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Grey solitons in a strongly interacting superfluid Fermi Gas

The Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS to BEC) crossover problem is solved for stationary grey solitons via the Boguliubov-de Gennes equations at zero temperature. These \emph{crossover solitons} exhibit a localized notch in the gap and a characteristic phase difference across the notch for all interaction strengths, from BEC to BCS regimes. However, they do not follow the well-known Josephson-like sinusoidal relationship between velocity and phase difference except in the far BEC limit: at unitary the velocity has a nearly linear dependence on phase difference over an extended range. For fixed phase difference the soliton is of nearly constant depth from the BEC limit to unitarity and then grows progressively shallower into the BCS limit, and on the BCS side Friedel oscillations are apparent in both gap amplitude and phase. The crossover soliton appears fundamentally in the gap; we show, however, that the density closely follows the gap, and the soliton is therefore observable. We develop an approximate power law relationship to express this fact: the density of grey crossover solitons varies as the square of the gap amplitude in the BEC limit and a power of about 1.5 at unitarity.