LETTERS

Synthetic magnetic fields for ultracold neutral atomsY.-J. Lin1, R. L. Compton1, K. Jimenez-Garcıa1,2, J. V. Porto1 & I. B. Spielman1

Neutral atomic Bose condensates and degenerate Fermi gases havebeen used to realize important many-body phenomena in theirmost simple and essential forms1–3, without many of the complex-ities usually associated with material systems. However, the chargeneutrality of these systems presents an apparent limitation—awide range of intriguing phenomena arise from the Lorentz forcefor charged particles in a magnetic field, such as the fractionalquantum Hall effect in two-dimensional electron systems4,5. Thelimitation can be circumvented by exploiting the equivalence ofthe Lorentz force and the Coriolis force to create synthetic mag-netic fields in rotating neutral systems. This was demonstrated bythe appearance of quantized vortices in pioneering experiments6–9

on rotating quantum gases, a hallmark of superfluids or super-conductors in a magnetic field. However, because of technicalissues limiting the maximum rotation velocity, the metastablenature of the rotating state and the difficulty of applying stablerotating optical lattices, rotational approaches are not able toreach the large fields required for quantum Hall physics10–12.Here we experimentally realize an optically synthesized magneticfield for ultracold neutral atoms, which is evident from the appear-ance of vortices in our Bose–Einstein condensate. Our approachuses a spatially dependent optical coupling between internal statesof the atoms, yielding a Berry’s phase13 sufficient to create largesynthetic magnetic fields, and is not subject to the limitations ofrotating systems. With a suitable lattice configuration, it should bepossible to reach the quantum Hall regime, potentially enablingstudies of topological quantum computation.

In classical electromagnetism, the Lorentz force for a particle ofcharge q moving with velocity v in a magnetic field B is v 3 qB. In theHamiltonian formulation of quantum mechanics, where potentialsplay a more central role than fields, the single-particle Hamiltonian isH~B2 kqA=Bð Þ2

2m, where A is the vector potential giving rise to

the field B 5 = 3 A, Bk is the canonical momentum and m is themass. In both formalisms, only the products qB and qA are import-ant. To generate a synthetic magnetic field B* for neutral atoms, weengineered a Hamiltonian with a spatially dependent vector potentialA* producing B* 5 = 3 A*.

The quantum mechanical phase is the relevant and significantquantity for charged particles in magnetic fields. A particle of chargeq travelling along a closed loop acquires a phase w 5 2pWB /W0 due tothe presence of magnetic field B, where WB is the enclosed magneticflux and W0 5 h/q is the flux quantum. A similar path-dependentphase, the Berry’s phase13, is the geometric phase acquired by a slowlymoving particle adiabatically traversing a closed path in aHamiltonian with position-dependent parameters. The Berry’s phasedepends only on the geometry of the parameters along the path, andis distinct from the dynamic contribution to the phase, whichdepends upon the speed of the motion.

The close analogy with the Berry’s phase implies that properlydesigned position-dependent Hamiltonians for neutral particlescan simulate the effect of magnetic fields on charged particles. We

created such a spatially varying Hamiltonian for ultracold atoms bydressing them in an optical field that couples different spin states. Theappropriate spatial dependence can originate from the laser beams’profile10,14,15 or, as here, from a spatially dependent laser–atomdetuning16. An advantage of this optical approach over rotating gasesis that the synthetic field exists at rest in the laboratory frame, allow-ing all trapping potentials to be time-independent.

The large synthetic magnetic fields accessible by this approachmake possible the study of unexplored bosonic quantum-Hall states,labelled by the filling factor n 5 WB /W0, the ratio of atom number tothe number of flux quanta. The outstanding open questions inquantum-Hall physics centre on systems whose elementary quasi-particle excitations are anyons: neither bosons nor fermions. In somecases these anyons may be non-abelian, meaning that moving themabout each other can implement quantum gates, which makes non-abelian anyons of great interest for this ‘topological’ quantum com-putation17. In electronic systems, the observed n 5 5/2 quantum-Hallstate may be such a system, but its true nature is still uncertain18. Incontrast, the n 5 1 bosonic quantum-Hall state with contact interac-tions has the same non-abelian anyonic excitations as the n 5 5/2state in electronic systems is hoped to have19.

