Extracting nonlocal interpolaron interactions from collisional dynamics

This study develops an experimental method of deducing the profile of interaction induced between impurities in a trapped gas of ultracold Fermi or Bose atoms, which are often referred to as Fermi or Bose polarons. In this method, we consider a two-body Fermi or Bose polaron collision experiment in which impurities and atoms interact only weakly. Numerical simulations of the quantum dynamics reveal the possibility to obtain information regarding the nonlocal-induced interaction between two polarons from a measured profile of the polaron wave packet at several snapshots. This is because the potential of the induced interaction is well balanced by the quantum potential whenever the WKB approximation for the relevant Schrödinger equation is applicable.

Figure 1

Probability density of the two-polaron relative wave packet (time vs relative position) for (a) $x_0/R_{\mathrm{F}}=0.5,\eta /R_{\mathrm{F}}=0.1$, (b) $x_0/R_{\mathrm{F}}=0.5,\eta /R_{\mathrm{F}}=0.05$, (c) $x_0/R_{\mathrm{F}}=0.25,\eta /R_{\mathrm{F}}=0.05$, and (d) $x_0/R_{\mathrm{F}}=0.25,\eta /R_{\mathrm{F}}=0.01$. Here, $R_{\mathrm{F}}$ and $E_{\mathrm{F}}$ denote the Thomas-Fermi radius and Fermi energy, respectively. The Fermi energy is given by $E_{\mathrm{F}}=\frac{1}{2}m{\omega }^2{R}_{\mathrm{F}}^2={\left(6N\right)}^{\frac{1}{3}}\hslash \omega$. The time $t_c$ at which the center of the relative wave packet reaches $x=0$ can be estimated from $t_c{\omega }_{\mathrm{F}}\simeq \frac{1}{4}\frac{2\pi }{\omega }{\omega }_{\mathrm{F}}\simeq 125,$ where ${\omega }_{\mathrm{F}}=E_{\mathrm{F}}/\hslash$.

Figure 2

Relative density profile $\rho (x,t)$ at three values of $t{\omega }_{\mathrm{F}}$ of the order of, or greater than, the collision time. The distinction between (a)–(d) corresponds to the difference in initial parameters as in Fig. 1.

Figure 3

Same as Fig. 2 for minus the quantum potential. For comparison, the RKKY interaction employed to solve the TDSE is also plotted as a dotted line. For detailed movies, see the Supplemental Material [45].

Figure 4

Minimum degree of deviation, $\min \left[I\right(t\left)\right]$, vs (a) initial Gaussian position $x_0/R_{\mathrm{F}}$ with $\eta /R_{\mathrm{F}}=0.01$ and (b) initial Gaussian width $\eta /R_{\mathrm{F}}$. The time written near each datum is the value that minimizes $I\left(t\right)$.

Figure 5

Degree of deviation, $I\left(t\right)$, vs time $t{\omega }_{\mathrm{F}}$ for (a) $\eta /R_{\mathrm{F}}=0.01$ [with $x_0/R_{\mathrm{F}}=0.20$ (blue dashed line), $x_0/R_{\mathrm{F}}=0.25$ (black solid line), and $x_0/R_{\mathrm{F}}=0.60$ (orange dash-dotted line)] and (b) $x_0/R_{\mathrm{F}}=0.25$ [with $x_0/R_{\mathrm{F}}=0.005$, $\eta /R_{\mathrm{F}}=0.010$ (black solid line), and $\eta /R_{\mathrm{F}}=0.025$ (orange dash-dotted line)].

Figure 6

Probability density of the two-polaron relative wave packet (time vs relative position) for the Yukawa-type (upper panels) and Efimov-type (lower panels) interactions with $U_0/\hslash \omega =U_1/\hslash \omega =-0.1$ and ${\kappa }_0{R}_{\mathrm{HO}}={\kappa }_1{R}_{\mathrm{HO}}=20$, where $R_{\mathrm{HO}}=\sqrt{\hslash /m_r\omega }$. (a) and (c) [(b) and (d)] The initial parameters are set to $x_0/R_{\mathrm{HO}}=0.3$ and $\eta /R_{\mathrm{HO}}=0.05$ ($x_0/R_{\mathrm{HO}}=1$ and $\eta /R_{\mathrm{HO}}=0.05$).

Figure 7

Relative density profile $\rho (x,t)$ at three values of $t\omega$ of the order of, or greater than, the collision time. The distinction between (a)–(d) corresponds to the difference in initial parameters as in Fig. 6.

Figure 8

Same as Fig. 7, for minus the quantum potentials. The dotted line in each panel denotes the mediated interaction involved. For detailed movies, see the Supplemental Material [45].

Figure 9

Minimum degree of deviation, $\min \left[I\right(t\left)\right]$, vs initial Gaussian position $x_0/R_{\mathrm{HO}}$ for the (a) Yukawa-type and (b) Efimov-type interactions with $\eta /R_{\mathrm{HO}}=0.05$. The time written near each datum is the value that minimizes $I\left(t\right)$.

Figure 10

Degree of deviation, $I\left(t\right)$, vs time $t\omega$ when (a) $\eta /R_{\mathrm{HO}}=0.3$ [with $x_0/R_{\mathrm{HO}}=0.30$ (blue dashed line), $x_0/R_{\mathrm{HO}}=0.70$ (black solid line), and $x_0/R_{\mathrm{HO}}=1.10$ (orange dash-dotted line)] for the Yukawa-type potential, as well as (b) $\eta /R_{\mathrm{HO}}=0.7$ [with $x_0/R_{\mathrm{HO}}=0.30$ (blue dashed line), $x_0/R_{\mathrm{HO}}=0.70$ (black solid line), and $x_0/R_{\mathrm{HO}}=1.10$ (orange dash-dotted line)] for the Efimov-type potential.