Complex-valued in-medium potential between heavy impurities in ultracold atoms

TL;DR

This work defines and analyzes a complex-valued in-medium potential between two heavy impurities (polarons) in finite-temperature cold-atom media, encapsulating decoherence via the imaginary part $V_{ ext{Im}}$. In the weak-coupling regime, both the real and imaginary parts are expressed through the medium's density-density retarded Green's function, enabling concrete calculations for a Fermi gas and a superfluid. A striking result is the universal long-range tail $V_{ ext{Im}}(r) \\propto r^{-2}$, arising from elastic scattering of medium excitations by the static impurities rather than gaplessness, with clear experimental pathways via RF interferometry, bipolaron spectroscopy, and impurity-quench density dynamics. The findings bridge open-quantum-system concepts with polaron physics, offering a framework applicable from cold atoms to quark-gluon plasmas and suggesting avenues for quantum simulation of dissipative impurity interactions.

Abstract

We formulate the induced potential in a finite temperature cold atomic medium between two heavy impurities, or polarons, which is shown to be \textit{complex-valued} in general. The imaginary part of the complex-valued potential describes a decoherence effect, and thus, the resulting Schrödinger equation for the two polarons acquires a non-Hermitian term. We apply the developed formulation to two representative cases of polarons interacting with medium particles through the $s$-wave contact interaction: (i) the normal phase of single-component (i.e., spin-polarized) fermions using the fermionic field theory, and (ii) a superfluid phase using the superfluid effective field theory, which is valid either for a Bose-Einstein condensate (BEC) of a single-component Bose gas or for the BEC-BCS crossover in two-component fermions at a low-energy regime. Computing the leading-order term, the imaginary part of the potential in both cases is found to show a universal $r^{-2}$ behavior at long distance. We propose three experimental ways to observe the effects of the universal imaginary potential in cold atoms.

Figures (5)

• Figure 1: Feynman diagram representing the exchange of the particle-hole fluctuation (solid lines), which induces the potential at $O(g^2)$ between two impurities (amputated bold solid lines).
• Figure 2: (a) The imaginary part of the induced potential for Fermi polarons with $T/T_F = 0.1,0.5$, and $1.0$ (solid blue, dashed-dotted green, and dashed red curves), normalized by $T$. The dotted black curve shows the zero-temperature limit of the imaginary potential. (b) The imaginary part of the induced potential at long distances for Fermi polarons with $T/T_F = 0.1,0.5$, and $1.0$ (solid blue, dashed-dotted green, and dashed red curves), normalized by $T_F$. The dotted black curve shows the power-law decay $r^{-2}$, which matches those of the induced potential.
• Figure 3: Feynman diagram representing the exchange of the phonon (solid lines), which induces the potential between two impurities (amputated bold solid lines) at $O(g^2)$. At low frequencies, the left diagram can be effectively reduced to the right one (see the main text).
• Figure 4: The imaginary part of the induced potential for polarons in the superfluid. (a) at short distances and (b) long distances. Solid blue, dashed-dotted green, and dashed red curves show results for $T/(mc_s^2) = 0.6,0.8$, and $1.0$. In panel (b), the dotted black curve shows the power-law decay $r^{-2}$, which matches the asymptotic behaviors of the induced potential.
• Figure 5: Scattering interpretation of the imaginary part of the induced potential. Taking the imaginary part results in cutting the diagram with the dashed line in the left panel. The imaginary part is thus given by the right panel, which gives a cross section of the on-shell medium excitation (solid lines) scattered by the static impurity (amputated bold lines).