Trapped two-dimensional condensates with synthetic spin-orbit coupling
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Trapped two-dimensional condensates with synthetic spin-orbit coupling
Subhasis Sinha, Rejish Nath, and Luis Santos Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia 741252, India. Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Strasse 38, D-01187 Dresden, Germany Institut f¨ur Theoretische Physik , Leibniz Universit¨at, Hannover, Appelstrasse 2, D-30167, Hannover, Germany (Dated: September 19, 2018)We study trapped 2D atomic Bose-Einstein condensates with spin-independent interactions inthe presence of an isotropic spin-orbit coupling, showing that a rich physics results from the non-trivial interplay between spin-orbit coupling, confinement and inter-atomic interactions. For lowinteractions two types of half-vortex solutions with different winding occur, whereas strong-enoughrepulsive interactions result in a stripe-phase similar to that predicted for homogeneous conden-sates. Intermediate interaction regimes are characterized for large enough spin-orbit coupling by anhexagonally-symmetric phase with a triangular lattice of density minima similar to that observedin rapidly rotating condensates.
Introduction.
The engineering of synthetic electro-magnetism in ultra-cold gases has recently attracted amajor attention [1]. Although an homogeneous effectivemagnetic field may be generated by simple rotation, re-cent techniques based on appropriate laser arrangementsand tailored dressed states allow for more flexible con-trol of effective artificial gauge fields [2, 3]. Interestingly,these techniques allow as well for the creation of non-Abelian gauge fields [4, 5], and more specifically spin-orbit coupling (SOC) [6–10], a crucial effect in solid-statephysics, essential for topological insulators [11]. Recentground-breaking experiments have explored this fascinat-ing possibility, demonstrating the creation of SOC withequal Rashba and Dresselhaus strengths [12].The creation of SOC in spinor gases opens fascinat-ing questions about the physics of ultra-cold gases withSOC, which have aroused a rapidly-growing theoreticalattention both in what concerns degenerated fermions [9,13, 14] and Bose-Einstein condensates(BECs) [7, 15–22].In particular, Wang et al. have recently shown that theground-state of an homogeneous, i.e. untrapped, two-component BEC with SOC is a single plane-wave phaseor a spin stripe phase depending on spin-dependent inter-actions [18]. On the other hand, non-interacting trappedspin-orbit coupled BECs are expected to present a half-quantum vortex configuration [7, 15].In this Letter, we show that the interplay between trapenergy, SOC and interactions leads to a rich ground-statecondensate physics. We consider in particular the com-plete phase diagram of trapped two-dimensional conden-sates with spin-independent interactions in the presenceof isotropic spin-orbit coupling. This phase diagram ischaracterized by four different phases. As expected, forlow interactions, a half-vortex solution (HV(1/2) below),with angular momentum l such that | l + 1 / | = 1 /
2, isrecovered. Increasing interaction leads to a second half-vortex solution (HV(3/2)) with a higher | l + 1 / | = 3 / FIG. 1. Total density for κ = 15 and g = 0 .
05 (a), g =0 . g = 0 .
85 (c), and g = 2 . g = 0 .
85, and Fig. (f) shows thespatial distribution of component 1 for g = 2 . minima in the total density, similar to a vortex lattice infastly rotating BECs [23]. Finally, for even larger interac-tions, a spin-stripe phase develops, similar to that foundin homogeneous BECs [18]. We discuss the existence andmain properties of these phases. Model.
We consider a BEC of atoms with mass m ,confined in two-dimensions on the xy plane, by a tightharmonic confinement along z . The atoms have two avail-able internal states, which in typical experiments are, asmentioned above, atom-light dressed states of Zeemansublevels with a proper laser configuration. Typically,inter-atomic interactions between Zeeman componentsare very similar. Hence, for simplicity we consider spin-independent interactions characterized by the 2D cou-pling constant ˜ g >
0, which hence determines as well theinteractions between the atoms in the dressed states.Both components are equally confined in an isotropicharmonic potential V ( r ) = mω r /
2, with r = x + y .We consider that an appropriate laser configuration ischosen such that the atoms experience an isotropic gaugefield A = ~ ˜ κ ( σ x e x + σ y e y ), where σ x,y are the Paulimatrices and e x,y are the unit vectors along the directions x and y [1]. The physics of the condensate in the presenceof SOC and interactions is described by the Hamiltonianˆ H = Z d r Ψ † (cid:20) m ( − i ~ ∇− A ) + V ( r )+ ˜ g Ψ † · Ψ (cid:21) Ψ , (1)where Ψ is a two component spinor. Homogeous solution.