To engineer a vector potential A~Ax xx, we illuminated a 87RbBose–Einstein condensate (BEC) with a pair of Raman laser beamswith momentum difference along xx (Fig. 1a). These coupled togetherthe three spin states, mF 5 0 and 61, of the 5S1/2, F 5 1 electronicground state (Fig. 1b), producing three dressed states whose energy–momentum dispersion relations Ej(kx) are experimentally tunable.Example dispersions are illustrated in Fig. 1c. The lowest ofthese, with minimum at kmin, corresponds to a term in theHamiltonian associated with the motion along xx, namelyHx<B2 kxkminð Þ2

2m~B2 kxqAx

B

2.2m, where Ax is an

engineered vector potential that depends on an externally controlledZeeman shift for the atom with synthetic charge q*, and m* is theeffective mass along xx. To produce the desired spatially dependentAx yð Þ (see Fig. 1d), generating Bzz~+|A, we applied a Zeemanshift that varied linearly along yy. The resulting B* was approximatelyuniform near y 5 0, at which point Ax~By. (Here, the microscopicorigin of the synthetic Lorentz force20 was optical along xx, dependingupon the velocity along yy; the force along yy was magnetic, dependingupon the xx velocity.) In this way, we engineered a Hamiltonian forultracold atoms that explicitly contained a synthetic magnetic field,with vortices in the ground state of a BEC. This is distinctly differentfrom all existing experiments, where vortices are generated by phaseimprinting21,22, rotation7–9, or a combination thereof23. Each of theseearlier works presents a different means of imparting angularmomentum to the system yielding rotation. Figure 1e shows anexperimental image of the atoms with B* 5 0. Figure 1f, withB* . 0, shows vortices. This demonstrates an observation of an optic-ally induced synthetic magnetic field.

We created a 87Rb BEC in a 1,064-nm crossed dipole trap, loadedinto the lowest-energy dressed state24 with atom number N up to

1Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899, USA. 2Departamento de Fısica, Centro deInvestigacion y Estudios Avanzados del Instituto Politecnico Nacional, Mexico DF, 07360, Mexico.

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2.5 3 105, and a Zeeman shift vZ/2p5 gmBB/h < 2.71 MHz, pro-duced by a real magnetic bias field Byy. The l 5 801.7 nm Ramanbeams propagated along yy+xx and differed in frequency by a constantDvL^vZ, where a small Raman detuning d 5DvL 2 vZ largelydetermined the vector potential Ax . The scalar light shift from theRaman beams, combined with the dipole trap, gave an approximatelysymmetric three-dimensional potential, with frequencies fx, fy, fz <70 Hz. Here, BkL~h

ffiffiffi2p

l

and EL~B2k2L

2m are the appropriate

units for the momentum and energy.The spin and momentum states jmF, kxæ coupled by the Raman

beams can be grouped into families of states labelled by themomentum Bkx. Each family Y(kx) 5 j21, kx12kLæ,j0, kxæ,j11,kx 2 2kLæ is composed of states that differ in linear momentumalong xx by 62BkL, and are Raman-coupled with strength BVR. Foreach kx, the three dressed states are the eigenstates in the presence ofthe Raman coupling, with energies24 Ej(kx). The resulting vectorpotential is tunable within the range 2kLvqAx

Bv2kL. In addi-

tion, Ej(kx) includes a scalar potential16 V9. Ax , V9, and m* are func-tions of Raman coupling VR and detuning d, and for our typicalparameters m* < 2.5m, reducing fx from about 70 Hz to about40 Hz. The BEC’s chemical potential m/h < 1 kHz is much smallerthan the ,h 3 10 kHz energy separation between dressed states, sothe BEC only occupies the lowest-energy dressed state. Further, itjustifies the harmonic expansion around qAx

B, valid at low energy.

Hence, the complete single-atom Hamiltonian is H~HxzB2 k2

y zk2z

.2mzV rð Þ, where V(r) is the external potential includ-

ing V9(VR, d).The dressed BEC starts in a uniform bias field B~B0yy, at Raman

resonance (d 5 0), corresponding to Ax~024. To create a syntheticfield B*, we applied a field gradient b9 such that B~ B0b’yð Þyy,

ramping in 0.3 s from b9 5 0 to a variable value up to 0.055 Tm21,and then held it constant for th to allow the system to equilibrate. Thedetuning gradient d0~gmBb0=B generates a spatial gradient in Ax . Forthe detuning range in our experiment, LAx

Ld is approximately

constant, leading to an approximately uniform synthetic field B*

given by B~LAxLy~d’LAx

Ld (see Fig. 1d). To probe the dressed

state, we switched off the dipole trap and the Raman beams in lessthan 1 ms, projecting each atom into spin and momentum compo-nents. We absorption-imaged the atoms after a time-of-flight (TOF)ranging from 10.1 ms to 30.1 ms (Fig. 1e, f).

