PHYSICAL REVIEW A 83, 063617 (2011)

Tunable dipolar resonances and Einstein-de Haas effect in a 87Rb-atom condensate

Tomasz Swisłocki,1 Tomasz Sowinski,1 Joanna Pietraszewicz,1 Mirosław Brewczyk,2

Maciej Lewenstein,3,4 Jakub Zakrzewski,5,6 and Mariusz Gajda1

1Instytut Fizyki PAN, Al. Lotnikow 32/46, PL-02-668 Warszawa, Poland2Wydział Fizyki, Uniwersytet w Białymstoku, ul. Lipowa 41, PL-15-424 Białystok, Poland

3ICFO, Institut de Ciences Fotoniques, Parc Mediterrani de la Tecnologia, E-08860 Castelldefels, Barcelona, Spain4ICREA, Institucio Catalana de Recerca i Estudis Avancats, E-08010 Barcelona, Spain

5Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiellonski, ul. Reymonta 4, PL-30-059 Krakow, Poland6Mark Kac Complex Systems Research Center, Jagiellonian University, ul. Reymonta 4, PL-30-059 Krakow, Poland

(Received 8 February 2011; published 14 June 2011)

We theoretically study a spinor condensate of 87Rb atoms in a F = 1 hyperfine state confined in an opticaldipole trap. Putting initially all atoms in an mF = 1, component we observe a significant transfer of atoms toother, initially empty Zeeman states exclusively due to dipolar forces. Because of conservation of a total angularmomentum the atoms going to other Zeeman components acquire an orbital angular momentum and circulatearound the center of the trap. This is a realization of the Einstein-de Haas effect in a system of cold gases. Weshow that the transfer of atoms via dipolar interactions is possible only when the energies of the initial and thefinal sates are equal. This condition can be fulfilled utilizing a resonant external magnetic field, which tunesenergies of involved states via the linear Zeeman effect. We found that there are many final states of differentspatial density, which can be tuned selectively to the initial state. We show a simple model explaining highselectivity and controllability of weak dipolar interactions in the condensate of 87Rb atoms.

DOI: 10.1103/PhysRevA.83.063617 PACS number(s): 03.75.Mn, 05.30.Jp, 75.45.+j

I. INTRODUCTION

Experimental achievement of Bose-Einstein condensationin a gas of 52Cr atoms [1,2] has launched a huge interestin studying properties of ultracold dipolar systems [3,4].Although polar molecules with their large electric dipolemoments seem to be perfect candidates to investigate dipolareffects, such systems have not been condensed so far. Alsothe necessity of an external field inducing the dipole inthese systems might suppress some intrinsic interaction effects(for instance, the electric dipole-dipole interactions do notconserve the total angular momentum, i.e., the sum of spin andorbital angular momentum). On the other hand, experimentswith chromium condensates have already showed spectacularfeatures characteristic for dipolar interactions. For example,in Ref. [5] the suppression (and even complete inhibition)of inversion of ellipticity during expansion of a cloud ofchromium atoms was observed. In Ref. [6], on the otherhand, the collapse dynamics of 52Cr condensate was studied.The scattering length was decreased by means of a Feshbachresonance below the critical value above which the condensateis stable. Then the system was held in a trap for some time, andeventually the trap was switched off. The specific cloverleafpatterns for the density caused by anisotropic dipole-dipoleinteractions were observed when the system was imaged aftertime of flight.

Some authors suggested, however, that the magnetic dipolarinteractions might already lead to observable effects incondensates of alkali-metal atoms [7–13], whose magneticdipole moment is an order of magnitude lower than that ofchromium atoms. Indeed, in experimental work of Ref. [14] itis demonstrated that a spin-1 87Rb spinor condensate exhibitsdipolar properties. The authors of Ref. [14] observe thespontaneous decay of helical spin textures toward a spatiallymodulated structure of spin domains and relate this effect to

the presence of magnetic dipolar interactions in the system.This claim is supported by an observation that the modulatedphase is suppressed when the dipolar interactions are reduced.There are, however, difficulties in understanding the results ofthis experiment within the frame of mean-field and Bogoliubovtheories at zero temperature [15].

One of the most spectacular manifestations of dipolarinteractions is the Einstein-de Haas effect [16], studied recentlytheoretically in gaseous systems of chromium [17,18] andrubidium condensates [9,19]. Interestingly, this phenomenonmight occur even in very weak dipolar systems. To observe theEinstein-de Haas effect in alkali-metal-atom condensates onehas to tune an external magnetic field to some resonant valueor for a fixed magnetic field to modify appropriately trappingparameters. Only when the Zeeman energy fits the energygap between different single-particle spatial states associatedwith mF = 1 and mF = 0 spin states is the transfer of atomsresonantly amplified.

We note that the resonant magnetic field is typically of theorder of tens or hundreds of microgauss, making theobservation of the Einstein-de Haas effect difficult at present.Since dipole-dipole interactions are very weak, the resonantcurves are also very narrow. This means that experimentalrealization needs high precision. On the other hand, itguarantees that resonances are highly selective. By choosingan appropriate value of the magnetic field one can tunethe transition of atoms to a particular spatial state. Indeed,controlling dipolar interactions is the crucial point in workingwith dipolar systems. Such a control has been recentlyimposed in a chromium condensate [20]. It was shown thatthe external static magnetic field strongly influences thedipolar relaxation rate: There exists a range of magneticfield intensities where this relaxation rate is strongly reduced,allowing for the accurate determination of S = 6 scattering

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TOMASZ SWISŁOCKI et al. PHYSICAL REVIEW A 83, 063617 (2011)

length for chromium atom. In Ref. [21], on the other hand,a two-dimensional optical lattice and a static magnetic fieldare used to control the dipolar relaxation into higher latticebands. In this work evidence for the existence of the relaxationthreshold with respect to the intensity of the magnetic field isgiven. As the authors of Ref. [21] claim, such an experimentalsetup might lead to the observation of the Einstein-de Haaseffect.

