Magnetic Vortices

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Synthetic magnetic fields for ultracold neutral atoms

Y.-J. Lin1, R. L. Compton1, K. Jimenez-Garcia1,2, J. V. Porto1, and I. B. Spielman11Joint Quantum Institute, National Institute of Standards and Technology,

and University of Maryland, Gaithersburg, Maryland, 20899, USA and2Departamento de Fsica, Centro de Investigacion y Estudios Avanzados

del Instituto Politecnico Nacional, Mexico D.F., 07360, Mexico(Dated: July 5, 2010)

Neutral atomic Bose condensates and degenerate Fermi gases have been used to realize impor-tant many-body phenomena in their most simple and essential forms [13], without many of thecomplexities usually associated with material systems. However, the charge neutrality of thesesystems presents an apparent limitation - a wide range of intriguing phenomena arise from theLorentz force for charged particles in a magnetic field, such as the fractional quantum Hall statesin two-dimensional electron systems[4, 5]. The limitation can be circumvented by exploiting theequivalence of the Lorentz force and the Coriolis force to create synthetic magnetic fields in rotat-ing neutral systems. This was demonstrated by the appearance of quantized vortices in pioneeringexperiments[69] on rotating quantum gases, a hallmark of superfluids or superconductors in amagnetic field. However, because of technical issues limiting the maximum rotation velocity, themetastable nature of the rotating state and the difficulty of applying stable rotating optical lattices,rotational approaches are not able to reach the large fields required for quantum Hall physics [1012]. Here, we experimentally realize an optically synthesized magnetic field for ultracold neutralatoms, made evident from the appearance of vortices in our Bose-Einstein condensate. Our approachuses a spatially-dependent optical coupling between internal states of the atoms, yielding a Berrysphase[13] sufficient to create large synthetic magnetic fields, and is not subject to the limitations ofrotating systems; with a suitable lattice configuration, it should be possible to reach the quantumHall regime, potentially enabling studies of topological quantum computation.

In classical electromagnetism, the Lorentz force for aparticle of charge q moving with velocity v in a magneticfield B is v qB. In the Hamiltonian formulation ofquantum mechanics, where potentials play a more cen-tral role than fields, the single-particle Hamiltonian isH = 2(k qA/)2/2m, where A is the vector potentialgiving rise to the field B = A, k is the canoni-cal momentum and m is the mass. In both formalisms,only the products qB and qA are important. To generatea synthetic magnetic field B for neutral atoms, we en-gineer a Hamiltonian with a spatially dependent vectorpotential A producing B = A.

The quantum mechanical phase is the relevant and sig-nificant quantity for charged particles in magnetic fields.A particle of charge q traveling along a closed loop ac-quires a phase = 2B/0 due to the presence of mag-netic field B, where B is the enclosed magnetic flux and0 = h/q is the flux quantum. A similar path-dependentphase, the Berrys phase[13], is the geometric phase ac-quired by a slowly moving particle adiabatically travers-ing a closed path in a Hamiltonian with position depen-dent parameters. The Berrys phase depends only on thegeometry of the parameters along the path, and is dis-tinct from the dynamic contribution to the phase whichdepends upon the speed of the motion.

The close analogy with the Berrys phase implies thatproperly designed position-dependent Hamiltonians forneutral particles can simulate the effect of magnetic fieldson charged particles. We create such a spatially-varyingHamiltonian for ultracold atoms by dressing them inan optical field that couples different spin states. Theappropriate spatial dependence can originate from the

laser beams profile[10, 14, 15] or, as here, a spatially-dependent laser-atom detuning[16]. An advantage of thisoptical approach compared to rotating gases is that thesynthetic field exists at rest in the lab frame, allowing alltrapping potentials to be time-independent.

