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PHYSICS for BEC confined by box traps), then v` is the self-induced vortex velocity arising from the presence of an image vortex. Finally, if the BEC density is smoothly varying (such as arises

for BECs confined by harmonic traps), then v∇ρ = C3(κ∇ρ/ρ) is precisely the individual velocity of a vortex induced by the density gradients, with C3 depending on the trap’s geometry (56, 57). In the next section we will see how these scalings, and the crossover, emerge in typical scenarios through numerical simulations.

Fig. 1. Reconnecting vortex lines exchanging strands. Shown are schematic vortex configurations before the reconnection (Left) and after (Right); the vortices’ shape is as determined analytically by Nazarenko and West (18). Color gradient along the vortices and blue/red arrows indicate the directions of the vorticity along the vortices and the direction of the flow velocity around them. Dashed black arrows indicate the vortex motion, first toward each other and then away from each other.

Dimensional Analysis We conjecture that, in the system under consideration (super- fluid helium, atomic BECs), δ depends only upon the following physical variables: the time t from the reconnection, the quan- tum of circulation κ of the superfluid, a characteristic length scale ` associated to the geometry of the vortex configuration, the fluid’s density ρ, and the density gradient ∇ρ. We hence postulate the following functional form:

f (δ, t, κ, `, ρ, ∇ρ)=0. [1]

Following the standard procedure of the Buckingham π-theorem (55) (see SI Appendix, section SI.1 for details), we derive the scalings

1/2 1/2 δ(t) = (C1κ) t interaction regime, [2]

 κ  δ(t) = C t driven regime, [3] 2 `

 ∇ρ  δ(t) = C κ t driven regime, [4] 3 ρ

where C1, C2, and C3 are dimensionless constants. Physi- cally, the δ ∼ t 1/2 scaling of Eq. 2 identifies the quantum of circulation κ as the only relevant parameter driving the reconnection dynamics (20); this scaling corresponds to vor- tex dynamics driven by the mutual interaction between vortex strands, as illustrated more in detail in Homogeneous Unbounded Systems. Eqs. 3 and 4, on the other hand, introduce the δ ∼ t scaling. Fig. 2. Minimum distance between reconnecting vortices: past and present This scaling suggests the presence of a characteristic veloc- results. (Top) All data reported describe the behavior of the rescaled ity which drives the approach/separation of the vortex lines. minimum distance δ* between vortices as a function of the rescaled tem- poral distance to the reconnection event t*. Open (solid) symbols refer to Indeed, we can offer some physical examples of these veloci- pre(post)reconnection dynamics. GP simulations: red ♦, ref. 50; blue , ref. ties. If ` is the radius of a vortex ring, then v` = C2(κ/`) is, 48; turquoise 4, ref. 38; green ◦ and , present simulations, ring–vortex to a first approximation, the self-induced velocity of the ring. collision and orthogonal reconnection, respectively. VF method simula- Alternatively, if ` is equal to the distance of a vortex to a sharp tions: purple C, ref. 44; green ♦, present simulations, ring–line collision. boundary in an otherwise homogeneous BEC (such as arises Experiments: F, ref. 52. (Bottom) Zoom-in on GP simulations.

