The Dynamics of a Tilted Vortex in a Trapped Quantum Fluid
TThe Dynamics of a Tilted Vortex in a Trapped Quantum Fluid
Chuanzhou Zhu , Mark E. Siemens , Mark T. Lusk Department of Physics, Colorado School of Mines, Golden, CO 80401, USA ∗ Department of Physics and Astronomy, University of Denver, Denver, CO 80208, USA ∗ (Dated: February 23, 2021)The nonequilibrium dynamics of vortices in 2D quantum fluids are shown to obey a kineticequation in which vortex tilt (ellipticity) is coupled to the gradient in background fluid density. Inthe absence of nonlinear interactions, a trapped, tilted vortex is analytically found to move in anelliptic trajectory that has the same orientation and aspect ratio as the vortex projection itself,allowing experimental measurement of vortex tilt via path observation. A combination of analysisand simulation is then used to show that nonlinear interactions cause the vortex tilt to precess, andthat the rate of vortex precession is once again mimicked by a precession of the elliptical trajectory.Both vortex tilt and rate of precession can therefore be inferred by observing its motion in a trap.An ability to anticipate local vortex structure is expected to prove useful in designing few-vortexsystems in which tilt is a ubiquitous, as-yet-unharnessed feature. Dark solitons [1] with orbital angular momentum havebeen widely studied in Bose-Einstein condensates (BEC)[2] and optical fluids [3]. Although these vortex quasi-particles typically have circular cross-sections in equi-librium, their non-equilibrium counterparts tend to beelliptical in both superfluid [4, 5] and optical [6, 7] set-tings. In fact, non-circular shapes are expected when-ever two or more vortices interact, as in the gener-ation/annihilation of vortex-antivortex pairs [7–9], themerging of co-rotating vortices [10–12], and the braidingof vortex pairs [13]. Such vortices do not move with theunderlying fluid, as in incompressible flows, nor can theirtrajectories be anticipated by accounting for the influenceof density gradients [14]. Ellipticity introduces two ad-ditional degrees of freedom that couple to the gradientsin the background quantum state , and a kinetic equationhas recently been derived that correctly incorporates thisand applies it to predict the motion of optical vortices inlinear media [7].In this paper, we elucidate the relationship betweentrap strength, nonlinear interaction, and the motion ofan isolated, tilted vortex in quantum fluids. We findthat a circular harmonic trap causes a tilted vortex tomove in an elliptical trajectory that, surprisingly, has thesame orientation and aspect ratio as the vortex projec-tion. Nonlinear interactions, on the other hand, inducea precession in the vortex tilt that is mimicked by ananalogous precession in the vortex trajectory.It will prove both mathematically convenient and phys-ically insightful to treat a 2D vortex as the projectionshown in Fig. 1. There a green disk represents a non-physical 3D structure that is projected onto the physical x - y plane along the Z axis. Tilt is then described byazimuthal orientation ξ and polar lean θ . The physicalvortex, shown in red, is thus viewed as the planar pro-jection of a 3D construct that is tilted with respect tothe physical plane. The line of projection is arbitrary,amounting to a choice of gauge in a theory with four de-grees of freedom: two position coordinates and two angles describing tilt.The wave function of a linear-core (LC) vortex withtilt angles, ξ and θ , offset from the center of the fluid by x , is given by ψ ( x, y ) = N e − ( x + y ) [( x − x ) a + yb ] , (1)where a = − cos ξ + i cos θ sin ξ and b = − sin ξ − i cos θ cos ξ , and N is a normalization factor. The enve-lope is a Gaussian function, in keeping with the groundstate of Bose-Einstein condensate (BEC) in a harmonictrap as well as the typical falloff for optical beams. Thefluid density, | ψ | , is plotted in Fig. 1(b), while the phasegradient of the state gives the fluid velocity plotted inFig. 1(c).We consider ψ ( x, y ) in Eq. (1) with ξ = ξ and θ = θ as the initial state in the time evolution. The evolvingquantum state, ψ ( x, y, t ) , is assumed to be governed bythe Gross-Pitaevskii equation (GPE): i∂ t ψ = (cid:16) H s + β | ψ | (cid:17) ψ, (2)where the single-particle Hamiltonian is H s = −