The large synthetic magnetic fields accessible by thisapproach make possible the study of unexplored bosonicquantum-Hall states, labeled by the filling factor =

B/0, the ratio of atom number to the number offlux quanta. The most outstanding open questions inquantum-Hall physics center on states whose elemen-tary quasiparticle excitations are anyons: neither bosonsnor fermions. In some cases these anyons may be non-abelian, meaning that moving them about each othercan implement quantum gates, thus non-abelian anyonsare of great interest for this topological quantumcomputation[17]. In electronic systems, the observed = 5/2 quantum-Hall state may be such a system, butits true nature is still uncertain[18]. In contrast, the = 1 bosonic quantum-Hall state with contact inter-actions has the same non-abelian anyonic excitations asthe = 5/2 state in electronic systems is hoped to[19].

To engineer a vector potential A = Axx, we illumi-nate a 87Rb BEC with a pair of Raman laser beams withmomentum difference along x (Fig. 1a). These coupletogether the three spin states, mF = 0 and 1, of the5S1/2, F = 1 electronic ground state (Fig. 1b), produc-ing three dressed states whose energy-momentum disper-sion relations Ej(kx) are experimentally tunable. Ex-ample dispersions are illustrated in Fig. 1c. The lowestof these, with minimum at kmin, corresponds to a termin the Hamiltonian associated with the motion along x,

a

rXiv:1007.0294v1

[cond-mat.quant-g

as]2Jul2010


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Z

Z

5S1/2

F=1

L+

L

x

B0- by

y

z

Raman Raman

BEC

L

e Dressed state,R= 8.20 E

L

Effective position y, or detuning/EL

Vectorpotential

q*A*/k

L

d Vector potential, R= 8.20 E

Lc Dispersion relation, = -2 E

La Geometry

b Level diagram

'=0.34 kHz m-1

-1

0

1

20-2

SpinprojectionmF

Momentum kx/k

L

20-2

-1

0

1

SpinprojectionmF

Momentum kx/k

L

'=0

kmin

f Dressed state, R= 8.20 E

L

10

5

0

-5

EnergyE/E

L

-3 -2 -1 0 1 2 3

Momentum kx/k

L

-2

-1

0

1

2

-10 -5 0 5 10

FIG. 1: Experiment summary for synthesizing magnetic fields. a, The BEC is in a crossed dipole trap in a magneticfield B = (B0 b

y)y. Two Raman beams propagating along y x (linearly polarized along y x) have frequencies L,L + L. b, Raman coupling scheme within the F = 1 manifold: Z and are the linear and quadratic Zeeman shifts,and is the Raman detuning. c, Energy-momentum dispersion relations. The grey curves represent the states absent Ramancoupling; the three colored curves are Ej(kx) of the dressed states. The arrow indicates the minimum at kmin. d, Vectorpotential qAx = kmin versus Raman detuning . The insets show the dispersion E1(kx) for = 0 (top inset) and 2EL(bottom inset). e,f, Dressed BEC imaged after a 25.1 ms TOF without (e) and with (f) a gradient. The spin components,mF = 0,1, separate along y due to the Stern-Gerlach effect.

Hx 2(kx kmin)2/2m = 2(kx qAx/)2/2m,where Ax is an engineered vector potential that dependson an externally controlled Zeeman shift for the atomwith a synthetic charge q, and m is the effective massalong x. To produce the desired spatially-dependentAx(y) (Fig. 1d), generating Bz = A, we applya Zeeman shift that varies linearly along y. The result-ing B is approximately uniform near y = 0, at whichpoint Ax = B

y. (Here, the microscopic origin of thesynthetic Lorentz force[20] is optical along x, depending

upon the velocity along y; the force along y is magnetic,depending upon the x velocity.) By these means, we en-gineer a Hamiltonian for ultracold atoms, that explicitlycontains a synthetic magnetic field, with vortices in theground state of a BEC. This is distinctly different fromall existing experiments, where vortices are generated byphase imprinting[21, 22], rotation[79], or a combinationthereof[23]. Each of these earlier works presents a differ-ent means to impart angular momentum to the systemyielding rotation. Fig. 1e shows an experimental image of

the atoms with B = 0. Fig. 1f, with B > 0, shows vor-tices. This demonstrates an observation of an opticallyinduced synthetic magnetic field.

