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7. Kundel, H. L. in Medical Imaging 2000: Image Perception andPerformance (ed. Krupinski, E. A.) 135–144 (2000).

8. Gur, D., Rockette, H. E., Warfel, T., Lacomis, J. M. & Fuhrman,C. R. Acad. Radiol. 10, 1324–1326 (2003).

doi:435439aSupplementary information accompanies thiscommunication on Nature’s website.Competing financial interests: declared none.

by charging the probed structure electrically.This measurement is more sensitive than theone discussed above because the trappingparameters can be adjusted independentlyfrom the measured potential landscape byusing a separate wire for holding the BEC.

The optimal potential sensitivity of a BECused as a field sensor, �B���N/z0

3, isachieved if the trapping parameters areadjusted so that the cloud’s transverse sizematches the desired spatial resolution z0. Here�N is the minimal atom-number variationresolved by the imaging system, and � containsall the atomic-physics parameters of the spe-cific atom (��8.63�10�29 tesla cubic metresfor the 87Rb used in our experiment). Currentlyavailable CCD (charge-coupled device) cam-eras allow atom-shot noise-limited detectionwith a �N value of better than 10 atoms perpixel in absorption imaging, so that a sensitiv-ity, �B, of 1 nanotesla is possible even at a highspatial resolution of 1 �m (or 1 picotesla at10 �m). By changing to a different atom withhigher mass and/or by reducing the inter-atomic interaction, a significant increase insensitivity can be achieved.

A comparison of different magnetic-fieldmeasurement techniques1–4,12 (see supplemen-tary information) shows that BECs as mag-netic sensors could reach unprecedentedsensitivity over a large range of spatial resolu-tion. The sample measurements we presenthere reach higher sensitivities than thoseobtained with established techniques operat-ing at the same spatial resolution.Stephan Wildermuth*, Sebastian Hofferberth*,Igor Lesanovsky*, Elmar Haller*,L. Mauritz Andersson*, Sönke Groth*†, Israel Bar-Joseph†, Peter Krüger*,Jörg Schmiedmayer**Physikalisches Institut, Universität Heidelberg,Philosophenweg 12, 69120 Heidelberg, Germany e-mail: schmiedmayer@atomchip.org†Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel

1. Bending, S. J. Adv. Phys. 48, 499–535 (1999).2. Freeman, M. R. & Choi, B. C. Science 294, 1484–1488

(2001).3. Faley, M. I. et al. Supercond. Sci. Technol. 17, 324–327 (2004).4. Kominis, I. K., Kornack, T. W., Allred, J. C. & Romalis, M. V.

Nature 422, 596–599 (2003).5. Fortagh, J., Ott, H., Kraft, S., Günther, A. & Zimmermann, C.

Phys. Rev. A 66, 041604 (2002). 6. Esteve, J. et al. Phys. Rev. A 70, 043629 (2004).7. Leanhardt, A. E. et al. Phys. Rev. Lett. 89, 040401 (2002).8. Jones, M. P. A. et al. Phys. Rev. Lett. 91, 080401 (2003).9. Dunjko, V., Lorent, V. & Olshanii, M. Phys. Rev. Lett. 86,

5413–5416 (2001).10. Folman, R., Krüger, P., Schmiedmayer, J., Denschlag, J. &

Henkel, C. Adv. At. Mol. Opt. Phys. 48, 263–356 (2002).11. Rous, P. J., Yongsunthon, R., Stanishevsky, A. &

Williams, E. D. J. Appl. Phys. 95, 2477–2486 (2004).12. Oral, A. et al. IEEE Trans. Magn. 38, 2438–2440 (2002).

doi:435440aSupplementary information accompanies thiscommunication on Nature’s website .Competing financial interests: declared none.

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➧ www.nature.com/bca see Contents pages.

BOSE–EINSTEIN CONDENSATES

Microscopic magnetic-field imagingToday’s magnetic-field sensors1 are not capa-ble of making measurements with both highspatial resolution and good field sensitivity.For example, magnetic force microscopy2

allows the investigation of magnetic structureswith a spatial resolution in the nanometrerange, but with low sensitivity, whereasSQUIDs3 and atomic magnetometers4 enableextremely sensitive magnetic-field measure-ments to be made, but at low resolution. Herewe use one-dimensional Bose–Einstein con-densates in a microscopic field-imaging tech-nique that combines high spatial resolution(within 3 micrometres) with high field sensi-tivity (300 picotesla).

