Density Modulations of Zero Sound

TL;DR

The paper addresses how a uniformly moving impurity induces density modulations in a neutral three-dimensional Fermi gas at $T=0$, focusing on the excitation of zero sound when the impurity speed $v$ exceeds the zero-sound velocity $c_0$. It employs linear response theory within the random-phase approximation using a finite-range interaction to decompose the density response into a coherent zero-sound contribution and an incoherent particle-hole background, with a polar representation $\chi(\mathbf{q},\omega+i0^+)=W(q,\Omega(q))\left[\frac{1}{\omega+i0^+-\Omega(q)}-\frac{1}{\omega+i0^++\Omega(q)}\right]$. A semi-analytic expression isolates the zero-sound signal (Eq. dnanal) which is then validated against full numerical calculations, revealing a long-range wake behind the moving impurity for strong interactions and $v>c_0$, while subsonic motion yields only localized perturbations. The results highlight the critical dependence on interaction strength, range, and shape, and discuss experimental routes in ultracold Fermi gases, including potential realization in 2D dipolar systems, to observe zero sound as a distinct density modulation.

Abstract

We study the density modulation of an interacting Fermi gas caused by the uniform motion of an impurity at zero temperature. For strong enough interaction among Fermi atoms, the modulation propagates thanks to the excitation of the collective zero sound mode if the impurity speed is above the zero sound threshold. We are able to assess, via a semi-analytic evaluation, the extent of the zero sound contribution to the density oscillation over and above the incoherent background of particle-hole excitations. Given the strong dependence of the results on the features of the gas interaction potential, we also analyze how they vary depending on its strength, range and shape.

Figures (8)

• Figure 1: (a) Dynamic structure form factor at $\omega>0$ showing the continuum of incoherent p-h excitations and the undamped collective excitation of the zero sound mode (sharp peaks). Gas parameters: $\tilde{V}_0=4,\, k_Fr_0=1,\,\alpha=8.$ (b) Zero sound dispersion relation for increasing values of $\tilde{V}_0=2,4,8$ (diamonds, triangles and circles respectively) and p-h continuum (gray region). Dotted line $\omega=v_F q$ in normalized units. The lower half of the spectrum is omitted for symmetry reasons.
• Figure 2: (a) Integration path of the cylindrical coordinate $q_{||}$ on the $(q,\omega)$ plane of Fig. \ref{['zs0']}(b). (b) Dependence of the pole $q_{||}^*$ on $q_\perp$. (c) Spectral weight $W$ of the pole $q_{||}^*$ as a function of $q_\perp$. Parameters: $\tilde{V}_0=4,\, k_Fr_0=1,\, {q_{\perp}}=0.3k_F,\, v=1.25 c_0$.
• Figure 3: (a) Density modulation induced by a mobile impurity along its line of motion $({R_{\perp}}=0)$ with velocity $v=1.25c_0$ and perturbing potential strength $\tilde{U}_0=0.1$. The dashed line reports the zero sound contribution given by Eq. (\ref{['equ:dnanal']}). Gas interaction parameters: $\tilde{V}_0=4,\, k_Fr_0=1,\,\alpha=8$. (b) Same as (a) but including the dimension perpendicular to the line of motion $({R_{\perp}} > 0)$.
• Figure 4: Density modulation for varying impurity velocity from below to above the zero sound threshold $c_0$. Gas parameters: $\tilde{V}_0=4,\, k_Fr_0=1,\,\alpha=8$.
• Figure 5: (a) Density modulation for varying strength $\tilde{V}_0$ of the gas interaction. The black dotted line corresponds to Eq. (\ref{['equ:dnanal']}) for $\tilde{V}_0=2$. (b) Density modulation for varying sharpness $\alpha$ of the gas potential $\tilde{V}(q)$. The black dotted line corresponds to Eq. (\ref{['equ:dnanal']}) for $\alpha=4$. In both panels all other parameters are as in Fig. \ref{['drho0']}.