In the absence of trapping, V ( r ) = 0, the condensate minimizes the energy by spon-taneously breaking the rotational symmetry, develop-ing a spatial spin modulation along an arbitrary direc-tion [18]. Chosing that direction as x , we may re-write Ψ = e i (˜ κx ) σ x Φ . For a constant Φ (with | Φ | = n ), boththe interaction energy, ˜ gn , and h ( − i ~ ∇ − A ) i = 0, ac-quire its minimum value. Chosing Φ T = √ n (1 , Ψ = √ n (cid:18) cos(˜ κx ) i sin ˜ κx (cid:19) (2)The presence of the trap modifies and enriches theground-state physics since it introduces an additionalenergy scale, ~ ω . The trapped BEC is hence best de-scribed by two dimensionless parameters, κ ≡ ˜ κl HO , with l HO = ~ /mω , and g ≡ ˜ gm/ ~ . Note that due to the 2Dnature of the problem, the dimensionless coupling con-stant g is actually independent of ω . Below we employdimensionless expressions, using oscillator units. HV(1/2) Phase.
The non-interacting case, g = 0,is best described in momentum representation [7]. Inthe absence of trap, the Hamiltonian acquires the form( ~ k − A ) /
2, which presents two eigen-energy branches, ǫ ± ( k ) = ( k ± ˜ κ ) /
2, with k = | k | . The correspondingeigen-spinors are of the form u ± ( ϕ ) = 1 √ (cid:18) ∓ e iϕ (cid:19) , (3)with ϕ the polar angle of the vector k . Assuming adominant SOC, such that κ / Ψ ( k ) = ψ ( k ) u − ( ϕ ).Introducing the trapping potential, V = −∇ k /
2, with ∇ k the gradient in momentum space, and using ψ ( k ) = P l k − / f l ( k ) e ilϕ , one obtains: − (cid:18) d dk f l − ( l + 1 / k f l (cid:19) + ( k − ˜ κ ) f l = E l ( k ) f l (4)Note that the original kinetic energy term becomesa mexican-hat-like “potential” in momentum space,whereas the trap results in a ”radial kinetic energy” termand a ”centrifugal barrier” [7]. For κ ≫
1, we may de-velop around k ≃ κ , obtaining the eigenenergies E nl = ( l + 1 / κ + n + 1 / , (5)where n characterizes the radial excitations of themexican-hat.Without interactions the lowest energy is given by n =0 and | l + 1 / | = 1 /
2, which are states of the form: Ψ l =0 , − ( k ) ∝ e − ( k − ˜ κ ) l HO / e ilϕ u − , (6)These states present a spatial dependence of the form: Ψ T ( r = ( r, α )) ∼ ( J ( κr ) , e iα J ( κr )), and Ψ T − ( r ) ∼ ( − e − iα J ( κr ) , J ( κr )), and hence constitute half-vortexsolutions [15]. In the following we denote this phase asHV(1/2). Solutions with l = 0 and l = − l = 0 and l = − l = 0 or l = − k ≃ κ , whereas linearcombinations result in a cos ϕ modulation along the ring. HV(3/2) Phase.
For κ ≫
1, angular excitationsalong the ring are much less energetic than radial ex-citations. Hence weak interactions result in the popu-lation of higher | l + 1 / | values. Simulations of Eq. (1)show that the HV(1/2) phase remains the ground statefor sufficiently small g (Fig. 1(a)). However, the totaldensity of the HV(1/2) solution has a maximum at thetrap center due to the J ( κr ) contribution. On the con-trary, the solutions with n = 0, | l + 1 / | = 3 /
2, dependas J l = − , ( κr ) and J l = − , ( κr ), and hence have a mini-mum of the total density at the trap center, presenting areduced interaction energy compared to HV(1/2). Solu-tions with | l + 1 / | = 3 / g = g cr , when the interaction energy balances the en-ergy difference between both solutions for g = 0: g cr = 2 πκ (cid:20) f (1 , κ ) f (1 , κ ) − f (0 , κ ) f (0 , κ ) (cid:21) (7)with f α = R rdre − αr (cid:2) J m ( κr ) + J m +1 ( κr ) (cid:3) α . For large κ , g cr ≃ . π/κ ).Decomposing ψ ( k ) = P a l Ψ l ( k ), we evaluate ξ ≡h| l + 1 / |i = P | a l | | l + 1 / | (Fig. 2(a)). For g < g cr , thephase HV(1/2) is characterized by a plateau at ξ = 1 / g ξ gg χ (b)(a) FIG. 2. (a) Value of ξ = h| l + 1 / |i as a function of g for κ = 15, the dashed lines indicate the value 1 / / χ of the cloud for the same case. The dashed curveindicates the expected aspect ratio from Eq. (10). At g = g cr a sudden jump occurs into ξ = 3 /
2. Inter-estingly, the numerical simulation of Eq. (1) shows thatup to a second critical g (2) cr , a second plateau at ξ = 3 / l = 1 and l = − l = 1 or l = − ϕ ) modulation along the ring. Lattice phase.