12 ( ∂ xx + ∂ yy ) + 12 (cid:0) x + y (cid:1) , (3)and β is the nonlinear interaction parameter. This modelfor quantum fluids can be applied to 2D BEC by settingthe atomic mass, the trap frequency, and ~ as character-istic units, regarding t as the time and β as the prod-uct of atom number and two-atom contact interactionstrength [15]. It also captures the dynamics of paraxialoptical fluid propagating through a nonlinear medium bytreating the wave number and dielectric trap strength [16]as characteristic units, interpreting the Poynting vectoraxis as time, and identifying β as the non-dimensionalthird-order susceptibility [17]. Because analytical solu-tions do not exist for non-equilibrium vortex dynamics,the GPE is solved numerically [18, 19]. We subsequently a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b -2.5 2.5-2.52.5 00 -2.5 2.50x xy 0.270 - L L z z ’ - p r o j e c t i o n x T il t e d V o r t e x z’ li n e o f p r o j e c t i o n (a) (b) (c) FIG. 1:
Tilted vortex with azimuthal angle ξ = 150 ◦ and polarlean θ = 60 ◦ located at x = 0 . and y = 0 in a quantumfluid. (a) A tilted vortex (red circle) formed by projectinga 3D circular vortex (green disk) onto the 2D plane along Z axis. (b) The fluid density profile | ψ | , where the shapeof the tilted vortex is highlighted by the white contour, theazimuthal angle ξ is denoted by the white arrow, and thepolar lean θ is given by cos ( θ ) = L /L . (c) The phase of ψ ,with the local fluid velocity shown with black arrows. use a recently derived algebraic methodology [7] to de-termine the tilt angles, ξ and θ .A variational argument [7] can then be used to showthat the total vortex velocity is described by ~v = ~v ϕ + ~v ρ , (4)where ~v ϕ = ∇ ⊥ ϕ bg , ~v ρ = − Λ (ˆ z × ∇ ⊥ log ρ bg ) . (5)Here ϕ bg and ρ bg are the phase and magnitude of thebackground field, ψ bg = ρ bg e iϕ bg , obtained by dividingout the contribution from the vortex itself. Their gra-dients each contribute to the vortex velocity denoted by ~v ϕ and ~v ρ , respectively. The influence of vortex tilt iscaptured by the 2D tensor, Λ . In the coordinate frameof Fig. 1, its elements are Λ xx = cos θ cos ξ + sec θ sin ξ, (6) Λ yy = sec θ cos ξ + cos θ sin ξ, (7) Λ xy = Λ yx = − sin(2 ξ ) sin θ θ . (8)This shows that the tilt of a vortex affects its motion be-cause it is coupled to the local gradient in the background density, a result valid for both linear and nonlinear quan-tum fluids. We are now in a position to explore the role oftrap structure and nonlinear interaction in characterizingand controlling vortex motion. Case 1: Linear Quantum Fluid.
A single, harmon-ically trapped vortex within a non-interacting quantumfluid ( β = 0 in Eq. (2)) moves within a background fieldgenerated by the trap by itself: ψ bg = e − ıt e − ( x + y ) . (9)Since this has no phase gradient, any vortex motion issolely due to an evolving gradient in the background den-sity. The background field can be applied to Eq. (4) toobtain the following prediction of vortex velocity: v x = − x (sin t + cos t cos ξ sin θ sin ξ tan θ ) v y = x cos t sec θ (cos ξ + cos θ sin ξ ) . (10)Here ξ and θ describe the initial vortex tilt. The ex-pressions for velocity can be easily integrated to obtainthe vortex trajectory, { x v ( t ) , y v ( t ) } . To more easily in-terpret this trajectory, consider a linear transformationthat rigidly rotates the trajectory clockwise by the angle ξ : (cid:2) e x v ( t ) e y v ( t ) (cid:3) T = R (cid:2) x v ( t ) y v ( t ) (cid:3) T , (11)where the rotation matrix is R = (cid:18) cos ξ sin ξ − sin ξ cos ξ (cid:19) . (12)This results in the relationship [ e x v ( t )] ( x cos θ ) + [ e y v ( t )] x = sin ξ + cos ξ cos θ , (13)which implies that the vortex trajectory is a fixed el-lipse with azimuthal orientation ξ tr = ξ and aspect ra-tio cos θ tr = cos θ . In addition, an explicit evaluation ofthe vortex tilt gives that the azimuthal angle and polarlean are both independent of time—i.e. ξ ( t ) = ξ and θ ( t ) = θ . This is shown in Fig. 2 for a specific initialvortex tilt. If the vortex is initially untilted, the resultingcircular orbit is consistent with earlier work [20, 21]. Case 2: Nonlinear Quantum Fluid.
