Magnetic tensor gradiometry using Ramsey interferometry of spinor condensates

A. A. Wood, L. M. Bennie, A. Duong, M. Jasperse, L. D. Turner,∗ and R. P. AndersonSchool of Physics and Astronomy, Monash University, Victoria 3800, Australia.

(Dated: September 24, 2018)

We have realized a magnetic tensor gradiometer by interferometrically measuring the relativephase between two spatially separated Bose–Einstein condensates (BECs). We perform simultaneousRamsey interferometry of the proximate 87Rb spin-1 condensates in freefall and infer their relativeLarmor phase – and thus the differential magnetic field strength – with a common-mode phase noisesuppression exceeding 50 dB. By appropriately biasing the magnetic field and separating the BECsalong orthogonal directions, we measure the magnetic field gradient tensor of ambient and appliedmagnetic fields with a nominal precision of 0.30 nT mm−1 and a sensor volume of 2 × 10−5 mm3.We predict a spin-projection noise limited magnetic energy resolution of order ∼ 10~ for typicalZeeman coherence times of trapped condensates with this scheme, even with the low measurementduty cycle of current BEC experiments.

PACS numbers: 37.25.+k,07.55.Ge,03.75.Mn,06.20.-f

I. INTRODUCTION

Precision measurement of magnetic fields under-pins applications as diverse as fundamental symmetrytests [1], magnetoencephalography [2] and geophysicalexploration [3]. Many of these applications require pre-cise and accurate measurements of the change in mag-netic field across a region of space: magnetic gradiome-try. On kilometer scales, magnetic gradiometers remotelysense the field of mineral deposits against the larger butlocally homogeneous field of the Earth dipole [4]. Onthe microscopic scale, optimal magnetic gradiometry ofbiomagnetic or surface-science sources demands magneticsensor volumes orders of magnitude smaller. We presentthe first microscopic tensor magnetic gradiometer basedon atomic magnetometry, measuring ∂Bi/∂xj (the gra-dient of vector components along orthogonal axes) witha dynamically configurable baseline.

Tensor measurements incisively probe magnetic sourcedistributions [3, 5]: the gradient tensor at a singlepoint in space determines the bearing, normalized source-strength, and orientation of a dipole [4]. Tensor gradientmagnetometers have to date been macroscopic devices– employing SQUID [6–8] or fluxgate [9, 10] sensors –primarily applicable to geophysics and ordinance detec-tion [11]. Magnetic fields also change on much smallerlength scales, necessitating equally small magnetic sen-sors to precisely characterize field variations from mi-croscopic magnetic sources [12]. Spinor condensates arehighly sensitive to magnetic field gradients, which typ-ically must be eliminated to observe their rich emer-gent phenomena: quantum phase transitions, dynam-ics of topological defects, and fragile macroscopic en-tangled states [13]. In general, inferring in vacuo mag-netic field profiles from ex vacuo measurements is pro-foundly difficult, with many applications demanding a

∗ lincoln.turner@monash.edu

direct atomic metric from which to diagnose the cancel-lation of stray magnetic fields and gradients. Measuringthe magnetic field gradient tensor in vacuo provides analmost complete measurement of the local magnetic fieldlandscape of a trapped quantum gas, an indispensabletool for characterizing dephasing mechanisms from inho-mogenous magnetic fields.

Precise measurement of magnetic field gradients onsmall length scales demands small magnetic sensors withhigh spatiotemporal sensitivity. Atomic magnetometry isa well-established alternative to SQUIDs and other solid-state magnetometers, delivering absolute, calibration-free measurement of magnetic fields by measurement ofthe Larmor precession frequency of atomic spins [14].Warm atomic vapor magnetometers have sensing vol-umes ranging from hundreds of cubic millimeters for themost sensitive magnetometers down to a few cubic mil-limeters achieved with micro-fabricated glass cells [15].Elongated clouds of ultracold atoms have been used astime-resolved magnetometers and gradiometers with sub-nT sensitivity and a spatial resolution of 50µm [16, 17].Colder, denser clouds of atoms in traps, such as Bose-Einstein condensates (BECs), offer the prospect of pre-cise magnetic measurement on the microscale. Wilder-muth et al. [18] used a highly elongated condensateto measure the magnetically-induced trapping poten-tial variations from a current-carrying wire. Vengalat-tore et al. [19] used a non-destructive phase contrastimaging technique to spatially resolve Larmor preces-sion in a spinor BEC, attaining sensitivities comparableto SQUID-based devices. This work establishes tensorgradiometry using highly-sensitive, small-volume atomicmagnetometers with a dynamically configurable orienta-tion.

