Geometric phases of a vortex in a superfluid
Rodney E. S. Polkinghorne, Andrew J. Groszek, Tapio P. Simula
GGeometric phases of a vortex in a superfluid
Rodney E. S. Polkinghorne, Andrew J. Groszek,
2, 3 and Tapio P. Simula Optical Sciences Centre, Swinburne University of Technology, Melbourne 3122, Australia ARC Centre of Excellence in Future Low-Energy Electronics Technologies,School of Mathematics and Physics, University of Queensland, St Lucia, QLD 4072, Australia ARC Centre of Excellence for Engineered Quantum Systems,School of Mathematics and Physics, University of Queensland, St Lucia, QLD 4072, Australia
We consider geometric phases of mobile quantum vortices in superfluid Bose–Einstein condensates. Haldaneand Wu [Phys. Rev. Lett. , 2887 (1985)] showed that the geometric phase, γ C = π N C , of such a vortexis determined by the number of condensate atoms N C enclosed by the vortex trajectory. Considering anexperimentally realistic freely orbiting vortex leads to an apparent disagreement with this prediction. We resolveit using the superfluid electrodynamics picture, which allows us to identify two additional contributions to themeasured geometric phase; (i) a topologically protected edge current of vortices at the condensate boundary, and(ii) a superfluid displacement current. Our results generalise to, and pave the way for experimental measurementsof vortex geometric phases using scalar and spinor Bose–Einstein condensates, and superfluid Fermi gases. The Aharonov–Bohm phase [1] and the Pancharatnam–Berry phase [2, 3] are classic examples of an ever growingfamily of geometric phases that influence many areas of physics[4, 5]. Topological invariants and geometric phases are closelyrelated [6, 7] and in two-dimensional systems [8] they havemajor consequences for prospecting future technologies suchas dissipationless conductors beyond quantum Hall systems[9] and topological quantum computers [10–13].Soon after Berry’s seminal result [3], Haldane and Wu [14]considered a quantum vortex being adiabatically transportedalong a closed path C in a two-dimensional superfluid and con-jectured that as a consequence the system acquires a geometricphase γ C = π N C , where N C is the number of atoms enclosedby the path of the vortex. Perhaps surprisingly, this profoundresult remains to be confirmed by experiments. Thouless, Ao,and Niu further showed that the geometric phase of such a vor-tex is related to the transverse force on a vortex [15, 16], whichis further linked to the vortex mass in a superfluid [17–20].Quantum vortices in superfluid helium are hard to con-trol and interrogate due to their subnanometer-scale vortexcore structure being beyond optical resolvability. By contrast,in atomic Bose–Einstein condensates (BECs) and superfluidFermi gases, quantised vortices can readily be both guided andimaged using lasers. Indeed, vortices in BECs [21, 22] andsuperfluid Fermi gases [18] have been observed to undergoadiabatic periodic orbital motion [23] and are therefore wellsuited for designing experiments to study the geometric phase,transverse force, and the vortex mass in a superfluid.Here we show that a straightforward application of the resultby Haldane and Wu [14] to experimentally realistic superfluidsleads to a seeming contradiction. To resolve this issue, we de-ploy a vortex electrodynamics description of two-dimensionalsuperfluids and show that two additional sources of geometricphase emerge in such systems. These are (i) a topologically pro-tected vortex edge current , and (ii) a superfluid displacementcurrent . Properly accounting for these additional contributions,we recover the Haldane and Wu prediction, opening a pathwayfor experimental measurements of the vortex geometric phasesin a superfluid.Let us begin with a demonstration of the result derived byHaldane and Wu [14]. They considered a superfluid order parameter of the form ψ ( r ) = | ψ ( r ) | e iS ( r ) with a real valuedphase function S ( r ) = arctan( x − x v , y − y v ) that describes aquantised vortex in two-dimensional space whose position attime t is parametrised by R ( x v , y v , t ). When the vortex phasesingularity R is transported along a closed path C that encircles N C atoms of the superfluid, the geometric phase γ C = i (cid:73) C (cid:104) n ; R |∇ R | n ; R (cid:105) dR = π N C , (S1)where | n ; R (cid:105) is an eigenstate parametrised by the