Trapped cold atoms are ideal magnetic

sensors as they are very sensitive to changes inmagnetic-field landscapes, even in the pres-ence of large homogeneous offset fields. Den-sity modulations in trapped thermal atomicclouds have already been used as a measure ofmagnetic field variation caused by irregularcurrent flow in nearby conductors5–8. We haveproduced a versatile, high-resolution sensorbased on Bose–Einstein condensates (BECs).Its sensitivity is not limited by the temperatureT of the cloud, but is rather determined by thechemical potential � of the condensate, whichcan be orders of magnitude lower than kBT(where kB is Boltzmann’s constant).

The principles of the technique are shown inFig. 1a. A BEC is trapped at the measurementsite so that its density profile can be directlyimaged. The spatially varying density is a mea-sure of the potential energy and hence of thelocal magnetic-field variation. To probe spatialmagnetic-field variations, we start by confininga one-dimensional BEC (in which � is smallerthan an energy quantum of transverse excita-tion)9 in an elongated magnetic micro-trapwith strong transverse and weak longitudinalconfinement, created by small conductorsmounted on the surface of an atom chip10.

As a demonstration, we measured the magnetic-field variations above the 100-�m-wide current-carrying wire used to create thetrap itself. Scanning the position of the BECenabled us to reconstruct a full two-dimen-sional magnetic-field profile (Fig. 1b) near the wire with unprecedented accuracy (sensi-tivity of 4 nanotesla) at the measurement resolution (3 �m). From this map, we recon-structed the local current flow in the wire11

and found extremely small angular deviations(2�10�4 root-mean-square radians) from astraight current path (for details, see supple-mentary information).

We also investigated an independent fieldlandscape by placing a BEC close (5 �m) to atest wire structure. As long as this structure isgrounded and carries no current, the atomic-density profile is homogeneous within thedetection sensitivity. This corresponds to anupper bound in potential roughness of lessthan 10�14 eV, corresponding to a temperatureof 200 picokelvin (field sensitivity of300 picotesla). As soon as a small current(about 5 mA) is passed through the wire, acharacteristic field profile is imaged.

The technique is applicable not only to mag-netic fields, but can also be used to detect vari-ations in electrostatic fields, as can be shown

Probedsample

Chip substrate

Trapping wire

Imaging laser beam

CCD camera

Probe BEC

yz

x

–400–200

0

B (n

T)

200

300200100

0

050

–50

400

a

b

Figure 1 | One-dimensional Bose–Einsteincondensate as a magnetic-field sensor.a, Experimental set-up. A Bose–Einsteincondensate is created and trapped by a current-carrying wire mounted on a silicon surface (atom chip) and is positioned above the sample to be probed. b, Two-dimensional scan of themagnetic landscape (field component along the wire direction) above a 100-�m-wide and 3.1-�m-tall gold wire. This profile has beenreconstructed from 28 equally spaced one-dimensional atomic-density traces measured10 �m above the current-carrying wire at ahomogeneous offset field of 2 millitesla, so that relative field variations of only 4 p.p.m.stemming from slightly irregular current flowcould be measured at a spatial resolution of 3 �m.

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Supplementary methods Microscopic magnetic imaging with Bose-Einstein condensates

Experimental setup

In our apparatus, we produce Bose Einstein Condensates (BEC) in a procedure described

in detail in ref1. We first accumulate >108 cold 87Rb atoms in the MOT near the surface of

an atom chip2 which directly serves as the mirror. The atoms are then transferred to a

magnetic micro-trap on the atom chip and evaporatively cooled so that a BEC of up to

105 atoms forms in the chosen micro trap on the chip. The atom number and the desired

chemical potential µ can be adjusted during the final cooling stage.

In magnetic traps, atoms in low field seeking states (V = mFgFµBB > 0, where µB is the

Bohr-magneton, gF is the g-factor associated to the atomic hyperfine level and mF is the

quantum number identifying the magnetic sublevel) are confined at a local field

minimum. In micro traps on atom chips the minimum forms at the position where the

magnetic field formed by a current through the wire is compensated by an external

homogeneous bias field. Choosing appropriate values for currents and bias field

(magnitude and direction) allows to precisely place the atom cloud over a wide range of

positions. A small field component BIP along the wire direction avoids Majorana spin flip

losses and a weak spatial variation of BIP provides longitudinal confinement. Micro traps

on an atom chip can be designed to form quasi one-dimensional traps with tight

transverse and weak longitudinal confinement. (For a review of atom chip techniques see

ref. 2)

In our experiment we use exceptionally smooth trapping wires3,4 fabricated by thermal

evaporation followed by a lithographic lift-off procedure5. These allow us to create

Bose-Einstein condensates with trap aspect ratios of up to ~ 5000. BECs with a low

chemical potential µ stretch over a large length on the order of 1mm.

