Quantum Dynamics of a Bose Superfluid Vortex

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 ρn(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 frequencyΩ = Ω/kBT , the standard Hall-Vinen-Iordanskii equations are valid whenΩ 1 (the "classical regime"), but elsewhere, the equations of motion become highly retarded, with significant experimental implications whenΩ 1. Remarkably, the fundamental question of how vortices and quasiparticles interact, and how vortices move is very controversial, notably for superfluids [5][6][7]. The argument is usually phrased in terms of forces acting on a vortex moving with respect to both the superfluid [having local velocity v s (r) and superfluid density ρ s ] and the normal fluid [with velocity v n (r) and density ρ n ]. Then, the classical equation of motion for the semiclassical vortex coordinate R v (t) (here taken to be a point in the plane -we discuss the 3D problem at the end) is usually written [8] aswhere F ac (t) is some driving force, M v is the vortex mass,Magnus force for a vortex with circulation κ =ẑh/m, and the quasiparticle force iswhereThese "HVI equations," due to Hall, Vinen, and Iordanskii [8,9], have been used to analyze thousands of experiments in superfluids and superconductors in the last 60 years [10]. However, there is no consensus on the value of either the mass M v (estimates in the literature range from zero to infinity [6,9,[11][12][13]. The resolution of these questions has become an important unsolved problem in physics. We briefly discuss the experimental situation below.Previous analyses have been restricted to a local (in space and time) description, derived from semiclassical or perturbative calculations of quasiparticle scattering off a static vortex, with no vortex recoil [6,9,[11][12][13]. This yields forces acting instantaneously on a quasiclassical vortex coordinate R v (t). There is no general agreement between these calculations (which are rendered difficult by the long-range vortex profile). Our tactic has been to formulate the problem fully quantum mechanically, in terms of an equation of motion for the vortex reduced density matrixρ(R, R ; t) which is obtained by integrating out all quasiparticle degrees of freedom. Here the vortex coordinate states |R , |R are defined by the position of the vortex node appearing in the many-body wave function. We then define a vortex "center of mass" coordinate R v = 1 2 (R + R ), and a "quantum fluctuation coordinate" ξ = R − R . Equations of motion are then found for these two quantum variables, with the vortex recoil automatically incorporated. Remarkably, in thermal equilibrium we find that the results largely depend on one key parameter, the ratioΩ = Ω/k B T , where Ω is the characteristic frequency of the vortex motion, and k B T is the thermal energy of ...

Energy spectra of finite temperature superfluid helium-4 turbulence

“…I am grateful to Andrei Golov for indicating to me the bottom-up approach to vortex dynamics of Thompson and Stamp, 16 and especially to Joe Vinen for numerous discussions on the nature of the quasiparticle forces on superfluid vortices.…”

“…This conclusion is consistent with the microscopic (quantum field theoretical) computation of Thompson and Stamp. 16 …”

“…15 f L is the second term that provides the coupling between Langevin and Navier-Stokes fields. It is a lift force per unit vortex length due to the Aharonov-Bohm effect for both phonon and roton parts of the normal-fluid quasiparticle spectrum, as first derived by Iordanskii (and recently rederived 16 ). Due to its topological nature, the lift force does not depend on core structure, hence scales only with material properties…”

Inertial and Fluctuational Effects on the Motion of a Bose Superfluid Vortex

“…This density matrix is conveniently rewritten in terms of a 'centre of mass' variable R = (r + r ′ )/2 and a quan-tum fluctuation variable ξ = (r − r ′ )/2. In the 'classical limit' where the characteristic frequency Ω of the vortex dynamics is low (such that Ω ≪ kT , where T is the temperature), the quantum fluctuations ξ(t) become negligible, and we then expect [10] a set of modified HVI equations to be valid for what is now a semiclassical vortex coordinate R(t). In a 2-dimensional Bose superfluid these take the form…”

“…[10]). Here F ap is the applied force on the vortex, ρ is the fluid density, κ = h m is the quantum of circulation, D 0 (T ) is the temperature-dependent longitudinal damping coefficient, and F fluc (t) is a fluctuating force with the high-T Markovian correlator…”