The Josephson relation for the superfluid density in the BCS-BEC crossover

TL;DR

This paper derives an exact Josephson relation for the superfluid density in Fermi superfluids across the BCS-BEC crossover, linking it to the pair order parameter and extending the relation from Bose to Fermi superfluids.

Contribution

It introduces a universal Josephson relation for superfluid density applicable throughout the BCS-BEC crossover, connecting it to the pair fluctuation propagator.

Findings

Provides a simple exact relation between superfluid density and order parameter.

Applicable to both weak-coupling BCS and Bose condensate regimes.

Extends the Josephson relation from Bose to Fermi superfluids.

Abstract

The Josephson relation for the superfluid density is derived for a Fermi superfluid in the BCS-BEC crossover. This identity extends the original Josephson relation for Bose superfluids. It gives a simple exact relation between the superfluid density $\rho_s$ and the broken-symmetry Cooper pair order parameter $\Delta_0$ in terms of the infrared limit of the pair fluctuation propagator. The same expression holds through the entire BCS-BEC crossover, describing the superfluid density of a weak-coupling BCS superfluid as well as the superfluid density of a Bose condensate of dimer molecules.