For a dilute BEC in low synthetic fields, we expect to observevortices. In this regime, the BEC is described by a macroscopic wave-function y(r) 5 jy(r)jeiw(r), which obeys the Gross–Pitaevskii equa-tion (GPE). The phase w winds by 2p around each vortex, withamplitude jyj5 0 at the vortex centre. The magnetic flux WB* resultsin Nv vortices and for an infinite, zero-temperature system, the vor-tices are arrayed in a lattice25 with density q*B*/h. For finite systemsvortices are energetically less favourable, and their areal density isbelow this asymptotic value, decreasing to zero at a critical field Bc .For a cylindrically symmetric BEC, Bc is given by qBc

h~

5= 2pR2ð Þln 0:67R=jð Þ where R is the Thomas–Fermi radius and jis the healing length26. Bc is larger for smaller systems. For ournon-cylindrically symmetric system, we numerically solve the GPEto determine Bc for our experimental parameters (see Methods).

For synthetic fields greater than the critical value, we observedvortices that enter the condensate and reach an equilibrium vortexnumber Nv after about 0.5 s. Owing to a shear force along xx when theRaman beams are turned off, the nearly symmetric in situ atom cloudtilts during TOF. Although the vortices’ positions may rearrange, anyinitial order is not lost. During the time of our experiment, the

ωZ

δ

δ

ωZ

5S1/2

F = 1

ωL + ∆ωL

x

B0 – b′yy

z

Raman Raman

BEC

ωL

Effective position ‘y’ or detuning, hδ/EL

Vec

tor

pot

entia

l,q

*A*/hk

L

Geometry

Level diagram

δ′/2π = 0.34 kHz µm–1

–1

0

1

20–2

Sp

in p

roje

ctio

n, m

F

Momentum, kx/kL

20–2

–1

0

1

Sp

in p

roje

ctio

n, m

F

Momentum, kx/kL

δ′ = 0

kmin

10

5

0

–5

Ene

rgy,

E/E

L

–3 –2 –1 0 1 2 3

Momentum, kx/kL

–2

-1

0

1

2

–10 –5 0 5 10

a

b

c dDispersion relation, hδ = –2EL

e f

Vector potential, h R= 8.20ELΩ

Dressed state, h R = 8.20ELΩ Dressed state, h R = 8.20ELΩ

–1

+1

0ε

Figure 1 | Experiment summary for synthesizing magnetic fields. a, TheBEC is in a crossed dipole trap in a magnetic field B~ B0b’yð Þyy. TwoRaman beams propagating along yy+xx (linearly polarized along yy+xx) havefrequencies vL and vL 1DvL. b, Raman coupling scheme within the F 5 1manifold: vZ and e are the linear and quadratic Zeeman shifts, and d is theRaman detuning. c, Energy–momentum dispersion relations. The greycurves represent the states without Raman coupling; the three coloured

curves represent Ej(kx) of the dressed states. The arrow indicates theminimum at kmin. d, Vector potential qAx~Bkmin versus Raman detuningd. The insets show the dispersion E1(kx) for Bd 5 0 (top inset) and 22EL

(bottom inset). e, f, Dressed BEC imaged after a 25.1-ms TOF without(e) and with (f) a gradient. The spin components mF 5 0 and 61 separatealong yy owing to the Stern–Gerlach effect.

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vortices did not form a lattice and the positions of the vortices wereirreproducible between different experimental realizations, consist-ent with our GPE simulations. We measured Nv as a function ofdetuning gradient d0 at two couplings, BVR 5 5.85EL and 8.20EL

(Fig. 2). For each VR, vortices appeared above a minimum gradientwhen the corresponding field Bh i~d’ LAx

Ld

exceeded the crit-

ical field Bc . (For our coupling, B* is only approximately uniformover the system and ÆB*æ is the field averaged over the area of theBEC.) The inset shows Nv for both values of VR plotted versusWB=W0~Aq Bh i=h, the vortex number for a system of areaA~pRxRy with the asymptotic vortex density, where Rx (or Ry) isthe Thomas–Fermi radius along xx or yyð Þ. The system size, and thusBc , are approximately independent of VR, so we expected this plot tobe nearly independent of Raman coupling. Indeed, the data forBVR 5 5.85EL and 8.20EL only deviated for Nv , 5, probably owingto the intricate dynamics of vortex nucleation27.