vortex position R [14]. Using Stokes’ theorem the line integral in Eq. (S1) mayalternatively be expressed as an areal integral γ C = (cid:82) A Ω n ( R ) · d a , in terms of the Berry curvature Ω n ( R ) = ∇ R × A n ( R ) , (S2)where A n = i (cid:104) n ; R |∇ R | n ; R (cid:105) is the Berry connection. We brieflymention the subtlety of gauge invariance of the Berry phase,which is not strictly satisfied in this vortex problem since inthis formulation the vortex state ψ is not an eigenstate of thesuperfluid evolution operator due to the embedded dynamicsof the Bogoliubov phonons. Nevertheless, we proceed by as-suming an approximate gauge invariance and associate themacroscopic condensate wavefunction ψ ( r ; R ) with the eigen-state | n ; R (cid:105) in Eq. (S1).We first consider a method of imprinting the vortex arounda closed trajectory, see the first column in Fig. S1, the detailsof which can be found in the Supplemental Material [24]. Thisyields a vortex geometric phase in agreement with the predic-tion by Haldane and Wu [14], and this result is compared witha direct simulation of the Gross–Pitaevskii equation (GPE) inthe second column. Frames S1(a) and (b) show the probabilitydensity | ψ ( r ; R ) | mapped onto the color intensity with blackcorresponding to zero density and the maximum density oc-curring near the centre of each image. The location R of thevortex core is visible as a black hole. The white circle showsthe trajectory C of the vortex, initially placed at the locationof the white marker. The colors in (a) and (b) correspond todifferent values of the phase function S as marked on the edgesof the image (a). Figure S1(d) shows the accumulated Berrycurvature, Eq. (S2), for the imprinted vortex, while the accumu-lated total geometric phase, Eq. (S1), is shown in Fig. S1(c) as a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n - 5.0 - 2.5 0.0 2.5 5.0 - 5.0- 2.50.02.55.0 - 5.0 - 2.5 0.0 2.5 5.0- 5.0- 2.50.02.55.0 t (2 ⇡/! v )
Imprinted vortex
Haldane & Wu theory
Imprinted vortex
2, where m is the mass of an atom, g is the coupling constant, ω osc - 5.0 - 2.5 0.0 2.5 5.0 - 5.0- 2.50.02.55.0 - 5.0 - 2.5 0.0 2.5 5.0- 5.0- 2.50.02.55.0 ( ⇡ N C )
100 is a dimensionless moat amplitude, σ = . a osc is the waist of the moat and R moat = . a osc is the moat radius.The motivation for the moat is to engineer a controllable phasereference on the outside of the inner harmonically trapped partof the condensate. Similar potential landscapes were employedin the experiments by Eckel et.al. [26]. The resulting conden-sate density and phase from the GPE are shown in Figs S2(a)and (b) for two instants in time, quantified by the vertical linesin frame (e). The moat potential for this calculation is chosensuch that there is no phase accumulation in the outer conden-sate annulus, which therefore defines a stable phase reference.The accumulated Berry curvatures corresponding to (a) and(b) are shown in (c) and (d), respectively. Similarly to Fig. S1,the path of the inner vortex is shown in (a)-(d). In this case,a second vortex with opposite sign of circulation is found toreside within the moat [24] and the path that it traces is shownin (a)-(d) by the outer dashed arcs. The presence of this an-tivortex in the moat is a topological necessity and we identifyits motion as a topological vortex edge current . We emphasisethat the inner part of the BEC (inside the moat) is qualitativelyequivalent (although having quantitatively different number ofatoms) to the whole condensate simulated in Fig. S1, wheretopological charge conservation also requires the existence ofan antivortex (not shown) in the periphery of the system.When the contributions to the Berry phase, shown inFig. S2(e), of both the inner vortex (orange line with a pos-itive slope) and the antivortex in the moat (pink dashed linewith a negative slope) are accounted for (blue line with a nega-tive slope) an excellent agreement with the Haldane and Wuresult [14] is recovered. The inner vortex covers a smallerarea (small number N inner of atoms) and accumulates a positiveBerry phase. The moat vortex travels in the same directionas the inner vortex but has an opposite sign of circulation andtherefore it accumulates a negative Berry phase whose magni-tude is greater because the number of atoms N outer it encirclesis greater. The sum of these two contributions, due to the vortexand