Our gradiometer measures the phase difference be-tween two spin-1 Ramsey interferometers formed fromspatially separated F = 1 87Rb BECs. The use of con-densed atoms as magnetic sensors in this case is inessen-tial, though not without benefit: BECs simultaneouslyoffer high atomic density with microscopic sensor vol-

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umes, making them ideally suitable as small volumeatomic magnetometers. Each condensate is formed ina separate potential well of a three-beam crossed opticaldipole trap and translated to a spatial separation of justless than 1 mm by an acousto-optic modulator (AOM).We can change our dipole trap configuration so that thegradiometer spans two separation axes in a plane (Fig. 1)and apply bias magnetic fields to make the gradiome-ter sensitive to field components in all spatial directions.A Ramsey pulse sequence probes the phase acquired byeach condensate over an interrogation time T due to themagnetic field B at the position of each condensate. Thephase difference between fringes from the two interferom-eters is proportional to the field gradient between the twocondensates. The differential mode of operation of ourgradiometer suppresses common-mode noise from tran-sient variations in the magnetic field as well as detectionproblems. Simultaneous interrogation of a dual atomicfountain interferometer demonstrated magnetic gradientmeasurement over a large spatial region [20], with sub-stantial common-mode rejection of noise from drifts andpulse errors. In this work, we describe the experimen-tal procedure for realizing a tensor gradiometer with apair of trapped atomic clouds. We then characterize thegradiometer by measuring the gradient tensor of ambientmagnetic fields in our laboratory, as well as measuring theresponse to applied magnetic field gradients. Later wedescribe how the gradiometer configuration can also beused as a prospective microscale co-magnetometer withsubstantial common-mode rejection.

II. DIFFERENTIAL RAMSEYINTERFEROMETRY

The differential interferometer output ∆φ is a measureof the difference in phase acquired by each condensateduring the Ramsey sequence, which begins with a π/2spin rotation pulse, followed by free evolution over T andclosed with a second π/2-pulse. The Ramsey sequencesimultaneously addresses both condensates, which canbe considered as two independent spin-1 interferometers.For static, spatially varying magnetic fields sampled bytwo condensates at positions r1 and r2 we may write

∆φ

γT= |B(r1)| − |B(r2)| ≈ ∇|B|r=r12 · (r1 − r2) , (1)

with γ = 2π × 6.996 kHzµT−1 is the gyromagnetic ratioand r12 = (r1 + r2)/2. The differential interferometer isthus sensitive to derivatives of the magnetic field strength

B(r) ≡ |B(r)| =√B2x +B2

y +B2z ,

∂B

∂xi=BxB

∂Bx∂xi

+ByB

∂By∂xi

+BzB

∂Bz∂xi

. (2)

To achieve vector magnetic field sensitivity along xfor example, we experimentally null magnetic field com-ponents along y and z, leaving a magnetic field with

AOM

AO

M

(RF

2)RF

1RF

2RF

1

bias coils

BECs

(a) (b)

(c)baseline

( baseline)

1064 nm laser

f = 300mm

FIG. 1. (Color online) Diagram of the experimental setup(a). The gradiometer spans a spatial baseline given by ∆x′ (b)or ∆z′ (c) in the horizontal plane (gravity along −y) by trans-lating the pair of condensates in opposite directions along oneof the dipole beams, which are oriented at ∼ 45◦ to the mag-netic bias field axes, x, z. This translation is achieved byfeeding a second rf frequency into the AOMs that control theposition and amplitude of the dipole beams.