We next con-sider the dynamics of a tilted vortex placed off-center ina nonlinear quantum fluid. Although the vortex kineticsare still described by Eq. (4), nonlinear interaction hasa significant effect on the background field. Eq. (2) issolved numerically with vortex position obtained using aroot finder. The evolving vortex tilt is then obtained us-ing the previously developed algebraic methodology [7].Strikingly, we find that the nonlinear interaction gen-erates a precession in the vortex azimuthal orientation,which is evident in the time slices of the fluid density,shown in Fig. 3(b,c,d), where highlighted white contours t t t t t t (b)(c)(d)(a) xyy -2.5002.50 y -2.5 0 2.5 xy FIG. 2:
Dynamics of a tilted vortex in a linear quantum fluid( β = 0 ). The initial state is as shown in Fig. 1 with x = 0 . , y = 0 , ξ = 150 ◦ , and θ = 60 ◦ . (a) The vortex trajectoryanalytically derived in Eq. (10). The vortex tilt ξ ( t ) = ξ andpolar lean θ ( t ) = θ are stationary as the vortex moves, andits path is an ellipse characterized by these same tilt angles,i.e. ξ tr = ξ and θ tr = θ . (b,c,d) The fluid density profilesfor t = π/ , t = π , and t = 3 π/ , respectively, with specificcontours highlighted to show that ξ and θ are stationary. show the vortex shape. As shown in Fig. 3(e), the vor-tex precesses clockwise as its azimuthal orientation ξ ( t ) (green line) is a linearly decreasing function of time, whilethe polar lean θ ( t ) (yellow line) is constant over the en-tire simulation. A typical vortex trajectory is shown inFig. 3(a), which exhibits several new features due to non-linear interactions. The vortex now moves on an ellipticaltrajectory that is precessing, quantified by its evolvingazimuthal orientation, ξ tr ( t ) , plotted by the red-blue linein Fig. 3(e). In addition, these ellipses grow and shrink,a new type of slow breathing mode. Nonlinear interac-tion is therefore manifested in cyclic outward and inwardspirals, and we have verified that a circular counterpartis observed even when the vortex is untilted, detailed inSupplementary Material. The comparison between ξ ( t ) and ξ tr ( t ) demonstrates that the rate of precession ofthe trajectory mimics that of the vortex itself: Whenthe vortex trajectory is in the smallest ellipse, the vortexand trajectory precess at the same rate just as in a linearfluid. This implies that structural character of the vortexcan be quantified by observing the shape of the smallesttrajectory. As the size of the ellipse glows, nonlinear ef-fects cause the vortex precesses faster than the trajectorydoes.In principle, the kinetic model of Eq. (4) should beable to predict even such complex vortex motion. Toverify this, the background field was obtained at eachtime step by numerically dividing out the vortex coreusing a methodology detailed in the Supplementary Ma-terial. Eq. (5) is then used to construct contributions (a) OutwardInward (e) () T il t A n g l e s TimeOutward Inward xyy -2.5002.50 y (b)(c)(d) t t t y x Start -2.5 0 2.5 t t t t t t Idealized Analytical Model for
FIG. 3:
Dynamics of a tilted vortex in a nonlinear quantumfluid.
