Resonant Einstein–de Haas Effect in a Rubidium Condensate

Krzysztof Gawryluk,1 Mirosław Brewczyk,1 Kai Bongs,2 and Mariusz Gajda3

1Instytut Fizyki Teoretycznej, Uniwersytet w Białymstoku, ulica Lipowa 41, 15-424 Białystok, Poland2Institut fur Laser-Physik, Universitat Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

3Instytut Fizyki PAN, Aleja Lotnikow 32/46, 02-668 Warsaw, Poland(Received 4 September 2006; published 25 September 2007)

We theoretically consider a spin polarized, optically trapped condensate of 87Rb atoms in F � 1. Weobserve a transfer of atoms to other Zeeman states due to the dipolar interaction which couples the spinand the orbital degrees of freedom. Therefore the transferred atoms acquire an orbital angular momentum.This is a realization of the Einstein–de Haas effect in systems of cold gases. We find resonances whichmake this phenomenon observable even in very weak dipolar systems, when the Zeeman energy differenceon transfer is fully converted to rotational kinetic energy.

DOI: 10.1103/PhysRevLett.99.130401 PACS numbers: 03.75.Mn, 03.75.Lm, 03.75.Nt

Magnetic effects in ultracold quantum gases have beensubject to intense theoretical and experimental studiesduring recent years. So far most of these investigationshave concentrated on short-range interactions as the domi-nant spin exchange process in spinor condensates of 23Na[1] and 87Rb [2–4]. These interactions result in rich mul-ticomponent physics as demonstrated by the observation ofphenomena like magnetic phases [1,3,4], coherent spindynamics [3–6], domain formation [7], and a magneticallytuned spin-mixing resonance [8]. Including magneticdipole-dipole interactions would further enhance the rich-ness of these systems and, in particular, their anisotropicnature is expected to add completely new aspects. Forrelatively weak dipolar interactions phenomena like theEinstein–de Haas effect [9], spontaneous magnetization[10,11], squeezing, and entanglement [11] have been pre-dicted. Most of these studies concentrate on the recentlyachieved case of a chromium Bose-Einstein condensate[12], as it is commonly believed that these effects arepractically unobservable in the widely available alkalicondensates due to the smallness of dipolar interactionsin these systems. However, as pointed out in [11], for 87Rbin the F � 1 state the size of the dipolar interactions ascompared to the spin-mixing part of the short-range inter-actions reaches 10%, such that dipolar effects might beobservable in this system.

In this Letter, we show that under the right conditions thedipolar interactions can even dominate the dynamics of a87Rb spinor condensate, making it a promising candidatefor the observation of the Einstein–de Haas effect [13].Our calculations demonstrate the existence of resonancesthat amplify the effect of dipolar interactions and can betuned by the magnetic field or by the trap geometry. Theyoccur when the Zeeman energy fits the rotational kineticenergy per particle. The resonances we find explore a newregime in comparison with that considered in Ref. [9] for a52Cr condensate, where the dipolar energy (not a kineticone) is related to the Zeeman energy.

In the second quantization notation, the Hamiltonian ofthe system we investigate is given by

H �Zd3r

� yi �r�

��

@2

2mr2 �Uext�r�

� i�r�

� � yi �r�BFi;j j�r� �c0

2 yj �r�

yi �r� i�r� j�r�

�c2

2 yk �r�

yi �r�FijFkl j�r� l�r�

��

1

2

�Zd3rd3r0 yk �r�

yi �r0�Vdij;kl�r� r0� j�r0� l�r�;

(1)

where repeated indices (each of them going through thevalues �1, 0, and �1) are to be summed over. The fieldoperator i�r� annihilates an atom in the hyperfine statejF � 1; ii at point r. The first term in (1) is the single-particle Hamiltonian (H0) that consists of the kinetic en-ergy part (with m being the mass of an atom) and thetrapping potential Uext�r�. The second term describes theinteraction with the magnetic field B with � being thegyromagnetic coefficient which relates the effective mag-netic moment with the hyperfine angular momentum (� ��F). The terms with coefficients c0 and c2 describe thespin-independent and spin-dependent parts of the contactinteractions, respectively—c0 and c2 can be expressedwith the help of the scattering lengths a0 and a2 whichdetermine the collision of atoms in a channel of total spin 0and 2. One has c0 � 4�@2�a0 � 2a2�=3m and c2 �4�@2�a2 � a0�=3m [14], where a0 � 5:387 nm and a2 �5:313 nm [15]. F are the spin-1 matrices. Finally, the lastterm describes the magnetic dipolar interactions. The in-teraction energy of two magnetic dipole moments �1 and�2 positioned at r and r0 equals

Vd ��1�2

jr� r0j3� 3��1�r� r0����2�r� r0��

jr� r0j5(2)

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0031-9007=07=99(13)=130401(4) 130401-1 © 2007 The American Physical Society

and since � � �F one has Vdij;kl�r� r0� � �2FijFkl=jr�r0j3 � 3�2�Fij�r� r0���Fkl�r� r0��=jr� r0j5.