In this article, first, we report on the existence of variousdipolar resonances in rubidium spinor condensates. As com-pared with our previous work [9], we demonstrate that, infact, many resonances are possible depending on the value ofthe external magnetic field. Second, we reveal the physics ofdipolar resonances. Based on this understanding we explainthe geometrical structures of states populated via dipolarresonances.

The paper is organized as follows. In Sec. II we describethe system under consideration. In particular, we discuss thecontact and dipolar interactions (see Appendix A for furtherdetails). Section III presents numerical results regarding theresonances and the Einstein-de Haas effect in a rubidium con-densate, and Sec. IV introduces a simple model explaining theorigin of the resonances as well as properties of spatial statespopulated via dipolar interactions. We end with conclusions inSec. V. Appendix B offers many technical details implementedin a numerical procedure used to obtain the results presentedin this paper as well as in our previous work [9].

II. DESCRIPTION OF THE SYSTEM

We investigate a spinor condensate of atoms in theF = 1 hyperfine state. In addition to the dominating binarycontact interactions, we consider long-range dipolar magneticinteractions. In the formalism of a second quantization, theHamiltonian of the system is given by

H =∫

d3r

{ψ

†i (r)

[− h2

2m∇2 + Vext(r)

]ψi(r)

− γ ψ†i (r) B · Fij ψj (r) + c0

2ψ

†j (r)ψ†

i (r)ψi(r)ψj (r)

+ c2

2ψ

†k (r)ψ†

i (r) Fij · Fkl ψj (r)ψl(r)

}

+ 1

2

∫d3r d3r ′ψ†

k (r)ψ†i (r ′)V d

ij,kl(r − r ′)ψj (r ′)ψl(r) ,

(1)

with repeated indices being summed over the values +1, 0, and−1. The field operator ψi(r) [ψ†

i (r)] annihilates (creates) anatom in the hyperfine state |F = 1,i〉 at point r . The first row in(1) represents the single-particle Hamiltonian that consists ofthe kinetic energy part and the trapping potential assumed to beindependent of the hyperfine state. The first term in the secondrow describes the interaction with the magnetic field B, whichis in our case always directed along the z axis. The magneticmoment of the atom µ is proportional to the dimensionlessspin-1 matrices F, µ = µF. Effective magnetic moment of theatom µ differs from the Bohr magneton by Lande giromagneticfactor γ , µ = γµB . The terms with coefficients c0 and c2

describe the spin-independent and -dependent parts of the

contact interactions, respectively. Coefficients c0 and c2 canbe expressed with the help of the scattering lengths a0 anda2, which determine the collision of atoms in a channel oftotal spin 0 and 2. One has c0 = 4πh2(a0 + 2a2)/3m andc2 = 4πh2(a2 − a0)/3m (m is the mass of an atom) [22], wherea0 = 5.387 nm and a2 = 5.313 nm [23]. Finally, the last termrepresents the magnetic dipolar interactions and originatesfrom the interaction energy of two magnetic dipole momentsµ1 and µ2:

V d = µ1 ·µ2

|r − r ′|3 − 3[µ1 ·(r − r ′)] [µ2 ·(r − r ′)]

|r − r ′|5 , (2)

where r and r ′ denote the positions of the dipoles.The equation of motion reads

ih∂

∂t

⎛⎜⎝

ψ1

ψ0

ψ−1

⎞⎟⎠ = (Hsp + Hc + Hd )

⎛⎜⎝

ψ1

ψ0

ψ−1

⎞⎟⎠ , (3)

where Hsp = − h2

2m∇2 + Vext − µmF B represents the single-

particle part (including the interaction with the externalmagnetic field); Hcc and Hd originate from the two-particleinteractions and are discussed in detail in Appendix A. Herewe just state that the diagonal part of Hc is responsible forelastic collisions, whereas other elements of Hc allow for thechange of spin projections of individual atoms, preserving,however, the projection of total spin.

The dipolar part of Eq. (3), Hd , is more complex. To betterunderstand the dipolar processes it is convenient to rewrite thedipolar interactions in the following form [24]:

V d ∝µ=2∑

µ=−2

Y �2µ(r)�2µ, (4)

where Y2µ(r) (with r denoting a unit vector in the directionof relative position of two atoms) is a spherical harmonics ofrank-2 and �2µ defined as

�2,0 = −√

3

2(F1zF2z − F1 · F2/3),

�2,±1 = ±1

2(F1zF2± + F1±F2z), (5)

�2,±2 = −1

2F1±F2±

is a rank-2 spherical tensor built of atomic spin operators(including raising and lowering operators F1± and F2±). Itis clear from (4) that when two atoms interact the total spinprojection MF (as well as the total spin itself) can change atmost by 2, whereas the spin projection of individual atoms (5)changes maximally by 1 not by 2. Therefore, the atom cannotbe transferred directly from the mF = 1 to −1 component(populating the mF = −1 state is a second-order process).When �MF = ±2, the last row in (5) shows that the projectionof the spin of each atom changes by +h or −h; i.e., both atomsinitially in the same state go simultaneously to the nearest (ina sense of magnetic number mF ) state, or when atoms are indifferent but neighboring components they are transferred tothe states shifted in number mF by +1 or −1. In addition tothe processes just described there are the atomic collisions in

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which only one atom is transferred to the other Zeeman state[�MF = ±1; see the middle row in (5)] or the collisions thatdo not change the spin projection [�MF = 0; the first rowin (5)]. In the latter case, however, the process transferringone atom from mF = 0 to 1 and the other from mF = 0 to−1 states is still allowed just like it is allowed for contactinteractions. According to (4) in each case the change of thetotal spin projection is accompanied by an appropriate changeof a relative orbital angular momentum of colliding atoms.