the (anti)vortex edge current, equals − π times the numberof atoms, N annulus = N outer − N inner , within the (cyan) annulus inFig. S2(d) swept by the vortices. Note that in general, the innervortex and the moat vortex do not travel at the same orbitalangular frequency.This observation prompts an interpretation in terms of two-dimensional vortex electrodynamics [20]. In this picture theelectric and magnetic fields are defined as E sf = ρ s v s × e z and B sf = (cid:126) mg ∂ S ∂ t e z , (S5)where ρ s = m | ψ ( r ) | , v s = (cid:126) ∇ S / m is the superfluid velocity.The superfluid Ampere–Maxwell law is ∇ ⊥ × B sf = µ v j v + µ v (cid:15) v ∂ E sf ∂ t , (S6)where µ v and (cid:15) v are the superfluid vacuum constants and j v is the vortex current density [20]. The full set of superfluidMaxwell equations are coupled with the exact vortex equationof motion [27] that in the superfluid electrodynamics picturereplaces the Lorentz force law. We then associate the Berry curvature with the superfluid magnetic field via B sf ˆ = m π Ω n , (S7)such that the closed loop Berry phase can be viewed as asuperfluid magnetic flux γ C = π m (cid:90) A B sf · d a , (S8)where the mass of the atom is interpreted as the magnetic fluxquantum.The result in Fig. S2 was obtained using a judicious choiceof the chemical potential µ such that the time derivative of theradial phase gradient across the moat, the last term µ v (cid:15) v ∂ t E sf in Eq. (S6), is zero and the moat vortex orbital frequencyequals that of the inner vortex. The Berry phase in this caseaccumulates solely due to the total vortex current j v of the twovortices in the system. Extending this to arbitrary number ofvortices yields γ HW = π (cid:88) v sign( v ) N v ( v ) , (S9)where N v ( v ) is the number of atoms swept across by the vor-tex v , and sign( v ) is + − µ , we consider the case where the vortex current j v is set to be zero. This corresponds to a vortex-free superfluid.Figure S3(a) shows a ground state BEC in the moat potentialwhere the effective local chemical potential in the condensateannulus outside the moat is zero. The inner part of the conden-sate has a nonzero chemical potential (with respect to µ ) suchthat there is a chemical potential difference in the radial direc-tion across the moat. The result is a superfluid displacementcurrent, the last term in Eq. (S6), illustrated with white arrowsin Fig S3(a). Figures S3(b), (c) and (d) show, respectively, theaccumulated Berry phase, the radial phase S ( r ) at three differ-ent times, and the magnitude of the effective superfluid electricfield, which is localised at the edge of the inner condensate,inside the moat. Figure S3(e) shows the accumulated Berryphase as a function of time (blue markers) and the predictedsuperfluid displacement current (solid sloping line) accordingto the last term in Eq. (S6). The three vertical lines correspondto the times sampled in (c) and (d).We note that as the condensate phase S ( t ) inside the moatperiodically cycles through its range [ − π, π ), both the directionof the electric field and the rate of change of its amplitudechange signs, resulting in a constant rate of Berry phase accu-mulation (effective magnetic flux) [24]. In other words, thesteadily changing condensate phase in the central region resultsin a uniform superfluid magnetic field B sf , which is understoodto be induced by the superfluid displacement current, the lastterm in Eq. (S6), localised within the moat as illustrated usingthe white arrows in Fig. S3(a). - 5.0 - 2.5 0.0 2.5 5.0- 5.0- 2.50.02.55.0 - 5.0 - 2.5 0.0 2.5 5.0- 5.0- 2.50.02.55.0 ( a )
143 and are similar to Fig. S2(a) and (c),respectively, except that the vortices are absent here. The colorbar inFig. S2(d) applies to Fig. S3(b) as well. The white arrows illustrate thesuperfluid displacement current that is localised in the moat. Frame(c) shows three samples of the evolution of the radial condensatephase across the (cylindrically symmetric) moat. Frame (d) shows thecorresponding amplitudes of the superfluid electric field (the verticalaxis has been scaled (divided) by a factor of 1000 for the sake ofpresentational clarity). The frame (e) shows the accumulated Berryphase as a function of time with the three vertical lines specifying thesampling times used for the data in frames (c) and (d). SupplementalMovie S3 shows the evolution of images (a)-(d) [24].
