Memory effects in superfluid vortex dynamics

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.