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Fluid Dynamics Research

On the rotational state of a Bose–Einstein condensate

Tsutomu Kambe

Published May 28, 2014• 1 citation
Publisher

Abstract

A quantum liquid of an almost ideal Bose gas brought into rotation is investigated from a physical and mechanical point of view on the basis of the Gross–Pitaevskii (GP) equation by applying a quantum-mechanical scenario, i.e. the London scenario. This scenario allows a superfluid to have rotational states. Considering that the equation governs an interacting Bose gas, it is proposed that the GP equation admits rotational flows of a superfluid. This is carried out without incurring essential change of the equation. By this reformulation, a superfluid placed in a rotating vessel is able to have a solid body rotation with the same angular velocity as its container and also to have a meniscus approximated by a parabolic profile. The solid body rotation is accompanied by a density increase proportional to the square of its angular velocity. These are consistent with experimental observations. In addition, this formulation allows a vortex of quantized circulation with coaxial rotational core whose density does not necessarily vanish at its center.

DOI
10.1088/0169-5983/46/3/031418

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A quantum liquid of an almost ideal Bose gas brought into rotation is investigated from a physical and mechanical point of view on the basis of the Gross–Pitaevskii (GP) equation by applying a quantum-mechanical scenario, i.e. the London scenario. This scenario allows a superfluid to have rotational states. Considering that the equation governs an interacting Bose gas, it is proposed that the GP equation admits rotational flows of a superfluid. This is carried out without incurring essential change of the equation. By this reformulation, a superfluid placed in a rotating vessel is able to have a solid body rotation with the same angular velocity as its container and also to have a meniscus approximated by a parabolic profile. The solid body rotation is accompanied by a density increase proportional to the square of its angular velocity. These are consistent with experimental observations. In addition, this formulation allows a vortex of quantized circulation with coaxial rotational core whose density does not necessarily vanish at its center.
We consider a strongly interacting Bose-Einstein condensate in a spherical harmonic trap. The system is treated by applying a slave-boson representation for hard-core bosons. A renormalized Gross-Pitaevskii theory is derived for the condensate wave function that describes the dilute regime (like the conventional Gross-Pitaevskii theory) as well as the dense regime. We calculate the condensate density of a rotating condensate for both the vortex-free condensate and the condensate with a single vortex and determine the critical angular velocity for the formation of a stable vortex in a rotating trap.
We investigate the properties of quantized vortices in a dipolar Bose-Einstein condensed gas by means of a generalized Gross-Pitaevskii equation. The size of the vortex core hugely increases by increasing the weight of the dipolar interaction, approaching the transition to the supersolid phase. The critical angular velocity for the existence of an energetically stable vortex decreases in the supersolid, due to the reduced value of the density in the interdroplet region. The angular momentum per particle associated with the vortex line is shown to be smaller than $\ensuremath{\hbar}$, reflecting the reduction of the global superfluidity. The real-time vortex nucleation in a rotating trap is shown to be triggered, as for a standard condensate, by the softening of the quadrupole mode. For large angular velocities, when the distance between vortices becomes comparable to the interdroplet distance, the vortices are arranged into a honeycomb structure, which coexists with the triangular geometry of the supersolid lattice and persists during the free expansion of the atomic cloud.
Phase transitions can modify quantum behavior on mesoscopic scales and give access to new and unusual quantum dynamics. Here we investigate the superfluid properties of a rotating two-component Bose-Einstein condensate as a function of changes in the interaction energy and in particular through the phase transition from miscibility to immiscibility. We show that the breaking of one of the hallmarks of superfluid flow, namely, the quantization condition on circulation, is continuous throughout an azimuthal phase-separation process and displays intriguing flow dynamics. We find that the resulting currents are stable for long times and possess a boundary between the two condensate components that exhibits classical solid-body rotation, despite the quantum nature of superfluid flow. To support this coexistence of classical and quantum behavior the system develops a unique velocity flow profile, which includes unusual radial flow in regions near the boundary.
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.
In this paper, we study the dynamics of rotating Bose--Einstein condensates (BEC) based on the Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and present an efficient and accurate algorithm for numerical simulations. We examine the conservation of the angular momentum expectation and the condensate width and analyze the dynamics of a stationary state with a shift in its center. By formulating the equation in either the two-dimensional polar coordinate system or the three-dimensional cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient time-splitting method which is time reversible, unconditionally stable, efficient, and accurate for the problem. Moreover, it conserves the position density. We also apply the numerical method to study issues such as the stability of central vortex states and the quantized vortex lattice dynamics in rotating BEC.
In the previous chapter we described quantized vortex lines, which are one of the characteristic features of superfluids. In a classical fluid, the circulation of vortex lines is not quantized and, in addition, vortex lines decay because of viscous processes. Another feature of a superfluid, the lack of response to rotation for a small enough angular velocity, was also demonstrated. This is analogous to the Meissner effect for superconductors. One characteristic common to superfluids and superconductors is the ability to carry currents without dissipation. Such current-carrying states are not the lowest-energy state of the system. They are metastable states, the existence of which is intimately connected to the nature of the low-lying elementary excitations. The word ‘superfluidity’ does not refer to a single property of the system, but it is used to describe a variety of different phenomena (see Ref. [1]). Historically, the connection between superfluidity and the existence of a condensate, a macroscopically occupied quantum state, dates back to Fritz London's suggestion in 1938, as we have described in Chapter 1. However, the connection between Bose–Einstein condensation and superfluidity is a subtle one. A Bose–Einstein condensed system does not necessarily exhibit superfluidity, an example being the ideal Bose gas for which the critical velocity vanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems may exhibit superfluid behaviour in the absence of a true condensate, as we shall see in Chapter 15.