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Fig. where BECs confined inhomogeneous, to model simula- the GP extend GP also the We in of ones). larger VF use in times larger times (5–20 100–2,000 studies and distance tions, numerical initial past larger in the that are is methods simulations these our of in details distance described minimum Technical the vortices. of time reconnecting with scaling the tigate “cut-and-paste” hoc whose ad an points 61). by (59, Lagrangian vor- algorithm performed and of are law, Biot–Savart reconnections set classical tex the vortex a by model, governed repre- using VF are dynamics the depletions discretized In density are themselves. cores the lines vortex and the by vortices sented moving by core, created vortex the than larger VF much fea- scales typically the the length at hand, probing flow other the model, of incompressible the tures mesoscopic On a itself. phase is equation topological method GP as the identified of solutions are wavefunction vortices condensate the within of model, defects is vor- GP superfluid density the the the In around where core tube line vortex cylindrical microscopic, tex the the a of than diameter is smaller fluctuations the equation scales density GP describing length The of and capable flow. model, the compressible of scales length space on concentrated given 60). is a (59, case of curves our field in the which velocity on distribution, the based vorticity describing is BEC latter law the interacting Biot–Savart filament and weakly classical (58), vortex limit a zero-temperature the describes the dynam- former in and vortex The model quantum method. (GP) (VF) of Gross–Pitaevskii models the established ics: two are There Simulations Numerical ntepeetsuy euebt PadV oest inves- to models VF and GP both use we study, present the In probed the is models VF and GP between difference main The 5 ξ ,tebu-ahdln hw the shows line blue-dashed the ), Psmltos ooeeu none Es hw seouino h ecldmnmmdistance minimum rescaled the of evolution is Shown BECs. unbounded homogeneous simulations: GP 10 4 a itntv of Distinctive SI.7. and SI.6 sections Appendix, SI 0 or 10 δ 0 5 * = a 0 0 n otxrn radii ring vortex and 100 elcigaydniyperturbation density any neglecting , t * 1/2 75% n eoncin are reconnections and Ψ, cln,adthe and scaling, f ( e R 0 ftebl density). bulk the of .I both In ). δ 0 R 0 * oprdwith compared a δ qa o5(orange 5 to equal 0 0 * dfie as (defined Left qa o1 (violet 10 to equal δ otmInsets Bottom t between .Oe sld ybl orsodt r(otrcneto yais (Left dynamics. pre(post)reconnection to correspond symbols (solid) Open *. and h oiotldse lc ieidctstewdho h otxcore vortex the of width the indicates line black dashed horizontal the Right, hwteiiilvre ofiuain oo rdeto otcsindicates vortices on gradient Color configuration. vortex initial the show ,75(yellow 7.5 ◦), egho h ytm(a system the of length te ueia Psuis(8 0,adn ute evidence further adding 50), (38, with inconsistent studies and GP (48) simulations numerical consistent GP other is recent This most dynamics. the postreconnection with and pre- both for less symmetrical a significantly density) is bulk the them con- than between the that region other the each for in to close density that so densate’s observe are lines we vortex BEC), two the homogeneous (when a in sound of and instant nection inves- the extend distance distances we initial Here to respectively. the tigations and bosons, interaction, of boson density of bulk strength repulsive the mass, boson curvatures the and where limit the K to vor- first corresponding orthogonal The and tices, straight reconnections. initially two of of types consists configuration fundamental which two configurations the vortex initial generate limiting two quantum identify we homogenous fluids, in reconnections vortex of Systems. understanding Unbounded Homogeneous with (21). experimentally resolution investigate unprecedented be now can reconnections vortex cigwt nioae otxln,wihi h iiigcase limiting (K the vatures (K is curvatures which different significantly line, K of vortex vortices two isolated of an with acting nta distances initial in an for to results refers sponding it cross-section. as small study the to present due the event, in unlikely extremely neglected is rings, vortex 2 1 h rhgnlrcneto ofiuainadtecorre- the and configuration reconnection orthogonal The rvosG iuain fti emtyused geometry this of simulations GP Previous S1–S4. 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PHYSICS 1/2 to a t ∗ symmetrical scaling at small distances for orthogonal reconnections. 