The agreement between the theory and the numerical experi-ments corroborate the vortex electrodynamics picture wherebythe Berry curvature that is associated with the superfluid mag-netic field has two sources: (i) a current of quantised vortices j v , and (ii) the displacement current ∂ E sf /∂ t due to the spatio-temporal condensate phase evolution. Once both of these termsare accounted for, the apparent disagreement between the re-sult of Haldane and Wu [14] for the vortex Berry phase andthat predicted by time-evolution of the non-linear Schrödingerequation, see Fig. S1, can be readily understood.Our numerical results have verified that the prediction byHaldane and Wu, under the assumptions made in [14], is cor-rect. However, the way those assumptions apply to the bound-ary conditions of a realistic superfluid order parameter is quitesubtle, and is obscured by the common but unphysical assump-tion of an infinite condensate of uniform density. In particu-lar, an infinitesimal perturbation to a ground state superfluidwave function with a chemical potential µ can transform adynamical phase rotating uniformly at infinity as e − i µ t / (cid:126) , intoa geometric phase associated with an edge current of vorticeswith static condensate phase at infinity, or into any combinationof dynamical and geometric phases depending on the specificdetails of the boundary condition that is realised. This ambiguity between the geometric and dynamical phasesin this interacting many-particle system bears similarity to theduality between particles and fields. The traditional interpre-tation of a superfluid Bose–Einstein condensate is that thesuperfluid atoms are particles, and the vortices are a feature(akin to magnetic flux) embedded in the order parameter field,and that each atom encircling a vortex picks up a 2 π phasewinding. But an alternative interpreation is possible, wherethe vortices are emergent particles, and the condensate wavefunction comprising the atoms is a dynamical gauge field thatmediates the vortex interactions via phonons of the superfluid.Such particle-vortex duality has been extensively discussedwithin the context of condensed matter systems [28–35], wheretypically the particle substrate is electrons, and the geomet-ric phase occurs due to Aharonov–Bohm type effects whencharged particles loop around singular magnetic vector poten-tials. In a superfluid, the dual picture assigns each vortex acharge equal to its quantised circulation, and the gradient andthe rate of change of the order parameter phase are identifiedwith effective electric and magnetic fields [20]. When the re-sults in Figs S2 and S3 are viewed through this window, thedynamics of the condensate phase is associated with a mag-netic field whose flux is quantised with the mass of an atomcorresponding to a quantum of flux, and the edge vortex isassociated with a current of charged particles flowing aroundthe superfluid magnetic field generated by the atoms that thevortices encircle.Interestingly, in the superfluid electrodynamics picture thevortex in a BEC, see Fig. S1(a), realises a dynamic Corbinodisk geometry [36], and opens a window for studies of aplethora of vortextronic applications familiar from electronicsystems [37–41]. Considering two vortices initially located ondiametrically opposite sides of the condensate, each of whichthen orbits half a circle, results in a vortex exchange phase of2 π times an integer number of atoms, reflecting the bosonicorigin of the vortices. These results also generalise to Fermi su-perfluids