Figure 3 illustrates a progression of images showing that vorticesnucleate at the system’s edge, fully enter to an equilibrium densityand then decay along with the atom number. The timescale for vortexnucleation depends weakly on B*, and is more rapid for larger B* withmore vortices. It is about 0.3 s for vortex number Nv $ 8, andincreases to about 0.5 s for Nv 5 3. For Nv 5 1 (B* near Bc ), the singlevortex always remains near the edge of the BEC. In the dressed state,spontaneous emission from the Raman beams removes atoms fromthe trap, causing the population to decay with a 1.4(2)-s lifetime, andthe equilibrium vortex number decreases along with the area of theBEC.

To verify that the dressed BEC has reached equilibrium, we pre-pared nominally identical systems in two different ways. First, wevaried the initial atom number and measured Nv as a function ofatom number N at a fixed hold time of th 5 0.57 s. Second, startingwith a large atom number, we measured both Nv and N, as they

X position after TOF (μm)

δ′ = 0 δ′/2π = 0.13 kHz µm–1 δ′/2π = 0.27 kHz µm–1

δ′/2π = 0.31 kHz µm–1 δ′/2π = 0.34 kHz µm–1 δ′/2π = 0.40 kHz µm–1

Y p

ositi

on a

fter

TO

F (μ

m)

Detuning gradient, δ′/2π (kHz µm–1)

g

–80 0 80–80 0 80

–80

–80

80

80

0

0

–80 0 80

10

5

020100

12

10

8

6

4

2

00.40.30.20.10.0

Vor

tex

num

ber

, Nv

a b c

d e fB*/ 0Φ Φ

Figure 2 | Appearance of vortices at different detuning gradients. Data wastaken for N 5 1.4 3 105 atoms at hold time th 5 0.57 s. a–f, Images of the| mF 5 0æ component of the dressed state after a 25.1-ms TOF with detuninggradient d0/2p from 0 to 0.43 kHzmm21 at Raman coupling BVR 5 8.20EL.g, Vortex number Nv versus d0 at BVR 5 5.85EL (blue circles) and 8.20EL (redcircles). Each data point is averaged over at least 20 experimental

realizations, and the uncertainties represent one standard deviation s. Theinset displays Nv versus the synthetic magnetic flux WB=W0~Aq Bh i=h inthe BEC. The dashed lines indicate d0, below which vortices becomeenergetically unfavourable according to our GPE computation, and theshaded regions show the 1s uncertainty from experimental parameters.

0

–100

100

0

–100

100

Y p

ositi

on a

fter

TO

F (µ

m)

0–100 100 0–100 100 0–100 100X position after TOF (µm)

th = –0.019 s th = 0.15 s th = 0.57 s

th = 1.03 s th = 1.4 s th = 2.2 s

15

10

5

0

Vort

ex n

umb

er, N

v

2.01.51.00.50.0Hold time, th (s)

2

1

0

δ′ (a

.u.)

g

N

Nv

0

Atom

numb

er, N (×

105)

a b c

d e f

Figure 3 | Vortex formation. a–f, Images of the | mF 5 0æ component of thedressed state after a 30.1-ms TOF for hold times th between 20.019 s and2.2 s. The detuning gradient d0/2p is ramped to 0.31 kHzmm21 at thecoupling BVR 5 5.85EL. g, Top panel shows time sequence of d0. (a.u.,

arbitrary units.) Bottom panel shows vortex number Nv (solid symbols) andatom number N (open symbols) versus th with a population lifetime of1.4(2) s. The number in parentheses is the uncorrelated combination ofstatistical and systematic 1s uncertainties.

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decrease with th (Fig. 3). Figure 4 compares Nv versus N measuredwith both methods, each at two detuning gradients corresponding tofields B1vB2 . The data show that Nv as a function of N is the same forthese preparation methods, providing evidence that for th $ 0.57 s,Nv has reached equilibrium. As the atom number N falls, the lastvortex departs the system when the critical field—increasing withdecreasing N—surpasses the actual field.