1/2 To map quantitatively the emergence of the t ∗ behavior in distinct intervals of δ, in Table 1 we report the scaling exponents α α of the power-law fits δ∗ ∼ t ∗ for the intermediate initial dis- ∗ ∗1/2 tance δ0 = 20. From Table 1 it clearly emerges that the t scaling also holds in the postreconnection dynamics at large distances, while in the intermediate region 2.5 < δ∗ < 10 (both pre- and postreconnection) and at large distances during the 1/2 approach, the scalings deviate from t ∗ . To investigate these deviations, we calculate the velocity contribution of the local vor- ∗ ∗ tex curvature, vγ , to the approach/separation velocity dδ /dt ∗ for the intermediate initial distance δ0 = 20, displaying the cor- responding results in Fig. 4, Top (see SI Appendix, section SI.2 for the calculation of vγ ). Fig. 4, Top clearly shows that the 1/2 observed deviations from the t ∗ scaling depend on the relative ∗ ∗ curvature contribution σ = vγ /(dδ /dt ): The larger σ is, the 1/2 more prominent the deviations are from the t ∗ behavior. This 1/2 implies that the t ∗ scaling observed at large distances in the separation dynamics originates from an interaction-dominated motion of the reconnecting strands. If the dynamics are gov- erned by the mutual interaction of the two vortices, in fact, ∗ ∗ ∗ dδ /dt ∝ κ/(4πδ ) (44), leading straightforwardly to the scal- Fig. 4. Curvature contribution. (Top) Temporal evolution of the ratio σ ∗ ∗1/2 * ing δ ∼ t , derived in Eq. 2. This argument corroborates for the orthogonal reconnection with initial separation distance δ0 = 20. ∗1/2 Red (blue) symbols correspond to pre(post)reconnection dynamics. (Bottom) the experimentally observed t scaling (46, 52). Concluding Temporal evolution of σ (green solid line) and rescaled minimum distance our study of orthogonal reconnections, we note that vortex lines δ* (orange circles) for the ring–line prereconnection, with R0* = 5. The 1/2 move faster after the reconnection than before it, as pointed out dashed blue line shows the t* scaling, while the dotted-dashed violet in past studies (38, 48–50), and that the postreconnection curves line indicates the t* scaling. ∗ show a slight sensitivity to the initial distance δ0 . The second homogeneous system which we consider is a vor- tex ring reconnecting with an isolated initially straight vortex line our knowledge, the linear scaling has not been reported in pre- (Movies S5–S8). This scenario is notably relevant in superfluid vious studies. We also note that the crossover between these two helium turbulence at low temperatures, where the low density of regimes occurs at a distance of δc ∼ 3 − 4 ξ; we will revisit the thermal excitations is not able to quickly damp out small vortex importance of this scale later. structures. In fact, the methods used to generate turbulence at The linear scaling implies that dδ∗/dt ∗ is constant: At large low temperatures involve either injecting vortex rings (62–65) or distances, the relative velocity between the two points at mini- vibrating small structures (e.g., spheres, wires, and grids) which mum distance, xring (on the vortex ring) and xline (on the vortex trigger a great number of ring–line collisions (66–68). Similarly, line), projected on the separation vector δ = xring − xline, is con- also inhomogeneous quantum turbulence (69, 70) and boundary stant with respect to time. We argue that, at large separation layer turbulence (71) display conspicuous vortex loop–vortex line distances, dδ∗/dt ∗ is approximately equal to the initial speed of collisions. the vortex ring (72): Fig. 3 (Right) illustrates this vortex setup and presents the ∗ behavior of δ(t). We vary the initial radius of the ring (R0 = ∗   dδ ∗ κ ∗ R0/ξ = 5, 7.5, and 10), while keeping constant its initial dis- ∗ = vring = ∗ ln(8R0 ) − 0.615 . [7] ∗ dt 4πξcR0 tance δ0 = 100 to the line. We first focus on the prereconnection evolution of δ(t). This clearly reveals the two distinct scalings v predicted by the dimensional analysis, The self-induced velocity of the vortex ring ring thus plays the role of the characteristic velocity v` in Eq. 3. We make the 1/2 ∗ ∗ ∗ ∗ ∗ notation compact and rewrite Eq. 7 as vring = Cf (Re0), where δ (t ) ∼ a1/2 t for t . 5, [5] C = κ/(4πcR0), f (Re0) = [ln(Re0) + 3.48]/Re0, R0 = 7.5 ξ is ¯ the average radius of the three simulations, and Re0 = R0/R0. ∗ ∗ ∗ ∗ ∗ ∗ δ (t ) ∼ a1 t for t & 20, [6] We then arrive at the result that Eq. 6 takes the form δ (t ) ∼ ∗ ∗ Cf (Re0) t . This is confirmed in Fig. 3, Right, Top Inset: When δ where a1/2 and a1 are constant prefactors corresponding, respec- ∗ 1/2 is rescaled as δ /f (Re0), the curves collapse onto a single curve tively, to (C1κ) in Eq. 2 and v` and v∇ρ in Eqs. 3 and 4. To ∗1/2 ∗ ∗ in the δ ∼ t regime. The clear t scaling for δ . δc is con- sistent with the interpretation put forward for the orthogonal Table 1. GP simulations: homogeneous unbounded BECs, reconnection, as at such small distances, the approach/separation orthogonal reconnection is likely to be governed by the mutual interaction of the δ* < 2.5 2.5 < δ* < 10 δ* > 10 two vortices given that δ is smaller than the ring’s radius of curvature. α− 0.48 0.40 0.43 We hence suggest that these two scalings correspond to a α+ 0.49 0.60 0.50 crossover from dynamics predominantly driven by mutual vortex ∗ ∗1/2 Shown are scaling exponents α− (α+) for power-law behavior δ* ∼ t*α interaction (δ ∼ t scaling) to the self-driven (curvature- ∗ ∗ for pre(post)reconnection dynamics for initial separation δ0* = 20. driven) motion of the ring (δ ∼ t scaling). To check this