Hence, the dipolar interactions do not conserve the pro-jection of total spin of two interacting atoms. Nor is theprojection of total orbital angular momentum preserved [see(6)]. However, the dipolar interactions couple the spin and theorbital motion of atoms (with regard to any axis) as revealedby the last relation in (6):

[V d,F1z + F2z] �= 0, [V d,L1z + L2z] �= 0,(6)

[V d,L1z + L2z + F1z + F2z] = 0 .

Therefore, going to mF = 0 and −1 states atoms ac-quire the orbital angular momentum and start to circulatearound the center of the trap. This is the essence of the famousEinstein-de Haas effect, and the discussion in Sec. III suggestsits possible realization in cold gases.

III. DIPOLAR RESONANCES AND THE EINSTEIN-DEHAAS EFFECT

In this section we investigate the properties of a dipolarspinor condensate of 87Rb atoms in the F = 1 hyperfine state(γ = 1/2) by solving the mean-field version of Eq. (3):

ih∂

∂t

⎛⎝ψ1

ψ0

ψ−1

⎞⎠ = (Hsp + Hc + Hd )

⎛⎝ψ1

ψ0

ψ−1

⎞⎠ , (7)

with ψi being the wave function (referred sometimes also toas an order parameter) of the ith spin component. Usuallydipolar properties of rubidium condensates are neglected.This is because the magnetic moment of rubidium atoms issmall. Comparing the typical energy related to the dipolarinteractions (given by µ2n, where µ is the magnetic momentof 87Rb atom and n is the atomic density) to the characteristiccontact interaction energy gn (g determines the strength of thecontact interactions), one obtains for the maximally stretchedF = 1 hyperfine state (i.e., for the spin-polarized case whereg = 4πh2a2/m) of 87Rb the ratio of the order of 10−4. Thisis two orders of magnitude lower than the correspondingratio calculated for condensate of chromium atoms, which iscommonly considered as a dipolar condensate. In this sectionwe show, however, that for a rubidium condensate there existresonances that significantly enhance the effect of dipolarinteractions and might lead to the manifestation of variousproperties in as strong a way as in the case of chromiumcondensate.

The numerical experiment we perform consists of preparingthe condensate initially in an mF = 1 Zeeman state (all atomicmagnetic moments aligned along the magnetic field). It is doneby evolving the mean-field version of Eq. (3) in imaginarytime at the presence of an external magnetic field that is largeenough to enforce almost all atoms to populate the mF = 1

0 0.05 0.1 0.15 0.2 0.25 0.3MAGNETIC FIELD (mG)

0

0.05

0.1

0.15

0.2

0.25

0.3

RE

LAT

IVE

PO

PU

LAT

ION

0 0.1 0.2 0.30

0.04

0.08

0.12

FIG. 1. Population of mF = 0 (main frame) and mF = −1 (inset)components as a function of the magnetic field (note that the magneticfield is pointed toward the negative z axis, and only its values aredepicted here). We observe many magnetic resonances correspondingto different values of the magnetic field. The initial number of atomsin the mF = 1 state is equal to N+1 = 5 × 104, and the aspect ratio isβ = ωz/ωr = 1/4. Maximal transfer is reached at t ∼ 0.1 s for eachresonance.

component. Next, we change both the direction and the valueof the magnetic field. What we usually observe is the transferof a small number of atoms to mF = 0 and −1 components.We would like to stress that in our numerical experimentthe initial condition excludes the short-range spin dynamics,and the depletion of mF = 1 state occurs only due to dipolarinteractions.

However, there exist particular values of the magnetic field(for example, B ≈ −0.11 mG for the axially symmetric trapwith ωz = 2π × 100 Hz and ωr = 2π × 400 Hz and initialnumber of atoms N = 5 × 104, as shown in Fig. 1) whenthe observed transfer is large: on the order of 10% of initialnumber of atoms in the mF = 1 state. The maximal transferhappens after the time on the order of 100 ms (see Fig. 2)and is consistent with the characteristic time scale related todipolar interaction, which is h/µ2n. This formula tells us thatincreasing the atomic magnetic moment (for example, by goingfrom rubidium to chromium condensate), the time neededfor reaching the maximal transfer gets shorter. Indeed, forchromium condensate (with 12 times larger magnetic momentin comparison with 87Rb atoms in F = 1 state) this time is onthe order of 1 ms [17]. Similar behavior is visible for largermagnetic fields.

0 0.1 0.2 0.3 0.4 0.5TIME (s)

0

0.05

0.1

0.15

0.2

0.25

0.3

RE

LAT

IVE

PO

PU

LAT

ION

FIG. 2. Population of mF = 0 (black) and mF = −1 (gray)components as a function of time for the first resonance. The magneticfield is equal to B = −0.12 mG.