|Bx| � |By| , |Bz|. The measured gradient of B is wellapproximated by

∂B

∂xi≈ Bx

B

∂Bx∂xi

= sign(Bx)∂Bx∂xi

. (3)

Two condensates can then be aligned along an axis xi,separated by ∆xi = |r1 − r2| to measure field differencesalong that axis. In general we measure components of themagnetic field gradient tensor via the differential Ramseysignal:

∂Bj∂xi

≈ sign(Bj)1

γ∆xi

d(∆φ)

dT. (4)

The differential interferometer is sensitive to any differ-ence in Zeeman energy between the two condensates. Weperform the interferometry sequence in freefall to preventspurious contributions to the measured magnetic fieldgradients from vector light shifts induced by the trappingbeams [21, 22]. Imperfect linear polarization of the trap-ping beams induces an atomic vector polarizability, thespatial variation of which appears as a synthetic magneticfield gradient across each condensate and would lead todephasing and loss of interferometric contrast. Addition-ally, if each condensate experienced a different overallvector light shift due to small intensity differences be-tween the two beams, it would contaminate the gradientmeasurement.

Freefall interferometry eliminates the vector shift atthe expense of introducing a trade-off between the maxi-mum Ramsey interrogation time T and the spatial reso-lution due to gravity-induced blurring of the sensor vol-ume. Alternatively, differential Ramsey interferometryof trapped clouds presents a means of precisely measur-ing and canceling vector light shifts, as discussed in Sec-tion VI.

3

1.0

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–1.0–1.0 –0.5 0.0 0.5 1.050 100 150 200 250 300 3500

pulse phase (degrees) interferometer 1

inte

rfer

omet

er 1

(

), 2

(

)

interferometer 2

T = 3 ms

T = 0.1 ms

FIG. 2. (Color online) Spin projections at two differentRamsey times T = 0.1 ms and 3 ms, plotted (left) as a func-tion of π/2-pulse phase ϕ, and (right) parametrically fromthe output of each interferometer. If the phase acquired byeach interferometer is stable (determined by the interrogationtime T and the shot-to-shot stability of |B(ri)|), scanning ϕthrough 360◦ maps one period of an interference fringe in Fz

(top left). Shot-to-shot fluctuations of |B(ri)| lead to an in-creasingly irreproducible phase acquired by each interferom-eter for longer interrogation times (bottom left). This onsetof absolute phase noise scrambles the phase domain fringesfor T > 1 ms, but does not affect the the uncertainty in therelative phase ∆φ calculated from fitting an ellipse, as thefluctuations in |B(ri)| are common-mode to each interferom-eter (right). Error bars correspond to the uncertainty in themeasured spin projection in a single absorption image.

III. EXPERIMENT

Our experiment begins by forming two 87Rb Bose-Einstein condensates in the |F = 1,m = −1〉 hyperfineground state in two 1064 nm crossed-beam optical dipoletraps. For technical reasons (atomic beam along z andimaging beam along x) our dipole trapping beams areoriented at ∼ 45◦ to the horizontal coordinates x andz. The propagation directions of the two intersectingdipole trapping beams (1/e2 radii of 75µm and 89µmrespectively) define near-perpendicular horizontal axesx′ and z′ as shown in Fig. 1. The amplitude and hor-izontal position of each beam is controlled using a sep-arate AOM. Driving either one of the AOMs with tworadiofrequency (rf) tones from an agile direct-digital syn-thesizer produces two diffracted orders, resulting in twocrossed-beam dipole traps separated along either x′ or z′.The rf frequencies determine the separation of the dipoletraps [23] along the intersecting beam.

