Ground-state multiquantum vortices in rotating two-species superfluids
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Ground-state multiquantum vortices in rotating two-species superfluids
Pekko Kuopanportti,
1, 2, ∗ Natalia V. Orlova, and Milorad V. Miloˇsevi´c † School of Physics and Astronomy, Monash University, Victoria 3800, Australia Department of Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Dated: August 17, 2018)We show numerically that a rotating, harmonically trapped mixture of two Bose–Einstein-condensed superfluids can—contrary to its single-species counterpart—contain a multiply quantizedvortex in the ground state of the system. This giant vortex can occur without any accompanyingsingle-quantum vortices, may either be coreless or have an empty core, and can be realized in a Rb– K Bose–Einstein condensate. Our results not only provide a rare example of a stable, soli-tary multiquantum vortex but also reveal exotic physics stemming from the coexistence of multiple,compositionally distinct condensates in one system.
PACS numbers: 67.85.Fg,03.75.Mn,03.75.LmKeywords: Bose–Einstein condensation, Superfluid, Vortex, Multicomponent condensate
I. INTRODUCTION
According to the conventional paradigm [1, 2], theground state in a rotating superfluid will involve onlysingly quantized vortices (SQVs). Vortices with largerquantum numbers are energetically unfavorable and donot occur—not even for rapid rotation, which insteadspawns a triangular Abrikosov lattice of SQVs [3]. Al-though this is well established [4–6] for a solitary super-fluid described by a single C -valued order parameter Ψ,vortex physics becomes much more diverse when multi-ple mutually interacting superfluids are rotated simulta-neously in the same container.Already for the simplest mixture, which consists of twosuperfluid species and is described by two C -valued or-der parameters Ψ and Ψ , a myriad of unusual ground-state vortex structures have been found in experimen-tal and theoretical studies [7]. Experimentally, a ver-satile platform to study vortices is provided by atomicBose–Einstein condensates (BECs) [8–10], in which two-component superfluid mixtures have been realized usingtwo different spin states of the same isotope [10–15], twodifferent isotopes of the same element [16, 17], or two dis-tinct elements [18–26]. The unconventional vortex struc-tures that were detected in these experiments comprisecoreless SQVs [10] and square vortex lattices [11]. The-oretical studies, however, have furnished the two-speciesBECs with many more ground-state vortex configura-tions than the aforementioned two [27–35]: Predictedbut hitherto unobserved ones include serpentine vortexsheets [36], triangular lattices of vortex pairs [35], and, ina pseudospin-1 / ∗ [email protected] † [email protected] ance of multiply quantized vortices (MQVs) in the ro-tating ground state of the harmonically trapped sys-tem [29, 35, 37]. So far, the MQVs, also known as gi-ant vortices, have been predicted only in complicatedstates involving a number of accompanying SQVs anda large total circulation, thereby requiring rotation fre-quencies close to the maximum set by the harmonic trapfrequency. Consequently, the states have eluded experi-mental observation and, due to the accompanying SQVs,might not be suitable for investigating the rarely encoun-tered ground-state MQV in a controlled fashion. Be-sides being exotic and interesting in their own right,MQVs could also be used to realize bosonic quantumHall states [38], initiate quantum turbulence [39–41], orimplement a ballistic quantum switch [42].In this article, we make the ground-state MQVs moreaccessible to experiments by showing theoretically thatan interacting mixture of two dilute superfluids, whenrotated at moderate speed, exhibits ground states thatcontain a solitary MQV in one of the superfluids. Wefind such states both for mutually attractive mixtures,where the MQV has a completely empty core, and formutually repulsive mixtures, where the core is occupiedby the other, vortex-free superfluid species. These statesrepresent a rare instance of a stable, solitary MQV inan atomic BEC and, as such, constitute a robust, well-isolated, and tunable environment for the experimentalexploration of MQV physics, in complement to earlierobservations in mesoscopic superconductors [43–46].All the discovered states share the property that thetwo superfluid species carry unequal numbers of circu-lation quanta under the same external rotation. Thisrequires the two superfluids to be composed of particleswith sufficiently different masses [47]. For concreteness,we will focus on the harmonically trapped two-speciesBEC of Rb and K because it has already been re-alized in several experiments [18–23], it enables a flex-ible control over its interaction strengths [21], and ithas a suitable atomic mass ratio of ∼
2. Although wepresent ground states only for this particular system, theessential features of our results apply generally to mass-imbalanced binary mixtures of dilute superfluids.
