Memory effects in superfluid vortex dynamics

TL;DR

This paper models the non-Markovian dissipative dynamics of superfluid vortex lines, incorporating scattering effects from fermion impurities, and compares theoretical predictions with experimental data.

Contribution

It introduces a non-Markovian equation of motion for vortex dynamics that includes impurity scattering effects and provides detailed calculations of the friction coefficient.

Findings

Non-Markovian effects are weak and grow with temperature and impurity concentration.

The model's predictions agree well with experimental data and established formulas.

Markovian and elastic scattering limits are incompatible at low frequencies.

Abstract

The dissipative dynamics of a vortex line in a superfluid is investigated within the frame of a non-Markovian quantal Brownian motion model. Our starting point is a recently proposed interaction Hamiltonian between the vortex and the superfluid quasiparticle excitations, which is generalized to incorporate the effect of scattering from fermion impurities ($^3$He atoms). Thus, a non-Markovian equation of motion for the mean value of the vortex position operator is derived within a weak-coupling approximation. Such an equation is shown to yield, in the Markovian and elastic scattering limits, a $^3$He contribution to the longitudinal friction coefficient equivalent to that arising from the Rayfield-Reif formula. Simultaneous Markov and elastic scattering limits are found, however, to be incompatible, since an unexpected breakdown of the Markovian approximation is detected at low cyclotron frequencies. Then, a non-Markovian expression for the longitudinal friction coefficient is derived and computed as a function of temperature and $^3$He concentration. Such calculations show that cyclotron frequencies within the range 0.01$-$0.03 ps$^{-1}$ yield a very good agreement to the longitudinal friction figures computed from the Iordanskii and Rayfield-Reif formulas for pure $^4$He, up to temperatures near 1 K. A similar performance is found for nonvanishing $^3$He concentrations, where the comparison is also shown to be very favorable with respect to the available experimental data. Memory effects are shown to be weak and increasing with temperature and concentration.