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= µ /ξ x ω 0 c /R ) y and where , δ IAppendix, SI = (t ω ,and Right), x ω x ) = and S10) z n the and along r = 0 /R ω ξ c ⊥ ⊥ is ). x , ξ BECs. in reconnections vortex region (n of values all from crossover a observe 6, Fig. in reported 5. Fig. prahadsprto rltdt nices ftelclvor- local the of increase an to (related 6, dynamics, separation and Fig. postreconnection approach in faster the seen is as geometries other and Buckingham’s applying when variables physical of distance the minimum characterizes Further- merging. which of density start scale dependence cores the length vortex more, correct when the dynamics approach is core) that tex implies result This corresponding of Inset). values curves distinct the to approximation), Thomas–Fermi the χ of values different value three parameter the orbit on the depending of ejections), reconnections, reconnec- vortex double rebounds, tions, (vortex occur can interactions vortex (long) the intersecting vortices outer Hence, of trap. values the larger of (with center the at length healing the length healing the trap. is the vorticity length of superfluid of center Unit the dark). of to direction light the green (from indicates Light vortices 5% configuration. on at vortex gradient density initial Color condensate of of views isosurfaces (B) are top surfaces and (A) Lateral B) r 0 = TF fw ecl h iiu distance minimum the rescale we If nte iiaiybtenrcnetosi amnctraps harmonic in reconnections between similarity Another planes, radial on imprinted are vortices orthogonal two If vlae ttercneto point reconnection the at evaluated (x 5 0. .35, Psmltos amnclytapdBC,iiilcniin.( A conditions. initial BECs, trapped harmonically simulations: GP r δ ρ ) ∗ = . en h odnaepril est codn to according density particle condensate the being mn h rrcneto vlto of evolution prereconnection The 0.6 5, ugsigta hsfauei neduieslfor universal indeed is feature this that suggesting 5, oevr the Moreover, χ. tefpasasgicn oei eemnn the determining in role significant a plays itself δ hsvlct ifrnebetween difference velocity This Inset. Left sfrtern–iercneto,we reconnection, ring–line the for As Left. ti seatywyw included we why exactly is —this χ oefaster. move χ) x χ vra for overlap l nedrn otxreconnections: vortex engendering all χ, xsa poiepositions opposite at axis 2) eut rsne eerfrto refer here presented Results (21). ξ t r ∗ 1/2 on t ξ ∗ r 1/2 mn to hneterdu ftevor- the of radius the (hence δ t TF cln gi cusi the in occurs again scaling r ∗ ∗ = δ (x cln.Ti cusfor occurs This scaling. x NSLts Articles Latest PNAS ihtehaiglength healing the with δ/ξ r r , ) ξ niae htmass that indicates r r fta-etrdensity. trap-center of . = ξ 5 c p vlae nthe in evaluated π Fg 6, (Fig. ±x 2gmn -theorem. δ ρ ∗ 0 nteset the in = distinct , ~ TF | δ/ξ Right f8 of 5 (x and c r is )

