Spectral functions and rf response of ultracold fermionic atoms

We present a calculation of the spectral functions and the associated rf response of ultracold fermionic atoms near a Feshbach resonance. The single-particle spectra are peaked at energies that can be modeled by a modified BCS dispersion. However, even at very low temperatures their width is comparable to their energy except for a small region around the dispersion minimum. The structure of the excitation spectrum of the unitary gas at infinite scattering length agrees with recent momentum-resolved rf spectra near the critical temperature. A detailed comparison is made with momentum integrated, locally resolved rf spectra of the unitary gas at arbitrary temperatures and shows very good agreement between theory and experiment. The pair size defined from the width of these spectra is found to coincide with that obtained from the leading gradient corrections to the effective-field theory of the superfluid.

Figure 1

(Color online) The dimensionless contact coefficient $s$ is shown as a function of the dimensionless coupling strength $v=1/k_{F}a$. The red solid line represents our numerical result obtained from Eq. (2.39). The left and right black dashed lines represent the asymptotic formulas $s_{wc}$ and $s_{BEC}$ given in the text, respectively.

Figure 2

(Color) Density plots of the spectral function $A\left(k,\epsilon \right)$ at temperature $T=0.01T_F$ for the interaction strengths $v=1/\left(k_{F}a\right)=-1,0,+1$ from left to right. The white horizontal lines indicate the chemical potential $\mu$. The gray lines are fits to the maxima of $A\left(\mathbf{k},\epsilon \right)$ using Eq. (4.1).

Figure 3

(Color) Density plots of the spectral function $A\left(\mathbf{k},\epsilon \right)$ at unitarity $\left[v=1/\left(k_{F}a\right)=0\right]$ for different temperatures. From top left to bottom right: $T/T_F=0.01$, 0.06, 0.14, $0.160\left(T_c\right)$, 0.18, and 0.30. The white horizontal lines mark the chemical potential $\mu$. At temperatures smaller than the superfluid transition temperature $T_c$ two quasiparticle structures with a BCS-like dispersion can be seen. The width of the spectral peaks is of the same order as the quasiparticle energy. With increasing temperature the two branches gradually merge into a single quasiparticle structure with a quadratic dispersion above $T_c$. Note, however, that the quadratic dispersion is shifted to negative frequencies compared to the bare fermion dispersion relation. This Hartree shift is of the order of $U=-0.46{\epsilon }_F$ and is essentially responsible for the shifted rf spectra in the normal phase in Fig. 6.

Figure 4

(Color online) The spectral function $A\left(\mathbf{k},\epsilon \right)$ as a function of $\epsilon$ for selected fixed values $k$ at unitarity $v=1/\left(k_{F}a\right)=0$ and at criticality $T/T_F=0.160\left(T_c\right)$. The selected values of the wave number $k$ are represented by the colors of the lines corresponding to the peaks from left to right: $k/k_F=0.00$ (black), 0.52 (red), 0.77 (orange), 1.00 (green), 1.26 (cyan), 1.51 (blue), and 2.02 (magenta). The different methods for calculating the spectral function are distinguished by the line styles: maximum-entropy method (solid lines) and Padé approximation (dashed lines).

Figure 5

(Color) Density plot of the hole part of the spectral function $A_{-}\left(\mathbf{k},\epsilon \right)$ at unitarity $v=1/\left(k_{F}a\right)=0$ and $T/T_F=0.150$ slightly below $T_c$ (left) and in the BEC regime at $v=1/\left(k_{F}a\right)=+1$ and $T/T_F=0.207$ at the superfluid transition temperature (right). The white horizontal line marks the chemical potential $\mu$. The color scheme is the same as in Figs. 2, 3.

Figure 6

(Color online) Comparison of the calculated rf spectra at unitarity with the experimental data of Schirotzek et al. [65] from MIT at different temperatures $T$. Our numerical result is shown by the red solid line. The experimental data are represented by open diamonds connected by straight thin dashed lines. Apart from adjusting the peak heights, no fitting parameters have been used.

Figure 7

Dominant contribution to the BCS-quasiparticle self-energy at zero temperature.