Reconnections of quantized vortex rings in superfluid $^4$He at very low temperatures

Collisions in a beam of unidirectional quantized vortex rings of nearly identical radii $R$ in superfluid $^4$He in the limit of zero temperature (0.05 K) were studied using time-of-flight spectroscopy. Reconnections between two primary rings result in secondary vortex loops of both smaller and larger radii. Discrete steps in the distribution of flight times, due to the limits on the earliest possible arrival times of secondary loops created after either one or two consecutive reconnections, are observed. The density of primary rings was found to be capped at the value $500{\rm \,cm}^{-2} R^{-1}$ independent of the injected density. This is due to collisions between rings causing piling-up of many other vortex rings. Both observations are in quantitative agreement with our theory.

Turbulence appears in various systems: fluids, plasmas, interstellar matter -with common properties such as the existence of long-lived regions of concentrated vorticity, whose reconnections facilitate the evolution of the flow field and redistribution of the kinetic energy between length scales. A paradigm of an isolated vortical structure is a vortex ring [1,2], and their pair interactions are a testbed of the physics of vortex reconnections. Numerical simulations of collisions of two vortex rings predict various outcomes: either a single ring or several rings, depending on the initial conditions [3,4]. There were experimental attempts to visualize these processes in classical fluids [5][6][7]; however, they are often hard to interpret because of the inevitable decay, core instabilities and poor characterization of vortex rings in viscous fluids.
In superfluid 3 He-B [38] and 4 He [39], collisions and subsequent reconnections, in a dense beam of vortex rings were found to result in QT. Tangles resulted from longer and more intensive beams of rings show large-scale velocity fluctuations [40,41] and late-time decay [38,39,42] characteristic of classical turbulence. This implies the existence of the inverse cascade of energy from the small length scales (of order ring radii) into which the initial energy is injected -up to the size of the resulting tangle. It was speculated [36] that the inverse cascade might be maintained by the merger of pairs of rings into larger loops. Yet, no direct quantitative observations of ringring reconnections have been reported so far. This Letter reports the first quantitative observations of either one or two consecutive reconnections, and the discovery of the ensuing universal state of depleted density -within a beam of unidirectional quantized vortex rings all of similar radii, with their number density n and radius R under our control. The resulting mechanism of seeding the large-scale velocity fluctuations out of a seemingly random beam of vortex rings is suggested.
In our experiments, to create vortex rings of a required size and to detect their arrival, each was tagged by one excess electron trapped on the vortex core. Applying an electric field along the x-axis allowed small seed charged vortex rings (CVRs), injected at x = 0, to grow to the desired radius R(x), and also to trace the location of the reconnection process that resulted in secondary charged vortex rings of a particular radius. The radius of a quantized vortex ring is directly related to its self-induced velocity, v ∼ κ/R -which determines its arrival time at the collector at x = d. The numbers of primary and secondary vortex rings as a function of their radii could be extracted from the time-dependence of the collector current I c (t) through "time of flight spectroscopy". The radius of primary rings at the collector, R(d), was varied within 1-6 µm, with number density n(d) between 10 4 and 10 7 cm −3 ; while the mean radius of the seed CVRs was estimated asR 0 ≤ 0.5 µm.
The energy of a CVR, subject to a potential φ(x), is E(x) = E 0 + eφ(x) in the absence of dissipation at T < 0.5 K. The velocity and energy depend on where κ = h/m 4 is the circulation quantum, ρ is the density of superfluid, Λ = ln 8R a0 and a 0 = 1.3Å [43]. A deeper insight can be achieved within the approximation for constant Λ ≈ 13 and uniform electric field The radius of an isolated CVR then grows linearly and the time for a CVR to travel from x = 0 to x = d is In what follows, unless specified, we will be using the approximation R 0 = 0. With increasing density of CVRs, their collisions become more frequent. These collisions are caused by small fluctuations in the direction and magnitude of the rings' velocities -mainly due to the variations in initial radii δR 0 and direction of the seed CVRs when injected at x = 0. Along with reconnections upon a direct collision, hydrodynamic dipole-dipole interactions between neighboring CVRs (that grows in strength with increasing n and R -and are hence the strongest near the collector at x → d) affect CVR's velocity. The Coulomb repulsion between neighboring CVRs of R > 1 µm is much weaker than their hydrodynamic interaction.