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X µm

Yµm

3 0 33

0

3

3 0 33

0

3

3 0 33

0

3

3 0 33

0

3

3 0 33

0

3

3 0 33

0

3

FIG. 3. Phase (upper panel) and density (lower panel) of mF =+1,0,−1 components (left, middle, and right columns, respectively)in the xy plane for N+1 = 5 × 104 and β = 1/4. The data correspondto the first maximum in the resonance structure (Fig. 1). The magneticfield is equal to B = −0.12 mG. N0 = 15 000 and N−1 = 6000 atomsboth at their maximal transfers.

An interesting observation is made when looking at thephase and the density of spinor components for all resonancesthat we found (Figs. 3 and 4). First, the inspection of the phaseof the wave functions of mF = 0, − 1 components proves theformation of quantized vortices in these components. For themF = 0 state the phase of the order parameter winds up by2π (upper middle frame in Fig. 3), whereas in the mF = −1component (upper right frame in Fig. 3) the phase winds upby 4π . At the same time the orbital angular momentum peratom equals h and 2h in the mF = 0 and −1 components,respectively. The circulation induced in the mF = 0 and −1states is, in fact, the realization of the Einstein-de Haas effectin cold gases. At the same time the densities exhibit nontypicalpatterns. The density of the mF = 0 state consists of an evennumber of rings when looking from a side (Fig. 4, upperpanel), whereas in the case of the mF = −1 component thenumber of rings is odd (Fig. 4, lower panel). The explanation

FIG. 4. Density of mF = 0 (upper panel) and mF = −1 (lowerpanel) components in the xz plane for N+1 = 5 × 104 atoms andβ = 1/4. Successive frames show density patterns characteristic forresonances depicted in Fig. 1. Note the increasing number of ringswith the order of the resonance as well as the presence of even (odd)number of rings in the mF = 0 (mF = −1) component.

of this property will be given in Sec. IV; here we present justarguments showing that the number of rings in componentsis strictly related to the channel the atoms choose when theycollide.

Since the number of rings in mF = 0 component is even(Fig. 4, upper panel), in the simplest case its wave functionfulfills the condition ψ0(r) ∼ Ylm, where the difference l − m

is odd. The dipole-dipole interactions are small and can betreated as a perturbation. Therefore, the wave function of themF = −1 component is determined by nonzero amplitudescorresponding to processes in which only one atom changes itsspin projection upon the collision (such amplitudes are givenby 〈Yl′m′ |Y21|Ylm〉) and processes when both atoms changetheir spin projections (here amplitudes are 〈Yl′m′ |Y22|Ylm〉).In the first case the amplitudes differ from zero only whenm′ = 1 + m and l′ = l − 2, l, l + 2 which means that l′ − m′is always even. Therefore, the density of the mF = −1 stateexhibits an odd number of rings. However, in the second casel′ − m′ is odd, which implies an even number of rings in themF = −1 component, which is not the case (see Fig. 4, lowerpanel). Evidently the second channel is somehow closed.

As was already mentioned in Sec. I, the dipolar resonancescan be tuned not only by the magnetic field, but also bythe trap geometry. In order to demonstrate that, we haveperformed calculations for the spherically symmetric trap.Two resonances found in this case are shown in Fig. 5.The properties of the first resonance (i.e., the one for lowermagnetic field) resemble those already discussed for axiallysymmetric trap. Looking, for example, at the densities in thexz planes, we see two and three rings in the mF = 0 andmF = −1 components, respectively. However, the situationchanges for the second resonance. Here we observe inner ringsstructures for both components (Fig. 6). This is a signature thatradial excitations as well as axial ones start to play a role. Theexplanation of this is given in Sec. IV.

IV. ORIGIN OF DIPOLAR RESONANCES

To understand the origin of dipolar resonances let us firstconsider the collision of two atoms, both initially in the F = 1

0 0.05 0.1 0.15 0.2 0.25MAGNETIC FIELD mG

0

0.05

0.1

0.15

0.2

0.25

RE

LAT

IVE

PO

PU

LAT

ION

0 0.05 0.1 0.15 0.2 0.250

0.001

0.002

0.003

0.004

FIG. 5. Population of the mF = 0 (main frame) and mF = −1(inset) components as a function of the magnetic field for the spher-ically symmetric trap with ω = 2π × 100 Hz. We observe magneticresonances corresponding to different values of the magnetic field.Initial number of atoms in the mF = 1 state is equal to N+1 = 5 × 104.Maximal transfer is reached at t ∼ 0.3 s for each resonance.

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X µm

Yµm

Zµm

7 0 77

0

7

7 0 77

0

7

7 0 77

0

7

7 0 77

0

7

7 0 77

0

7

7 0 77

0

7

FIG. 6. Phase (left column) and density (middle and rightcolumns for xy and xz planes, respectively) in the mF = 0 (upperpanel) and mF = −1 (lower panel) components corresponding to thesecond resonance in Fig. 5 at maximal population. Initial number ofatoms is N+1 = 5 × 104, B = −0.16 mG, and the trap is sphericallysymmetric with frequency ω = 2π × 100 Hz.

and mF = +1 internal state and the ground state of the axiallysymmetric harmonic oscillator:

ϕ0(r) ∼ e−(x2+y2+βz2)/2, (8)

with corresponding single-particle energy E0 = 1 + β/2.Here β = ωz/ω⊥ is the aspect ratio of the harmonic trap. Theoscillatory units based on the radial frequency ω⊥ are usedthroughout. We use a basis of single-particle states of harmonicoscillator and assume that contact repulsive interactions donot modify these states substantially. This assumption can bejustified if occupation of single-particle states is small. Directevaluation of the two-body dipolar matrix elements leads tothe following possible outcomes of the collision:

1. No atom changes its spin projection. In such a case bothatoms stay in the initial state (8). The elastic dipolar collisionscontribute effectively to the contact interactions.

