Ns = ηα/β. As-suming constant trap parameters, α and β are both pro-portional to vapor pressure. Therefore, similar to canon-ical MOT loading, the ratio α/β and thus the total atomnumber are constant. Note that with r0 = 1, η = 1,and in the limit of small Tc, Nn becomes the usual MOTloading equation [16],

N(t) =α

β

(

1− e−βt)

. (4)

We test the efficacy of this model with two experi-ments. In the first, we explore the recapture dynamics ofour apparatus, for a 20 ms cycle time. This is revealed bymonitoring the atom number in the trap after the startof recapture until just prior to launch. This experimentis done in steady-state, where the number of atoms lostin recapture is replenished by loading from vapor. Fig. 3demonstrates a large increase in atom number during therecapture process over a millisecond timescale. The largeinitial increase represents ensemble recapture from theprevious shot of the experiment, with rNs atoms loaded.To achieve steady-state, an additional αηTc atoms are

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FIG. 4. (Color online) Inset: Loading of the exchange MOTafter starting with the traps off, at β = 3.3 1/s. Each datapoint represents a single shot of the experiment. Fitting thedata to Eq. (2) gives values for r and Ns at any given va-por pressure. Figure: Recapture efficiency r and steady-stateatom number Ns as a function of loss rate β. Circles rep-resent observed data during ensemble-exchange. Solid linesare calculated from MOT loading parameters α and β, us-ing the base recapture efficiency r0 as a free parameter. Thedata show that sacrificing a moderate amount of recaptureefficiency optimizes the atom number in the cycle, due to ahigher loading rate from vapor while the trap is active. Thedashed circles show typical operating regimes, as well as theparameters for Fig. 3

.

loaded from vapor. By subtracting out the linear contri-bution and dividing by Ns, we calculate a recapture effi-ciency of r = 85%. From the atom number Ns = 7× 106

and the size of the ensemble from Fig. 2, we calculate thepeak atom density to be 8× 109 atoms/cm3. This is wellbelow the limit for density-induces losses [16], validatingthe assumption made Eq. (1).As a second experiment, we measure the recapture ef-

ficiency by fitting Eq. (2) to the total atom number Nn

over multiple shots of the experiment, while the processapproaches steady-state. Beginning with zero atoms, en-semble exchange will build the population to a steady-state over 40 cycles. This is depicted in the inset ofFig. 4. Occasionally triggering the trapping coils on andoff allows us to monitor the loading of atoms until steadystate. Here, recapture efficiency is revealed by the num-ber of cycles required to reach steady-state.The recapture efficiency and steady-state atom number

are strongly affected by vapor pressure. Fig. 4 depicts theobserved and calculated recapture efficiency and steady-state atom number over a range of vapor pressures, char-acterized by atom loss rate, β. We demonstrate a recap-ture efficiency ranging from 85% to 92%. The recaptureefficiency decreases at higher vapor pressure due to an in-creased rate of collisions with background atoms duringthe ensemble’s trajectory. However, the increased vapor

loading rate results in a net increase of steady-state atomnumber Ns, until an optimum of approximately 85% re-capture efficiency is achieved at β = 5 1/s. From this fit,we extrapolate the expected recapture efficiency at zerovapor pressure (no background losses) to 96%.By minimizing the recapture time, we can increase the

bandwidth of our experiment and achieve a higher sen-sitivity. One way to accomplish this is to decrease thetrap time-constant τMOT by employing a large magneticfield gradient. However, this limits the maximum atomnumber in the trap. Accordingly, we employ a dynamicgradient of the quadrupole field. At the start of the re-capture, a high field gradient lowers the time constantof the trap allowing for atoms to quickly relax into thetrap. The field gradient is slowly ramped down over theremaining recapture time to increase the trap volume asatoms are loaded from vapor. For typical operating pa-rameters, this results in an increase of 10% in steady-state atom number.

IV. INTERFEROMETER

We create the atom interferometer using stimulatedRaman transitions. The Raman transitions arise fromresonant pulses of light applied at three equispaced timesduring the ensemble’s ballistic trajectory. The Ramanbeams are seeded from an external cavity diode laserlocked 900 MHz red of the |F = 2〉 → |F ′ = 3〉 transitionin 87Rb. They are generated from the phase coherent zeroand first-order sidebands of a fiber-based electro-opticmodulator operated at the hyperfine frequency. Follow-ing the modulator, the beams are amplified and shutteredby an acousto-optic modulator. Both sidebands are de-livered to the apparatus via common optical paths toavoid unwanted relative phase noise. At the apparatus,we separate this light into three beams that are retro-reflected from a shared 2 inch diameter mirror that de-fines a common inertial reference. The positions of thesethree beams coincide with the positions of the ensemblesat each of the three pulses in the π/2− π − π/2 interfer-ometer pulse sequence. The polarization configuration islin ⊥ lin to suppress the Doppler-free transition.The beams are delivered to the atoms at an angle of 10

