Traveling dark solitons in superfluid Fermi gases

Families of dark solitons exist in superfluid Fermi gases. The energy-velocity dispersion and number of depleted particles completely determine the dynamics of dark solitons on a slowly varying background density. For the unitary Fermi gas, we determine these relations from general scaling arguments and conservation of local particle number. We find solitons to oscillate sinusoidally at the trap frequency reduced by a factor of $1/\sqrt{3}$. Numerical integration of the time-dependent Bogoliubov–de Gennes equation determines spatial profiles and soliton-dispersion relations across the BEC-BCS crossover, and proves consistent with the scaling relations at unitarity.

Figure 1

Soliton energy $E_s$ as a function of velocity $v_s$ for different couplings: (a) $\eta =-0.5$ (BCS), (b) $\eta =0$ (unitary), and (c) $\eta =1$ (BEC), where the solid lines show the analytical relations given by Eq. (4) and $E_s^{\mathrm{GP}}=\frac{4}{3}\hslash n_{B}c[1-v_s^2/c^2]{}^{3/2}$, respectively.

Figure 2

Soliton properties at unitarity $\eta =0$ as a function of velocity: (a) the magnitude of the order parameter in the center, (b) the phase difference $\delta \varphi =\arg [\Delta (\infty )]-\arg [\Delta (-\infty )]$, (c) the density in the center, and (d) the number of particles in the soliton. Dots show numerical data and lines plot Eqs. (8) and (2), respectively.

Figure 3

The spatial structure of the soliton order parameter $\Delta$ at different velocities by its (a) imaginary part and (b) magnitude in the BCS regime at $\eta =-0.5$ [panels (1a) and (1b)], unitarity limit $\eta =0$ [panels (2a) and (2b)], and BEC regime $\eta =1$ [panels (3a) and (3b)].

Figure 4

The density profile at different velocities for (a) $\eta =-0.5$ (BCS), (b) $\eta =0$ (unitarity), and (c) $\eta =1$ (BEC).