Nonlinear interaction β = 1 and initial state is as shownin Fig. 1 with x = 0 . , y = 0 , ξ = 150 ◦ , and θ = 60 ◦ . (a)The vortex trajectory for time period t = 0 to , in whichthe vortex spirals out (red) before spiraling back in again(blue). (b,c,d) The fluid density profiles for t = 16 . , t = 51 ,and t = 85 . , respectively, corresponding to the black dots in(a,e), with specific contours highlighted to show the precessionof vortex orientation. (e) Time evolution of azimuthal andpolar angles of the vortex, ξ ( t ) (green) and θ ( t ) (yellow),over one cycle of outward/inward spiraling shown along withevolution of azimuthal orientation of the trajectory, ξ tr ( t ) (red-blue), and the idealized analytical model, Eq. (19), forthe evolution of vortex orientation (magenta), with h h i =0 . . to the vortex velocity associated with background gra-dients in the phase, ~v ϕ , and density, ~v ρ . The magentaarrows in Figs. 4(a,b,c) depict these velocity contribu-tions as well as their sum, ~v . In each case, the vectorsoriginate at the position of the vortex. Figs. 4(b) showsthat the extremum of the background density clearly de-viates from the trap center, and the entire backgroundfield actually spirals cyclically in sync with the vortex.The background phase gradient shown in Figs. 4(c) isparticularly interesting because it changes only slightlyover the entire domain–i.e. it has a global character. Allof these features are departures from what is observed forthe linear media of Eq. (10), where the background fieldis on-center, without any phase gradient, and the vortexvelocity is purely from ~v ρ .Gradients of the background fields (Figs. 4(b, c)) wereused to calculate a radial velocity that was subsequentlyintegrated to obtain the prediction for the radial evolu-tion of the vortex (red) shown in Fig. 4(d). This com-pares favorably with the vortex position measured witha root finder (blue). In both cases, low-pass filtering toremove rapid cyclical oscillations (darker curves) helps tomore easily compare prediction with measurement overlonger time scales. The background fluid density gra-dient is an essential contributor to the vortex velocity,and this was quantified in terms of the mean value of itscontribution to the total vortex velocity of Eq. (5): (cid:28) | ~v ρ | p | ~v ϕ | + | ~v ρ | (cid:29) = 36% . (14)Panel (d) also shows curve (black) of what would be pre-dicted for the radial position if the coupling between tiltand fluid density were not accounted for. This was pro-duced by setting Λ = in Eq. (5), and its poor predictiondemonstrates how crucial it is to account for the newlyidentified coupling.While new kinetic equation quantitatively predicts vor-tex motion, precession of the vortex can be explainedwith an idealized analytical model using variational anal-ysis. Consider a tilted vortex placed at the center of atrapped quantum fluid and assume that the fluid wavefunction preserves the linear-core vortex structure: ψ ( x, y, t ) = N e − i t e − x y ψ LC ( x, y, t ) . (15)Here N is a normalization factor, and the wave functionof the linear-core vortex is ψ LC ( x, y, t ) = − x cos ξ ( t ) − iy cos ξ ( t ) cos θ ( t ) − y sin ξ ( t ) + ix cos θ ( t ) sin ξ ( t ) , (16)with ξ ( t ) and θ ( t ) to be determined through varia-tional analysis. Substituting ψ ( x, y, t ) in Eq. (15) intothe GPE given by Eq. (2), we obtain i∂ t ψ LC ( x, y, t ) = β | ψ ( x, y, t ) | ψ LC ( x, y, t ) . Note that the β | ψ ( x, y, t ) | term has a local minimum at the vortex center and hencecan be treated as a self-trapping potential. If we considera nearly circular vortex with small θ ( t ) and subtract theirrelevant overall portion from the self-trapping potential,then the self-trap can be idealized as a bucket potentialformed by a flat-bottomed cylinder with trap depth h ( t ) .Under this approximation, the field local to the vortex isgoverned by i∂ t ψ LC ( x, y, t ) = − βh ( t ) ψ LC ( x, y, t ) , (17)where we estimate that h ( t ) = max( | ψ ( x, y, t ) | ) / .Substituting ψ LC ( x, y, t ) given by Eq. (16) into Eq. (17)and assuming that θ ( t ) is small, we obtain d t ξ ( t ) = − β h ( t ) . (18) (c) (b) (a) -2.5-2.5 00 MeasuredPredicted
Time V o r t e x R a d i a l P o s i t i o n Predicted ( ) (d) y xx x1.00.80.60.40.20 50 100 150 (Eq. (4))(Fig. 3)
FIG. 4:
Prediction versus measurement of the radial positionof a vortex.