2. Both atoms flip their spin projection by one. To ensureconservation of total angular momentum, the two-atom systemmust gain two quanta of orbital angular momentum. Thisangular momentum can be shared by both atoms, and theneach one goes to the state with mF = 0 and orbital angularmomentum mL = 1, or only one atom gains the angularmomentum, i.e., it goes to the state with mF = 0 and mL = 2.The second atom remains in the mL = 0 state.

Evidently, in our numerical calculations we do not observetransitions to the state with angular momentum mL = 2. Webelieve that this is due to the bosonic nature of atoms, whichprefers transitions with two atoms going to the same final state.

3. Only one atom flips its spin projection. Conservation oftotal angular momentum requires that one of the interactingatoms has to gain one quanta of angular momentum, i.e., itswave function has mL = 1. Due to the symmetry of dipolarinteractions the final wave function in the z coordinate ofrelative motion must be antisymmetric.

For resonances observed in our GP simulations additionalrotational quanta are shared by the atom that enters the mF = 0state after the collision. This can be checked by comparing thephases of the mF = 1 and 0 wave functions. There is no vortex

(a) (b) (c)

FIG. 7. Spatial densities of the three lowest states coupled bydipolar interactions. (a) Ground state of the trap with mF = +1.(b) Lowest excited state with one quanta of angular momentum andsymmetric in the z direction. It is accessible when two interactingatoms flip their spins simultaneously. (c) Lowest excited state with onequanta of angular momentum and antisymmetric in the z direction. Itis accessible when one ore two interacting atoms flip their spins.

in the mF = 1 component, but there is a single vortex in themF = 0 one; see Fig. 3, for example.

To gain a better understanding of observed density profiles,we here introduce a simplistic model where all accessiblesingle-particle states are limited to the lowest energy statesof the symmetry consistent with numerical results, i.e., to thefollowing states:

ϕ0(r) ∼ e−(x2+y2+βz2)/2, (9a)

ϕ1(r) ∼ (x + iy)e−(x2+y2+βz2)/2, (9b)

ϕ2(r) ∼ z(x + iy)e−(x2+y2+βz2)/2. (9c)

Three-dimensional graphs showing characteristic densitydistributions of all these states are shown in Fig. 7. Note that|ϕ2|2 has two density rings located symmetrically with respectto the xy plane, while |ϕ1|2 has only one ring in the z = 0plane.

Treating dipolar interactions in the first order of theperturbation, we can expand the spinor field operator inthe basis of these single-particle trap states. In the case ofthe elongated cigar-shaped trap the lowest excited states areϕ1(ρ,φ,z) and ϕ2(ρ,φ,z), and we can assume the followingform of spinor components of the field operator:⎛

⎜⎝ψ1

ψ0

ψ−1

⎞⎟⎠ =

⎛⎜⎝

ϕ0(r)a0

ϕ1(r)a1 + ϕ2(r)a2

0

⎞⎟⎠ . (10)

The second quantization Hamiltonian, limited by the expan-sion Eq. (10), consists of three parts: the single-particle, thecontact, and the dipolar contributions. They can be written as

H = H SP + HC + HD, (11a)

where the single-particle component is

H SP = (E0 − µB) a†0a0 + E1 a

†1a1 + E2 a

†2a2, (11b)

and the term proportional to B corresponds to the linearZeeman shift. The contact two-body term is of the form

HC = U00 a†0a

†0a0a0 + U11 a

†1a

†1a1a1 + U22 a

†2a

†2a2a2

+U01 a†0a

†1a0a1 + U02 a

†0a

†2a0a2 + U12 a

†1a

†2a1a2

+UNR a†1a

†1a2a2 + U ∗

NR a†2a

†2a1a1, (11c)

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and finally the two-body dipolar interaction is

HD = D00a†0a

†0a0a0 + D01a

†0a

†1a0a1 + D02a

†0a

†2a0a2

+ T(2)

1 (a†1a

†1a0a0 + H.c.) + T

(1)2 (a†

0a†2a0a0 + H.c.)

+ T(2)

2 (a†2a

†2a0a0 + H.c.). (11d)

All U , D, and T depend on the trap geometry and canbe easily evaluated. Exact formulas are given in Appendix C.Parameters T

(P )i are dipolar integrals for processes when P =

1 or 2 particles are transferred from the initial state to the stateof the wave function ϕi (i = 1,2).

Since the total number of particles N = a†0a0 + a

†1a1 +

a†2a2 is conserved, the Hamiltonian separates into diagonal

blocks corresponding to a given number of particles. The“initial” state,

|N,0,0〉 = 1√N !

(a†0)N |0〉, (12)

couples, in the first-order perturbation theory, with

|N − 2,2,0〉 = 1√2(N − 2)!

(a†0)N−2(a†

1)2 |0〉, (13a)

|N − 1,0,1〉 = 1√(N − 1)!

(a†0)N−1a

†2 |0〉, (13b)

|N − 2,0,2〉 = 1√2(N − 2)!