We create a 87Rb BEC in a 1064 nm crossed dipoletrap, loaded into the lowest energy dressed state[24] withatom number N up to 2.5 105, and a Zeeman shiftZ/2 = gBB/h 2.71 MHz, produced by a realmagnetic bias field By. The = 801.7 nm Ramanbeams propagate along y x and differ in frequencyby a constant L

Z, where a small Raman de-

tuning = L Z largely determines the vectorpotential Ax. The scalar light shift from the Ramanbeams, combined with the dipole trap gives an approxi-mately symmetric three-dimensional potential, with fre-quencies fx, fy, fz 70 Hz. Here, kL = h/(

2) and

EL = 2k2L/2m are the appropriate units for the momen-

tum and energy.

The spin and momentum states |mF, kx coupled bythe Raman beams can be grouped into families of stateslabeled by the momentum kx. Each family (kx) =


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Xposition after TOF (m)

a =0 b =0.13 kHz m-1 c =0.27 kHz m-1

d =0.31 kHz m-1 e =0.34 kHz m-1 f =0.40 kHz m-1

YpositionafterTOF(m)

Detuning gradient (kHz m-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

B*

/0

VortexnumberN

v

FIG. 2: Appearance of vortices at different detuning gradients. Data was taken for N = 1.4 105 atoms at hold time

th = 0.57 s. a-f, Images of the |mF = 0 component of the dressed state after a 25.1 ms TOF with detuning gradient

from

0 to 0.43 kHz m1 at Raman coupling R = 8.20 EL. g, Vortex number Nv versus

at R = 5.85 EL (blue circles), and8.20 EL (red circles). Each data point is averaged over 20 experimental realizations, and the uncertainties represent standarddeviations (). The inset displays Nv versus the synthetic magnetic flux B/0 = Aq

B/h in the BEC. The dashed lines

indicate

below which vortices become energetically unfavorable according to our GPE computation, and the shaded regionsshow the 1- uncertainty from experimental parameters.

{| 1,kx + 2kL,|0, kx,| + 1, kx 2kL} is composed ofstates that differ in linear momentum along x by 2kL,and are Raman-coupled with strength R. For each kx,the three dressed states are the eigenstates in the pres-ence of the Raman coupling, with energies Ej(kx)[24].The resulting vector potential is tunable within the range2kL < qAx/ < 2kL. In addition, Ej(kx) includes ascalar potential V

[16]. Ax, V

, and m are functions ofRaman coupling R and detuning , and for our typical

parameters m

2.5m, reducing fx from 70 Hz to 40 Hz. The BECs chemical potential /h 1 kHzis much smaller than the h 10 kHz energy sepa-ration between dressed states, therefore the BEC onlyoccupies the lowest energy dressed state. Further, it jus-tifies the harmonic expansion around qAx/, valid atlow energy. Hence, the complete single-atom Hamilto-nian is H = Hx + 2(k2y + k2z)/2m + V(r), where V(r) isthe external potential including V

(R, ).

The dressed BEC starts in a uniform bias field B =B0y, at Raman resonance ( = 0), corresponding toAx = 0[24]. To create a synthetic field B

, we apply

a field gradient b

such that B = (B0

b

y)y, ramp-

ing in 0.3 s from b = 0 to a variable value up to0.055 Tm1, and then hold for th to allow the systemto equilibrate. The detuning gradient

= gBb

/ gen-erates a spatial gradient in Ax. For the detuning rangein our experiment, Ax/ is approximately constant,leading to an approximately uniform synthetic field B

given by B = Ax/y =

Ax/ (Fig. 1d). To probethe dressed state, we switch off the dipole trap and theRaman beams in less than 1 s, projecting each atominto spin and momentum components. We absorption-

image the atoms after a time-of-flight (TOF) rangingfrom 10.1 ms to 30.1 ms (Fig. 1ef).

