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arXiv:1110.6386 · Physical Review Letters

Quantum Dynamics of a Bose Superfluid Vortex

L. Thompson, Philip C. E. Stamp

Published May 3, 2012• 28 citations
arXivHTMLPDF

Abstract

We derive a fully quantum-mechanical equation of motion for a vortex in a 2-dimensional Bose superfluid in the temperature regime where the normal fluid density ${\ensuremath{\rho}}_{n}(T)$ is small. The coupling between the vortex ``zero mode'' and the quasiparticles has no term linear in the quasiparticle variables---the lowest-order coupling is quadratic. We find that as a function of the dimensionless frequency $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}=\ensuremath{\hbar}\ensuremath{\Omega}/{k}_{B}T$, the standard Hall-Vinen-Iordanskii equations are valid when $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}\ensuremath{\ll}1$ (the ``classical regime''), but elsewhere, the equations of motion become highly retarded, with significant experimental implications when $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}\ensuremath{\gtrsim}1$.

DOI
10.1103/physrevlett.108.184501

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We derive a fully quantum-mechanical equation of motion for a vortex in a 2-dimensional Bose superfluid in the temperature regime where the normal fluid density \ensuremathρn(T){\ensuremath{\rho}}_{n}(T)\ensuremathρn​(T) is small. The coupling between the vortex zero mode'' and the quasiparticles has no term linear in the quasiparticle variables---the lowest-order coupling is quadratic. We find that as a function of the dimensionless frequency $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}=\ensuremath{\hbar}\ensuremath{\Omega}/{k}_{B}T$, the standard Hall-Vinen-Iordanskii equations are valid when $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}\ensuremath{\ll}1$ (the classical regime''), but elsewhere, the equations of motion become highly retarded, with significant experimental implications when \ensuremathΩ\texttildelow\ensuremath≳1\stackrel{\texttildelow{}}{\ensuremath{\Omega}}\ensuremath{\gtrsim}1\ensuremathΩ\texttildelow​\ensuremath≳1.
We model vortex dynamics in a 2-dimensional Bose superfluid using the Thompson-Stamp (TS) equations of motion, which describes both the classical Hall-Vinen-Iordanskii (HVI) dynamical regime and the fully developed quantum regime, and the crossover between them. The TS equations can be written in the form of a quantum Langevin equation. Analytic solutions are given for all of these regimes for a single vortex in a 2-dimensional system. In the classical regime we include the vortex inertial and Langevin noise terms, which are dropped in the usual HVI analysis. In the quantum and crossover regimes the effect of memory terms is important, and leads to clear differences from the classical predictions.
A hydrodynamic description is used to study the normal modes of a vortex in a zero-temperature Bose-Einstein condensate. In the Thomas-Fermi limit, the circulating superfluid velocity far from the vortex core provides a small perturbation that splits the originally degenerate normal modes of a vortex-free condensate. The relative frequency shifts are small in all cases considered (they vanish for the lowest dipole mode with ∣m∣=1|m|=1∣m∣=1), suggesting that the vortex is stable. The Bogoliubov equations serve to verify the existence of helical waves, similar to those of a vortex line in an unbounded weakly interacting Bose gas. In the large-condensate (small-core) limit, the condensate wave function reduces to that of a straight vortex in an unbounded condensate; the corresponding Bogoliubov equations have no bound-state solutions that are uniform along the symmetry axis and that decay exponentially far from the vortex core.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.
After delineating the physical regimes that vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate $\ensuremath{\Omega}$ increases, we derive the normal modes of the vortex lattice in two dimensions at zero temperature. Taking into account effects of the finite compressibility, we find an inertial mode of frequency $\ensuremath{\geqslant}2\ensuremath{\Omega}$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\ensuremath{\Omega}$ is small compared with the lowest compressional mode frequency, to quadratic in the wave vector in the opposite limit. We calculate the correlation functions of vortex displacements and phase, density and superfluid velocities, and find that the zero-point excitations of the soft quadratic Tkachenko modes lead in a large system to a loss of long-range phase correlations, growing logarithmically with distance, and hence lead to a fragmented state at zero temperature. The vortex positional ordering is preserved at zero temperature, but the thermally excited Tkachenko modes cause the relative positional fluctuations to grow logarithmically with separation at finite temperature. The superfluid density, defined in terms of the transverse velocity autocorrelation function, vanishes at all temperatures. Finally we construct the long-wavelength single-particle Green's function in the rotating system and calculate the condensate depletion as a function of temperature.
The formation of a regular lattice of quantized vortices in a fluid under rotation is a smoking-gun signature of its superfluid nature. Here we study the vortex lattice in a dilute superfluid gas of bosonic atoms at zero temperature along the crossover from the weak-coupling regime, where the inter-atomic scattering length is very small compared to the average distance between atoms, to the unitarity regime, where the inter-atomic scattering length diverges. This study is based on high-performance numerical simulations of the time-dependent nonlinear Schrödinger equation for the superfluid order parameter in three spatial dimensions, using a realistic analytical expression for the bulk equation of state of the system along the crossover from weak-coupling to unitarity. This equation of state has the correct weak-coupling and unitarity limits and faithfully reproduces the results of an accurate multi-orbital microscopic calculation. Our numerical predictions of the number of vortices and root-mean-square sizes are important benchmarks for future experiments.
We introduce a nonlinear Schr\"odinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a{n}^{1/3}\ensuremath{\ll}1$ with aaa the interatomic s\mathit{s}s-wave scattering length and nnn the bosonic density, to the unitarity limit, where $a\ensuremath{\rightarrow}+\ensuremath{\infty}$. We call this equation the unitarity Schr\"odinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.