The simulation of Fig. 3 is used to assess the ac-curacy of the kinetic model of Eq. (4). (a,b,c) t = t = 85 . ,corresponding to Fig. 3(d): fluid density | ψ | , backgroundmagnitude ρ bg , and background phase φ bg with magenta ar-rows denoting two times of velocities ~v , ~v ρ , and ~v ϕ , respec-tively, and a white dot denoting the center of backgroundfield. (d) Prediction of vortex radial position (red), obtainedby integration of Eq. (4) compared to the position measured(blue) using a root finder, as used in Fig. 3(a), to identify theevolving zero of the wave function. Also shown is the pre-dicted radial position if coupling with tilt is not accountedfor (black) by setting Λ = in Eq. (5). Darker curves are theresult of low-pass filtering. The magenta line in Fig. 3(e) is plotted for ξ ( t ) = − β h h i t + ξ (0) , (19)where the average trap depth, h h i , is estimated fromour numerical simulation. Combined with the observa-tion that vortex evolution depends only weakly on θ (de-tailed in Supplementary Material), the ansatz of small θ amounts to an easily satisfied requirement that there beat least some ellipticity to the self trap. This perspectiveis verified by the quantitatively reasonable estimate forthe rate of change of vortex orientation shown by the ma-genta line in Fig. 3(e). This variational analysis can alsobe applied to the case of a vortex placed at the centerof a freely-expanding quantum fluid, detailed in Supple-mentary Material.In conclusion, we have shown that it is possible toquantitatively predict vortex trajectories in quantum flu-ids by accounting for the coupling between vortex tilt andthe background quantum state. This coupling is negligi-ble in regimes that are well-approximated as incompress-ible, but they are particularly relevant when the vortexhealing length is on the order of vortex separation in few-body systems. In the absence of nonlinear interactions,the background field depends only on the trap, and iso-lated tilted vortices move in an elliptical path that isself-similar to their own projection. Nonlinear effects al-low for richer dynamics, though, since the vortex cannow contribute to its own background field. The effectamounts to the vortex being able to influence its own mo-tion and tilt. This is relevant to optical quantum fluidsas well, where it is possible to trap vortices using mediawith radially varying dielectric character. Significantly,vortex tilt and its rate of precession are mimicked in thetrajectory observed, allowing these important local fea-tures to be measured with relative ease. This capability,in turn, is expected to be useful in developing on-the-flymanipulation of trap strength and atomic interaction asa means of controlling few-body vortex interactions suchas nucleation, annihilation, scattering, and braiding.The authors acknowledge useful discussions with Jas-mine Andersen and Drew Voitiv. We are grateful to theW.M. Keck Foundation and the National Science Foun-dation (DMR 1553905) for supporting this research. ∗ Electronic address: [email protected],[email protected][1] D. J. Frantzeskakis, Journal of Physics A: Mathematicaland Theoretical , 213001 (2010).[2] A. L. Fetter, Reviews of Modern Physics , 647 (2009).[3] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M.Gong, and X. Yuan, Light: Science & Applications , 90(2019).[4] A. T. Powis, S. J. Sammut, and T. P. Simula, Physical Review Letters , 165303 (2014).[5] P. C. Haljan, B. P. Anderson, I. Coddington, and E. A.Cornell, Physical Review Letters , 2922 (2001).[6] P. Zhao, S. Li, Y. Wang, X. Feng, C. Kaiyu, L. Fang, W.Zhang, and Y. Huang, Scientific Reports , 7873 (2017).[7] J. M. Andersen, A. A. Voitiv, M. E. Siemens, and M. T.Lusk, arXiv , 1 (2021).[8] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis,and B. P. Anderson, Physical Review Letters , 160401(2010).