II. MODEL
We assume that the two-species BEC is rotated withangular velocity Ωˆ z . In the zero-temperature mean-field regime, the ground-state order parameters Ψ (as-signed to Rb) and Ψ ( K) satisfy the coupled time-independent Gross–Pitaevskii equations in the rotatingreference frame [7]: (cid:0) H j + g jj | Ψ j | + g | Ψ − j | − µ j (cid:1) Ψ j ( r, φ ) = 0 , (1)where j ∈ { , } , H j = − ~ m j ∇ + 12 m j ω j r + i ~ Ω ∂∂φ , (2)and the chemical potentials µ j ensure that R | Ψ j | d r = N j . Here N j , m j , and ω j denote, respectively, the to-tal number, the mass, and the radial harmonic trappingfrequency of species j atoms. We only consider quasi-two-dimensional configurations pertaining to, e.g., highlyoblate (prolate) traps with strong (weak) axial confine-ment and Ψ j approximately Gaussian (constant) in theaxial direction. The intraspecies interaction strengths g jj are assumed to be positive, whereas for the interspeciesparameter g we also consider negative values.We parametrize the interactions by the three dimen-sionless quantities U = ( g + g ) m N / ~ , g /g ,and Γ = g / √ g g > −
1. The ground states, i.e.,the lowest-energy solutions of Eqs. (1), are then uniquelyspecified by these three and the following four other pa-rameters: m /m , N /N , ω /ω , and Ω /ω . Focusingon the Rb– K BEC, we fix m /m = 0 . III. GROUND-STATE MULTIQUANTUMVORTICES
In order to understand why MQVs emerge in the ro-tating two-species BEC, we begin with a scenario whereonly Γ is varied while the other parameters are held con-stant. Furthermore, for κ j ∈ Z , let h κ , κ i denote asufficiently pointlike phase defect about which arg (Ψ )winds by κ × π and arg (Ψ ) by κ × π . For interspeciesrepulsion (Γ > h , i vortex, whereas for Γ <
0, thesimplest one corresponds to h , i . Below, we investigatethese two cases separately.Consider first the mutually repulsive mixture. Fig-ures 1(a)–1(d) depict ground states at different Γ ≥ Rb– K BEC in which there are two cir-culation quanta in Rb and none in K. Figure 1(e) showsthe relevant energy terms as a function of Γ. When Γ = 0 [Fig. 1(a)], the two off-centered h , i vortices are sepa-rated by a distance of ∼
10 times their core radius. AsΓ increases, the condensates move apart, with Rb shift-ing outward and K inward; this behavior is manifestedin the trap potential energy, which increases for Rb anddecreases for K. Consequently, Rb is depleted from theregion between the two h , i vortices, enabling them tomerge into a h , i vortex without the kinetic-energy in-crease typical of MQV formation; indeed, the kinetic en-ergy T of Rb decreases with Γ ∈ [0 , . ≥ .
5, we observe an axisymmetric h , i vortex, aboutwhich arg (Ψ ) winds by 2 × π . It is a coreless vor-tex [27–29, 49–52] in the sense that the total atomic den-sity n tot = | Ψ | + | Ψ | does not vanish at the phasesingularity. We stress that the h , i vortex is a uniqueexample of a ground-state MQV in a purely harmonictrap that occurs as a solitary topological defect withoutany accompanying SQVs.The emergence of the ground-state h , i vortex forΓ < h , i vorticesand one central h , i vortex. As Γ approaches −
1, thetwo h , i vortices move closer to each other, so thatat Γ = − .
