Experimentalists use particles as tracers in liquid helium. The intrusive effects of particles on the dynamics of vortices remain poorly understood. We implement a study of how basic, well understood vortex states, such as a propagating pair of oppositely signed vortices, change in the presence of particles by using a simple model based on the Magnus force. We focus on the two-dimensional case, and compare the analytic and semianalytic models with simulations of the Gross-Pitaevskii (GP) equation with particles modeled by dynamic external potentials. The results confirm that the Magnus force model is an effective way to approximate vortex-particle motion either with closed-form simplified solutions or with a more accurate numerically solvable ordinary differential equations. Furthermore, we increase the complexity of the vortex states and show that the suggested semianalytical model remains robust in capturing the dynamics observed in the GP simulations.Experimentalists use particles as tracers in liquid helium. The intrusive effects of particles on the dynamics of vortices remain poorly understood. We implement a study of how basic, well understood vortex states, such as a propagating pair of oppositely signed vortices, change in the presence of particles by using a simple model based on the Magnus force. We focus on the two-dimensional case, and compare the analytic and semianalytic models with simulations of the Gross-Pitaevskii (GP) equation with particles modeled by dynamic external potentials. The results confirm that the Magnus force model is an effective way to approximate vortex-particle motion either with closed-form simplified solutions or with a more accurate numerically solvable ordinary differential equations. Furthermore, we increase the complexity of the vortex states and show that the suggested semianalytical model remains robust in capturing the dynamics observed in the GP simulations.

Physical Review AMay 1, 202031 citations

Particles and fields in superfluids: Insights from the two-dimensional Gross-Pitaevskii equation

Vishwanath Shukla, Rahul Anil Pandit, Marc Brächet

We carry out extensive direct numerical simulations to investigate the interaction of active particles and fields in the two-dimensional Gross-Pitaevskii superfluid, in both simple and turbulent flows. The particles are active in the sense that they affect the superfluid even as they are affected by it. We tune the mass of the particles, which is an important control parameter. At the one-particle level, we show how light, neutral, and heavy particles move in the superfluid, when a constant external force acts on them; in particular, beyond a critical velocity, at which a vortex-antivortex pair is emitted, particle motion can be periodic or chaotic. We demonstrate that the interaction of a particle with vortices leads to dynamics that depends sensitively on the particle characteristics. We also demonstrate that assemblies of particles and vortices can have rich, and often turbulent, spatiotemporal evolution. In particular, we consider the dynamics of the following illustrative initial configurations: (a) one particle placed in front of a translating vortex-antivortex pair; (b) two particles placed in front of a translating vortex-antivortex pair; (c) a single particle moving in the presence of counter-rotating vortex clusters; (d) four particles in the presence of counter-rotating vortex clusters. We compare our work with earlier studies and examine its implications for recent experimental studies in superfluid helium and Bose-Einstein condensates.We carry out extensive direct numerical simulations to investigate the interaction of active particles and fields in the two-dimensional Gross-Pitaevskii superfluid, in both simple and turbulent flows. The particles are active in the sense that they affect the superfluid even as they are affected by it. We tune the mass of the particles, which is an important control parameter. At the one-particle level, we show how light, neutral, and heavy particles move in the superfluid, when a constant external force acts on them; in particular, beyond a critical velocity, at which a vortex-antivortex pair is emitted, particle motion can be periodic or chaotic. We demonstrate that the interaction of a particle with vortices leads to dynamics that depends sensitively on the particle characteristics. We also demonstrate that assemblies of particles and vortices can have rich, and often turbulent, spatiotemporal evolution. In particular, we consider the dynamics of the following illustrative initial configurations: (a) one particle placed in front of a translating vortex-antivortex pair; (b) two particles placed in front of a translating vortex-antivortex pair; (c) a single particle moving in the presence of counter-rotating vortex clusters; (d) four particles in the presence of counter-rotating vortex clusters. We compare our work with earlier studies and examine its implications for recent experimental studies in superfluid helium and Bose-Einstein condensates.

arXiv:1710.10107Physical Review AJan 24, 201818 citations

Dynamics of active particles in superfluids and their interaction with quantum vortices

Umberto Giuriato

Superfluids are inviscid flows in which vorticity is supported on filaments with quantized circulation. Such objects, known as quantum vortices, exhibit a hydrodynamical behavior. Experimentally, the dynamics of superfluids has been studied by using particles, which nowadays have become the main tool for visualizing quantum vortices. In this Thesis, we study numerically and analytically the dynamics of active and finite-size particles in superfluids. The superfluid is modeled with the Gross–Pitaevskii equation, while the particles are implemented as moving repulsive potentials coupled with the macroscopic wave function describing the superfluid. Firstly, the model is used to investigate the interaction between particles and quantum vortices at very low temperatures. This part aims to give a theoretical background to the current experiments in which macroscopic particles are used to sample superfluid vortices and quantum turbulence. Specifically, we address the following problems: the capture of a particle by a quantum vortex, the reconnections of vortex filaments and the propagation of Kelvin waves in presence trapped particles and the dynamics of particles in decaying quantum turbulence. In the last part of the manuscript, finite temperature effects are studied in the Fourier-truncated Gross–Pitaevskii model. The goal is to characterize the dynamics of impurities immersed in a thermal bath and how their presence modifies the statistical properties of the fluid. In particular, the random motion of the impurities and the temperature dependence of the friction coefficient are studied. Finally, the clustering of impurities and its effect on the phase transitions of the condensate are investigated.Superfluids are inviscid flows in which vorticity is supported on filaments with quantized circulation. Such objects, known as quantum vortices, exhibit a hydrodynamical behavior. Experimentally, the dynamics of superfluids has been studied by using particles, which nowadays have become the main tool for visualizing quantum vortices. In this Thesis, we study numerically and analytically the dynamics of active and finite-size particles in superfluids. The superfluid is modeled with the Gross–Pitaevskii equation, while the particles are implemented as moving repulsive potentials coupled with the macroscopic wave function describing the superfluid. Firstly, the model is used to investigate the interaction between particles and quantum vortices at very low temperatures. This part aims to give a theoretical background to the current experiments in which macroscopic particles are used to sample superfluid vortices and quantum turbulence. Specifically, we address the following problems: the capture of a particle by a quantum vortex, the reconnections of vortex filaments and the propagation of Kelvin waves in presence trapped particles and the dynamics of particles in decaying quantum turbulence. In the last part of the manuscript, finite temperature effects are studied in the Fourier-truncated Gross–Pitaevskii model. The goal is to characterize the dynamics of impurities immersed in a thermal bath and how their presence modifies the statistical properties of the fluid. In particular, the random motion of the impurities and the temperature dependence of the friction coefficient are studied. Finally, the clustering of impurities and its effect on the phase transitions of the condensate are investigated.