from the orthogonal of the launch trajectory and are thenretro-reflected. This projects a component of the Ramanbeams along the velocity of the atoms, and the resultingDoppler shift of 1.1 MHz allows us to resolve both di-rections of the Raman wavevector ke [17]. Selecting oneof these wavevectors, we drive Doppler-sensitive Ramantransitions between the hyperfine levels at three pointsalong the trajectory of the atoms. Due to the asymmetryof the velocity vector of the ensembles, a single detuningresults in one interferometer with a vector of +ke andanother with −ke. Thus, the acceleration and rotationphase shifts are given by φ+ = ke · aT 2 = (φa + φb)/2and φ− = ke · (2v ×Ω)T 2 = (φa − φb)/2, respectively.In our apparatus, gravity is oriented 10 from the Ra-

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FIG. 5. (Color online) (a) Transition probability as a function of the two-photon Raman detuning with respect to the hyperfinetransition. The 10 projection of the Raman beam axis along the launch trajectory results in a Doppler shift, causing theresonance conditions of the two ke vectors to be separated by 2.2 MHz. For a given choice of resonance, one interferometer isoperated at +ke while the other is at −ke. (b) Gravitational acceleration causes the resonance conditions to shift oppositelyfor each interferometer, resulting in the second and third pulses being driven off resonance. (c) Sample fringes from theinterferometer, generated by scanning the electro-optical phase of the third pulse. The interferometer uses an interrogationtime of T = 4.1 ms, and data is taken at 60 shots/second.

man beam axis, causing a chirp of the Raman transitionresonance at a rate of 24.7 kHz/ms. However, due to thevelocity vector asymmetry, this chirp is opposite for thetwo interferometers. This causes the resonances to sepa-rate in frequency space for the second and third pulses.Choosing an intermediate frequency allows both transi-tions to be resonant, although at a 20% lower efficiencyfor the final pulse. Raman beam arrangements that cre-ate identical ke for both interferometers would eliminatethis effect for a stationary device. However, a rotationrate of 2 rad/s in this latter case would cause the sameeffect.

A typical experiment cycle takes place in 16.66 ms, cor-responding to a data rate of 60 Hz. Atoms are launchedto 2.5 m/s and cooled over 1.5 ms to 35 µK. In additionto the repump beam, a π-polarized, D1, optical pumpingbeam resonant with the |F = 2〉 → |F ′ = 3〉 transitiontransfers 92% of the atoms into the magnetically insen-sitive |F = 2,mF = 0〉 state. A 1.5 G magnetic fieldparallel to the Raman beams defines a quantization axis.The atoms are then interrogated by the Raman beamswith T = 4.1 ms at three points along their trajectory.The π-pulse length is 1.6 µs, allowing for the beams toaddress a wide velocity class of the atoms for each inter-ferometer.

The population in |F = 2〉 and the total atom numberare measured by fluorescence detection. The detectionregion is smaller than the ensemble size as a means ofvelocity selection. Thus, only 30% of the total atoms inthe ensemble exchange contribute to the interferometersignal, while the remaining 70% facilitate loading for the

next shot. We collect 0.5% of the atoms’ photon scatterfrom two 100 µs pulses. The first probe pulse measuresthe population in |F = 2〉. This is followed by bothprobe and repumping to measure the total population.The light is coupled into a 1 mm core multi-mode fiberthat delivers it to an APD (Hamamatsu C5460-01). Thecollimators are aligned off-axis by 2 to minimize crosstalk between the signals. From the ratio of the measuredsignals, we determine the probability to be in the excitedstate for each interferometer, from which the phase shiftsφa and φb may be calculated. Given a fixed interrogationtime, the sensitivity (per

√Hz) is optimized with a 60 Hz

(Tc = 16.66 ms) cycle. This restricts the recapture timeto 6 ms, and reduces the steady-state atom number to4× 106. Loading this many atoms in our apparatus frompurely vapor would take 40 ms.