[9] S. W. Seo, B. Ko, J. H. Kim, and Y. Shin, ScientificReports , 4587 (2017).[10] C. Josserand and M. Rossi, European Journal of Mechan-ics B , 779 (2007).[11] P. Meunier, U. Ehrenstein, T. Leweke, and M. Rossi,Physics of Fluids , 2757 (2002).[12] S. Le Dizès and A. Verga, Journal of Fluid Mechanics , 389 (2002).[13] A. A. Voitiv, J. M. Andersen, M. E. Siemens, and M. T.Lusk, Optics Letters , 1321 (2020).[14] H. M. Nilsen, G. Baym, and C. Pethick, Proceedings ofthe National Academy of Sciences of the United Statesof America , 7978 (2006).[15] W. Ketterle, Reviews of Modern Physics , 1131 (2002).[16] T. Iadecola, T. Schuster, and C. Chamon, Physical Re-view Letters , 073901 (2016).[17] R. Boyd, Nonlinear optics (Academic Press, AmsterdamBoston, 2008).[18] X. Antoine and R. Duboscq, Computer Physics Commu-nications , 2969 (2014).[19] X. Antoine and R. Duboscq, Computer Physics Commu-nications , 95 (2015).[20] A. A. Svidzinsky and A. L. Fetter, Physical Review A , 063617 (2000).[21] A. A. Svidzinsky and A. L. Fetter, Physical Review Let-ters , 5919 (2000). upplementary Material:The Dynamics of a Tilted Vortex in a Trapped Quantum Fluid Chuanzhou Zhu , Mark E. Siemens , Mark T. Lusk Department of Physics, Colorado School of Mines, Golden, CO 80401, USA ∗ Department of Physics and Astronomy, University of Denver, Denver, CO 80208, USA ∗ (Dated: February 23, 2021) S1. THE OUTWARD AND INWARD SPIRALSOF A CIRCULAR VORTEX IN NONLINEARQUANTUM FLUID
The trajectory containing outward and inward spiralscan also be observed when the vortex is untilted withpolar angle θ = 0 , which is equivalent to saying that thevortex is circular in the 2D plane. With weak nonlin-earity ( β = 1 ), the circular vortex moves in a circulartrajectory plotted in Fig. S1, with the red line denotingthe outward spiral and the blue line denoting the inwardspiral. Note that the weak nonlinearity ( β = 1 ) doesnot induce a visible tilt and the vortex keeps its circularshape with θ ( t ) = 0 during the time evolution. We haveverified that this measured trajectory of a circular vortexis in agreement with the trajectory predicted by taking Λ = 1 in the vortex kinetic equation shown in Eq. (5) inthe main text. xy -1 Start
OutwardInward
FIG. S1: The trajectory of an untilted, circular vortex fortime period t = 0 to , in which the vortex spirals out (red)before spiraling back in again (blue). Interaction strength β = 1 and initial state is with x = 0 . , y = 0 , and θ = 0 . S2. PRECESSION OF AN ON-CENTER TILTEDVORTEX IN NONLINEAR QUANTUM FLUID
This section provides an extended discussion of theprecession of an on-center tilted vortex, which, in thetrapped nonlinear quantum fluid, is characterized by thetwo-color red and blue line in Fig. 3(b) in the main textand Eq.(19) in the main text. Here we provide additionaldetails on the numerical simulation and analytical ideal-izations of vortex precessions in trapped and untrappedquantum fluids with larger nonlinearity ( β = 20 ). Notethat although an on-center vortex does not move, its tiltdoes precess.As shown in Fig. S2(a), with the presence of a har-monic trap, the vortex precesses clockwise as its az-imuthal orientation, ξ ( t ) , decreases at a relatively con-stant rate. This precession is only weakly dependent onthe initial vortex lean θ , and the polar lean itself, θ ( t ) , isrelatively constant with time. These features are also ev-ident in the time slices of the field density, shown in Fig.S2(b), where highlighted white contours show the vortexshape and associated white arrows denote the evolvingazimuthal angle. In comparison with the numerical re-sult, the magenta line in Fig. S2(b) is plotted for theanalytical result estimated by Eq. (19) in the main text.Another