Detecting $d$-wave pairing and collective modes in fermionic condensates with Bragg scattering

We show how the appearance of $d$-wave pairing in fermionic condensates manifests itself in inelastic light scattering. Specifically, we calculate the Bragg scattering intensity from the dynamic structure factor and the spin susceptibility, which can be inferred from spin-flip Raman transitions. This information provides a precise tool with which we can identify nontrivial correlations in the state of the system beyond the information contained in the density profile imaging alone. Due to the lack of Coulomb effects in neutral superfluids, this is also an opportunity to observe the Anderson-Bogoliubov collective mode.

Figure 1

The Fermi surface for a square lattice in the tight-binding limit including only nearest-neighbor hoppings at $\mu =-t$. The nodal lines and sign of the gap are marked.

Figure 2

Plot of $\mathrm{Im}{[{\Pi }_{22}(0,0)-{\Pi }_{22}(\mathbf{Q},\Omega )]}^{-1}$ in the $s$-wave state for (a) $\Omega =0.5{\Delta }_0$, (b) $\Omega =1.5{\Delta }_0$, and (c) $\Omega =2.0{\Delta }_0$. The poles in the denominator of the second term in Eq. (18) illustrate the Anderson-Bogoliubov mode. (b) shows the approximate dispersion captured from (a) in the ($1,0$) and ($1,1$) directions compared to the result $\Omega =\frac{v_F}{\sqrt{2}}Q$.

Figure 3

The imaginary part of the zero temperature spin susceptibility, $\mathrm{Im}\left[{\Pi }_{00}^0\right]$, Eq. (12) in the $d$-wave state, for frequencies $\left(a\right)\Omega =0.5{\Delta }_0,\left(b\right){\Delta }_0,\left(c\right)1.5{\Delta }_0$, with ${\Delta }_0=0.2t$. Notice that there is zero response at $\mathbf{Q}=(0,0)$ consistent with the fact that the spin structure factor vanishes in the zero momentum limit [6].

Figure 4

The imaginary part of the zero-temperature density-density response, $\mathrm{Im}\left[{\Pi }_{33}\right]$, Eq. (18), in the $d$-wave states for frequencies (a) $\Omega =0.5{\Delta }_0$, (b) ${\Delta }_0$, and (c) $1.5{\Delta }_0$. We have artificially filled in the point (0,0) where the response exceeded the scale of the plot ${\Delta }_0=0.2t$.

Figure 5

Momentum-resolved density of states for fixed frequency $\Omega$. (a), (c) For $\Omega =0.1{\Delta }_0$ we see nodal excitations. (b), (d) For $\Omega ={\Delta }_0$ the region of highest DOS corresponds to minima in ${\nabla }_{\mathbf{k}}{E}_{\mathbf{k}}$. The solid black line is the equal-energy contour for the same energies. $k_x$ and $k_y$ are measured in 1/a where a is the lattice constant and $\mathrm{a}=1$.

Figure 6

(a), (b) Joint density of states $D_{\mathrm{Joint}}(\mathbf{Q},\Omega )$ for two-particle excitations over the range (0,$\pi$) and (c), (d) contour plots of the same.

Figure 7

Plot of $\text{Im}{[{\Pi }_{22}(0,0)-{\Pi }_{22}(\mathbf{Q},\Omega )]}^{-1}$ in the $d$-wave state for (a) $\Omega ={\Delta }_0$, (b) $\Omega =1.5{\Delta }_0$, and (c) $\Omega =2.0{\Delta }_0$. The poles in the denominator of the second term in Eq. (18) illustrate the Anderson-Bogoliubov mode. This result differs from the $s$-wave Anderson-Bogoliubov mode (Fig. 2) due to the presence of a $d$-wave gap.

Figure 8

Positions of the maxima in $\mathrm{Im}{\Pi }_{33}$ around the point $(\pi ,\pi )$ in $\mathbf{Q}$-$\Omega$ space.