Magnetic field reconstruction

The local density of the trapped BEC is imaged in situ by high resolution (3 µm)

absorption imaging. This experimentally determined density profile n1d (z) is a map of

the spatial profile of the field component BIP. The highest sensitivity for detecting

variations in BIP is reached by minimizing the chemical potential µ. In our elongated

traps, µ is smaller than the energy level spacing in the transverse (tight confinement)

direction ( trωµ < ), but much larger than the energy level spacing in the longitudinal

(weak confinement) direction ( lωµ >> ). In this so called one-dimensional Thomas-

Fermi (1dTF) regime6, the external potential energy and hence the longitudinal field

component profile can be computed according to

)(2)( 10 zngm

azBB d

BFF

scattr µ

ω=−

where ascat is the s-wave scattering length (5.2 nm for 87Rb ref7) and B0 is an arbitrary

homogeneous offset field.

Figure S1 shows a typical n1d(z) profile and illustrates how a one-dimensional BIP(z) trace

is derived from it. The field sensitivity of a single shot measurement is given by the

detection noise of n1d which in principle is limited by atom shot noise. In our

measurements shown in figure S1 we find a single shot sensitivity for a field

measurement of ∆B = 4 nT which corresponds to a potential energy change of ∆U ~ 10-13

eV. This sensitivity is reached at an offset field of 10 G, thus the relative detected

inhomogeneities are ∆B/B = 4·10-6 (inset in figure S1). The trap used for these

measurements had an oscillation frequency of ωtr=2π·3kHz, corresponding to a transverse

extension of the condensate (single particle ground state size) of ~ 200 nm.

In a second series of experiments we have measured the field profile near an independent

wire structure. Here, we have employed a more relaxed trap (ωtr=2π·200Hz), leading to

an order of magnitude higher sensitivity of ~ 300 pT in magnetic field units.

Mapping magnetic fields and current flow reconstruction

The image of the longitudinal magnetic field component depicted in figure 1b of the main

text is obtained by scanning the position of the BEC over the entire width of the 100 µm

wide 3.1 µm thick trapping wire in a plane parallel to the wire (chip) surface at a distance

of 10 µm. The magnetic trap and the cooling are configured in such a way that the

condensates form in the 1dTF regime and extend over a large longitudinal stretch of the

wire (~1 mm) while they can be transversely positioned with a precision better than 1

µm. This is achieved by using additional images taken from a direction close to

perpendicular to the atom chip surface to calibrate the positions relative to defined

markers (wire edges). In order to derive a full two-dimensional magnetic image, we have

formed condensates at 28 equally spaced transverse positions above the wire covering a

width that slightly exceeds that of the wire. The fact that the nominal wire field is exactly

compensated at the trap position makes our technique applicable for arbitrary wire

currents.

From this two-dimensional magnetic landscape the transverse component of the two-

dimensional current density in the trapping wire can be reconstructed. The current

reconstruction is only reliable at a resolution level comparable to the distance from the

wire where the magnetic field is scanned (10 µm). More details can be revealed and the

full resolution of the sensor can be exploited when the atom-surface distance is further

reduced. It has been shown that a surface approach of single microns is possible4.

Constraints are the limited trap depth in the presence of attractive surface potentials8 and

the finite trap lifetime9. Trap lifetimes have to be sufficiently long for the condensate to

reach thermodynamic equilibrium. This is typically not problematic as lifetimes on the

order of seconds can be achieved down to single micron distance from insulators10 or thin

conducting layers11 that are prone to be probed by BEC sensors.

1 Wildermuth, S. et al. Optimized magneto-optical trap for experiments with ultracold

atoms near surfaces. Phys. Rev. A 69, 030901(R) (2004).

2 Folman, R., Krüger, P., Schmiedmayer, J.,Denschlag, J. & Henkel, C., Microscopic

atom optics: from wires to an atom chip (and references therein). Adv. At. Mol. Opt. Phys. 48, 263 (2002).

3 Krüger, P., Coherent matter waves near surfaces. PhD-thesis, University of Heidelberg (2004).

4 Krüger, P. et al. Disorder Potentials near Lithographically Fabricated Atom Chips, arXiv:cond-mat/0504686 (2004).

5 Groth, S. et al. Atom chips: fabrication and thermal properties. Appl. Phys. Lett. 85, 2980 (2004).

6 Dunjko, V., Lorent, V. & Olshanii, M., Bosons in cigar-shaped traps: Thomas-Fermi regime, Tonks-Girardeau regime, and in between. Phys. Rev. Lett. 86, 5413 (2001).

7 van Kempen, E. G. M., Kokkelmans, S. J. J. M. F., Heinzen, D. J. & Verhaar, B. J., Interisotope determination of ultracold rubidium interactions from three high-precision experiments. Phys. Rev. Lett. 88, 093201 (2002).