In conclusion, we have demonstrated optically synthesized mag-netic fields for neutral atoms resulting from the Berry’s phase, afundamental concept in physics. This novel approach differs fromexperiments with rotating gases, in which it is difficult to add opticallattices and rotation is limited by heating, metastability, and thedifficulty of adding large angular momentum, preventing access tothe quantum-Hall regime. A standout feature in our approach is theease of adding optical lattices. For example, the addition of a two-dimensional (2D) lattice makes it immediately feasible to study thefractal energy levels of the Hofstadter butterfly28. Further, a one-dimensional lattice can divide the BEC into an array of 2D systemsnormal to the field. A suitable lattice configuration allows access tothe n < 1 quantum-Hall regime, with an ensemble of 2D systems eachwith approximately 200 atoms, and with a realistic interaction energyof about kB 3 20 nK.

METHODS SUMMARYDressed state preparation. We created a 87Rb BEC in a crossed dipole trap29,

with N < 4.7 3 105 atoms in jF 5 1, mF 5 21æ. The quadratic Zeeman shift was

Be~0:61EL for vZ/2p5 gmBB/h < 2.71 MHz, where g is the Lande g-factor. To

maintain d 5 0 at the BEC’s centre as we ramped the field gradient b9, we changed

gmBB0 by as much as 7EL. Simultaneously, we decreased the dipole beam power

by 20%, producing our approximately 40-Hz trap frequency along xx.

Additionally, the detuning gradient d’yy made the scalar potential V9 anti-trap-

ping along yy, reducing fy from 70 Hz to 50 Hz for our largest d0. Spontaneous

emission from the Raman beams decreased the atom number to N < 2.5 3 105

for th 5 0, with a condensate fraction of 0.85.

Numerical method. We compared our data to a finite temperature 2D stochastic

GPE30 simulation including the dressed state dispersion E(kx, y) that depends on y

through the detuning gradient d0. We evolved the time-dependent projected GPE:

iBLy x, tð Þ

Lt~P E iB

LLx

, y

B2

2m

L2

Ly2zg2D y x, tð Þj j2

y x, tð Þ

P projects onto a set of significantly occupied modes, and g2D parameterizes the

2D interaction strength. The stochastic GPE models interactions between the

highly occupied modes described by y and sparsely occupied thermal modes

with dissipation and an associated noise term. We approximately accounted for

the finite extent along zz by making g2D depend on the local 2D density. For low

temperatures this 2D model correctly recovers the three-dimensional Thomas–

Fermi radii, and gives the expected 2D density profile. These quantitative details

are required to compute correctly the critical field or number for the first vortex

to enter the system, which are directly tied to the 2D condensate area.

Received 7 August; accepted 20 October 2009.

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10

8

6

4

2

0150100500

Atom number, N (×103)

Vor

tex

num

ber

, Nv

B1 < B2* *

*B2

*B1

Figure 4 | Equilibrium vortex number. Vortex number Nv versus atomnumber N at detuning gradient d’1=2p~0:26 kHz mm1 (red circles) andd’2=2p~0:31 kHz mm1 (black circles), corresponding to synthetic fieldsB1vB2, at Raman coupling BVR 5 5.85EL. The two data points with thelargest N show representative 1s uncertainties, estimated from data inFig. 2g. We vary N by its initial value with a fixed hold time th 5 0.57 s (solidsymbols), and by th with a fixed initial N (open symbols). The vertical dashedlines indicate N, below which vortices become energetically unfavourablecomputed using our GPE simulation. The shaded regions reflect the 1suncertainties from the experimental parameters.

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Acknowledgements We thank W. D. Phillips for discussions. This work waspartially supported by ONR, ARO with funds from the DARPA OLE programme,and the NSF through the JQI Physics Frontier Center. R.L.C. acknowledges theNIST/NRC postdoctoral programme and K.J.-G. thanks CONACYT.

Author Contributions All authors contributed to writing of the manuscript. Y.-J.L.led the data collection effort (with assistance from R.L.C. and K.J.-G.). I.B.S. andJ.V.P. designed the original apparatus, which was largely constructed by I.B.S., and

Y.-J.L. implemented the specific changes required for the present experiment. I.B.S.conceived the experiment and performed numerical and analytic calculations. Thiswork was supervised by I.B.S. with consultations from J.V.P.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. Correspondence and requests for materials should beaddressed to I.B.S. (ian.spielman@nist.gov).

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