A reconnection of two CVRs results in secondary vortex loops, which are generally non-circular. The two trapped electrons are now carried by either one or by two (if any) of the secondary rings. One special case allows an exact analysis of the consecutive trajectory of one of the electrons -when a secondary CVR has a small initial radius (R 0 ≪ R). Then its initial deformation and direction of motion can be disregarded, because, under the pull of the electric field, it quickly gains sufficient energy and impulse along the x-direction, so to a good accuracy can be treated as a circular vortex ring [44,45]. If such singly-charged loops are created after collisions at some x = x 1 , their arrival at the collector (x = d) at time τ 2 (x 1 ) = τ 1 x1 d 2 + 1 − x1 d 2 will be earlier than of any other CVRs with either larger initial size (slower) or double charge (more energetic, hence, slower). The earliest arrival time, will be for collisions at x 1 = d/2. Furthermore, if such a secondary small ring grows and then reconnects with another vortex loop at some point x = x 2 (x 1 < x 2 < d), and this creates a new small singly-charged ring, the latter will arrive at the collector at time The earliest arrival, at time of these second-generation secondary CVRs will correspond to reconnections at x 1 = d/3 and x 2 = 2d/3. The experimental cell [46], a cube-shaped volume of side d = 4.5 cm, was filled with isotopically-pure liquid 4 He [47] at pressure 0.1 bar and temperature 0.05 K (see inset in Fig. 1). Seed CVRs were injected through a gridded opening in the center of one plate. They then traveled along the axis of the container (x-axis) towards the center of the opposite plate to the collector electrode, placed behind a Frisch grid of radius r = 6.5 mm and geometric transparency θ = 0.92. All currents and potentials are quoted with the opposite sign as if electrons had positive charge e. CVRs were subject to the propelling field set by potentials of plates φ(0) = 0 and φ(d) = U , thus gaining energy eU while traveling between the injector and collector grids [48]. The dependence φ(x) was close to the linear φ = U x d (see inset in Fig. 2). The seed CVRs resulted from reconnections within the dense vortex tangle, generated by the current of electrons emitted from a tungsten tip [51] behind the injector grid through the voltage U tip . These seed CVRs are injected in a broad range of angles; however, the impulse gained from the strong driving field quickly forces them to travel in nearly the same x-direction with a relatively narrow distribution of radii. The intensity and duration of the injected pulse were controlled by adjusting U tip and its duration ∆t (all data presented here are for ∆t = 0.2 s). For the same U tip and ∆t, the total charge injected through the grid was increasing with increasing drive voltage U nearly linearly for all studied voltages. To quantify the time of flight τ 1 of CVRs, we take the time interval between the middle of the tip voltage pulse and the position of the maximum of I c (t) (and subtract the electronics response time of 0.03 s).

cm
Typical records of the collector current, I c (t), following the injection of a pulse of CVRs are shown in Fig. 1. These are all for the same drive voltage U = 135 V but several different injection currents. There is a welldefined peak at time τ 1 = 1.0 s corresponding to the ar- rival of primary CVRs. With increasing density of CVRs, this peak initially grows in magnitude while maintaining its position, τ 1 , and width, ∼ ∆t. At higher numbers of injected CVRs, however, a broad pedestal begins to grow, coexisting with the original peak (whose magnitude is now saturated). This broad peak-pedestal is due to the secondary CVRs that result from collisions between primary CVRs. The current at t > τ 1 reflects the arrivals of larger secondary vortex loops, while that at the earlier arrival times t < τ 1 are from smaller secondary CVRs. A sharp step builds up at τ * 2 = τ 1 /2, coinciding with the earliest possible arrival of the first generation of secondary CVRs (Eq. 4). At the highest intensity of injection, another sharp step begins to form at τ * 3 = τ 1 /3, corresponding to the earliest possible arrival of the second generation of secondary CVRs (Eq. 5).
In Fig. 2, we show I c (t), similar to those in Fig. 1, but now for the same tip voltage U tip = 440 V and four different drive voltages U . With increasing U , the position of the peak τ 1 increases as expected for isolated CVRs, Eq. 3. The peak's magnitude I m (U ) initially grows with U but then, above U = 68 V, decreases -even though the total collected charge Q c = ∞ 0 I c (t)dt keeps increasing. Simultaneously, the broad pedestal due to secondary CVRs progressively overgrows the primary peak until completely swamping it at U = 270 V. The sharp steps due to the earliest possible arrivals of secondary CVRs of first generation at τ 1 /2 (Eq. 4) and second generation at τ 1 /3 (Eq. 5) are labeled by arrows. We thus obtained quantitative evidences of either single or two consecutive reconnections of CVRs during their motion from the injector to collector. Furthermore, the substantial contribution to the collector current right after the cut-offs indicates that very small CVRs are created with high probability. This might contradict the expectations that reconnections should mostly result in vortex loops of size comparable to the radius of curvature of the initial vortex lines [28,33]. The question whether trapped ions might help to pinch-off small CVRs requires further research.