(a†0)N−2(a†

2)2 |0〉, (13c)

where |0〉 is the vacuum. Note that dipolar matrix elementcorresponding to two particles going to the ϕ2(r) state isproportional to

√2N (N − 1) T

(2)2 , which in the limit of large

N scales as ∼NT(2)

2 . The corresponding matrix elementassociated with only one particle jumping to this state is√

N (N − 1) T(1)

2 and scales with the number of particles as∼N3/2T

(1)2 . Evidently in the limit of large N atoms enter the

state ϕ2(r) one by one: Only one atom per every two-bodydipolar collision flips its spin. The processes driven by a termproportional to T

(1)i dominate the collisions driven by the T

(2)i

for large N .For N = 2 the four states listed above span the whole sub-

space. The spectrum obtained from the exact diagonalizationis presented in Fig. 8. The figure explains the origin of theresonant behavior of the system. For B = 0 the energy ofeach state is a sum of kinetic energy, potential energy, andcontact interaction energy. Energy related to the dipole-dipoleinteraction is very small and can be ignored. When themagnetic field is switched on, the additional Zeeman energymust be taken into account for atoms that are in states withmF = +1. As we see, for some magnetic fields the energiesof different states become equal, and then dipolar interactionsstart to play a crucial role. First, they cancel degeneracy ofthe states, and, second, they can lead to noticeable transferbetween them. This is how the dipolar resonances appear.

Three avoided crossings seen in Fig. 8 correspond tothree different resonances. The width of every resonance isdetermined by the magnitude of the dipolar energy and isvery small. For two rubidium atoms in a magnetic trap offrequency �100 Hz it is on the order of �B � 10−4 mG. Suchprecision in controlling the magnetic field is extremely difficultand beyond experimental reach. However, the dipolar energy

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

Ene

rgy

[osc

. uni

ts]

B [mG]

|020⟩

|002⟩

|101⟩

|200⟩

1.14353 1.14361

1.4275 1.431

FIG. 8. Energy of two-atom quantum states as a function of anexternal magnetic field for ωz/ω⊥ = 1/4. For some values of theexternal magnetic field energies of some states become equal due tothe linear Zeeman effect. Note, however, that, according to (11b), thecrossing of Zeeman levels happens for magnetic fields oriented alongthe negative z direction. Two insets show real and avoided crossings.Note that avoided crossings appear only for states coupled by dipolarinteractions.

grows with the number of atoms, as discussed above. Thereforefor N ∼ 104 (as studied in Sec. III), widths of resonancesincrease by several orders of magnitude �B � 10−1 mG,giving hope for an experimental realization. Thus for largesystems the effects discussed could be observed in a rubidiumcondensate in experiments performed at ultra-low magneticfields requiring a special shielding of all external magneticfields. On the other hand, large atomic densities can bereached in optical lattices with only few atoms per site. Insuch a situation chromium atoms are preferred because of therelatively large magnetic moment. For typical optical latticesthe width of resonances for two chromium atoms in a single siteis about �B � 10−1 mG. Moreover, in the case of an opticallatice, the width of the resonance is significantly increaseddue to the finite energy width of excited bands as seen inRef. [21]. Again special shielding is required for experimentalobservation of resonant dipolar interactions. If the magneticfield does not match a resonant value, the dipolar interactionscan be totally ignored.

Weakness of dipolar interactions is responsible for a verysmall width of the resonances, making them difficult to detect.On the other hand, weak dipolar interactions have one veryimportant advantage: All the resonances shown in Fig. 8 arewell separated. Choosing an appropriate value of the magneticfield a particular resonance can be addressed, and dipolarinteractions can be tuned to couple the desired center-of-massstates. Obviously far from all the resonances the dipolarinteraction can be set to zero.

The exact value of the resonant magnetic field depends onthe energy difference of resonantly coupled states at B = 0,and it is a linear function of number of atoms in the initial state.For realistic values of scattering lengths the contact interactionin the initial |N,0,0〉 state grows with N faster then the contactinteraction in three other states considered here. Therefore, thetotal energy of this state can be larger than the energy of theresonantly coupled state. It clearly follows from Fig. 8 that insuch a case the corresponding resonant magnetic field becomes

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TUNABLE DIPOLAR RESONANCES AND EINSTEIN–DE . . . PHYSICAL REVIEW A 83, 063617 (2011)

positive; i.e., its direction is along the direction of the magneticmoment of atoms. For numerical calculations discussed inSec. III the resonant magnetic field is still negative, whichmeans that it is inverted with respect to the direction used forthe preparation of the ground state.

V. CONCLUSIONS

In conclusion, we have shown the existence of dipolarresonances in rubidium spinor condensates. The resonancesoccur when the Zeeman energy of atoms in the mF = 1 com-ponent matches the excitation energy of a spatial state that ispopulated upon the transfer. Symmetries of dipolar interactionsforce the atoms in mF = 0,−1 states to circulate around thequantization axis. In fact, singly and doubly quantized vorticesare formed in mF = 0,−1 components, respectively. Sincedipolar interactions are weak, the resonances are narrow. Itmeans that dipolar resonances are very selective; i.e., onecan choose the transition to the proper state by choosing themagnetic field. Hence, dipolar resonances seem to be a routeto observation of the Einstein-de Haas effect, as well as otherphenomena related to dipolar interactions, in weak dipolarsystems.