and spinor Bose–Einstein condensates. In the formercase [18, 42] Cooper-like pairing results in the effective vortexcharge being κ/ κ forthe bosonic scalar superfluid, while in the latter case the frac-tional vortices are anyons that in general acquire a non-abelianexchange Berry phase upon braiding [13, 43].In summary, we have studied geometric phases of a vortexin a superfluid having identified two new contributions: (i) avortex edge current, and (ii) a superfluid displacement current.Our results open the path for experimental measurements ofthese geometric phases of vortices in superfluids. ACKNOWLEDGMENTS
This work was performed on the OzSTAR national facility atSwinburne University of Technology. The OzSTAR programreceives funding in part from the Astronomy National Collab-orative Research Infrastructure Strategy (NCRIS) allocationprovided by the Australian Government. The simulations madeextensive use of the open source libraries Optim.jl [44] andDifferentialEquations.jl [45]. This research was funded by theAustralian Government through the Australian Research Coun- cil (ARC) Discovery Project DP170104180 and the FutureFellowship FT180100020. [1] Y. Aharonov and D. Bohm, Physical Review , 485 (1959).[2] S. Pancharatnam, Proc. Indian Acad. Sci. , 398 (1956).[3] M. V. Berry, Proc. R. Soc. Lond. A , 45 (1984).[4] E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen,and E. Karimi, Nature Reviews Physics , 437 (2019).[5] F. Wilczek and A. Shapere, Geometric Phases in Physics (WorldScientific, 1989).[6] M. Nakahara,
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The numerical simulations were implemented using a di-mensionless form of the two-dimensional Gross–Pitaevskiiequation (GPE), i ψ t = (cid:16) −∇ + r (cid:17) ψ + C | ψ | ψ, (S10)where the length and time are expressed in units of the har-monic oscillator length a osc = (cid:112) (cid:126) / m ω osc , and the inversetrap frequency τ = /ω osc , respectively. The dimension-less condensate order parameter was normalised accordingto (cid:82) | ψ | dx dy =
1. In the two-dimensional formalism usedhere, the coupling constant is C = gN / ( (cid:126) ω osc a l z ) , where l z is the thickness of the sheet of condensate. All simulationsused C = l z =
10 µm thick condensate madeof Rb, this would correspond to a condensate of approxi-mately 77000 atoms in a radial harmonic oscillator trap offrequency ω osc = π ×
10 Hz. The moat potential had an ampli-tude a = σ = . a osc , and radius R moat = . a osc .When initialising the orbiting vortices, to constrain a vortexat r v , the order parameter was relaxed with a standard conju-gate gradient routine. After each step of relaxation, the value ψ ( r v ) was interpolated, and the order parameter was projectedonto the subspace where ψ ( r v ) =
0. The projection was per-formed by subtracting a Gaussian whose width was adjustedto cover the distance moved by the vortex in a relaxation step. - 5.0 - 2.5 0.0 2.5 5.00.00.51.01.52.02.5 x ( a osc )
Vortex states ψ GPE ( r , R ) were first obtained by solvingthe GPE. For each location R of the vortex along its or-bit, an imprinted condensate wavefunction ψ imprinted ( r , R ) = | ψ GPE ( r , R ) | e iS ( r ; R ) , where S ( r ; R ) = arctan( x − x v , y − y v ) wasconstructed. In words, the GPE phase functions S GPE werereplaced by the cylindrically symmetric phase functions whileretaining the true condensate densities. These vortex stateswere then used for calculating the geometric phases for theimprinted vortex case shown in Fig. 1 of the main text.
