Quantized Vortex Rings Research Articles

Curvature conservation and conduction modulation for symmetric charged ferroelectric domain walls

Curvature conservation is a common feature for the physical properties with topological constraints, from the macroscopic cosmic string loops to the microscopic quantized vortex rings in the superfluid. Interestingly, ferroelectric domain-wall curvature is important to determine its conduction, which is the key parameter for designing domain-wall nanoelectronic devices. Here, we demonstrate the curvature conservation of the charged domain walls confined in ferroelectric topological structures with specific domain symmetry by combining piezoelectric force microscopy, conducting atomic force microscopy, and phase-field simulations. Significantly, the intrinsic conductivity of the charged domain walls exhibits an inverse functional relationship with the local domain-wall curvature in the ferroelectric BiFeO3 nano-islands, which can also be precisely and continuously modulated by applying an electric field. This work provides us with an insightful understanding of the conduction mechanism of charged domain walls confined in ferroelectric topological structures with a high degree of spatial symmetry, which paves the way for the application of charged domain walls as flexible conducting channels with continuously changing resistance states to be used for ferroelectric domain-wall memristors.

Vortex wake patterns in superfluid 4He

Excitations in the form of quantized vortex rings are known to exist in superfluid 4He at energies and momenta exceeding those of the Landau phonon–roton spectrum. They form a vortex branch of elementary excitations spectrum which is disconnected from the Landau spectrum. Interference of vortex ring excitations determines wake patterns due to uniformly traveling sources in bulk superfluid at low speeds and pressures. The dispersion law of these excitations resembles that of gravity waves on deep water with infrared wave number cutoff. As a result, vortex wake patterns featuring elements of the Kelvin ship wake are predicted. Specifically, at lowest speeds the pattern with fully developed transverse and diverging wavefronts is present. At intermediate speeds transverse wavefronts are absent within a cone whose opening angle increases with the source velocity. At largest speeds only diverging wavefronts confined within a cone whose opening angle decreases with the source velocity are found. When experimentally observed, these changes in appearance of wake patterns serve as indicators of the beginning part of the vortex branch of elementary excitations.

Imaging quantized vortex rings in superfluid helium to evaluate quantum dissipation

The motion of quantized vortices is responsible for many intriguing phenomena in diverse quantum-fluid systems. Having a theoretical model to reliably predict the vortex motion therefore promises a broad significance. But a grand challenge in developing such a model is to evaluate the dissipative force caused by thermal quasiparticles in the quantum fluids scattering off the vortex cores. Various models have been proposed, but it remains unclear which model describes reality due to the lack of comparative experimental data. Here we report a visualization study of quantized vortex rings propagating in superfluid helium. By examining how the vortex rings spontaneously decay, we provide decisive data to identify the model that best reproduces observations. This study helps to eliminate ambiguities about the dissipative force acting on vortices, which could have implications for research in various quantum-fluid systems that also involve similar forces, such as superfluid neutron stars and gravity-mapped holographic superfluids.

Friction-enhanced lifetime of bundled quantum vortices

Experiments in the early 1980s have shown that a compact bundle of quantum vortex rings in superfluid helium remains coherent and travels a significant distance compared to its size. This is surprising because a single vortex ring, under the effect of friction with the background of thermal excitations (the so-called normal fluid), would quickly lose energy and decay. The observation of these long-lived vortex structures has remained an unsettled question since their experimental detection. In this work, by taking into account the fully coupled dynamics of superfluid vortices and the normal fluid, we show that the motion of the superfluid bundle generates a large-scale wake in the normal fluid which reduces the overall friction experienced by the bundle, enhancing its lifetime. This collective effect is similar to the drag reduction observed in systems of active, hydrodynamically cooperative agents such as bacteria in aqueous suspensions, fungal spores in the atmosphere and cyclists in pelotons.