idealization for explaining the dynamics of theon-center tilted vortex is to combine the effects of the trapterm and the nonlinear interaction term into a nonlocal,elliptical, and harmonic self-trap. The numerical resultsof Fig. S2(a) indicate that atomic interaction has a rela-tively small effect on the polar angle while its azimuthalorientation decreases at a relatively constant rate. Bothof these characteristics can be explained with a simpleidealization in which the nonlinear term in GPE is ab-sorbed into the trap strength using β | ψ | ≈ β | ψ LC , init | . (S1)Vortex evolution is then governed by a linear system withan elliptical trap, H ellip = − ~ m ( ∂ xx + ∂ yy ) + 12 mω (cid:0) x + γ y (cid:1) , (S2)where γ and ω are functions of β and the initial vor-tex tilt angles. The associated eigenmodes are productsof Hermite polynomials, and the initial condition is rea-sonably approximated as the sum of the lowest pair of a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b t t t t t ξ ( o ) θ ( o ) t t t t Analy ti cal (a)(c) (d) slope = 0 − − slope < 0 (b)t t ξ ( o ) θ ( o ) t -0.80.8-0.8 0.80 000 00.20.8 -0.80.004 0.0010 0 t t t t t t t t θ ( o ) ξ ( o ) Analy ti cal (e) (f) yy x x FIG. S2:
Dynamics of an on-center tilted vortex in a nonlin-ear quantum fluid.
Interaction strength β = 20 and initialazimuthal angle ξ = 150 ◦ . (a) Time evolutions of azimuthalangle ξ ( t ) and polar angle θ ( t ) with trap frequency ω = 1 ,where the numerical result is for θ = 10 ◦ and ◦ , and theanalytical estimate is with h h i = 0 . valid for small θ ;(b) Time sequence of fluid densities corresponding to blackpoints in panel (a), where white lines show density contoursand white arrows indicate ξ ( t ) ; (c, d) Time evolution of vor-tex tilt for an elliptical trap idealization with γ = 0 . ; (e, f)Time evolution of vortex tilt within a freely expanding quan-tum fluid without the harmonic trap term in Gross-Pitaevskiiequation, where the analytical estimate, Eq. (19) of the maintext, is with h (0) = 0 . valid for small θ . excited modes, | i and | i . The evolving field is thenof the form | ψ ( t ) i = N ( | i + αe ı ( ε − ε ) t | i ) . (S3)The ellipticity-induced difference in mode energies, ε − ε , results in a beating phenomenon that is seen as thevortex lean bobbing up and down as shown in Fig. S2(d).Likewise, the ellipticity-induced weighting coefficient, α ,imbalances what would otherwise be an azimuthal stand-ing mode, and the vortex orientation oscillates back andforth. The magenta lines in panel Figs. S2(c,d) empha-size that the rate of change of vortex orientation, ˙ ξ , isinitially constant and negative, while the rate of changeof polar lean, ˙ θ , is initially zero. The nonlinear inter-action in the GPE, though, amounts to a self-trap that rotates with the vortex, implying that the idealized tiltdynamics hold for all times in a Zeno-like manner. Thisexplains the trends observed in the numerical results ofFig. S2(a).Our simulation and analysis can also be applied to thecase of a vortex placed at the center of a freely-expandingquantum fluid without the harmonic trap term in Gross-Pitaevskii equation, with the result summarized in Fig.S2(e, f). As for the trapped quantum fluid, the vortexorientation precesses clockwise with a relatively constantspeed. Without a constraining trap, though, the preces-sion rapidly decelerates, because expansion reduces thefluid density, | ψ ( x, y, t ) | , and thus the effect of the non-linear term, β | ψ ( x, y, t ) | . The deceleration can be ana-lytically estimated by once again using a variational anal-ysis in which the tilt is described by unknown functionsof time: ψ ( x, y, t ) = N − i + t ) e i x y − i + t ) ψ LC ( x, y, t ) . (S4)Here ψ LC ( x, y, t ) is given by Eq. (16) in the main