Universal Bound States with Bose-Fermi Duality in Microwave-Shielded Polar Molecules

TL;DR

This work uncovers universal bound states in microwave-shielded polar molecules by leveraging an effective 1D description of long-range dipolar interactions coupled via a highly elliptic microwave field. Through a Born-Oppenheimer framework and a hierarchy of 1D reductions, the authors show that two- and three-molecule sectors host universal bound states governed by the dipolar length $l_d$ and the microwave-induced length $l_\Omega=\sqrt{\hbar/(m\Omega)}$, with the hexatomic bound states exceeding twice the tetratomic binding energy. A robust Bose-Fermi duality arises from a large repulsive core produced by angular fluctuations, enforcing identical energies and densities for bosons and fermions in the effective 1D sector. The results further predict elongated, crystalline self-bound droplets in large ensembles for both statistics, highlighting a pathway to novel quantum phases in dipolar molecular systems. Overall, the work demonstrates that microwave ellipticity and long-range dipolar interactions yield universal few-body clusters independent of short-range details, with explicit connections between two-, three-, and many-body states.

Abstract

We report universal bound states of microwave-shielded ultracold molecules that solely depend on the strengths of long-range dipolar interaction and microwave coupling. Under a highly elliptic microwave field, few-molecule scatterings in three dimension are shown to be governed by effective one-dimensional (1D) models, which well reproduce the tetratomic bound state and the Born-Oppenheimer potential in three-molecule sector. For hexatomic systems comprising three identical molecules, we find much deeper bound state than the tetratomic one, with binding energy exceeding twice of the latter. Strikingly, these bound states display Bose-Fermi duality as facilitated by the effective 1D scattering with a large repulsive core from angular fluctuations. For large molecule ensembles, our results suggest the formation of elongated self-bound droplets with crystalline patterns in both bosonic and fermionic molecules.

Figures (12)

• Figure 1: (Color Online). Schematics of interaction potentials and bound states of polar molecules shielded by a highly elliptic microwave field. (a) Interaction potential $V({\mathbf{r}})$ at $xy$ plane ($z=0$). (b1) Slices of $V$ at different $y$ ($y_1>y_2$, as marked by arrows in (a)). Fluctuations along $x$ ($\sim\delta\phi$) lead to a zero-point energy and effectively upshift the potential to horizontal level. The resulted effective potential $U(r\equiv |y|)$ is plotted in (b2), showing a long-range attraction $(-r^{-3})$ and a repulsive core $(r^{-4})$ from angular fluctuations. Its minimum is located at $r_m$. (c) Ground state distribution of bosonic or fermionic molecules with number $N_M$. They are all bound states aligned along $y$ with typical inter-molecule distance $r_m$.
• Figure 2: (Color Online). Binding energies of two and three identical molecules as functions of microwave coupling $\Omega$. Red solid and yellow dot lines show exact tetratomic energies of bosonic ($E_b^{(2)}$) and fermionic ($E_f^{(2)}$) systems, in comparison to pink dashed lines from effective 1D model ($E^{(2)}_{\rm 1D}$). Blue dash-dot lines show hexatomic energies from effective 1D model ($E^{(3)}_{\rm 1D}$). For each case we show two lowest energy levels. The units of $E$ and $\Omega$ are $E_u$ and $E_u/\hbar$.
• Figure 3: (Color Online). Bose-Fermi duality of tetratomic and hexatomic bound states at $\hbar\Omega/E_u=52$. (a1,a2) and (b1,b2) show wavefunctions of the lowest tetratomic states, respectively, from exact solution and effective 1D model. (c1,c2) are wavefunctions of the lowest hexatomic states from effective 1D model. (a1,b1,c1) are for bosonic molecules and (a2,b2,c2) are for fermionic ones. (d) and (e) are density correlation functions $G_2(y)$ for tetratomic and hexatomic states. In (d), exact results of bosonic (red solid) and fermionic (yellow dot) systems are compared with those from effective 1D model (gray solid). Here the length unit is $l_u$.
• Figure 4: (Color Online). Light-induced heavy-heavy potential $V_{\rm BO}(R\equiv|{\mathbf{R}}|)$ in the Born-Oppenheimer limit. Here $\hbar\Omega/E_u=52$, and the gray horizontal line marks the binding energy of one heavy-light pair. For ${\mathbf{R}}$ along $y$, we show two orthogonal levels of $V_{\rm BO}$ (blue and red triangles), while for ${\mathbf{R}}$ along $x$ or $z$ (equivalent), we show the lowest $V_{\rm BO}$ (green circle). Dashed lines are results from effective 1D model (Eq.\ref{['U_L']}). Dotted lines at large $R$ show mean-field energies between a heavy-light pair and the rest light moleculesupple. The length and energy units are respectively $l_u$ and $E_u$.
• Figure 5: (Color Online). Effective potential and wavefunctions of three identical molecules. (a) Effective 1D potential $U^{(3)}$ (Eq.\ref{['U0']}) in $(y_r,y_{\rho})$ plane at $\hbar\Omega/E_u=52$. (b) $U^{(3)}$ and $\tilde{U}^{(3)}$ as functions of $y_r$ at a fixed $y_{\rho}=0$ (corresponding to three equally spaced molecules). (c) Energy spectra from $U^{(3)}$ (solid) and $\tilde{U}^{(3)}$ (dashed). (d) Wavefunction overlap for the ground and excited states of two models. The units of length, energy, and $\Omega$ are respectively $l_u$, $E_u$ and $E_u/\hbar$.