98, all three phase singularities lie at theorigin and make up an axisymmetric h , i vortex. Toexplain the movement, we note that the kinetic energy T increases when the two vortices approach each other,whereas the interspecies interaction energy E decreasesdue to the increasing overlap R n n d r . Since E gainsin importance when the attraction becomes stronger, iteventually begins to dominate over T , and thus the h , i vortex forms [Fig. 2(d)].The ground-state h , i vortex constitutes a rare in-stance of a stable MQV with a genuinely empty, self-supporting core. Typically, such vortices are renderedunstable against splitting by quasiparticle excitationsthat are highly localized within the core [53–65]. In ourcase, however, the doubly quantized vortex in Ψ is heldtogether by the indivisible SQV in Ψ .The h , i vortices are most readily found for relativelysmall values of U . This is because small U implies alarge size of the vortex cores, which suppresses the kineticenergy near the phase singularities and leads to strongdependence of E on the vortex positions.To produce the MQVs of Figs. 1 and 2, it is desirableto have control over the parameter Γ = g / √ g g .In experiments, g jk may be tuned with Feshbach res-onances [66], which have been demonstrated for Rb– Rb [67–69], K– K [70, 71], and Rb– K [21] in-teractions. However, ground-state MQVs can also beobtained in the Rb– K BEC without employing Fesh-bach resonances. To demonstrate this for an axially uni-form system, we use the bare s -wave scattering lengths a /a B = 99 [72], a /a B = 60 [73], and a /a B =163 [18], where a B is the Bohr radius, and accordinglyset g /g = 1 .
29 and Γ = 2 .
27. The remaining parame-ters are fixed after the Rb– K experiment of Ref. [23].Figure 3 shows the resulting ground states at two dif- | Ψ | (cid:0) Rb (cid:1) | Ψ | (cid:0) K (cid:1) Arg (Ψ ) (a) Γ = 0 . . . . π A r g ( Ψ ) x y a r a r (e) E n e r g y ( un i t s o f N ¯ h ω ) V V E T T Γ Figure 1. Emergence of a ground-state two-quantum vortexin a rotating, mutually repulsive two-species Rb– K BEC.(a)–(d) Atomic densities | Ψ | and | Ψ | and the complexphase of the order parameter Ψ in the ground state at theindicated value of the interspecies interaction strength Γ = g / √ g g . In all four cases, Arg (Ψ ) ≡ const (not shown).(e) Interspecies interaction energy E = g R | Ψ Ψ | d r ,kinetic energies T j = ~ R |∇ Ψ j | d r/ m j , and trap ener-gies V j = m j ω j R r | Ψ j | d r/ m /m = 0 . g /g = 4, ω /ω = 10, N /N = 1, Ω /ω = 0 .
4, and U = ( g + g ) m N / ~ =300. The length unit is a r = p ~ /m ω . ferent rotation frequencies, Ω /ω = 0 . .
8. AtΩ /ω = 0 . | Ψ | (cid:0) Rb (cid:1) | Ψ | (cid:0) K (cid:1) Arg (Ψ ) (a) Γ = 0 . − . − . − . π A r g ( Ψ ) x y a r a r Figure 2. Formation of a ground-state vortex with specieswisequantum numbers h κ , κ i = h , i in a rotating, mutuallyattractive Rb– K BEC. (a)–(d) Atomic densities | Ψ | and | Ψ | and the complex phase of Ψ in the ground state at theindicated value of Γ = g / √ g g ; Arg [Ψ ( r, φ )] ≃ φ forall Γ (not shown). Here g /g = ω /ω = N /N = 1,Ω /ω = 0 .
7, and U = 50. π A r g ( Ψ ) | Ψ | (cid:0) Rb (cid:1) | Ψ | (cid:0) K (cid:1) Arg (Ψ ) (a) Ω = 0 . ω (b) Ω = 0 . ω x y a r a r Figure 3. Ground states of a Rb– K BEC with interactionparameters Γ = 2 .
27 and g /g = 1 .
29 corresponding tothe unmodified scattering lengths in a highly prolate trap,shown for two different rotation frequencies. In both states,Arg (Ψ ) ≡ const (not shown). Furthermore, ω /ω = 2 . N /N = 0 .