The equation of motion reads

i@@@t

1

0

�1

0B@

1CA � �H c �H d�

1

0

�1

0B@

1CA; (3)

where the operator H c originates from the the first fourterms in the Hamiltonian Eq. (1) whereas H d correspondsto the dipole-dipole interactions [the last term in Eq. (1)].The diagonal part of H c is given by H c11 � H0 � �c0 �

c2� y1 1 � �c0 � c2�

y0 0 � �c0 � c2�

y�1 �1, H c00 �

H0 � �c0 � c2� y1 1 � c0

y0 0 � �c0 � c2�

y�1 �1,

H c�1�1 �H0��c0� c2� y1 1��c0� c2�

y0 0��c0�

c2� y�1 �1. The off-diagonal terms that describe the

collisions not preserving the projection of spin of eachatom (although the total spin projection is conserved)equal H c10 � c2

y�1 0, H c0�1 � c2

y0 1. Moreover,

H c1�1 � 0. On the other hand, for the H d term onehas H dij �

Rd3r0 yn �r0�Vdij;nk k�r

0�. This term is respon-sible for the change of total spin projection of collidingatoms. It turns out that when two atoms interact the totalspin projection (�MF) can change at most by 2. In par-ticular, the diagonal elements of H d lead to the processeswith �MF � 1. In addition to such processes, the off-diagonal terms of H d introduce the interaction thatchanges the spin projection �MF by 2. It happenswhen both atoms initially in the same state go simulta-neously to the nearest (in a sense of magnetic number mF)state or in the case when atoms in different but neighboringcomponents transfer to the states shifted in number mF by�1 or �1. There is no way for the atom to be transferreddirectly from the mF � 1 to the mF � �1 state; therefore,the populating of the mF � �1 component is a secondorder process.

Hence, the dipolar interaction does not conserve theprojection of total spin of two interacting atoms, nor isthe projection of total orbital angular momentum preserved[see (4)]. However, the dipolar interaction couples the spinand the orbital motion of atoms as revealed by the lastrelation in (4)

�Vd; F1z � F2z� � 0; �Vd; L1z � L2z� � 0;

�Vd; L1z � L2z � F1z � F2z� � 0:(4)

Therefore, going to mF � 0;�1 states atoms acquire theorbital angular momentum and start to circulate around thecenter of the trap. This is the realization of the famousEinstein–de Haas effect in cold gases.

To solve Eq. (3) we neglect the quantum fluctuations andreplace the field operator i�r� by an order parameter i�r�for each component and apply the split-operator method.All integrals appearing in H dij are the convolutions andwe use the Fourier transform technique to calculate them.

To find analytical formulas for the Fourier transforms ofthe components of the convolutions that do not changeduring the evolution we apply the regularization proceduredescribed in Ref. [16].

The gyromagnetic coefficient for 87Rb atoms in an F �1 hyperfine state is positive and equals � � 1

2�B=@. Weprepare an initial state of the condensate as the one with allmagnetic moments aligned along the magnetic field; i.e.,all atoms are inmF � 1 component. To this end, we run themean-field version of Eq. (3) in imaginary time while themagnetic field is turned on (and equal to B � 0:73 mG fora spherically symmetric trap with the frequency ! �2�� 100 Hz). Then we reverse the direction of the mag-netic field and look for the transfer of atoms to otherZeeman states.

Our starting condition (all atoms in mF � 1 state) sup-presses the short-range spin dynamics and initially themF � 1 state is depleted only due to the dipolar interac-tion. Usually we observe a small number of atoms goingfrom the Zeeman state mF � 1 to the mF � 0;�1 states.However, on resonance (see Fig. 1) the transfer to the otherstates can be of the order of the initial population of mF �1 component. This transfer is as large as in the case ofchromium condensate [9] despite the fact that the dipolarenergy (�2n, where n is the atomic density) is approxi-mately 100 times smaller. The only difference is that thetime scale corresponding to the maximal transfer is about100 ms, i.e., 100 times longer than for chromium. This canbe understood as follows. For 87Rb the dipolar energy is the

