We study numerically the formation of a vortex lattice inside a rotating bucket containing superfluid helium, paying attention to an important feature which is practically unavoidable in all experiments: the microscopic roughness of the buckets surfa
ce. We model this using the Gross-Pitaevskii for a weakly-interacting Bose gas, a model which is idealised when applied to superfluid helium but captures the key physics of the vortex dynamics which we are interested in. We find that the vortex lattice arises from the interaction and reconnections of nucleated U-shaped vortex lines, which merge and align along the axis of rotation. We quantify the effects which the surface roughness and remanent vortex lines play in this process.
We study the elementary characteristics of turbulence in a quantum ferrofluid through the context of a dipolar Bose gas condensing from a highly non-equilibrium thermal state. Our simulations reveal that the dipolar interactions drive the emergence o
f polarized turbulence and density corrugations. The superfluid vortex lines and density fluctuations adopt a columnar or stratified configuration, depending on the sign of the dipolar interactions, with the vortices tending to form in the low density regions to minimize kinetic energy. When the interactions are dominantly dipolar, the decay of vortex line length is enhanced, closely following a $t^{-3/2}$ behaviour. This system poses exciting prospects for realizing stratified quantum turbulence and new levels of generating and controlling turbulence using magnetic fields.
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the sets underlying structure. We begin by investigating finite sets of perfect squ
ares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in finite fields of prime order. Finally, for sufficiently small finite fields we computationally calculate the minimum for all prime orders.
Understanding quantum dynamics in a two-dimensional Bose-Einstein condensate (BEC) relies on understanding how vortices interact with each others microscopically and with local imperfections of the potential which confines the condensate. Within a sy
stem consisting of many vortices, the trajectory of a vortex-antivortex pair is often scattered by a third vortex, an effect previously characterised. However, the natural question remains as to how much of this effect is due to the velocity induced by this third vortex and how much is due to the density inhomogeneity which it introduces. In this work, we describe the various qualitative scenarios which occur when a vortex-antivortex pair interacts with a smooth density impurity whose profile is identical to that of a vortex but lacks the circulation around it.
Using the classical field method, we study numerically the characteristics and decay of the turbulent tangle of superfluid vortices which is created in the evolution of a Bose gas from highly nonequilibrium initial conditions. By analysing the vortex
line density, the energy spectrum and the velocity correlation function, we determine that the turbulence resulting from this effective thermal quench lacks the coherent structures and the Kolmogorov scaling; these properties are typical of both ordinary classical fluids and of superfluid helium when driven by grids or propellers. Instead, thermal quench turbulence has properties akin to a random flow, more similar to another turbulent regime called ultra-quantum turbulence which has been observed in superfluid helium.
We model the superfluid flow of liquid helium over the rough surface of a wire (used to experimentally generate turbulence) profiled by atomic force microscopy. Numerical simulations of the Gross-Pitaevskii equation reveal that the sharpest features
in the surface induce vortex nucleation both intrinsically (due to the raised local fluid velocity) and extrinsically (providing pinning sites to vortex lines aligned with the flow). Vortex interactions and reconnections contribute to form a dense turbulent layer of vortices with a non-classical average velocity profile which continually sheds small vortex rings into the bulk. We characterise this layer for various imposed flows. As boundary layers conventionally arise from viscous forces, this result opens up new insight into the nature of superflows.
We use classical field simulations of the homogeneous Bose gas to study the breakdown of superflow due to vortex nucleation past a cylindrical obstacle at finite temperature. Thermal fluctuations modify the vortex nucleation from the obstacle, turnin
g anti-parallel vortex lines (which would be nucleated at zero temperature) into wiggly lines, vortex rings and even vortex tangles. We find that the critical velocity for vortex nucleation decreases with increasing temperature, and scales with the speed of sound of the condensate, becoming zero at the critical temperature for condensation.
We reinvestigate numerically the classic problem of two-dimensional superfluid flow past an obstacle. Taking the obstacle to be elongated (perpendicular to the flow), rather than the usual circular form, is shown to promote the nucleation of quantize
d vortices, enhance their subsequent interactions, and lead to wakes which bear striking similarity to their classical (viscous) counterparts. Then, focussing on the recent experiment of Kwon et al. (arXiv:1403.4658) in a trapped condensate, we show that an elliptical obstacle leads to a cleaner and more efficient means to generate two-dimensional quantum turbulence.
In a recent experiment, Kwon et. al (arXiv:1403.4658 [cond-mat.quant-gas]) generated a disordered state of quantum vortices by translating an oblate Bose-Einstein condensate past a laser-induced obstacle and studying the subsequent decay of vortex nu
mber. Using mean-field simulations of the Gross-Pitaevskii equation, we shed light on the various stages of the observed dynamics. We find that the flow of the superfluid past the obstacle leads initially to the formation of a classical-like wake, which later becomes disordered. Following removal of the obstacle, the vortex number decays due to vortices annihilating and reaching the boundary. Our results are in excellent agreement with the experimental observations. Furthermore, we probe thermal effects through phenomenological dissipation.
We show that an elliptical obstacle moving through a Bose-Einstein condensate generates wakes of quantum vortices which resemble those of classical viscous flow past a cylinder or sphere. The role of ellipticity is to facilitate the interaction of th
e vortices nucleated by the obstacle. Initial steady symmetric wakes lose their symmetry and form clusters of like-signed vortices, in analogy to the classical Benard-von Karman vortex street. Our findings, demonstrated numerically in both two and three dimensions, confirm the intuition that a sufficiently large number of quanta of circulation reproduce classical physics.