Creation of quantum knots and links driven by minimal surfaces

Using an improved numerical code we investigate the creation and evolution of quantum knots and links as defects of the Gross–Pitaevskii equation. The particular constraints put on quantum hydrodynamics make this an ideal context for application of geometric and topological methods to investigate dynamical properties. Evolutionary processes are classified into three generic scenarios representing (i) direct topological cascade and collapse, (ii) structural and topological cycles, and (iii) inverse topological cascade of complex structures. Several examples and test cases are studied; the head-on collision of quantum vortex rings and the creation of a trefoil knot from initially unlinked, unknotted loops are realized for the first time. Each type of scenario is studied by carrying out a detailed evaluation of fundamental geometric and dynamical properties associated with evolution. Direct topological cascade that governs the decay of complex structures to small-scale vortex rings is identified by writhe measures, while picks of total curvature are found to provide a clear signature of reconnection events. We demonstrate that isophase minimal surfaces spanning knots and links have a privileged role in the decay process by detecting surface energy relaxation of complex structures. Minimal surfaces are shown to be critical markers for energy and prove to be appropriate detectors for the evolution of complex systems.

Hydrodynamic generation of skyrmions in a two-component Bose-Einstein condensate

When an obstacle is moved in a superfluid faster than a critical velocity, quantized vortices are generated behind the obstacle. Here we propose a method to create more complicated topological excitations, three-dimensional skyrmions, behind a moving obstacle. We numerically show that, in a two-component Bose-Einstein condensate, component-dependent obstacle potentials can generate skyrmions in the wake, made up of quantized vortex rings in different components that are linked with each other. The lifetime of generated skyrmions can be prolonged by a guiding potential, which enables the formation of a skyrmion train.

Density functional theory of superfluid helium at finite temperatures.

A density functional theory-based method is developed to describe the static and dynamic response of superfluid helium at finite temperatures. The model relies on the well-established 0K Orsay-Trento functional, which has been extensively used to study the response of bulk superfluid helium as well as superfluid helium droplets. By including a phenomenological stochastic term in this model, it is possible to obtain thermodynamic equilibrium corresponding to a given temperature by propagating the system in imaginary time. The temperature dependence of thermodynamic quantities, such as the internal energy and entropy, is computed and is compared well with experimental reference data for the bulk liquid up to about 2K, but begins to gradually deviate above that temperature. Inspection of pseudovorticity during real-time evolution of the system near 2K reveals the presence of roton quasiparticles, which are suggested to be precursors for quantized vortex rings (Onsager's ghosts), as well as weaker analogs of extended vortex loops.

Hydrodynamic decay of decorated quantum vortex rings

The decay of quantum vortex rings in counterflow regimes, visualized in Helium II with the help of solid hydrogen particles trapped to their cores, has been a puzzling issue within the usual description of superfluid vortex dynamics, grounded on the hypothesis that a vortex filament is, effectively, an extended massless object subject to a canceling superposition of Magnus and mutual friction forces. We discuss, from a general energy-budget point of view, a phenomenological solution of this problem, which relies on viscous and quantum dissipation mechanisms, the later associated to the backreaction of vortex singular structures on the surrounding two-component fluid mixture.

The time-development of energy spectrum of a quantized vortex ring

The present work is devoted to the study of time dependence of the energy spectrum that is associated with quantized vortex ring movement in the superfluid helium at different temperatures. Temperature changes from 1.3 K to 1.9 K. The full Biot-Savart equation is used to calculate velocity fields. The energy spectrum is calculated by using the method of structure functions. It is established that the structure function amplitude is associated with energy dissipation, while the energy spectrum is independent of temperature and time. The relationship between the energy dissipation rate and the vortex ring length is also discussed.

Accurate numerical determination of a self-preserving quantum vortex ring

We compute simultaneously the translational speed, the magnitude and the phase of a quantum vortex ring for a wide range of radii, within the Gross–Pitaevskii model, by imposing its self preservation in a co-moving reference frame. By providing such a solution as the initial condition for the time-dependent Gross–Pitaevskii equation, we verify a posteriori that the ring’s radius and speed are well maintained in the reference frame moving at the computed speed. Convergence to the numerical solution is fast for large values of the radius, as the wavefunction tends to that of a straight vortex, whereas a continuation technique and interpolation of rough solutions are needed to reach convergence as the ring tends to a disk. Comparison with other strategies for generating a quantum ring reveals that all of them seem to capture quite well the translational speed, whereas none of them seems to preserve the radius with the accuracy reached in the present work.