0 0.1 0.2 0.3 0.4TIME s

0

2

4

6

AT

OM

S10

3

mF 1

0 0.1 0.2 0.3 0.4TIME s

0

1

2

3

L Z10

0 0.1 0.2 0.3 0.4TIME s

0123456

AT

OM

S10

4 mF 0

0 0.1 0.2 0.3 0.4TIME s

1.4

1.6

1.8

2.0

AT

OM

S10

5

mF 1

FIG. 1 (color online). Transfer of atoms to mF � 0;�1Zeeman states as a function of time. Initially, N � 2� 105

atoms were prepared in the mF � 1 component in a sphericallysymmetric trap with the frequency ! � 2�� 100 Hz. Theresidual magnetic field equals �0:029 mG (solid black lines,on-resonance case) and B � �0:015 mG, �0:036 mG (bluedotted lines and red dashed lines, respectively, off-resonancecase). The lower panel shows the number of atoms in mF � 1state and the time dependence of the orbital angular momentumper atom. The maximum of the latter (0:35@) is consistent withthe transfer of 30% of atoms to mF � 0 state (where a singlyquantized vortex is formed) and 3% of atoms to mF � �1 statewith the doubly quantized vortex.

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smallest energy of the problem and is only a perturbationas compared to the kinetic rotational energy and theZeeman energy. An efficient population transfer frommF � 1 to mF � 0 due to the dipolar interactions is onlypossible when the total energies (the sum of mean-field,trap, kinetic, and Zeeman energies) in these states areapproximately equal. Figure 2(a) clearly shows that thetotal energy tends to equalize on resonance, which is nottrue in the off-resonant case. Numerics also shows[Fig. 2(b)] that the above condition can be fulfilled onlywhen the Zeeman energy fits the rotational kinetic energy:�Bres � �rot. It means that the resonant magnetic field isinversely proportional to the magnetic moment of an atom,

Bres / 1=�: (5)

Surprisingly, the smaller the atomic magnetic moment thelarger the value of the resonant magnetic field. For chro-mium condensate, however, the dipolar energy is largerthan the kinetic energy. Therefore, in this case the reso-nance condition should be derived by relating the Zeemanand the dipolar (not the kinetic one) energies, i.e., �Bres ��2n [9]. This results in a condition Bres / � which differsqualitatively from (5).

The maximal transfer is reached at a time which is of theorder of the characteristic time scale determined by thedipolar interactions (@=�2n). Since the magnetic momentof the 87Rb atom is 12 times smaller than that of 52Cr wehave to wait hundreds of milliseconds (not a fraction of amillisecond as in Ref. [9]) to see the action of resonance.

Figure 2 illustrates the ideas just discussed. In the leftframe the total energies of the mF � 1; 0 components areplotted as a function of time both in on- and off-resonancecases showing that the resonances we find are dynamicalphenomena. The transfer gets maximal when the energiesapproach each other (perhaps crossing both curves wouldrequire the dynamical tuning of the resonance by changingthe magnetic field). On the contrary, almost no transfer ofatoms occurs when the energy curves keep away.Simultaneously, the right frame proves that on resonancethe Zeeman energy (in fact, together with the kinetic

energy) is transferred to the rotational kinetic energy ofatoms in the mF � 0 component.

Huge transfer of atoms to mF � 0;�1 states is therealization of the Einstein–de Haas effect in cold gases.Numerical analysis of phases of spinor components showsthat the vortices are generated in mF � 0;�1 spin statesand atoms in mF � 0;�1 rotate around the quantizationaxis. In mF � 0 and mF � �1 components singly anddoubly quantized vortices are formed, respectively, as aresult of total angular momentum conservation. The den-sity in these states is fragmented and the number of ringsresults from the symmetry of dipolar interaction, Fig. 3.Similar fragmentation was already predicted in the case of52Cr condensate in Ref. [9].

Figure 4 (upper frame) shows the position and the widthof the resonance displayed in Fig. 1. Similar behavior isobserved when the value of the reversed magnetic field iskept constant and the trap geometry is changed (lowerframe). Here, the maximal transfer of atoms is obtainedin a cigar trap with the aspect ratio!�=!z � 4 with almost50% efficiency at B � �0:073 mG. The inset shows theresonance at an experimentally easier to control value ofmagnetic field B � 0:3 mG but still detectable number ofatoms in mF � 0 state. Note that in all these cases themaximal atomic density is of the order of 1014 cm�3

making the three-body losses low enough and hence allow-ing the observation of population transfer on a time scale ofthe order of hundreds of milliseconds. To understandquantitatively the resonance we start from the conditiondiscussed earlier: �Bres � Erot=N0, where Erot is the rota-tional energy of the mF � 0 component which is assumedto be a singly quantized vortex, given within the Thomas-Fermi approximation by 0��;�; z� � f���m!2��2 �

z2�=2� @2=�2m�2��=c0g

1=2ei�. Here, the chemical poten-tial � is obtained by the requirement that the number ofatoms in the 0 state equals N0. One can tune to theresonance in two ways: (i) by adjusting the magnetic fieldB, and (ii) by changing the trap geometry that influencesthe rotational energy entering the resonance condition andkeeping the magnetic field constant. The curve resultingfrom the condition �Bres � Erot=N0 in the case of thespherically symmetric trap is plotted in Fig. 5. The resonant

FIG. 2 (color online). Total energy for mF � 1; 0 componentsas a function of time (left frame). All parameters are the same asin Fig. 1. The resonance happens for B � �0:029 mG (solidblack lines) whereas the off-resonance cases are represented byB � �0:015 mG (blue dotted lines) and B � �0:036 mG (reddashed lines). The right frame shows �E � �E�1

kin=N�1 ��B��rot�=�rot, where �rot � Erot=N0.

