Past exam of the superconductivity and Quantum Coherence course of the University of Cambridge / 2016 / Paper 1 / 1 / c

The two-fluid model of superfluidity separates mass current into a superfluid fraction and a normal component of a superfluid:The normal component carries entropy and responds to shear viscosity; an ideal superfluid flow is entropy-free. These are interpenetrating hydrodynamic responses, not two permanently distinguishable sets of atoms. Thermal elementary excitations carry momentum relative to the condensate and generate the normal response.

The Andronikashvili experiment gives an operational picture. A closely spaced stack of disks oscillates torsionally in He-II. Viscous coupling entrains the normal response between disks, changing the oscillator's inertia, while vortex-free superfluid need not follow the rotation. The temperature-dependent inertia measures the normal component under the apparatus's coupling conditions. Counterflow also permits second sound, primarily an entropy/temperature wave rather than the ordinary density wave of first sound.

Condensate depletion is a different quantity. The condensate fraction counts occupation of a single-particle mode, while ${\rho }_s$ measures the response to phase gradients or motion. In a weak three-dimensional homogeneous Bose gas at zero temperature,so interactions produce quantum depletion even without thermal excitations. In a translation-invariant Galilean system at zero temperature, the superfluid mass density can nevertheless equal the total density. He-II has strong interaction-induced depletion but a nearly complete zero-temperature superfluid response. Lattices, disorder and finite temperature require their own response calculation; condensate and superfluid fractions should not be equated in general.

The Landau criterion follows by considering an excitation of momentum $\mathbf{p}$ in a moving fluid. Its laboratory-frame creation cost can be $ϵ(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$. A necessary energetic condition against this decay channel isFor the low-momentum Bogoliubov spectrum of a weakly interacting Bose gas, $ϵ\simeq c_{s}p$, giving the sound-speed threshold for that channel. The phonon–roton spectrum of He-II can instead have its minimum ratio near the roton region. Vortex nucleation, boundaries and a moving obstacle can lower a measured critical velocity; the ideal excitation criterion is not a guarantee of dissipation-free flow at every smaller speed in every apparatus. An ideal noninteracting Bose gas has a condensate but quadratic excitations with $v_L=0$, another distinction between condensation and robust superfluidity.

A cold-gas test moved a blue-detuned laser beam through a trapped condensate and measured heating as its speed changed. The repulsive optical obstacle supplies a controlled perturbation; strong heating appeared above a threshold. Its interpretation must account for the inhomogeneous cloud and excitation/nucleation mechanism, rather than identify every observed threshold with the central sound speed.^[ref]