PHYSICS Fig. 6. GP simulations: harmonically trapped BECs. Shown is evolution of the minimum distance δ* between reconnecting vortices as a function of the temporal distance to reconnection t*. Open (solid) symbols correspond to pre(post)reconnection dynamics. (Left) Prereconnection scaling of δ* for initially imprinted orthogonal vortices with corresponding orbit parameter χ = 0.35 (yellow ◦), χ = 0.5 (red ), and χ = 0.6 (blue 5). Inset shows short-time prereconnection (open symbols) and postreconnection (solid symbols) scaling of δ* for χ = 0.35 (yellow ◦) and χ = 0.5 (red ). (Right) Temporal evolution of the minimum distance δ* rescaled with f(χ). Symbols are as in Left panel. Inset shows short-time prereconnection scaling of rescaled minimum distance 1/2 δ*r = δ/ξr . In both Left and Right, the dashed blue and dotted-dashed violet lines show the t* and t* scalings, respectively. The horizontal dashed line indicates the width of the vortex core at the center of the trap (≈ 5 ξ). Green arrows indicate the direction of time.

tex curvature in the reconnection process and to an emission tems, the motion of individual vortices is found to be driven by of acoustic energy) seems a universal feature of quantum vor- vortex images, leading to a linear scaling at large distances. At 1/2 tex reconnections (48) and is also observed in simulations of small distance we again recover the δ ∼ t ∗ scaling. The results reconnecting classical vortex tubes (51). obtained in all of the trapped BECs investigated in this work, dδ∗ Fig. 6, Left shows that is constant for t ∗ 20 before dt ∗ & the reconnection, increasing with increasing values of the orbital parameter χ (this is not surprising since isolated vortices move faster on outer orbits). It seems reasonable to assume that dδ∗ χ = Cf (χ), where f (χ) = and C is a constant which dt ∗ 1 − χ2 depends on the trap’s geometry. Indeed, the magnitude of the vortex velocity induced by both density gradients (57, 82, 89) and vortex curvature (assuming, for simplicity, that the shape of the vortex is an arc of a circle) is proportional to f (χ). As a ∗ ∗ ∗ ∗ consequence, we expect that δ (t ) ∼ Cf (χ) t for t & 20. This conjecture is confirmed in Fig. 6, Right: When plotted as δ∗/f (χ), the curves for different χ collapse onto a universal curve in this region. We stress that the observed linear scaling at large distances is a result that, to our knowledge, has not been observed previously in literature. However, although we have numerically identified the dependence of dδ∗/dt ∗ on χ at large distances, we still lack a simple physical justification of this result. In harmonic traps, the predominant effect driving the approach of the vortices at large distances is hence the individual vortex motion driven by curvature and density gradients (the role of vor- tex images still remains unclear (83) in this trap geometry), inde- pendent of the presence of the other vortex. The scaling crossover in harmonic traps is thus governed by the balance between the Fig. 7. The role of density depletions. Shown is a plot of the condensate density along the line containing the separation vector δ between the col- interaction of the reconnecting strands and the driving of the liding vortices, as a function of the distance r* to the midpoint of the individual vortices, as it occurs for the ring–line reconnection in separation segment and the rescaled temporal distance to reconnection t*, homogeneous BECs described previously. for the vortex ring–vortex line prereconnection dynamics with R0* = 5. In the The nature of this scaling crossover is confirmed by the inves- initial phase of the approach (top part), δ ∼ t*; the crossover to the δ ∼ t*1/2 tigation of vortex reconnections in box-trapped BECs, outlined scaling occurs when the vortex cores start to merge (bottom part) for t* . 5 in SI Appendix, section SI.5 and Fig. S4. In these trapped sys- (dashed line).