We Bose condense 5× 104 atoms in each trap, initiallyseparated by 100µm to maximize loading efficiency froma precursor hybrid optical dipole-magnetic quadrupoletrap [24]. The condensates are further separated over 2 swith a smooth frequency ramp, achieving a maximumseparation of ∆x′ = 680µm (∆z′ = 840µm) when split-ting the beam propagating along the z′ direction (−x′

–1

+10 +1

0

–1

T = 0.1 ms 0.5 ms 1 ms 2 ms 3 ms

interferometer 1

inte

rfer

omet

er 2

Ramsey interrogation time T (ms)

phas

e d

i�er

ence

1.5

1.0

0.5

0.00.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0

FIG. 3. (Color online) Measurement of a magnetic fieldgradient using differential atom interferometry. Varying thephase of the second π/2-pulse of the Ramsey sequence for eachfixed interrogation time T traces an ellipse (top). From eachellipse we extract the magnitude of the phase difference |∆φ|,and determine the field gradient using Eq. (4); ∂By/∂z

′ =−53.3(3) nT mm−1 for these data. Statistical uncertainties of|∆φ| and Fz,i are smaller than the data points.

direction).Using three orthogonal coil pairs, we ensure the mag-

netic field is oriented along one of the (x, y, z) axes withmagnitude in the range 30–60µT. We extinguish thedipole trapping light and the two condensates begin tofall. After 100µs freefall we initiate Ramsey interferome-try between all Zeeman statesm = −1, 0,+1 of the F = 1hyperfine ground state using a resonant rf π/2-pulse. Thetwo falling condensates comprise two independent inter-ferometers. The interferometers are closed with a secondπ/2-pulse after an interrogation time T . Spin compo-nents (m = 0,±1) of each condensate are separated afterfurther freefall by pulsing a 0.50 T m−1 magnetic fieldgradient for 3 ms. The number of atoms Nm,α in statem of condensate α = 1, 2 is determined by absorptionimaging with a resonant laser after a total drop time of23 ms. This constitutes a single realization, or shot, ofthe experiment from which we compute the normalizedspin projection Fz,α =

∑mmNm,α/

∑mNm,α for each

interferometer.

IV. RESULTS

Ramsey fringes in the phase domain, Fz,1(ϕ) andFz,2(ϕ), are clearly resolved for interrogation times T <500µs, and are subsequently dominated by phase noiseinduced by magnetic field fluctuations common to bothinterferometers. Plotting the interferometer outputsparametrically yields an ellipse, which is immune tocommon-mode phase noise. This has been utilized ingravity gradiometry where atomic momentum states areinterfered, and the common-mode phase noise derivesfrom vibration of the reference platform [25, 26]. To our

4

knowledge this work is the first application of the ellip-tical data reduction to exclusively spin interferometry.The general form of a conic is aX2 + bXY + cY 2 + dX +eY + f = 0, with an ellipse satisfying b2 − 4ac < 0. Therelative phase is given by [25]

cos ∆φ =b

2√ac. (5)

We fit an ellipse [27, 28] to a parametric dataset(X,Y ) = (Fz,1(ϕ), Fz,2(ϕ)) to extract |∆φ|; by repeatingthis process for different interrogation times we computea magnetic field gradient via d(∆φ)/dT in Eq. (4). Wedetermined the sign of ∆φ from phase domain fringesat short interrogation times. A measurement of a gra-dient in the y-component of the magnetic field is shownin Fig. 3. We infer a nominal precision of 0.30 nT mm−1

from the combined statistical and systematic uncertaintyin the slope of such linear fits. Potential sources of sys-tematic error are T 3 corrections to ∆φ inherent to inter-rogation during freefall, and slow drifts in the gradientmeasurand. We emphasize that measurements with vary-ing interrogation times as shown here are performed toensure the correct number of phase cycles are accountedfor; in principle we may extract the field sensitivity froma single ellipse at the longest possible interrogation time.This will be discussed in more detail in section V.

We quantified the common-mode rejection of the gra-diometer by comparing the phase noise from a singleinterferometer to the uncertainty in the relative phaseextracted from the elliptical fits. Assuming our mag-netic field noise is baseband, the measured phase noiseyields a standard deviation of the Larmor frequency ofσωL

= 2π × 192(11) Hz, representing a common-moderejection ratio exceeding 50 dB. Fig. 2 shows the dete-rioration of phase domain fringes at long interrogationtimes, while the corresponding parametric plots do notexhibit discernible degradation.