Fig. 5 shows a sample fringe from the interferometer,with data taken at 60 Hz. We measure the phase noiseof each interferometer to be δφa = δφb = 22 mrad/shot.The corresponding acceleration and rotation compo-nents of the phase noise are δφ+ = 19 mrad/shot andδφ− = 11 mrad/shot, respectively. The resulting accel-

eration and rotation sensitivities are 0.92 µg/√Hz and

1.07 µrad/s/√Hz at 60 Hz. The rotation measurement

benefits greatly from common mode noise rejection ofthe vibrations of the retro-reflecting mirror, with a phasenoise approximately 2 times lower than the accelerationnoise. It is worth noting the potential performance ofthis approach under optimal conditions. If we assumeshot-noise limited signal with 107 atoms and 100% fringecontrast, with identical interrogation time and band-

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Con

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(%)

Tilt Angle (°)

0

1

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Ato

m N

umbe

r (×1

06 )

FIG. 6. (Color online) Immunity of the atom interferometerto a range of tilt. The tilt direction corresponds to θy , thuschanging the projection of gravity on the Raman beam axis,as well as the launch axis. Error bars represent the long-termdrift seen in the apparatus over 1 hr when the interferometeris flat on the table. The fringe contrast and steady-state atomnumber are largely unaffected by the orientation.

width, we calculate optimal sensitivities of 21 ng/√Hz

and 44 nrad/s/√Hz. Further optimization of timing pa-

rameters is likely to improve sensitivity even further.

The noise in our system derives from reduced fringecontrast and detection noise. The size of the ensemblerelative to the size of the Raman beams causes Rabi fre-quency inhomogeneities, which reduce the overall pop-ulation transfer efficiency. Additionally, the width ofthe Raman transition in frequency space must be largeenough to address the Doppler shifts from the thermalvelocity distribution of the atoms. These effects resultin a π-pulse efficiency of 55% for the first pulse. It islikely that a combination of an expanding position dis-tribution and the 24.7 kHz/ms gravity-induced Dopplershift reduces the π-pulse efficiency of the third pulse to35%, resulting in a fringe contrast of 20%. Accountingfor the fraction of the ensemble imaged and the fringecontrast, 1× 106 atoms contribute to the measured noisewhile 2× 105 atoms contribute to the signal. From this,we expect a non-inertial phase noise of 5 mrad per in-terferometer, corresponding to δφ± = 3.5 mrad. Elec-tronic noise, background scatter, and photon shot noiseare negligible. Thus, we attribute the noise in our systemto stability of the inertial reference, and frequency andintensity noise of the probe.

The most direct route towards enhancing the sensitiv-ity is by improving fringe contrast. It is possible to imple-ment independent detunings for the third pulse by addinga second, independently controllable Raman laser. Ad-ditionally, the increased Raman power enables one to re-duce Rabi frequency inhomogeneity by increasing Ramanbeam waist, further improving pulse efficiency. Pulse ef-ficiency can also be improved by using composite pulses

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100%

Rec

aptu

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Distance from Trap Center (mm)

Ensemble A Ensemble B

EnsembleTrajectory

FIG. 7. (Color online) Recapture efficiency over a range ofensemble positions relative to the trap center, at the onset ofballistic recapture. Dashed lines represent the 1/e2 diameterof the trapping volume. For early onset (negative position),the ensemble’s momentum still carries it into the trap. Recap-ture is still efficient over a large fraction of the trap volume.

[5, 18, 19].

V. DYNAMICS

Operating the interferometer in a dynamic environ-ment beyond benign laboratory conditions of 1 g and100 µrad/s demands robustness. Additionally, the appa-ratus must be able to handle large changes in acceleration(jerk) and rotation (angular acceleration). We motivatefurther research in this vein by exploring the more dy-namic features of our approach compared to conventionalatom interferometers.Gravitational acceleration in conventional atom inter-

ferometers with T ≈ 100 ms causes an atomic ensembleto traverse tens of centimeters. Thus, they are designedaccounting for a fixed orientation of gravity. Under anarbitrary orientation of gravity, the apparatus will notfunction due to the ensemble’s deflection. A short inter-rogation time minimizes the deflection and allows for ourinterferometer to operate in any orientation. To demon-strate this, we rotate the apparatus over a range of an-gles in the plane defined by v and ke. This changesthe projection of gravity on these vectors. As shown inFig. 6, the fringe contrast and the atom number remainunchanged to within the measurement uncertainty overa range of angles. The apparatus is designed to exhibit acertain detection and recapture volume. Under a rever-sal of gravity orientation, the ensemble does not violatethese boundaries.To estimate the acceleration and rotation rate the en-

semble exchange can endure, we measure the recaptureefficiency as a function of ensemble distance from the trap

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0.75

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Con

trast

, (n

orm

)

FIG. 8. (Color online) Contrast loss under high rotation rate.The black points are the observed fringe contrast as the appa-ratus was rotated in the ±g direction, up to ±1.5 rad/s. Thered line is a Gaussian fit, with a width of σ = 0.60(1) rad/s.