Quantized Vortex Rings and Loop Solitons

The vortex filament model is used to investigate the interaction of a quantized vortex ring with a straight vortex line and also the interaction of two solitons traveling in opposite directions along a vortex. When a ring reconnects with a line, we find that a likely outcome is the formation of a loop soliton. When they collide, loop solitons reconnect as they overlap each other producing either one or two vortex rings. These simulations are relevant for experiments on quantum turbulence in the zero temperature limit where small vortex rings are expected to be numerous. It seems that loop solitons might also commonly occur on vortex lines as they act as transient states between the absorption of a vortex ring before another ring is emitted when the soliton is involved in a reconnection.

Symmetry-breaking instability of leapfrogging vortex rings in a Bose-Einstein condensate

Three coaxial quantized vortex rings in a Bose-Einstein condensate exhibit aperiodic leapfrogging dynamics. It is found that such circular vortex rings are dynamically unstable against deformation breaking axial rotational symmetry. The dynamics of the system are analyzed using the Gross-Pitaevskii and vortex-filament models. The dependence of the instability on the initial arrangement of the vortex rings is investigated. The system is found to be significantly unstable for a specific configuration of the three vortex rings.

Three-dimensional stability of leapfrogging quantum vortex rings

It is shown by numerical simulations within a regularized Biot-Savart law that dynamical systems of two or three leapfrogging coaxial quantum vortex rings having a core width ξ and initially placed near a torus of radii R0 and r0 can be three-dimensionally (quasi-)stable in some regions of parameters Λ = ln(R0/ξ) and W = r0/R0. At fixed Λ, stable bands on W are intervals between non-overlapping main parametric resonances for different (integer) azimuthal wave numbers m. The stable intervals are most wide (ΔW ∼ 0.01–0.05) between m-pairs (1, 2) and (2, 3) at Λ ≈ 4–12, thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid 4He. With four and more rings, at least for W > 0.1, resonances overlap for all Λ and no stable domains exist.

Electric polarization of He II caused by second and first sound

It is shown that in a three-component gas of superfluid helium (He II) quasiparticles made up of phonons, rotons, and quantized vortex rings (QVR), equilibrium is established instantaneously on experimental time scales. This results in the formation of a common normal component which can move relative to the superfluid component with a velocity w. When a velocity w is present, an electric field develops in the He II that has been observed experimentally. The reasons for this are an anisotropic dependence of the QVR energy on its momentum when w is present and the existence of a QVR dipole moment. The case where the dipole moment of a QVR is made up of its intrinsic dipole moment plus the dipole moment created by an external field is considered. The problem of the electric potential in a standing wave of second sound in a constant external electric field is solved. The electric potentials in a first sound wave owing to the flexoelectric effect, acceleration of the liquid, and an external electric field are obtained. These formulas are used in numerical calculations for comparing theory and experiments, and they can stimulate the setting up of new experiments.

Parametric instability of oscillations of a vortex ring in a z-periodic Bose condensate and return to the initial state

The dynamics of deformations of a quantum vortex ring in a Bose-Einstein condensate with periodic equilibrium density $\rho(z)= 1-\epsilon\cos z$ has been considered within the local induction approximation. Parametric instabilities of the normal modes with azimuthal numbers $\pm m$ have been revealed at the energy integral $E$ near values $E_m^{(p)}=2m\sqrt{m^2-1}/p$, where $p$ is the resonance order. Numerical simulations have shown that already at $\epsilon\sim 0.03$ a rapid growth of unstable modes with $m=2$, $p=1$ to magnitudes of order of unity is typical, which is then followed, after a few large oscillations, by fast return to a weakly excited state. Such behavior corresponds to an integrable Hamiltonian of the form $H\propto \sigma(E_2^{(1)}-E)(|b_+|^2 + |b_-|^2) -\epsilon(b_+ b_- + b_+^* b_-^*) +u(|b_+|^4 +|b_-|^4) + w |b_+|^2|b_-|^2$ for two complex envelopes $b_\pm(t)$. The results have been compared to parametric instabilities of vortex ring in condensate with density $\rho(z,r)=1-r^2-\alpha z^2$, which take place at $\alpha\approx 8/5$ and at $\alpha\approx 16/7$.