Nov 20, 20200 citations

Magnus force in discrete and continuous two-dimensional superfluids

Z. Gecse, Sergei Khlebnikov

Motion of vortices in two-dimensional superfluids in the classical limit is studied by solving the Gross-Pitaevskii equation numerically on a uniform lattice. We find that, in the presence of a superflow directed along one of the main lattice periods, vortices move with the superflow on fine lattices but perpendicular to it on coarse ones. We interpret this result as a transition from the full Magnus force in a Galilean-invariant limit to vanishing effective Magnus force in a discrete system, in agreement with the existing experiments on vortex motion in Josephson junction arrays.Motion of vortices in two-dimensional superfluids in the classical limit is studied by solving the Gross-Pitaevskii equation numerically on a uniform lattice. We find that, in the presence of a superflow directed along one of the main lattice periods, vortices move with the superflow on fine lattices but perpendicular to it on coarse ones. We interpret this result as a transition from the full Magnus force in a Galilean-invariant limit to vanishing effective Magnus force in a discrete system, in agreement with the existing experiments on vortex motion in Josephson junction arrays.

arXiv:cond-mat/0503717Physical Review BAug 29, 20051 citation

Short-range intervortex interaction and interacting dynamics of half-quantized vortices in two-component Bose-Einstein condensates

Kenichi Kasamatsu, Minoru Eto, Muneto Nitta

We study the interaction and dynamics of two half-quantized vortices in two-component Bose-Einstein condensates. Using the Pad\'e approximation for the vortex core profile, we calculate the intervortex potential, whose asymptotic form for a large distance has been derived by Eto et al. [Phys. Rev. A 83, 063603 (2011)]. Through numerical simulations of the two-dimensional Gross-Pitaevskii equations, we reveal different kinds of dynamical trajectories of the vortices depending on the combinations of signs of circulations and the intercomponent density coupling. Under the adiabatic limit, we derive the equations of motion for the vortex coordinates, in which the motion is caused by the balance between Magnus force and the intervortex forces. The initial velocity of the vortex motion can be explained quantitatively by this point vortex approximation, but understanding the long-time behavior of the dynamics needs more consideration beyond our model.We study the interaction and dynamics of two half-quantized vortices in two-component Bose-Einstein condensates. Using the Pad\'e approximation for the vortex core profile, we calculate the intervortex potential, whose asymptotic form for a large distance has been derived by Eto et al. [Phys. Rev. A 83, 063603 (2011)]. Through numerical simulations of the two-dimensional Gross-Pitaevskii equations, we reveal different kinds of dynamical trajectories of the vortices depending on the combinations of signs of circulations and the intercomponent density coupling. Under the adiabatic limit, we derive the equations of motion for the vortex coordinates, in which the motion is caused by the balance between Magnus force and the intervortex forces. The initial velocity of the vortex motion can be explained quantitatively by this point vortex approximation, but understanding the long-time behavior of the dynamics needs more consideration beyond our model.

arXiv:1510.00139Physical Review AJan 19, 201648 citations

Propagating and annihilating vortex dipoles in the Gross-Pitaevskii equation

C. Rorai, Katepalli R. Sreenivasan, Michael Ellis Fisher

Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Pad\'e approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance ${d}_{0}$ apart, form a vortex dipole, which, in the GP model, can either annihilate $or$ propagate indefinitely as a ``solitary wave.'' We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance ${d}_{0}$ (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the ``effective wave function''), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or ``relaxed'') initial profiles are imposed. The initial ``circular'' phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by ``tuning'' the elliptical shape at fixed ${d}_{0}$ , a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a ``phase diagram'' for this ``AnSol'' transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Pad\'e approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance ${d}_{0}$ apart, form a vortex dipole, which, in the GP model, can either annihilate $or$ propagate indefinitely as a ``solitary wave.'' We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance ${d}_{0}$ (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the ``effective wave function''), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or ``relaxed'') initial profiles are imposed. The initial ``circular'' phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by ``tuning'' the elliptical shape at fixed ${d}_{0}$, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a ``phase diagram'' for this ``AnSol'' transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.

arXiv:1212.6389Physical Review BOct 29, 201322 citations

{d}_{0}

apart, form a vortex dipole, which, in the GP model, can either annihilate

or

propagate indefinitely as a ``solitary wave.'' We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance

{d}_{0}

(as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the ``effective wave function''), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or ``relaxed'') initial profiles are imposed. The initial ``circular'' phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by ``tuning'' the elliptical shape at fixed

{d}_{0}

, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a ``phase diagram'' for this ``AnSol'' transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.