)b()a(

FIG. 3. Density in the xz plane (the z axis goes vertically) ofmF � 0 (a) and mF � �1 (b) spin components in on-resonancecase at 140 ms.

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magnetic field is uniquely related to the number of atoms inthe mF � 0 component. However, this number depends onthe initial number of atoms in the mF � 1 state and can befound only numerically. To verify the resonance conditionwe compare the numerical results (marked by solid circlesand with additional information regarding the initial num-ber of atoms in the mF � 1 component) with the Thomas-Fermi results. The agreement is good, for example, wheninitially one has N1 � 2� 105 atoms in the mF � 1 com-ponent (in this case the maximal transfer of atoms to themF � 0 state equals 6� 104). The solid line in Fig. 5 givesfor the value of N0 � 6� 104 the value of the resonantmagnetic field 0:03 mG which is very close to thenumerical value (see upper frame in Fig. 4). For othersystems, e.g., 52Cr, the condition �Bres � Erot=NmF

sug-gests that the value of the resonant magnetic field is even10 times smaller since �Cr=�Rb � 12 and Erot=N�2 forchromium looks similar to Erot=N0 in the rubidium case.

In conclusion, we have shown the existence of dipolarresonances in rubidium spinor condensates. The reso-nances occur when the Zeeman energy of atoms in themF � 1 component, while transferring to themF � 0 state,is fully converted to the rotational kinetic energy. This is sofar an unexplored regime. Symmetries of the dipolar inter-

action force the atoms in mF � 0;�1 states to circulatearound the quantization axis and form singly and doublyquantized vortices, respectively. Therefore, dipolar reso-nances is a route to the observation of the Einstein–de Haas effect (as well as other phenomena related to thedipolar interaction) in weak dipolar systems.

We are grateful to J. Kronjager, J. Mostowski, andK. Rzazewski for helpful discussions. M. B. and M. G.acknowledge support by the Polish KBN GrantNo. 1 P03B 051 30. K. G. thanks the Polish Ministry ofScientific Research Grant Quantum Information andQuantum Engineering No. PBZ-MIN-008/P03/2003.

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(2004).[4] M.-S. Chang et al., Phys. Rev. Lett. 92, 140403 (2004).[5] M.-S. Chang et al., Nature Phys. 1, 111 (2005).[6] J. Kronjager et al., Phys. Rev. A 72, 063619 (2005).[7] L. E. Sadler et al., arXiv:cond-mat/0605351.[8] J. Kronjager et al., Phys. Rev. Lett. 97, 110404 (2006).[9] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. 96,

080405 (2006).[10] R. Cheng, J.-Q. Liang, and Y. Zhang, J. Phys. B 38, 2569

(2005).[11] S. Yi and H. Pu, Phys. Rev. A 73, 023602 (2006).[12] L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404 (2006).[13] A. Einstein and W. J. de Haas, Verh. Dtsch. Phys. Ges. 17,

152 (1915).[14] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998); T. Ohmi and

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(2002).[16] K. Goral and L. Santos, Phys. Rev. A 66, 023613 (2002).

0 200 400 600 800ATOMS IN mF 0 103

0.02

0.03

0.04

0.05

0.06

Bm

Gs

N 1 2x105

N 1 8x105

N 1 2x106

FIG. 5. Comparison between numerics (solid circles) and theThomas-Fermi approximation. Solid line indicates the value ofthe magnetic field at resonance (for spherically symmetric trap)as a function of number of atoms in the mF � 0 state.

0.04 0.03 0.02 0.01 0B mG

0102030405060

mF 0 N 1 2x105

ωρ ωz 1ωz 2 100Hz

mF 1AT

OM

S [1

03 ]A

TO

MS

[103 ]

2.5 5 7.5 10 12.5 15 17.5ρ z

0

100

200

300

400mF 0

mF 1

30 40 50 600246 B 0.3 mG

mF 0

ω ω

FIG. 4. Maximal transfer of atoms to mF � 0;�1 states as afunction of the residual magnetic field (upper frame) and the trapgeometry (lower frame). For lower frame !z � 2�� 100 Hz,B � �0:073 mG, and N�1 � 8� 105. Inset shows the reso-nance at B � �0:3 mG for !z � 2�� 20 Hz and N�1 � 105.

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