For g > g (2) cr , ξ departs from the 3 / g , being ≃ π/ κ , such that the border of the firstBrillouin zone of the lattice lies on the k = κ ring. Onthe contrary, as discussed below, the width of the cloudenvelope p h x i is independent of κ for dominant SOC,being only dependent on g . Hence enhancing interactionsjust increases the number of observed density minima( ∝ κ h x i ) keeping invariant the lattice structure. Theindividual components present in typical numerical simu-lations an involved density and phase distribution, char-acterized by vortices and anti-vortices of different quan- tizations. In this Letter we are not interested in them,since the particular distribution among the componentsmay depend on details of spin-dependent interactions.An interesting insight about the lattice phase may begained from its momentum distribution characterized bythe appearance of six maxima at angles ϕ j = jπ/ κ (Fig. 1(d)). The latticehence results from the combination of three pairs of op-posite momenta. Major features of the overall densityprofile may be obtained from these six-peaked structureby means of a Gaussian ansatz of the form: ψ ( k ) = X j a j e − h l r ( k ( j ) r − κ ) + l ϕ k ( j )2 ϕ i e − i ( ϕ j + k ( j ) ϕ /κ ) e i ( ϕ j + k ( j ) ϕ /κ ) ! (8)with a j = 1 / √ k ( j ) r = k x cos ϕ j + k y sin ϕ j and k ( j ) ϕ = − k x sin ϕ j + k y cos ϕ j . This solution is char-acterized by h x i = h y i = ( l r + l ϕ ) /
4. For a suf-ficiently large κ , the energy may be approximated as E ( η, g ) = 2 h x i = p (1 + η )(1 + gf ( η ) / π ), with f ( η ) = 1 /η + 8 / p η + 3 η , and η = l ϕ /l r . Mini-mizing the energy we obtain h x i in excellent agreementwith our numerical simulations (since the orientation ofthe cloud is arbitrary, the x direction is determined nu-merically as that with the larger width). Interestingly,the kinetic energy E kin = 1 / l r just contains the radialwidth l r , since the l ϕ dependence is exactly cancelled bythe SOC. This cancellation leads to a global shrinking ofthe cloud for the lattice phase, and plays a major role inthe stripe phase discussed below. Stripe phase.
All the phases mentioned above ful-fill χ ≡ h y i / h x i = 1, either due to a full rota-tional symmetry or due to hexagonal symmetry in thecase of the lattice. However, at a sufficiently large g ,the ratio χ departs from 1 (Fig. 2(b)), i.e. the cloudbreaks the hexagonal symmetry, becoming elongatedalong x (Fig. 1(e)). After a transient, in which χ de-creases abruptly (Fig. 2(b)), the condensate acquires fora sufficiently large g a stripe form along the major axis x (Fig. 1(f)), similar to that obtained in the case of ho-mogeneous condensates with SOC [18].The properties of the trapped stripe, and in particularits elongation along the direction of the stripe modula-tion, may be understood from the ansatz (8), but justconsidering two peaks on the ring at ϕ j = 0 , π . In thiscase, h x i = l r / h y i = l ϕ /
2. For a sufficiently large κ the energy functional may be approximated as E ≃ l r + l r + l ϕ g πl r l ϕ . (9)As mentioned above, the kinetic energy, which in absenceof SOC is of the form lr + l ϕ , reduces to l r , since the l ϕ contribution is exactly cancelled by the SOC. Thisbreaks the symmetry between x and y axis, leading to κ g FIG. 3. Phase diagram as a function of κ and g . The dashedline indicates the result of Eq. (7). an effective compression along y . Energy minimizationleads to an equation for the aspect ratio g π = χ − χ , (10)which provides an excellent agreement with our numeri-cal calculations as shown in Fig. 2(bottom). Phase diagram.