The Electrical Activity of He II with Relative Motion of Normal and Superfluid Components

It is shown that in the relative motion of the normal and superfluid components of He II with velocity w, an electric potential difference arises. The causes of this phenomenon are electrical properties of quantum vortex rings (QVR) and the anisotropic QVR energy dependence on its momentum due to w. Explicit analytical expressions for the potentials arising from the second sound wave in an external electric field are derived. The comparison of calculated values with observed ones is made, and new experiments are proposed.

Dynamics of Nonplanar Quantized Vortex Rings Before Reconnection at Finite Temperatures

The paper presents a numerical study of the dynamics of nonplanar quantized vortices at finite temperatures on their route to reconnection. We perform numerical simulations using the vortex filament method, solving the full Biot–Savart equation at a wide range of temperatures and initial conditions. We consider the dynamics of the two rings, lying initially in different planes and at different distances. The angles between planes are taken as equal to 30\(^{\circ }\), 45\(^{\circ }\), 60\(^{\circ }\), and 90\(^{\circ }\). It is observed that the temperature and the initial position of the vortices strongly affect the dynamics of the vortices on their route to reconnection. However, when the distances between the vertices of the vortices become smaller than the distances satisfying the Schwarz reconnection criterion, the dynamics of the system change drastically, and this trend is universal. The universality is expressed in the shapes and velocities of the vertices of the vortices.

Dynamics of Laser Ablation in Superfluid $$^4\hbox {He}$$

Pulsed laser ablation of metal targets immersed in superfluid $$^4\hbox {He}$$ is visualized by time-resolved shadowgraph photography and the products are analyzed by post-experiment atomic force microscopy (AFM) measurements. The expansion dynamics of the gaseous ablation half-bubble on the target surface appears underdamped and follows the predicted behavior for the thermally induced bubble growth mechanism. An inherent instability of the ablation bubble appears near its maximum radius and no tightly focused cavity collapse or rebound events are observed. During the ablation bubble retreat phase, the presence of sharp edges in the target introduces flow patterns that lead to the creation of large classical vortex rings. Furthermore, on the nanometer scale, AFM data reveal that the metal nanoparticles created by laser ablation are trapped in spherical vortex tangles and quantized vortex rings present in the non-equilibrium liquid.

Interactions between unidirectional quantized vortex rings

We have used the vortex filament method to numerically investigate the interactions between pairs of quantized vortex rings that are initially traveling in the same direction but with their axes offset by a variable impact parameter. The interaction of two circular rings of comparable radii produce outcomes that can be categorized into four regimes, dependent only on the impact parameter; the two rings can either miss each other on the inside or outside, or they can reconnect leading to final states consisting of either one or two deformed rings. The fraction of of energy went into ring deformations and the transverse component of velocity of the rings are analyzed for each regime. We find that rings of very similar radius only reconnect for a very narrow range of the impact parameter, much smaller than would be expected from geometrical cross-section alone. In contrast, when the radii of the rings are very different, the range of impact parameters producing a reconnection is close to the geometrical value. A second type of interaction considered is the collision of circular rings with a highly deformed ring. This type of interaction appears to be a productive mechanism for creating small vortex rings. The simulations are discussed in the context of experiments on colliding vortex rings and quantum turbulence in superfluid helium in the zero temperature limit.

Polarization of He II films upon the relative motion of the superfluid component and the quantized vortices

Theoretical study of the electrical activity of the saturated superfluid helium (He II) film upon the relative motion of the normal and superfluid components in the film was performed. The polarization vector due to the dipole moments of the quantized vortex rings in He II in the field of van der Waals forces was calculated taking into account the relative motion of the normal and superfluid components. An explicit analytical expression for the electric potential difference arising upon the relative motion of the normal and superfluid components in a torsional oscillator was derived. The obtained time, temperature and relative velocity dependences of the potential difference were in agreement with the experimental data.