IV. EXTRACTING THE VORTEX BERRY PHASE
The vortex path C is sampled in time as [ R , R , . . . ], so theintegral in Eq. (1) of the main text can be approximated as γ C = (cid:88) j ( γ j + − γ j ) , (S11)where γ j + − γ j = − Im (cid:104) n ; R j + | n ; R j (cid:105) . (S12)For a condensate wavefunction ψ ( r , R ), this yields γ j = (cid:90) j − (cid:88) k = Im (cid:0) − ψ ∗ ( r , R k + ) ψ ( r , R k ) (cid:1) d r . (S13)The Berry phases shown in the main text are the γ j obtainedwhen the order parameter is substituted into Eq. (S13). AllBerry phases are normalised by N C = (cid:82) A | ψ ( r ) | d r , where A is the area bounded by C . The Berry curvatures are theintegrands Ω j = j − (cid:88) k = Im (cid:0) − ψ ∗ ( r , R k + ) ψ ( r , R k ) (cid:1) . (S14)Measuring the Berry phase requires knowledge of the con-densate phase in addition to the condensate density that canbe obtained by standard absorption imaging. This could beachieved for instance using propagation based phase retrievalmethods such as the Gerchberg–Saxton algorithm or the Pa-ganin’s method. A proposed protocol would then be to (i)measure the interference patterns as in [26] (ii) extract the con-densate phase using a phase retrieval method of choice, and(iii) use Eq. (S13) to calculate the Berry curvature and phase. V. VORTEX CORE ASYMMETRY
Figure S6 shows the effect of an asymmetric density on thegeometric phase accumulated by an orbiting vortex. The vortexcore has been displaced from the phase singularity. The effectis greatest when the core is displaced in the radial direction,along the white dashed line in Fig. S6(b), corresponding to theradial asymmetry that affects a vortex at the edge of the trap.
VI. DETECTING THE MOAT VORTEX
Detecting the location of the inner vortex in Figs 1 and 2of the main text is straightforward using standard numericalmethods of measuring the phase winding. The presence andlocation of the vortex is also obvious from visual inspection ofimages (a) and (b) in Figs 1 and 2. Since the phase (dark redcolor) is constant in the annulus outside the moat, the topolog-ical charge conservation requires there to be an antivortex ofopposite sign of circulation with respect to the inner visible vor-tex, somewhere within the moat region. However, determining - 3 - 2 - 1 0 1 2 3- 2- 10123 (b) / C
The numerical computations in this work were carried outwith version 1.5 of the Julia programming language, andthe scripts that were used for generating the figures are in-cluded as separate supplemental material files. The figuresmay be reproduced by installing Julia, then unpacking gpvor-tex.tgz to create the gpvortex directory. In that directory, run julia -e ’using Pkg; Pkg.instantiate()’ to down-load and install the necessary software libraries. Then, forexample, julia -l figone.jl will run the code in the file figone.jl that generates Fig. 1 of the main text. This willload simulation results from a file orbit.jld2 , which canbe reproduced by running orbit.jl (this will take sometime). Finally, the data can be inspected and other figuresplotted using the command line. The scripts rely on the library
Superfluids.jl to do the numerical computations.
VIII. SUPPLEMENTAL VIDEOS
There are five included animations (animated gif file for-mat). The Berry curvatures in the animations show the relativedistribution only, and the absolute color scale changes slightlybetween frames. The axis labels are omitted from the moviesfor the sake of visual clarity. The first four animate the snap-shot images shown in the main text. The animation SV5 showsthe condensate ground state in a rotating frame as a functionof the radial position of the vortex. The orbiting vortex trans-forms into an edge mode as the radius of its orbit approachesthe Thomas–Fermi radius of the condensate. When the vortexis near the condensate edge, the vortex core localised kelvonquasiparticle mode hybridizes with the lowest energy surfonquasiparticle, resulting in the observed ripples.•
SV1.gif:
Imprinted vortex case corresponding toFig. 1(a) and 1(d).•
SV2.gif:
GPE vortex case corresponding to Fig. 1(b)and 1(e). •
SV3.gif:
Moat vortex case corresponding to Fig. 2.•
SV4.gif:
One phase cycle of the displacement current inFig. 3(a),(b),(c) and 3(d).•