Efimov Effect in Ultracold Microwave-Shielded Polar Molecules

Abstract

A quantum-mechanical description is presented for the three-body physics of shielded dipolar molecules, including a prediction of observable Efimov physics. Despite the anisotropic and long-range nature of the interaction, shielding enables a regime in which universality emerges already at the two-body level and extends to the three-body sector, where Efimov physics emerges. On the negative side of the scattering-length resonance, computed trimer binding energies display the characteristic scaling expected for Efimov resonances. Finally, the sudden approximation can be used to create trimer bound states, starting from positive energy trap states as a way to create or detect these molecular trimers. Moreover, the three-body parameter expressed in dipolar units is found to be universal.

Figures (4)

• Figure 1: (a) Typical variation of the scattering length $a_s$ (dashed) and dipolar length $a_d$ (solid) in scheme $\text{I}$ for NaCs as function of $\Omega_\pi$ with microwave field parameters $(\Omega_\sigma,\Delta_\sigma,\Delta_\pi)=2\pi\times(8,8,10)$ MHz . $a_d$ goes to zero at the compensation point where the $\sigma$ and $\pi$ field balance each other. On either side of the compensation point, there are field linked resonances. Increasing (Decreasing) $\Omega_\pi$ leads to $\pi$- ($\sigma$)-dominated resonances. (b) Variation of $a_s/|a_d|$ for CaF in scheme $\text{II}$ for different $\delta_{\pi,\sigma}$ values with $\mathcal{G}=2$ and $a_d=-(3840.6,~3639.4,~4900)a_0$ for $\delta_{\pi,\sigma}=(2,0.5)$, $(1.8,0.5)$ and $(2,0.3)$ respectively.
• Figure 2: (a) Detuning scheme for double microwave shielding. (b)Variation of the dimensionless ratio $a_s/|a_d|$ in the vicinity of the first field linked resonance for CaF at field parameters $(\Omega_\sigma,\Delta_\sigma,\Delta_\Pi)=2\pi\times(20,15,18)$ MHz. Grey shaded area shows the region where scattering length becomes the largest length scale and signatures of Efimov Physics start appearing. (c) Calculated three-body adiabatic potential curves for shielded CaF at unitarity $|a_s|\rightarrow\infty$. Adiabatic hyperspherical curves $U_{\nu}(R)$ (black solid) and $W_{\nu}(R)$ (red dashed) are shown together. Inset: Signature of the universal Efimov potential (d) Lowest hyperradial curves for different scattering lengths rescaled to reveal the effect of increasing $a_s/|a_d|$. Starting from the dashed-dot-dot curve, $a_s/|a_d|=-(3, ~7, ~26, ~83, ~273, ~864, ~4418)$ and the last solid one corresponds approximately to unitarity. Horizontal dashed line shows the non-interacting eigenvalue of the adiabatic Hamiltonian corresponding to $l_\text{eff}=3/2$.
• Figure 3: Two- and three-body universality. (a) $a_s/|a_d|$ for NaCs (blue circles) and CaF (orange squares) as a function of the reduced Rabi frequency $\hbar\Omega_\pi/E_d$ (middle x-axis), and the physical Rabi frequency for NaCs (top x-axis) and CaF (bottom x-axis) under scheme $\text{II}$ with $\delta_{\sigma,\pi}=(0.5,~2)$ and $\mathcal{G}=2$. For these field parameters, $a_d$ for NaCs is $-22794.5~a_0$, while $a_d$ for CaF is $-3840.6~a_0$. (b) Three-body trimer spectrum for NaK (blue circles), CaF (orange squares) and NaCs (green diamonds) versus the inverse scattering length $|a_d|/a_s$
• Figure 4: (a) Calculated probability of trimer formation $P_T$ with $a_s^\text{in}=-2.5~|a_d|$ as a function of the final scattering length $a_s^\text{fin}/|a_d|$ for different values of the oscillator length $a_\text{osc}/|a_d|$. (b) Comparison of the absolute value of the calculated fractional difference $\epsilon_T$ as a function of $a_s^\text{fin}/|a_d|$ for the same set of oscillator lengths used in the the left panel: $a_\text{osc}/|a_d|=2$ (blue circles), 3 (yellow squares), 4 (green diamonds), 5 (red triangles).