27, and U = 2800 after the experiment of Ref. [23]. harmonic trap potentials [74–82]. IV. PSEUDOSPIN TEXTURES
In this section, we analyze our results in thepseudospin-1 / n tot = 0, we define the local unit-lengthpseudospinˆ s ( r, φ ) = 1 n tot ( r, φ ) X jk Ψ ∗ j ( r, φ ) σ jk Ψ k ( r, φ ) , (3)where σ is a vector of the three Pauli matrices. Now con-sider a h κ , κ i vortex about which the atomic densitiesare locally axisymmetric. After transforming to shiftedpolar coordinates ( r ′ , φ ′ ) with the vortex core at r ′ = 0,we can write Ψ j , for small r ′ , in terms of a spin rotation Z and a U (1) gauge transformation acting on a unit-lengthreference spinor χ ∈ C :Ψ j ( r ′ , φ ′ ) = | Ψ j ( r ′ ) | e i ( κ j φ ′ + C j ) (4)= p n tot ( r ′ ) e i κ g φ ′ X k Z jk ( κ s φ ′ ) χ k ( r ′ ) , where κ s = κ − κ and κ g = κ + κ are integers thatdetermine, respectively, the number of 2 π rotations of ˆ s about the unit vector ˆ z and the number of π windingsof the U (1) gauge along a contour enclosing the core, Z ( κ s φ ′ ) = exp ( − iκ s φ ′ σ z / C j ∈ R are constants.Figure 4 shows ˆ s for some of the ground states inFigs. 1–3. The h , i vortex in Fig. 1(d) is interpretedas a doubly quantized skyrmion [29, 37] located at thecircular interface of the two species, where | Ψ | = | Ψ | .Since κ s = −
2, ˆ s rotates by − π about ˆ z when the inter-face is traversed azimuthally; additionally, the projectionˆ z · ˆ s changes from − h , i vortex in Fig. 2(d), the U (1) gaugewinds by 3 π and the spin ˆ s by − π . However, becausenow ˆ z · ˆ s vanishes everywhere, this state is not a skyrmionbut instead corresponds to a singly quantized spin vortex .The defect is structurally similar to the so-called crossdisgyration in the fermionic superfluid He- A [85, 86].Finally, the states in Fig. 3 feature giant skyrmions with(a) κ s = − κ s = − V. CONCLUSION
In this study, we have demonstrated that two-speciesBECs in rotating harmonic traps are able to host ther-modynamically stable multiquantum (or giant) vortices.Such topological entities rarely exist in the ground stateand have thus been elusive in BECs, whereas they areobserved and useful in, e.g., mesoscopic superconductiv-ity [42–46]. In the present case, their stability is notinduced by elaborate external potentials [74–82, 87–89]but is an inherent property of the harmonically confined,mass-imbalanced two-species system: The giant vortexin the heavier species is stabilized by its coupling to thelighter, giant-vortex-free species.Experimentally, the presence of the MQV could be ver-ified, e.g., by measuring the orbital angular momentumusing surface wave spectroscopy [90–92] or by detectingthe κ j -dependent concentric density ripples that wouldform in free expansion [93]. Due to its ground-state na-ture, the MQV is expected to be highly reproducible, bc bcbc bc bc bc bcbc bc bcbc bc (a) (b)(c) −
10 2 4 a r . a r . a r ˆ z · ˆ s r/a r Figure 4. Pseudospin textures of a (a) two-quantumskyrmion [for the state in Fig. 1(d)], (b) single-quantumspin vortex [Fig. 2(d)], and (c) nine-quantum giantskyrmion [Fig. 3(a)]. The arrows represent the projectionof the local pseudospin ˆ s = P jk Ψ ∗ j σ jk Ψ k /n tot onto the xy plane. Here σ is a vector of the Pauli matrices and n tot = | Ψ | + | Ψ | . The dashed circle is the species interface,where | Ψ | = | Ψ | . The inset in (c) shows the z projection ofˆ s as a function of the radial coordinate r . long lived, and therefore amenable to extensive measure-ments.We also classified the discovered states into spin-skyrmion (coreless) and spin-vortex (cored) variants,both of which can be realized in a Rb– K BEC [18–23].The similarities of these vortices with fractional [94] andskyrmionic [95] vortex states in multiband superconduc-tors, as well as the rich possibilities for the creation andtuning of multispecies BECs [18–26], open a wide avenuefor exploring emergent physics in multicomponent quan-tum systems consisting of inherently nonidentical com-ponents.
ACKNOWLEDGMENTS
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