To demonstrate the tensor sensitivity of the gradiome-ter, we measured multiple field derivatives as a functionof the current imbalance ∆Iz in the z-bias coils (Fig. 4).This allows us to explicitly evaluate the response of thegradiometer to an applied gradient when biased differ-ently; with a field along z, we measure ∂Bz/∂z and∂Bz/∂x proportional to and independent of ∆Iz, respec-tively, in agreement with a numerical Biot-Savart calcu-lation. Orienting the magnetic field along the y-axis ren-ders the gradiometer insensitive to the applied gradient,as this measures ∂By/∂x and ∂By/∂z (Eq. (3)), whichwe find to be < 5.1(6) nT mm−1 A−1 due to imperfectalignment of the magnetic field along the y-axis.

The background magnetic field environment of our ap-paratus has predominantly linear magnetic field gradi-ents originating from equipment within 1 m of the atoms.The magnetic field gradient tensor can be represented bya matrix Gij = ∂Bi/∂xj . Using three bias field orienta-tions and baselines that span the horizontal x′, z′ planeis sufficient to calculate the full gradient tensor G in the

200

100

0

–100

–200

–300–0.4 –0.2 0.0 0.2 0.4 0.6

(A)

grad

ient

(nT

mm

–1)

FIG. 4. (Color online) Response of the gradiometer to a gra-dient applied by driving a differential current ∆Iz through thez-bias coils. Four field derivatives ∂By/∂x, ∂Bz/∂x, ∂By/∂z,and ∂Bz/∂z are measured by biasing the gradiometer alongy or z with baselines along x′ or z′. The dominant gradient is∂Bz/∂z = −328(1)∆Iz nT mm−1; the relative insensitivity ofthe other measured gradients to ∆Iz quantifies the alignmentof the bias coils along the Cartesian axes.

x, y, z frame from Maxwell’s laws for magnetic fields ina vacuum. Our experimental conditions (in-vacuum, nosignificant electric fields) permit the use of ∇ × B = 0and ∇ · B = 0 when applying Maxwell’s equations. Wecan thus determine gradients ∂Bi/∂y from the gradientsmeasured in the horizontal plane, resulting in the gradi-ent tensor

G =

−57.1(7) −69.2(4) 147.0(7)−69.2(4) 151.8(8) 26.6(4)149.5(3) 26.6(4) −94.7(3)

nT mm−1 , (6)

where the inferred values are inside the dashed box. Weobserve that measurements of gradients over the courseof a month do not vary beyond their uncertainties. Thegradient tensor has applications in localizing magneticsources, which has seen widespread use in geophysics andsurveying [3, 7, 29]. The power of the gradient tensor isexemplified by the ability to localize a source from a ten-sor measurement at a single point in space; the threeorthogonal eigenvectors of the matrix G span a coordi-nate system in which off-diagonal gradient terms van-ish, with the eigenvector corresponding to the largestmagnitude eigenvalue pointing at the dominant dipolesource [4]. For the gradient tensor in Eq. (6), this vectorpoints in the direction of our dominant gradient source,the unshielded permanent magnets of an ion pump. Ho-mogeneous magnetic fields are required to preserve thecoherence or robust entanglement of spins in many sys-tems, e.g. the macroscopically entangled singlet state of aspinor quantum gas. Here the knowledge of the gradienttensor is paramount: by applying diagonal gradients (i.e.those produced by two anti-Helmholtz coil pairs) alongthe eigenaxes of G, all components of the gradient ten-sor can be zeroed. In many applications, it is sufficient

5

to achieve a uniform field strength, i.e. to zero |∇B|.Biasing the magnetic field along one of the eigenvectorsof G reduces this problem to canceling a single diagonalgradient. Elsewhere, merely minimizing |∇B| may besufficient, and large reductions in |∇B| can be achievedby changing the direction of the bias field alone, e.g. forthe gradient tensor in Eq. (6) |∇B| varies by a factor of4 depending on the field direction.