ACKNOWLEDGMENTS

We are grateful to K. Bongs, K. Gawryluk, B. Laburthe-Tolra, and K. Rzazewski for helpful discussions. JZ andMG acknowledge hospitality from ICFO and partial supportfrom the advanced ERC-grant QUAGATUA. This work wasalso supported by the UE project NAME-QUAM and thePolish Ministry for Science and Education for 2009–2011and via grant N202 124736 for 2009-2012 (JZ). ML ac-knowledges also ERC Grant QUAGATUA, Spanish MICINNprojects (FIS2008-00784 and QOIT), EU grants (AQUTE,NAMEQUAM), and the Humboldt Foundation.

APPENDIX A: EQUATION OF MOTION

The equation of motion (3) is obtained from the Heisenbergequations for the field operators ψi(r) for the system describedby the Hamiltonian H (1). Since the Hamiltonian (1) splits ina single-particle part and two-particle parts related to contactand dipolar interactions, the commutators [ψi(r),H ] as well asthe right-hand side of Eq. (3) get the form of the sum of threeterms of different physical origin. The single-particle part isdiagonal and given by Hsp = − h2

2m∇2 + Vext − µmF B.

The term Hc in Eq. (3), which is related to the contactinteractions, has the following diagonal elements:

Hc11 = (c0 + c2) ψ

†1ψ1 + (c0 + c2) ψ

†0ψ0

+ (c0 − c2) ψ†−1ψ−1,

Hc00 = (c0 + c2) ψ

†1ψ1 + c0 ψ

†0ψ0

(A1)+ (c0 + c2) ψ

†−1

ˆψ−1,

Hc−1−1 = (c0 − c2) ψ

†1ψ1 + (c0 + c2) ψ

†0ψ0

+ (c0 + c2) ψ†−1ψ−1.

These elements describe the collisions of atoms that preservethe projection of the spin for each atom. The off-diagonalelements, on the other hand, are responsible for collisionschanging separately the atomic spin projections but conservingthe projection of total spin. They yield

Hc10 = c2ψ

†−1ψ0, Hc

0−1 = c2ψ†0ψ1, Hc

1−1 = 0. (A2)

For dipolar interactions, Hd , in Eq. (3) one has

Hdij (r) =

∫d3r ′ψ†

k (r ′)V dij,kl(r − r ′)ψl(r ′), (A3)

where

V dij,kl(r − r ′) = µ2

|r − r ′|3 Fij Fkl − 3µ2

|r − r ′|5× [Fij (r − r ′)][Fkl(r − r ′)]. (A4)

Only two elements of Hd matrix are independent, namely,

Hd11(r) = µ2

∫d3r ′

[1

|r − r ′|3 − 3(z − z′)2

|r − r ′|5]

× (ψ†1ψ1 − ψ

†−1ψ−1)

− 3µ2

√2

∫d3r ′ z − z′

|r − r ′|5 [(x − x ′) − i(y − y ′)]

× (ψ†1ψ0 + ψ

†0ψ−1)

− 3µ2

√2

∫d3r ′ z − z′

|r − r ′|5 [(x − x ′) + i(y − y ′)]

× (ψ†0ψ1 + ψ

†−1ψ0) (A5)

and

Hd10(r) = −3

µ2

√2

∫d3r ′ [(x − x ′) − i(y − y ′)](z − z′)

|r − r ′|5× (ψ†

1ψ1 − ψ†−1ψ−1)

− 3

2µ2

∫d3r ′ [(x − x ′) − i(y − y ′)]2

|r − r ′|5× (ψ†

1ψ0 + ψ0†ψ−1)

+µ2∫

d3r ′[

1

|r − r ′|3 −3

2

(x − x ′)2 + (y − y ′)2

|r − r ′|5]

× (ψ†0ψ1 + ψ

†−1ψ0). (A6)

For other elements one has

Hd0−1 = Hd

10, Hd−1−1 = −Hd

11,(A7)

Hd1−1 = Hd

00 = 0 .

The Hdij terms (both diagonal and off-diagonal) are respon-

sible for the change of the total spin projection of collidingatoms. For example, even if components mF = 0 and −1 areinitially empty, then due to the first part in Hd

10, there exists aprocess when one of two atoms initially in the mF = 1 statepopulates the mF = 0 state after collision.

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TOMASZ SWISŁOCKI et al. PHYSICAL REVIEW A 83, 063617 (2011)

APPENDIX B: NUMERICAL DETAILS

To solve Eq. (3) we neglect the quantum fluctuations andreplace the field operator ψi(r) [ψ†

i (r)] by an order parameterψi(r) [ψ∗

i (r)] for each component. Next, we split the right-hand-side operator of (3) in the following way:

H = Hsp + Hc + Hd = T + V, (B1)

where T is the kinetic energy operator, which is a diagonalmatrix, and V represents the potential energy including thetrapping potential, the interaction with the external magneticfield, and the contact and dipolar interactions. Now we willapply the split-operator method to solve the mean-field versionof Eq. (3). First, we calculate the evolution of an orderparameter according to the potential energy. To this end, wediagonalize the matrix V at each spatial point V = PDP −1,where the matrix P consists of the eigenvectors written incolumns, and the matrix D is diagonal with the eigenvalueson the diagonal. The evolution by the time step �t is givenby

e−iV �t/h

⎛⎝ψ1

ψ0

ψ−1

⎞⎠ = Pe−iD�t/hP −1

⎛⎝ψ1

ψ0

ψ−1

⎞⎠ (B2)

and is calculated in three steps: First, the spinor orderparameter is multiplied by the matrix P −1, next by e−iD�t/h,which is a diagonal matrix since D is diagonal, and finallyby the matrix P . Then we calculate the evolution due to thekinetic part. Since the kinetic energy operator T is diagonal,we apply the Fourier technique to each component in a wayas is usually done in the case of a scalar Gross-Pitaevskiiequation.