8 Lin, Y-J, Teper, I., Chin, C. & Vuletic, V., Impact of the Casimir-Polder potential and Johnson Noise on Bose-Einstein Condensate stability near surfaces. Phys. Rev. Lett. 92, 050404 (2004).

9 Henkel, C. & Pötting, S., Coherent transport of matter waves. Appl. Phys. B 72, 73 (2001).

10 Sinclair, C.D.J. et al. Bose-Einstein Condensation on a permanent-magnet atom chip. arXiv:cond-mat/0503619 (2005).

11 Zhang, B. et al. Relevance of sub-surface chip layers for the lifetime of magnetically trapped atoms, submitted to Europ. J. Phys. D (2005).

Supplementary Figure 1

Figure S1 Magnetic field reconstruction from a density profile of a BEC. (a) From imaging

the density variations in a BEC onto a CCD-camera using resonant laser light a typical one-

dimensional density profile n1d(z) can be obtained. (b) From this the longitudinal magnetic

potential can be calculated. A section of a 1dTF BEC gives the magnetic field change over the

range of 200µm within one experimental shot. The three solid curves (red, green and blue)

have been taken at the same experimental parameters (10G homogeneous offset field) and

show the good reproducibility of our field measurement. The black (dashed) curve has been

reconstructed from a density profile taken with three times higher current in the wire (and

three times larger homogeneous offset field). The potential trace has been plotted according to

the calculated B/B (right axis), which corresponds to a scaling with the wire current (offset

field). The agreement between the B/B measurements using two largely different currents

confirms that the potential variations indeed originate from magnetic fields caused by

irregular current flow through the wire. The inset shows the shot to shot pixelwise variations

of the reconstructed magnetic field for several repetitions of the experiment yielding a single

pixel, single shot field sensitivity of 4nT. The spatial resolution of our diffraction limited

imaging system is 3µm. The transverse trapping frequency of the trap used in these

measurements was 32tr kHz.

Supplementary Figure 2

Figure S2 Field sensitivity versus spatial resolution for state-of-the-art magnetic microscopes. MFMi,ii and hall probesiii,iv operate at a high spatial resolution whereas SQUIDsv,vi,vii and thermal atom magnetometersviii offer high field sensitivity. Typical operation temperatures of the field sensor are indicated. The demonstrated BEC sensor allows mapping of magnetic fields in an intermediate parameter range (4nT at 3µm; figure S1). Positioning the BEC over an independent sample and thus lowering the transverse confinement has allowed to measure magnetic fields with one order of magnitude higher sensitivity. On the right axis sensitivity is given in units of eV since a BEC can be used to also map other local potentials (for example electric fields) with high spatial resolution. The dark grey shaded region indicates the sensitivity-resolution range currently accessible only to the BEC sensor. The solid (dashed) line displays the estimated sensitivity for a single shot measurement if the trapping potential is optimally adjusted to the desired resolution. Maximal spatial resolution is limited to 500nm by the imaging technique using blue laser light. Field sensitivity is estimated to be 1pT at a spatial resolution of 10µm (corresponding to 1Hz oscillation frequency). For higher field sensitivity (dashed line) and thus low oscillation frequencies, long trap lifetimes are required to reach equilibrium. To estimate the potential sensitivity of the BEC sensor (solid line) a shot noise limited atom detection of 100 atoms (∆N=10) has been assumed.

i McVitie, S., Ferrier, R.P., Scott, J., White, G.S. & Gallagher, A., Quantitative field measurements from magnetic force microscope tips and comparison with point and extended charge models. J. Appl. Phys. 89, 3656 (2001).

ii Proksch, R. et al. Quantitative magnetic field measurements with the magnetic force microscope. Appl. Phys. Lett. 69, 2599 (1996). iii Oral, A. et al. Room-temperature scanning hall probe microscope imaging of garnet films using new high-preformance InSb sensors. IEEE Trans. Magn. 38, 2438 (2002). iv Brook, A.J. et al. Integrated piezoresistive sensors for atomis force-guided scanning hall probe microscopy. Appl. Phys. Lett. 82, 3538 (2003). v Faley, M.I. et al. High temperature superconductor dc SQUID micro-susceptometer for room temperature objects. Supercond. Sci. Technol. 17, 324 (2004). vi Baudenbacher, F., Fong, L.E., Holzer, J.R. & Radparvar, M., Monolithic low-transition-temperature superconducting magnetometers for high resolution imaging magnetic fields of room temperature samples. Appl. Phys. Lett. 82, 3487 (2003). vii Zhang, Y. et al. Substrate resonator for HTS rf SQUID operation. Physica C 372-376, 282 (2002). viii Kominis, I.K., Kornack, T.W., Allred, J.C. & Romalis, M.V., A subfemtotesla multichannel atomic magnetometer. Nature 422, 596-599 (2003).