Tensor gradiometers make use of Maxwell’s laws to in-fer the complete tensor from as few as five independentgradient terms [6, 9, 10, 30]. Gradients measured overbaselines in the x-z plane quantify a systematic error inour inference; the equality of ∂Bz/∂x and ∂Bx/∂z is vi-olated by < 2.5(7) nT mm−1. We attribute this discrep-ancy to asymmetric sampling of residual field curvaturefrom the bias coils and imperfect cancellation of trans-verse field components when aligning B along a givenaxis (Eq. (3)). These two systematics may be reduced bylarger Helmholtz coils, and by applying larger bias fields.While an in-plane measurement is sufficient to determinethe gradient tensor, two-axis acousto-optic deflection ofthe trapping beams could be used to measure all termsindependently.

V. OPERATIONAL SENSITIVITY AS ACO-MAGNETOMETER

Our scheme could also function as a precision co-magnetometer, where one BEC placed in the vicinity of asmall magnetic source acts as a sensor and the other con-densate a reference interferometer, providing substantialcommon-mode rejection of ambient noise. This is themode of operation we have in mind when specifying fieldsensitivities per unit bandwidth or per unit spatiotem-poral bandwidth below, the standard metrics for mag-netometry. No analagous metric is in use for gradientsensitivity per spatiotemporal bandwidth.

The sensitivity of atomic magnetometers is determinedby the ability to detect the Larmor precession of spinsin a magnetic field. For large atom numbers, as inwarm vapor magnetometers, spin relaxation and pho-ton shot noise ultimately limit sensitivity. Cold atomsystems interrogate smaller, trapped samples, and thespin projection noise at the standard quantum limit(δFz)SQL = 1/

√2N for N spin-1 atoms is relatively

more important. As multiple experimental shots arerequired to impute a differential phase from the ellip-tical data reduction, a single-shot phase sensitivity is ill-defined here. Nonetheless, the differential phase uncer-tainty from fitting an ellipse with M points scales with1/√M , and thus the quantum limited field sensitivity

is δBSQL ∼ 1/(γ√NTDTint

)for N/2 atoms per con-

densate, a total integration time Tint = MTshot, a dutycycle D = T/Tshot, and a single-shot duration of Tshot.From repeated absorption images we determine that theuncertainty in our measurements of spin projection δFzare ∼ 3 times that of the standard quantum limit for

N = 105 atom condensates, and this ultimately deter-mines the relative phase uncertainty extracted from el-lipse fits. The corresponding field sensitivity per unitbandwidth is δB

√Tint = 360 pT Hz−1/2 for Tshot = 25 s

and T = 3 ms.

VI. IN-TRAP INTERFEROMETRY:PROSPECTIVE SENSITIVITY

A gradiometer or co-magnetometer formed from a pairof trapped condensates offers a smaller sensor volumeand a significant improvement in sensitivity compared toone formed from atoms in freefall. A prospective in-trapco-magnetometer can attain spatiotemporal sensitivitiescomparable with established forms of small-volume mag-netic sensing, as we show in this Section.

For an in-trap magnetometer to be possible, it is neces-sary to suppress the vector light shift induced by residualelliptical polarization of the trapping beams. These ‘fic-titious’ magnetic fields can be minimized by orienting themagnetic bias field to be perpendicular to the wavevec-tor of both dipole beams, although this limits vector fieldsensitivity to only one spatial direction. The vector lightshift may be reduced by several orders of magnitude byensuring the dipole trapping beams are near-linearly po-larized in vacuo with conventional ex vacuo polarimetry.The glass vacuum cell is inevitably birefringent, and su-perlative linearity of polarization at the atoms requiresan atomic measurement [22, 31]. Differential Ramseyinterferometry can provide such a measurement by sens-ing the phase difference between condensates exposed totrapping light of different intensities, but of common po-larization. Careful adjustment of a quarter-wave platebefore the vacuum cell will be sufficient to achieve linearpolarization at the atoms and thus a vanishing vectorlight shift. This will be the focus of a forthcoming pub-lication.