text.The effect of not having a trap is manifested in the factor / ( − i + t ) of Eq. (S4), the term that characterizes therate of decrease of fluid density. By following along thesame lines as for the variational analysis in the main text,we obtain ξ ( t ) = − βh (0) (cid:18) t t + arctan ( t ) (cid:19) + ξ (0) , (S5)where h (0) = max( | ψ (0 , x, y ) | ) / . This analytical re-sult is plotted by the magenta line in Fig. S2(e), whichcharacterizes the deceleration of the precession due to theremoval of trap. S3. PREDICTION OF VORTEX VELOCITYFROM VORTEX KINETIC EQUATION
This section sketches our techniques for calculating thetwo velocity components, ~v ϕ and ~v ρ in Eq. (5) of themain text, that are plotted in Fig. 4 in the main textfor the case of nonlinear quantum fluid. Defined by Eq.(6) in the main text, ~v ϕ and ~v ρ depends on the gra-dients of the phase ϕ bg and the magnitude ρ bg of thebackground field, ψ bg = ρ bg e iϕ bg , respectively. Note thatthese background gradients need to be calculated at thevortex center.To obtain the background field, we divide out the con-tribution from the vortex itself by ψ bg = ψψ LC , (S6)where ψ is the total wave function of the quantumfluid obtained from solving the Gross-Pitaevskii equation (b) (a) ρ bg x xy FIG. S3: (a) Background magnitude ρ bg and (b) backgroundphase factor ϕ bg at t = 85 . with magenta arrows denotingtwo times of velocities ~v ρ and ~v ϕ , respectively. In (a), thewhite dot denotes the center of the background field and thewhite circle denotes the shape of the approximated Gaussianprofile. In (b), the black arrows denote two times of ~v ϕ cal-culated near the vortex center. (GPE), and ψ LC is the wave function of a linear-core vor-tex given by ψ LC ( x, y ) = ( x − x v ) a + ( y − y v ) b, (S7)with a = − cos ξ + i cos θ sin ξ and b = − sin ξ − i cos θ cos ξ . Note that the vortex-center coordinates, x v and y v , and the vortex tilt angles, ξ and θ , are all ob-tained from the total wave function ψ . Since the rela-tively small nonlinear factor ( β = 1 ) only causes a slightnonlinearity in the vortex core shape, it is reasonable toapproximately divide out a linear-core vortex here. Theresulting magnitude ρ bg and phase ϕ bg of the backgroundfield are shown in Fig. S3.The challenge in calculating the gradients of ϕ bg and ρ bg at the vortex center is that the above dividing-outprocess causes an inevitable numerical error for ψ bg nearthe vortex center, because the value of ψ bg at the vor-tex center is infinity. To avoid this problem, we use two different approaches for estimating ~v ρ and ~v ϕ :For estimating ~v ρ , we approximate the backgroundmagnitude by a Gaussian profile, ρ bg = e − [ ( x − x b ) +( y − y b ) ] , (S8)where { x b , y b } is the numerically estimated center of thebackground field, denoting by the white dot in Fig. S3(a).Note that the center deviating from the origin is a mani-fest of the inward/outward spiral of the whole system.Thecircular shape of this Gaussian profile is plotted by thewhite circle in Fig. S3(a), in comparison to the real shapeof the background magnitude. Since we have obtainedthe vortex position, { x v , y v }, and the function of ρ bg ,from Eq. (6) in the main text, we get v ρx = − y v + y b ; (S9) v ρy = x v − x b , (S10)where v ρx and v ρy are x and y components of ~v ρ , respec-tively. The magenta arrow in Fig. S3(a) denotes ~v ρ .For estimating ~v ϕ , we calculate the gradients of ϕ bg at the nearby points of the vortex center, denoted bythe black arrows in Fig. S3(b). Since the color mapshows that ϕ bg is a slowly varying, global function in thereal space, we use the average of these black arrows toestimate the gradient at the vortex center, and thus thevortex velocity ~v ϕ denoted by the magenta arrow in Fig.S3(b). ∗∗