For a dilute BEC in low synthetic fields, we expect toobserve vortices. In this regime, the BEC is described bya macroscopic wave function (r) = |(r)|ei(r), whichobeys the Gross-Pitaevskii equation (GPE). The phase winds by 2 around each vortex, with amplitude || = 0at the vortex center. The magnetic flux B results inNv vortices and for an infinite, zero temperature sys-

tem, the vortices are arrayed in a lattice[25] with den-sity qB/h. For finite systems vortices are energeti-cally less favorable, and their areal density is below thisasymptotic value, decreasing to zero at a critical fieldBc . For a cylindrically symmetric BEC, B

c is given byqBc/h = 5/(2R

2)ln(0.67R/) where R is the Thomas-Fermi radius and is the healing length[26]. Bc is largerfor smaller systems. For our non-cylindrically symmetricsystem, we numerically solve the GPE to determine Bcfor our experimental parameters (see Methods).

For synthetic fields greater than the critical value, weobserve vortices that enter the condensate and reach anequilibrium vortex number Nv after 0.5 s. Due to ashear force along x when the Raman beams are turnedoff, the nearly-symmetric insitu atom cloud tilts duringTOF. While the vortices positions may rearrange, anyinitial order is not lost. During the time of our experi-ment, the vortices do not form a lattice and the positionsof the vortices are irreproducible between different ex-perimental realizations, consistent with our GPE simula-tions. We measure Nv as a function of detuning gradient

at two couplings, R = 5.85 EL and 8.20 EL (Fig. 2).For each R, vortices appear above a minimum gradientwhen the corresponding field B = Ax/ exceeds


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-100

100

0

-100

100

YpositionafterTOF(m)

0-100 100 0-100 100 0-100 100

Xposition after TOF (m)

a th= -0.019 s b t

h= 0.15 s c t

h= 0.57 s

d th= 1.03 s e t

h= 1.4 s f t

h= 2.2 s

15

10

5

0

Vortexnumbe

rN

v

2.01.51.00.50.0

Hold time th

(s)

2

1

0

10

x

5

(a.u.)

g

N

Nv

0

Ato

mnumberN

FIG. 3: Vortex formation. a-f images of the |mF = 0 component of the dressed state after a 30.1 ms TOF for hold times

th between -0.019 s and 2.2 s. The detuning gradient

/2 is ramped to 0.31 kHz m1 at the coupling R = 5.85 EL. g,

top: time sequence of

. bottom: vortex number Nv (solid symbols) and atom number N (open symbols) versus th with apopulation lifetime of 1.4(2) s. The number in parenthesis is the uncorrelated combination of statistical and systematic 1-uncertainties.

the critical field Bc . (For our coupling B is only approx-

imately uniform over the system and B is the field av-eraged over the area of the BEC.) The inset shows Nv forboth values of R plotted versus B/0 = AqB/h,the vortex number for a system of area A = RxRywith the asymptotic vortex density, where Rx (Ry) isthe Thomas-Fermi radius along x (or y). Since the sys-

tem size, and thus B

c , are approximately independentof R, we expect this plot to be nearly independent ofRaman coupling. Indeed, the data for R = 5.85ELand 8.20EL only deviate for Nv < 5, likely due to theintricate dynamics of vortex nucleation[27].

Figure 3 illustrates a progression of images showingvortices nucleate at the systems edge, fully enter to anequilibrium density and then decay along with the atomnumber. The time scale for vortex nucleation dependsweakly on B, and is more rapid for larger B with morevortices: It is 0.3 s for vortex number Nv 8, and in-creases to 0.5 s for Nv = 3. For Nv = 1 (B near Bc ),the single vortex always remains near the edge of the

BEC. In the dressed state, spontaneous emission fromthe Raman beams removes atoms from the trap, caus-ing the population to decay with a 1.4(2) s lifetime, andthe equilibrium vortex number decreases along with theBECs area.

