Effects of periodic potentials on the critical velocity of superfluid Fermi gases in the BCS-BEC crossover

We study the effects of an external periodic potential on the critical velocity of a superfluid Fermi gas in the crossover between the Bardeen-Cooper-Schrieffer (BCS) phase and Bose-Einstein condensation (BEC). We numerically solve the Bogoliubov–de Gennes equations to model a three-dimensional gas of ultracold atoms in the superfluid phase flowing through a one-dimensional optical lattice. We find that when the recoil energy is comparable to the Fermi energy, the presence of the periodic potential reduces the effect of pair-breaking excitations. This behavior is a consequence of the peculiar band structure of the quasiparticle energy spectrum in the lattice. When the lattice height is much larger than the Fermi energy, the periodic potential makes pairs of atoms to be strongly bound, even in the BCS regime, and pair-breaking excitations are further suppressed. We have also found that when the recoil energy is comparable to or larger than the Fermi energy, the critical velocity due to long-wavelength phonon excitations shows a nonmonotonic behavior along the BCS-BEC crossover.

Figure 1

Density profile $n(z)$ at $P=0$ in the BCS-BEC crossover for $E_{\mathrm{F}}/E_{\mathrm{R}}=1$ and $s=1$. Here, $n(z)$ is normalized by the average density $n_0.$

Figure 2

Amplitude $|\Delta (z)|$ of the gap parameter at $P=0$ in the BCS-BEC crossover: $1/k_{\mathrm{F}}{a}_s=-1$ (a), 0 (b), and 1 (c). The horizontal dotted lines show the amplitude of the gap parameter for the uniform system at the same value of $1/k_{\mathrm{F}}{a}_s$.

Figure 3

Critical velocity $v_c$ of the energetic instability for $E_{\mathrm{F}}/E_{\mathrm{R}}=2.5$ (a), $1$ (b), and $0.1$ (c) with $s=1$ in the BCS-BEC crossover. Open circles and filled squares show the critical velocity due to long-wavelength phonons and fermionic pair-breaking excitations, respectively. The horizontal dotted line in panel (c) represents the value of the sound velocity $c_s^{(0)}$ of uniform system at unitarity, $c_s^{(0)}/v_{\mathrm{F}}=(1+\beta ){}^{1/2}/\sqrt{3}\simeq 0.443$. The red solid lines and the black dashed lines are guides to the eye.

Figure 4

Lowest band of the quasiparticle energy spectrum ${\epsilon }_i$ for $P=0$ and $1/k_{F}a=-0.5$. Here, we show the first radial branch with $k_{\perp }^2\equiv k_x^2+k_y^2=0$. For $E_{\mathrm{F}}/E_{\mathrm{R}}=1$, the minimum of ${\epsilon }_i$ is located close to the Brillouin zone edge $k_z=Q_{\mathrm{edge}}\equiv q_{\mathrm{B}}/2$.

Figure 5

Critical current $j_c=n_0{v}_c$ for the same cases in Fig. 3. Here we show the lowest value of $j_c$, given either by long-wavelength phonon excitations or by fermionic pair-breaking excitations. The curves connecting symbols are guides to the eye.

Figure 6

Lowest band of the quasiparticle energy spectrum ${\epsilon }_i$ for large lattice height with $s=5$ and $E_{\mathrm{F}}/E_{\mathrm{R}}=0.1$ (i.e., $E_{\mathrm{F}}/V_0=0.02$) in the BCS regime at $1/k_{\mathrm{F}}{a}_s=-1$. Here, we show the first radial branch with $k_{\perp }^2\equiv k_x^2+k_y^2=0$, which always gives the smallest values of ${\epsilon }_i$ in this case. The inset shows the amplitude $|\Delta (z)|$ of the order parameter at $P=0$. The horizontal dotted line shows the amplitude of the order parameter for the uniform system at the same values of $1/k_{\mathrm{F}}{a}_s=-1$.

Figure 7

Critical velocity $v_c$ for dynamical instability due to long-wavelength excitations in the BCS-BEC crossover. The solid lines are guides to the eye. The inset shows the magnification of the region of $v_c/v_{\mathrm{F}}=0$ – $0.3$.