Measuring the One-Particle Excitations of Ultracold Fermionic Atoms by Stimulated Raman Spectroscopy

We propose a Raman spectroscopy technique which is able to probe the one-particle Green function, the Fermi surface, and the quasiparticles of a gas of strongly interacting ultracold atoms. We give quantitative examples of experimentally accessible spectra. The efficiency of the method is validated by means of simulated images for the case of a usual Fermi liquid as well as for more exotic states: specific signatures of, e.g., a $d$-wave pseudogap are clearly visible.

Figure 1

Intensity plots of the Raman rate $R(\mathbf{q},\Omega )$, for $\Omega$ close to threshold ${\Omega }_T$ ($\Delta \Omega =0.01t_{\alpha }$). (a) Noninteracting fermions on the homogeneous 2D square lattice with density $n_{\alpha }=0.22$ and a Lorentzian broadening of the spectral function $\Gamma =0.4t_{\alpha }$ uniform in $\mathbf{k}$ space. (b) Model $d$-wave pseudogap state (see text), with ${\Delta }_0=0.1t_{\alpha }$, ${\Gamma }_0=0.05t_{\alpha }$, ${\Gamma }_1=0.4t_{\alpha }$. The plot is for a hole-doped system ($n_{\alpha }=0.45$) with a nearest ($t_{\alpha }$) and next-nearest neighbor ($t_{\alpha }^{\prime }$) hopping, with $t_{\alpha }^{\prime }/t_{\alpha }=-0.3$ (typical for cuprates, but similar effects are expected also for smaller $|t_{\alpha }^{\prime }/t_{\alpha }|$). (c),(d) Same as (a) and (b) in the presence of a harmonic trap (${\omega }_0=0.02t_{\alpha }$). The pseudogap and nodal-antinodal differentiation are clearly visible in both (b) and (d).

Figure 2

Comparison between the spectral function $A$ and the Raman rate $R/\Delta \Omega$ for two points in momentum space indicated in Fig. 1. In the nodal direction ($N$, left), the spectrum displays a quasiparticle peak, while in the antinodal direction ($A$, right) a depletion of the signal is observed at low energy, corresponding to the pseudogap.

Figure 3

(a) Color plot illustrating the selective addressing of $\mathbf{k}$ space by a proper choice of $\Omega$ (cf. color scale on the left). (b) Time of flight $\mathbf{k}$ map obtained by integrating the Raman intensity for $\Delta \Omega =\Omega -{\Omega }_T$ varied in the range $[2.4t_{\alpha },6.8t_{\alpha }]$. The dispersion relation of the $\beta$ atoms is taken as ${\epsilon }_{\mathbf{k}\beta }={\epsilon }_{\beta }^0-2t_{\beta }(2+\cos k_x+\cos k_y)$ with $t_{\beta }=1.5t_{\alpha }$ (note that interactions will renormalize downwards the effective $t_{\alpha }$ even if bare values are equal). Parameters are as in Fig. 1c and $\mathbf{q}=0$.