To verify the dressed state has reached equilibrium,we prepare nominally identical systems in two differentways. First, we vary the initial atom number and mea-sure Nv as a function of atom number N at a fixed holdtime th = 0.57 s. Second, starting with a large atomnumber, we measure both Nv and N, as they decrease

with th (Fig. 3). Figure 4 compares Nv versus N mea-sured with both methods, each at two detuning gradientscorresponding to fields B1 < B

2 . The data show Nv as afunction ofN is the same for these preparation methods,providing evidence that for th 0.57 s, Nv has reachedequilibrium. As the atom number N falls, the last vor-tex departs the system when the critical field increasing

with decreasing N surpasses the actual field.In conclusion, we have demonstrated optically synthe-

sized magnetic fields for neutral atoms resulting from theBerrys phase, a fundamental concept in physics. Thisnovel approach differs from experiments with rotatinggases, where it is difficult to add optical lattices and ro-tation is limited by heating, metastability, and the diffi-culties to add large angular momentum, preventing ac-cess to the quantum-Hall regime. A standout feature inour approach is the ease to add optical lattices. For ex-ample, the addition of a 2D lattice makes it immediatelyfeasible to study the fractal energy levels of the Hofs-tadter butterfly[28]. Further, a 1D lattice can divide the

BEC into an array of 2D systems normal to the field. Asuitable lattice configuration allows access to the 1quantum-Hall regime, with an ensemble of 2D systemseach with 200 atoms, and with a realistic kB20 nKinteraction energy.

We thank W.D. Phillips for discussions. This workwas partially supported by ONR, ARO with funds fromthe DARPA OLE program, and the NSF through theJQI Physics Frontier Center. R.L.C. acknowledges theNIST/NRC postdoctoral program and K.J.G. thanksCONACYT.


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Atom number N x103

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*


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Phys. Rev. A 79, 063618 (2009).[28] D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).[29] Y.-J. Lin, A. R. Perry, R. L. Compton, I. B. Spielman,

and J. V. Porto, Phys. Rev. A 79, 063631 (2009).[30] P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh,

and C. W. Gardiner, Advances in Physics 57, 363 (2008).

Methods Summary

A. Dressed state preparation

We create a 87Rb BEC in a crossed dipole trap[29],with N 4.7 105 atoms in |F = 1, mF = 1. Thequadratic Zeeman shift is = 0.61EL for Z/2 =gBB/h 2.71 MHz, where g is the Lande g-factor. Tomaintain = 0 at the BECs center as we ramp the fieldgradient b

, we change gBB0 by as much as 7EL. Simul-

taneously, we decrease the dipole beam power by 20%,producing our 40 Hz trap frequency along x. Addition-ally, the detuning gradient

y makes the scalar potentialV

anti-trapping along y, reducing fy from 70 Hz to 50 Hz

for our largest

. Spontaneous emission from the Ramanbeams decreases the atom number to N 2.5 105 forth = 0, with a condensate fraction of 0.85.

B. Numerical method

We compare our data to a finite temperature 2Dstochastic GPE (SGPE[30]) simulation including thedressed state dispersion E(kx, y) that depends on ythrough the detuning gradient . We evolve the time-dependent projected GPE

i(x, t)t

= P

Ei

x, y 2

2m2

y2+ g2D |(x, t)|2

(x, t)

.

P projects onto a set of significantly occupied modes,and g2D parameterizes the 2D interaction strength. TheSGPE models interactions between the highly-occupiedmodes described by and sparsely occupied thermalmodes with dissipation and an associated noise term. Weapproximately account for the finite extent along z bymaking g2D depend on the local 2D density . For low

temperatures this 2D model correctly recovers the 3DThomas-Fermi radii, and gives the expected 2D densityprofile. These quantitative details are required to cor-rectly compute the critical field or number for the firstvortex to enter the system, which are directly tied to the2D condensate area.

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