center at the onset of recapture. By triggering the trapat different delay times following the launch, we simulaterecapture at different locations along the launch vector,x (Fig. 7). While the launch velocity direction causes anasymmetry where early activation still allows the atomsto relax into the trap, results for a delayed activationshould mirror deflections along the y and z axes. Weempirically measure the recapture radius along x to be5 mm, where the recapture efficiency falls below 50%.Dynamic conditions as large as 10 g of acceleration and20 rad/s of rotation would be necessary to replicate thislevel of excursion.Although the ensemble exchange tolerates a large

range of inertial conditions, successfully tracking the in-ertial signals requires that jerk and angular accelerationbe limited to an amount equivalent to a π/2 interferome-ter phase shift over one interferometer cycle. This corre-sponds to a jerk of |j| = π/(2|ke|T 2Tc), and an angularacceleration of |α| = π/(4|ke||v|T 2Tc). Exceeding theserates can be enabled by using complementary auxiliarysensors.An alternative rotation rate limit arises from phase

gradients in the ensemble due to the thermal velocity dis-tribution, leading to loss of contrast. Each atom derivesan independent rotational phase shift due to its ther-mal velocity coupled with the rotation rate. For a givencharacteristic velocity width σv, when the rotation rateexceeds the condition |Ω| > π/(2|ke|σvT

2), large inho-mogeneous dephasing occurs [13]. Given a rotation of

magnitude Ω at an angle of θ from ke, and an ensembletemperature of T , the decay of the fringe contrast χ isgiven by,

χ(Ω) = exp

(−Ω2

2σ2

)

,

σ =1

2

√

m

kBT1

|ke|T 2 sin (θ), (5)

where m is the mass of 87Rb, and kB is the Boltzmannconstant.We demonstrate this effect by placing the apparatus

on a 22 cm diameter rotation platform located outsideof the magnetic shielding. The experiment is configuredto rotate about ±g in Fig. 1, or θ = 10. The angu-lar position of the platform is recorded via a rotary en-coder. Fig. 8 shows measurements of the fringe contrastwhile rotating the apparatus in the ±g direction up to±1.5 rad/s. We observe a minimal change in atom num-ber of a couple percent at 1 rad/s for a duration of 1 s.At sustained rates where the ensemble is allowed to hitsteady-state, we expect the atom number to be slightlyless. As expected, the contrast loss is Gaussian with re-spect to the rotation rate Ω. The characteristic width ofthe distribution is σ = 0.60(1) rad/s, which correspondsto a temperature of 36 µK. This is in agreement with in-dependent temperature measurements. The most severelimit would arise if we rotated the apparatus perpen-dicular to ke, which would cause the contrast to vanishsin (90)/ sin (10) ≈ 6 times faster. This limit can be ex-tended by cooling to lower temperature or by placing theapparatus on gimbals. Although the ensemble contrastis diminished, the individual atoms still maintain phasecoherence and inertial purity. If the ensemble meets thecondition 2Tσv ≫ σx0

, where σx0is the initial charac-

teristic position width, the position of each atom duringdetection corresponds strongly with its velocity. Then,the inertial information for each velocity class may beretrieved with a sufficiently well-resolved detector [20].

VI. CONCLUSION

We have demonstrated a high data-rate atom inter-ferometer capable of simultaneous acceleration and ro-tation measurements at 60 Hz, with sensitivities suit-able for a wide application space. The system makesuse of the ensemble exchange technique to launch andrecapture atoms at a high data-rate, orders of magni-tude higher than typical LPAI systems. We modeled theatom number during ensemble exchange assuming a fixedrecapture fraction, and validated the model with experi-ments. Additionally, we characterized our interferometerand estimated the potential performance of our design.We showed immunity of recapture efficiency and interfer-ometer contrast over a range of acceleration and rotationvalues.Our ensemble exchange method enables a path towards

a compact atomic inertial measurement unit. Furtherwork is needed to miniaturize the remainder of the systemand achieve the ultimate sensitivity. We consider a devicewith sensitivities of ∼10 ng/

√Hz and ∼10 nrad/s/

√Hz,

and robust up to ∼10 g and ∼10 rad/s to be feasible inan engineered package that is significantly smaller thanour apparatus. Through careful study of the apparatus’dynamics, we may model the sensor response in variousscenarios and enable a fielded device.

8

ACKNOWLEDGMENTS

We gratefully thank G. Burns for experimental sup-port and Y.-Y. Jau for helpful discussions. We wouldalso like to thank D. Butts, D. Johnson, and R. Stonerat Draper Laboratory for helpful discussions regardingdevelopment of an inertial measurement unit using this

technique. This work was supported by the LaboratoryDirected Research and Development (LDRD) program atSandia National Laboratories. Sandia National Labora-tories is a multi-program laboratory managed and oper-ated by Sandia Corporation, a wholly owned subsidiaryof Lockheed Martin Corporation, for the U.S. Depart-ment of Energy’s National Nuclear Security Administra-tion under Contract No. DE-AC04-94AL85000.

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