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arXiv:1212.6389 · Physical Review B

Propagating and annihilating vortex dipoles in the Gross-Pitaevskii equation

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

Published Oct 29, 2013• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.88.134522

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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 d0{d}_{0}d0​ apart, form a vortex dipole, which, in the GP model, can either annihilate ororor 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.
In two dimensional superfluid, annihilation processes of vortices are investigated by numerical simulation within the dissipative Gross-Pitaevskii equation (GPE) model. First, quantum vortex solution is obtained and its fitting function is found. Second, the simulation show that positive and negative vortices accelerate in both x,y directions, until they annihilate into a soliton and then a crescent-shaped shock wave. The processes are found to be controlled by the dissipative parameter and the general Magnus force. For the behavior of separation distance between vortices d(t), an universal scaling exponent 1/2 is found which is same with the three dimensional cases. Third, system's energy is surprisingly found to be determined by system's configuration and their relation are obtained. Then we derive the general Magnus force which decreases with the increases of d(t) for large d(t) and increases with the increases of d(t) for small d(t).
Superfluids with strong spatial modulation can be experimentally produced in the area of cold atoms under the influence of optical lattices. Here we address 87Rb^{87}\mathrm{Rb}87Rb bosons at T=0T=0T=0 K in a flat geometry under the influence of a periodic potential with the Gross-Pitaevskii theory. The statics and dynamics of vortex excitations are studied in the case of one-dimensional (1D) and of two-dimensional (2D) optical lattices, as function of the intensity of the optical lattice. We compute how the vortex energy depends on the position of its core and the energy barrier that a vortex has to surmount in order to move in the superfluid. The dynamics of a vortex dipole, a pair of vortices of opposite chirality, differ profoundly from the case of a uniform superfluid. In the 1D case, when parallel ridges of density are present, the dynamics depends on the positions of the two vortices. If they are in the same channel between two ridges, then the two vortices approach each other until they annihilate each other in a short time. If the two vortices are in distinct channels, the dipole undergoes a rigid translation but with a velocity depending on the intensity of the optical lattice and this translation velocity can even change sign with respect to the case of the uniform superfluid. Superimposed on this translation an oscillatory motion is also present. A superposition of translation along a channel and an oscillation is also found with a single vortex when the system is bounded inside a circular trap. These oscillatory motions can be both longitudinal, i.e., along the channel, as well as transverse. In all cases the transverse motions are one sided, in the sense that the vortex core never crosses the equilibrium position nearest the starting position. In the case of the 2D lattices we study (square, triangular, and honeycomb), the two vortices of a dipole move mainly by jumps between equilibrium positions and approach each other until annihilation. This behavior has some similarity with what has been found for a vortex dipole in the supersolid state of dipolar bosons. We show that a rapid ramp down of the optical potential improves the visibility of the density holes at the vortex core.
We study vortex-vortex and vortex-wave collisions in two-dimensional weakly interacting Bose-Einstein condensates, processes that play a central role in decaying quantum turbulence. Using numerical simulations of the Gross-Pitaevskii equation, we show that during collisions of vortex-antivortex dipoles, the kinetic energy is transferred from incompressible to compressible modes by two distinct mechanisms. Below the critical vortex separation for annihilation, the transfer is mediated by quantum energy released during annihilation events, while above the threshold it arises from vortex acceleration. In wave-vortex collisions, an incoming solitary wave splits into transient phase slips that interact with the vortex, one of the phase slips contributes to vortex annihilation, and the other phase slip acquires a stable core and forms a new vortex. By analyzing vortex trajectories and energy spectra, we provide new insights into energy transfer mechanisms in quantum turbulence and offer broader implications for topological interactions mediated by vortices.
We study the properties of singly quantized linear vortices in the supersolid phase of a dipolar Bose-Einstein condensate at zero-temperature modeling 164Dy^{164}\mathrm{Dy}164Dy atoms. The system is extended in the $x\text{\ensuremath{-}}y$ plane and confined by a harmonic trap in the polarization direction zzz. Our study is based on a generalized Gross-Pitaevskii equation. We characterize the ground state of the system in terms of spatial order and superfluid fraction and compare the properties of a single vortex and of a vortex dipole in the superfluid phase (SFP) and in the supersolid phase (SSP). At variance with a vortex in the SFP, which is free to move in the superfluid, a vortex in the SSP is localized at the interstitial sites and does not move freely. We have computed the energy barrier for motion from an equilibrium site to another. The fact that the vortex is submitted to a periodic potential has a dramatic effect on the dynamics of a vortex dipole made of two counter-rotating parallel vortices; instead of rigidly translating as in the SFP, the vortex and antivortex approach each other by a series of jumps from one site to another until they annihilate in a very short time and their energy is transferred to bulk excitations.
One of the key issues related to superfluidity is the existence of quantized vortices. Following recent experiments on Bose–Einstein condensates exhibiting vortices, we investigate the behavior of the wave function which solves the Gross–Pitaevskii equation. For a rotating Bose–Einstein condensate in a harmonic trap, we give a simplified expression of the Gross–Pitaevskii energy in the Thomas–Fermi regime, which only depends on the number and shape of the vortex lines. This allows us to study in detail the structure of the line of a single quantized vortex, which has a U or S shape. S type vortices exist for all values of the angular velocity Ω but are not minimizers of the energy while U vortices are minimizers and exist only for Ω sufficiently large. Finally, we study the drag created by the movement of a detuned laser beam in a condensate and the nucleation of vortices in the low density region close to the boundary layer of the cloud.
The Gross-Pitaevskii (GP) equation admits a two-dimensional solitary wave solution representing two mutually self-propelled, anti-parallel straight line vortices. The complete sequence of such solitary wave solutions has been computed by Jones and Roberts (J. Phys. A, 15, 2599, 1982). These solutions are unstable with respect to three-dimensional perturbations (the Crow instability). The most unstable mode has a wavelength along the direction of the vortices of the same order as their separation. The growth rate associated with this mode is evaluated here, and it is found to increase very rapidly with decreasing separation. It is shown, through numerical integrations of the GP equation that, as the perturbations grow to finite amplitude, the lines reconnect to produce a sequence of almost circular vortex rings. 1