The sensitivity of each interferometer scales with theevolution time and the atomic density. A Zeeman co-herence time approaching one second was observed ina spin-1 87Rb condensate [32], limited by losses due todensity-dependent collisions. For a 87Rb BEC with apeak number density of 1014 atoms cm−3 (correspondingto N = 106 atoms for the current trap), an interrogationtime of T = 200 ms is foreseeable, an order of magnitudelower than the three-body limited lifetime. The prospec-tive in-trap magnetometer could thus achieve a differen-tial field sensitivity per unit bandwidth of 600 fT Hz−1/2

at the standard quantum limit, even with the same trapand single-shot duration used here (corresponding to anon-unity duty cycle of D = 0.008).

Spatial resolution is conventionally quantified bythe sensing volume V ; a vapor magnetometer withV = 300 mm3 attained sub-femtotesla sensitivities [33]whereas NV-center magnetic probes deliver nanoscopicresolution but at much lower field sensitivities [34]. Forthe freefall measurements described herein, the sen-

6

sor volume V = 2 × 10−5 mm3 is that swept outby a falling, expanding condensate during the Ram-sey interrogation. Using the metric of spatiotem-poral sensitivity, our demonstrated measurement hasδB√Tint√V = 51 fT cm3/2 Hz−1/2. The above prospec-

tive in-trap magnetometer has V = (20µm)3, corre-

sponding to δB√Tint√V = 0.05 fT cm3/2 Hz−1/2. This

large prospective improvement is not unprecedented inmicroscale magnetometry where first demonstrations arefar from fundamental limits; warm vapor magnetometryin microfabricated cells (V ∼ mm3) was first demon-strated at a sensitivity of 5500 fT cm3/2 Hz−1/2 [35],rapidly developed to 5.0 fT cm−3/2 Hz−1/2 [36], and morerecently 0.16 fT cm3/2 Hz−1/2 [15]. The spatiotemporalsensitivity can also be expressed in units of energy perunit bandwidth ε = (δB)2T V/2µ0. While the low dutycycle and number of spins in BEC based measurementslimits their field sensitivity per unit bandwidth, the farsmaller volume results in a ε comparable to warm vapormagnetometers, with ε ∼ 50–100~ for vapor magnetome-ters [15, 37], and ε ∼ 10~ for the prospective magnetome-ter described above.

During the preparation of this manuscript, we becameaware of related work also employing an array of con-densates simultaneously addressed with a Ramsey se-quence [38]. These authors focused on the applicationof spin-squeezing to realize phase sensitivities below theatomic shot noise (at short interrogation times), in con-trast to our work which focuses on dynamic reconfigura-bility of two condensates to achieve tensor sensitivity ofmagnetic field gradients.

VII. CONCLUSIONS

We have demonstrated magnetic tensor gradiometryusing differential Ramsey interferometry of spatially sep-arated BECs in freefall. The gradiometer senses vectorcomponents of the magnetic field, rejecting gradient com-ponents orthogonal to the biasing direction. The gra-diometer is immune to common-mode magnetic noise or-ders of magnitude larger than the field difference, andoperates without field cancellation or screening. The dy-namic reorientability of the gradiometer baseline withmicroscale resolution allows for precision surveys of mag-netic microstructures and the ambient magnetic environ-ment of trapped quantum gases. The gradiometer couldbe used as a high-precision co-magnetometer with sub-stantial common-mode rejection, allowing for microscalemagnetic sensing in vacuo.

ACKNOWLEDGMENTS

This work was supported by the Australian ResearchCouncil (DP1094399) and the Australian PostgraduateAward Scheme. We are grateful to Z. Szpak and W. Cho-jnacki for assistance with ellipse fitting, Y. Levin for use-ful conversations, and K. Helmerson for a careful readingof the manuscript.

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