All integrals appearing in formulas (A5) and (A6) are theconvolutions, and we use the Fourier transform techniqueto calculate them. The Fourier transform of the convolutionof two functions is the product of the Fourier transformsof these functions. Therefore, it is useful to have analyticalformulas for the Fourier transforms of the components ofthe convolutions that do not change during the condensateevolution. The appropriate Fourier transforms are found in thefollowing way. First, two rotations of coordinates system (byangles: β around the z axis and α around the y axis) are appliedto make the vector k, the argument of the Fourier transform,parallel to the z axis, and next, the regularization procedure asdescribed in Ref. [25] is used. The result reads

F[

1

|r|3 − 3z2

|r|5]

= −4π

3(1 − 3 cos2α),

F[

1

|r|3 − 3

2

x2 + y2

|r|5]

= 2π

3(1 − 3 cos2α),

F[

(x − iy)z

|r|5]

= −2π

3e−iβ sin 2α, (B3)

F[

(x + iy)z

|r|5]

= −2π

3eiβ sin 2α,

F[

(x − iy)2

|r|5]

= −4π

3e−i2β sin2α,

with the angles α and β defined as follows: cos α = kz/k andsin β = ky/

√k2 − k2

z .

Even with a help of formulas (B3) the evolution of the mean-field version of Eq. (3) is a time-consuming computationaltask. We point out that each time step consists of taking ninenumerical fast Fourier transforms (three for determining theevolution of the spinor wave function governed by the kineticenergy and six for calculating the Hd matrix) and doing thediagonalization of matrix V at each spatial point. Thereforethe parallel computing with OpenMP directives has been usedin our calculations.

The size of the grid was adjusted to the trap geometry; wehave chosen 25 × 25 × 25 grid points in the case of sphericallysymmetric traps (with the spatial steps equal to �x = �y =�z = 0.55 osc. units, where osc. unit=√

h/mω) and 25 ×25 × 26 grid points for cigar-like traps (with the spatial stepsequal to �x = �y = 0.275 and �z = 0.375 osc. units). Thetime step was chosen to �t = 2.5 × 10−5 osc. units, whereosc. unit=1/ω. Part of the results was recomputed with atwice smaller spatial step (a two times bigger grid size inevery dimension) as a self-test.

The ground state was obtained by evolving the mean-fieldversion of Eq. (3) in imaginary time at the value of the magneticfield B = 0.73 mG. Such a magnetic field was large enoughto result in having almost all atoms in the mF = 1 Zeemansublevel.

APPENDIX C: TWO-BODY MATRIX ELEMENTS

In this appendix we present explicit formulas for two-bodymatrix elements of the Hamiltonian (1) in the basis definedby (9).

Contact integrals:

U00 = c0 + c2

2

∫d3r |ϕ0(r)|4, (C1a)

U11 = c0

2

∫d3r |ϕ1(r)|4, (C1b)

U22 = c0

2

∫d3r |ϕ2(r)|4, (C1c)

U01 = (c0 + c2)∫

d3r |ϕ0(r)|2|ϕ1(r)|2, (C1d)

U02 = (c0 + c2)∫

d3r |ϕ0(r)|2|ϕ2(r)|2, (C1e)

U12 = 2c0

∫d3r |ϕ1(r)|2|ϕ2(r)|2, (C1f)

UNR = c0

2

∫d3r [ϕ∗

1 (r)ϕ2(r)]2. (C1g)

Dipolar interactions that do not change the spin of atoms:

D00 = µ2

2

∫ ∫d3r d3r ′ |ϕ0(r)|2f1(r − r ′)|ϕ0(r ′)|2, (C2a)

D01 = −µ2

2

∫ ∫d3r d3r ′ ϕ∗

1 (r)ϕ0(r)f1(r − r ′)ϕ∗1 (r ′)ϕ0(r ′),

(C2b)

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TUNABLE DIPOLAR RESONANCES AND EINSTEIN–DE . . . PHYSICAL REVIEW A 83, 063617 (2011)

D02 = −µ2

2

∫ ∫d3r d3r ′ ϕ∗

2 (r)ϕ0(r)f1(r − r ′)ϕ∗2 (r ′)ϕ0(r ′).

(C2c)

Dipolar interactions that change the spin of the atoms:

T(2)

1 = µ2

2

∫ ∫d3r d3r ′ ϕ∗

1 (r)ϕ0(r)f2(r − r ′)ϕ∗1 (r ′)ϕ0(r ′),

(C3a)

T(1)

2 = µ2∫ ∫

d3r d3r ′ ϕ∗0 (r)ϕ0(r)f3(r − r ′)ϕ∗

2 (r ′)ϕ0(r ′),

(C3b)

T(2)

2 = µ2

2

∫ ∫d3r d3r ′ ϕ∗

2 (r)ϕ0(r)f2(r − r ′)ϕ∗2 (r ′)ϕ0(r ′).

(C3c)

Functions fi have the following form:

f1(r) = 1 − 3n2z

r3, (C4a)

f2(r) = −3

2

(nx − iny)2

r3, (C4b)

f3(r) = −3√

2

2

nz(nx + iny)

r3. (C4c)

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