In Fig. 3, the experimental arrival times τ 1 (U ) for several intensities of injection are plotted. For small drive voltages U (i. e. when the radii of CVRs R(d) and density of CVRs n(d) are small) the experimental points agree with the theory for isolated CVRs (from Eqs. 1-2). To characterize the range of the distribution of times of flight, the vertical bars show the width of I c (t) at the 0.5I m level. One can see that at small U the width is constant, being equal to the injection duration ∆t = 0.2 s. Above a certain value of U ∼ 100 V (that decreases with increasing injection intensity, U tip ), the collector pulse broadens and the position of the maximum of I c (t) no longer agrees with the theoretical prediction for isolated CVRs (this coincides with the complete disappearance of the sharper peak due to primary CVRs, as on the trace for U = 270 V in Fig. 2); secondary CVRs dominate I c (t) in these conditions of high n and R.
The density of primary rings reaching the collector, n 1 ≡ n(d), can be found from the value of the collector current at its maximum (but only for the records I c (t) that have a sharp peak at t = τ 1 , dominant over the pedestal due to the secondary CVRs), where the relation (from Eqs. 1-2) has been used. In Fig. 4, we plot n 1 (calculated from the experimental values of I m (U ) using Eq. 6) vs. R 1 ≡ R(d), the radii of primary CVRs near the collector, (calculated from Eq. 2 with R 0 = 0). Again, there are two distinct regimes. While at low R 1 , n 1 increases with an increase of either U tip or U , at large R 1 , n 1 becomes a decreasing function of U , independent of U tip . Yet, the total injected charge (as measured by the integral of the collector current) increases monotonically at all U . In other words, only at small R 1 and n 1 , the density of rings can be controlled by varying the injection current. At high R 1 , just a small fraction of charge arrives with primary CVRs of now universal density n 1 ≈ 500 cm −2 R −1 1 ; the rest (secondary CVRs) contribute to the broad pedestal in I c (t) (Figs. 1-2).
The density of primary rings in the beam n(x, t), along the trajectory x(t), evolves according to [52]: where f is the rate of losses, per unit volume and time, due to ring-ring collisions. For small injected density and radii, the collisions can be neglected, f = 0, and the solution for the density near collector is i. e. it can be varied by changing either the injected density of CVRs (characterized by I m ) or drive voltage U -as observed in the experiment. When collisions become rife at higher densities and radii, accounting for the removal of primary CVRs due to their binary collisions results in where we used δR = 0.5 µm and the geometric crosssection for collisions σ 1 = σ ′ 1 R 2 with σ ′ 1 = 4π. Furthermore, the subsequent removal of primary CVRs that bump into the slower vortex loops left after the described collisions of primary CVRs will result in the solution, where the geometric cross-section for these pile-ups is estimated as σ 2 = σ ′ 2 R 2 with σ ′ 2 = 4 √ 2π. Both Eq. 9 and Eq. 10 reproduce the experimental universal dependence n 1 R 1 = 500 cm −2 qualitatively, while Eq. 10 is actually quite close quantitatively (our 1-dimensional model underestimates f by disregarding the transverse component of the relative motion of primary CVRs and hence overestimates the value of n 1 R 1 by a factor of 2 or so [52]).
The fact of occasional piling-up of many primary CVRs, in turn, helps to explain the appearance of largescale velocity fluctuations in the ensuing vortex tangle. In the initial random beam of primary CVRs, the fluctuations of the coarse-grained velocity on the length scales greater than ∼ n −1/3 ("quasi-classical" flow) are small; the energy spectrum is concentrated around the small length scale of order R. The small secondary vortex rings, observed in this work, and Kelvin waves excited by reconnections are evidences of the direct cascade of energy towards smaller length scales [28-30, 32, 33, 37]. However, following any of the pile-ups of many vortex rings, strong fluctuations of the coarse-grained velocity field on the "quasi-classical" length scales ≫ n −1/3 are being created. This is the inverse cascade of energy in this strongly anisotropic system. [34,36].
To summarize, we obtained quantitative evidence for collisions and reconnections of pairs of unidirectional vortex rings of similar radii that result in the creation of vortex loops of unequal size, including many small ones. We observed discrete steps at the time dependence of the collector current, that correspond to the earliest arrivals of the first and second generations of secondary CVRs. As each collision can cause a removal of many primary vortex rings, increasing the density of injected CVRs results in a new state in which the density of primary vortex rings is maintained at the critically depleted level independent of their initial density. The larger loops produced in the collisions become the seeds of quasi-classical QT with large-scale flow structures, which appear out of a seemingly random beam of small quantized vortex rings.
This work was supported by the Engineering and Physical Sciences Research Council (Grants No. GR/R94855, EP/H04762X/1 and EP/I003738/1).