Visualizing the dispersions of Fermi polaron and molecule via spin-orbit coupling

Abstract

We propose to measure the dispersions of Fermi polaron and molecule by engineering spin-orbit coupling (SOC) on the impurity, which induces spin flip with finite momentum transfer. The polaron dispersion can be probed at small SOC momentum from the linear response of impurity spin. For molecule, we show that it can be prepared through an adiabatic steady-state evolution when setting SOC momentum as the Fermi momentum of majority bath. By gradually reducing SOC strength to zero, the steady state smoothly evolves to a molecular state with directional symmetry breaking. The corresponding dispersion can then be probed experimentally through the center-of-mass momentum distribution of molecules at finite density. Our scheme reveals a fundamental momentum difference between Fermi polaron and molecule, thereby offering a clear physical picture for their first-order transition in single-impurity system.

Figures (4)

• Figure 1: (Color Online). Illustration of using spin-orbit coupling (SOC) to detect the dispersions of Fermi polaron (a) and molecule (b). For a specific example we take the SOC momentum along x direction, $\bar{\mathbf{k}}=\bar{k}{\mathbf{e}}_x$, and only show energy dispersion along x. In (a), SOC with small momentum $\bar{k}$ ($\ll k_F$) is applied to couple the initially non-interacting impurity (gray point) with Fermi polaron at finite momentum $Q(=\bar{k})$ (blue point). Polaron energy $E_Q$ can be detected through the linear response of impurity spin at short time. This scheme, however, fails to detect molecule at $Q\sim k_F$ because of vanishingly small effective coupling induced by SOC. An alternative way to detect molecule is illustrated in (b), where $\bar{k}$ is set to be $k_F$ and the impurity is initially relaxed to the ground (steady) state at large SOC strength $\Omega$ (green point). As gradually reducing $\Omega$, the steady state evolves adiabatically and eventually at $\Omega\rightarrow 0$ recovers molecule state (red point). The dispersion near $Q\sim k_F$ can then be probed through center-of-mass momentum distribution of finite-density molecules.
• Figure 2: (Color Online). Linear response of impurity spin under spin-orbit coupling (SOC) at $\Omega/E_F=0.03$ and $1/(k_Fa_s)=1.3$. (a) Time evolution of impurity magnetization $M(t)$ at a fixed detuning $\delta=-3.86E_F$ and different SOC momenta $\bar{k}/k_F=0.1,\ 0.3,\ 0.9$. (b) $M$ at a given short time $t_0=50/E_F$ as a function of $\delta$. $M(t_0)$ shows a maximum at $\delta=E_{\bar{k}}$, as located by vertical lines for various $\bar{k}$. In (a,b), circles and lines are respectively from numerical evaluations using (\ref{['psi_t']}) and simplified two-level model (\ref{['H_22']}).
• Figure 3: (Color Online). Steady-state preparation of molecule state under spin-orbit coupling (SOC) with $\bar{k}=k_F$. Here $1/(k_Fa_s)=1.3$ and $\delta=E_{k_F}+0.05E_F$. (a) Eigen-state dispersion $\bar{E}_{Q}$ of spin-orbit coupled polaron at different $\Omega/E_F=1.2, 1, 0.8, 0.6, 0.4, 0$ (from bottom to top). Red and blue points respectively mark the locations of steady-state momentum $k_{\rm min}$ and the other disconnected minimum $k'_{\min}$. (b) Evolutions of $k_{\rm min}$ and $k'_{\min}$ as changing $\Omega$. (c) Evolution of impurity magnetization $M$ as changing $\Omega$. The red and blue data respectively correspond to the minima at $k_{\min}$ and $k'_{\min}$.
• Figure 4: (Color Online). (a1,b1) Same as Fig.\ref{['fig_mol']}(a,b) except for $1/(k_Fa_s)=1.2$ and $\delta=E_{k_F}+0.02E_F$, where the molecule is a metastable state with a higher energy than polaron. (a2,b2) Same as Fig.\ref{['fig_mol']}(a,b) except for a larger $\delta=E_{k_F}+0.4E_F$, where the steady-state evolution from large to zero $\Omega$ produces polaron state at $k_{\min}=0$.