We have obtained the ground-statephase diagram by numerically solving Eq. (1) for a largenumber of g and κ values. Our results are summarized inFig. 3. The border g (3) cr between the lattice and the stripephase is determined as the g value at which χ ≃ . /κ dependence of the angular modes, forlarge κ values the HV(1/2) and HV(3/2) shrink, beingconfined to a regime of progressively smaller g values as κ increases. On the contrary, the lattice phase becomesprogressively more favorable extending for very large κ almost to g = 0. The border between lattice and stripeapproaches g ≃ κ . When κ decreases, thelattice phase shrinks until disappearing below κ c ≃ . κ c there is a direct HV(3/2) to stripe transition,at which the rotational symmetry is broken. Conclusions.
In summary trapped spin-orbit coupledBECs present intriguing ground-state properties as a re-sult of the interplay between trap energy, spin-orbit cou-pling and interactions. The corresponding phase diagramincludes two rotationally symmetric phases, HV(1/2) andHV(3/2), a hexagonally-symmetric phase with a trian-gular lattice of minima, which resembles a vortex lattice,and a stripe phase, similar to that predicted for homoge-neous BECs. In experiments the phases should be clearlydistinguisable either by monitoring in situ the densityprofile, or by time-of-flight imaging. The latter maps themomentum distribution, and, in particular, the latticephase will be characterized by six out-going clouds.Finally, let us point that in our paper we have consid-ered for simplicity spin-independent interactions. In typ-ical experiments spin-dependent interactions are much weaker than spin-independent ones, and hence should notaffect the overall density profiles, although the particulardistribution among the components will change. How-ever, due to the dressed nature of the states in actual ex-periments, spin-dependent interactions do not simply re-duce to a different inter-component density-density inter-action, as in usual two-component condensates. On thecontrary, phase-dependent terms will occur [20], whichmay considerably complicate and enrich the physical pic-ture. This issue, and anisotropic effects, both in the trapand in the SOC, will be the focus of further investigation.
Note added:
When completing this paper we becameaware of two recent works in which a similar problemis considered. In Ref. [24], which considers the effectof spin-dependent interactions, phases IIA (and IA), IA’(and part of IIB) and IB (and part of IIB) correspondto the HV(1/2), HV(3/2) and lattice phases. A new re-cent version of Ref. [15] discusses the border of the stripephase at small κ <
4, showing a direct half-vortex tostripe transition, in agreement with our results.We acknowledge support from the Center for QuantumEngineering and Space-Time Research QUEST. [1] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. ¨Ohberg,arXiv:1008.5378.[2] Y.-J. Lin et al, Phys. Rev. Lett. , 130401 (2009).[3] Y.-J. Lin et al , nature , 628 (2009).[4] J. Ruseckas, G. Juzeliunas, P. ¨Ohberg, and M. Fleis-chhauer, Phys. Rev. Lett. , 010404 (2005).[5] K. Osterloh et al. , Phys. Rev. Lett. , 010403 (2005).[6] G. Juzeliunas et al. , Phys. Rev. A 77, 011802(R) (2008).[7] T. D. Stanescu, B. Anderson and V. Galitski, Phys. Rev.A , 023616 (2008).[8] X.-J. Liu, X. Liu, L. C. Kewk, and C. H. Oh, Phys. Rev.Lett. , 026602 (2007).[9] J. D. Sau et al. , Phys. Rev. B , 140510(R) (2011).[10] D. L. Campbell, G. Juzeliunas, and I. B. Spielman,arXiv:1102.3945.[11] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[12] Y.-J. Lin et al. , Nature , 83 (2011).[13] T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev.Lett. , 110403 (2007).[14] Z.-Q. Yu and H. Zhai, arXiv:1105.2250.[15] C. Wu and I. Mondragon-Shem, arXiv:0809.3532.[16] M. Merkl et al. , Phys. Rev. Lett. , 073603 (2010).[17] J. Larson, J. P. Martikainen, A. Collin, and E. Sjoqvist,arXiv:1001.2527.[18] C. Wang, C. Gao, C.-M. Jian and H. Zhai, Phys. Rev.Lett. , 160403 (2010).[19] S. K. Yip, Phys. Rev. A , 043616 (2011).[20] Y. Zhang, L. Mao and C. Zhang, arXiv:1102.4045.[21] Z.-F. Xu, R. L¨u, and L. You, Phys. Rev. A , 053602(2011).[22] T. Kawakami, T. Mizushima, and K. Machida,arXiv:1104.4179.[23] C. Raman et al., Phys. Rev. Lett.87