Cooper pairs versus Bose condensed molecules: The ground-state current in superfluid ^{3}He-A

We present a new calculation of the current $\overrightarrow{\mathrm{g}}$ flowing in a ground state of the Bardeen-Cooper-Schrieffer (BCS) form for a weakly inhomogeneous superfluid with the symmetry of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-$A$. When the structure of the order parameter not determined by symmetry is appropriate to ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-$A$ and when the mass density $\rho$ of the helium is essentially uniform, our current reduces to that calculated by Cross. If the mass density is allowed to vary, we find a generalization of the Cross current which shows that when ${\overrightarrow{\mathrm{v}}}_s=0\mathrm{}$ and the anisotropy axis $\overrightarrow{\mathrm{l}}$ is uniform, then the current is simply $\left(\frac{\hslash }{4M}\right){\overrightarrow{\nabla }}_{\rho }\times \overrightarrow{\mathrm{l}}$. We show that this property of the BCS ground state, which taken with the Cross definition leads to an "intrinsic angular momentum density" of $\frac{\rho \hslash }{2M}$ at zero temperature, also follows directly from the Gor'kov equations. If the range of the order parameter is taken to be small compared with the interatomic separation, then the ground state does not describe ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-$A$, but a Bose-Einstein condensate of tightly bound diatomic molecules. In this limit our current reduces to the form calculated by Ishikawa et al. We indicate why their analysis is only valid in this limit, and offer some rather more general remarks on the differences between Cooper pairing and the Bose-Einstein condensation of diatomic molecules.