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(in established is behavior * repcieo h nta odto.Teeffect The condition. initial the of irrespective 5, ∗ 0–0 (2013). 501–503 493, n h distance the and Science δ antcRcneto:MDTer n Applications and Theory MHD Reconnection: Magnetic 3614 (1991). 1336–1342 251, * δ ∼ ∼ t * t 1/2 ∗ δ 1/2 * t cln i e)i bevda h dynam- the as observed is red) (in scaling ∼ 1/2 t to * urn n rvosG sim- GP previous and Current r δ 1/2 Nature cln:teinteraction-driven the scaling: ∗ δ ∼ = ∼ cln i e)silhls fthe if holds; still red) (in scaling t r ∗ t 8–8 (2011). 184–187 474, 1/2 /ξ ∗ t ∗ cln rsoe which, crossover scaling otemdon fthe of midpoint the to . eairspot the supports behavior hc swe the when is which 5, t ∗ 1/2 cln nthe in scaling a 1/2 t 1/2 in h iiu itnebtenrcnetn otxlnsmay observed already δ lines the regimes: vortex scaling-law reconnecting fundamental two between obey distance that minimum found larger. have the we magnitude arguments, of dimensional order rigorous one applying By distances over atomic behavior consid- have the (trapped we ered second, vorticity and, helium) quantized phys- superfluid main and display condensates two the which studied systems have we ical from first, work that, our is distinguishes studies previous What equation method). Gross–Pitaevskii filament an vortex (the and performing main available two by univer- models the using reconnections a mathematical simulations is vortex numerical of there quantum campaign whether extensive to of route question the sal addressed have We Conclusions consistent scales, length simulations. small at the valid with both are arguments The oto h niern n hsclSine eerhCucl(Grant Council Research Sciences Physical and Engineering the EP/R005192/1). of port are problem ACKNOWLEDGMENTS. this literature. to VF the peculiar the details in and technical described in equation some described been and GP already features the have main use, The and we standard which are methods method, numerical two The Methods and the Materials putting available, reach. experimental be within soon crossover this will of reconstructions detection suggests condensates 3D trapped full in abil- lines that technological vortex image current directly the to scaling ity Indeed, the condensates arises. trapped always in crossover that weakly find and we orthogonal vortices), initially curved for (e.g., separations arbitrary to been yet not literature. has the and in reported mechanisms the physical from illustrated inhomogeneous, and different dis- homogeneous in arises both it systems, Instead, direction. tinct same the the in that vortices stress both advect also We waves. Kelvin δ e.g., physics, due laws arise additional scaling can fundamental to scalings two Intermediate these behaviors: that limiting stress represent We schemati- 8. summarized Fig. is in behavior cally scaling This the the dynamics. individually by case, driven and determined motion latter is interaction-dominated regimes between the scaling balance two In den- the curvature, boundaries/images. between as crossover and such gradients, factors, extrinsic sity by linear driven a individually or interaction mutual two scales, the length of t continuation larger the At universal appear: cases reconnection. this limiting to of fundamental close existence law the system scaling for the evidence adds of further configuration, GP vortex nature initial all precise and in trapped) the or between scaling (homogeneous of this interaction independent of mutual observation simulations, the The in the region. density/nonlinearity either depleted reconnection the from or strands arises reconnecting this ing; .M ens .Kn,B ak .OHlea,M agt,Ioae pia otxknots. vortex optical Isolated Padgett, M. O’Holleran, K. Jack, of B. King, recombination. action R. site-specific Dennis, hidden gin M. the of 7. analysis probe Tangles to Sumners, D. topology Vazques, Using M. curtain: 6. the Lifting Sumners, D. 5. .M er,M ens eoncin fwv otxlines. vortex wave of Reconnections Dennis, M. Berry, M. 8. ∗ ∗ ∗ 1/2 hl nhmgnosssesthe systems homogeneous in While the observe always we scales, length small At (2012). Phys. Nat. Soc. Philos. Camb. Proc. enzymes. ∼ ∼ t t ∗ ∗ cln ftednmc r tl oendb h vortex the by governed still are dynamics the if scaling 1/2 antaiefo nfr o,wihwudsimply would which flow, uniform a from arise cannot t oieAMS Notice 1–2 (2010). 118–121 6, 1/2 cln n loa also and scaling IAppendix SI cln bevdcoet eoncini l GP all in reconnection to close observed scaling 2 (1995). 528 42, 6–8 (2004). 565–582 136, .. ... n ...akoldetesup- the acknowledge N.G.P. and C.F.B., L.G., . δ ∗ ∼ δ ∗ t ∗ ∼ scaling. t ∗ t ∗ eairi otcsare vortices if behavior NSLts Articles Latest PNAS 1/2 eaircnpersist can behavior u.J Phys. J. Eur. δ ∗ ∼ t δ ∗ ∗ 723–731 33, 1/2 ∼ | t δ f8 of 7 scal- Math. ∗ ∗ 1/2 ∼

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