Interaction-induced Dimension Reduction for Bound States in Microwave-Shielded Ultracold Molecules

TL;DR

This work develops an effective 1D framework to describe bound states of microwave-shielded ultracold molecules in 3D, demonstrating a dimensional reduction driven purely by anisotropic long-range interactions. By incorporating high-order angular fluctuations, the authors achieve quantitative accuracy for tetratomic and hexatomic states across a wide range of ellipticity $\xi$ and coupling $\Omega$, revealing a Bose-Fermi duality in real and spectral spaces while preserving distinguishable momentum distributions. They derive explicit 1D Hamiltonians along the attractive $y$ direction, including $U_{\rm eff}^{(2)}$ and $U_{\rm eff}^{(4)}$ corrections, and validate them against exact 3D solutions; the Born-Oppenheimer treatment further confirms the hexatomic binding structure, with ground states well described by linked tetratomic units. Significantly, the hexatomic binding is deeper and supports a self-bound, crystalline-like one-dimensional array in larger systems, offering a robust route to exploring universal few- to many-body phases in dipolar molecular gases.

Abstract

We investigate tetratomic and hexatomic bound states of ultracold molecules dressed by an elliptic microwave field. We show that these bound states can be accurately described by effective one-dimensional (1D) models incorporating high-order angular fluctuations, despite the physical system is in three-dimensional (3D) free space. By comparing with exact solutions of the full 3D system, we identify the validity region of such 1D description in the parameter plane of ellipticity and coupling strength of microwave field. The hard-core character of these effective models enables a duality between bosonic and fermionic molecules in real and spectral space, while their momentum distributions remain distinct. Our results have demonstrated an effective dimension reduction in microwave-shielded molecular systems, which is purely due to the intrinsic interaction anisotropy rather than any external confinement. Extending to large systems, our results suggest a self-bound single-molecule array as the ground state of both bosonic and fermionic molecular gases.

Figures (8)

• Figure 1: Interaction potential and tetratomic bound state of microwave-shielded molecules with elliptic angle $\xi=\pi/12$. (a) Typical interaction potential $V({\mathbf{r}})$ at $xy$ plane ($z=0$). (b1,b2) Schematics of angular fluctuations of $V({\mathbf{r}})$ along $x$ ($\sim \delta\phi$) and $z$ ($\sim \delta\theta$) directions. The red points denote the minimum of $V$ when ${\mathbf{r}}$ locates at $y$ axis ($x=z=0$). (c) Comparison of different 1D potentials (along $y$) at $\hbar\Omega/E_u=157$. $V^{(0)}$ is the bare potential, and $U_{\rm eff}^{(2)},\ U_{\rm eff}^{(4)}$ are the effective potentials from the lowest- and fourth-order angular fluctuations, respectively. (d) Tetratomic binding energies as functions of $\Omega$. $E_{2}^{B}$ and $E_{2}^{F}$ are from exact solutions of bosonic and fermionic molecules, and $E^{{\rm 1D};(n)}_2$ is from effective 1D model up to the $n$-th order angular fluctuations. The units of length, energy and $\Omega$ are respectively $l_u$, $E_u$ and $E_u/\hbar$.
• Figure 2: Validity of effective 1D description. (a) Tetratomic binding energy as a function of $\Omega$ at a given $\xi=\pi/12$. $E_{2}^B$ and $E_{2}^F$ are exact solutions of bosonic and fermionic systems, and $E_{2}^{\rm 1D}$ is from effective 1D model (\ref{['Heff_2']}) with high-order angular fluctuations. Inset plot shows $\eta$, defined in (\ref{['eta']}), as a function of $\Omega$. The horizontal line marks the location when $\eta$ reduces to $5\%$, as also shown by the green arrow in the main plot. (b) Diagram for the validity of 1D description in $(\Omega,\xi)$ plane. Solid line shows the critical boundary when $\eta=5\%$, and the 1D description is a good approximation in the green area above this boundary. The units of energy and $\Omega$ are respectively $E_u$ and $E_u/\hbar$. The upper axis of $\Omega$ in (b) shows its absolute value [in unit of $(2\pi)$MHz] for NaK system.
• Figure 3: Bose-Fermi duality of tetratomic bound states. (a1,a2) show the real-space wavefunctions of bosonic ($\Psi_2^{B}$) and fermionic ($\Psi_2^{F}$) systems from exact solutions, and (b) shows their reduced density correlation functions $G_2(y)$ (Eq.\ref{['G2']}), in comparison with results from the effective 1D model. Here we take $\xi=\pi/12, \hbar\Omega/E_u=157$, and the length unit as $l_u$.
• Figure 4: Momentum-space distributions of tetratomic bound states. (a1,a2) show momentum-space wavefunctions for the lowest tetratomic bound states in bosonic ($\Psi_2^{B}$) and fermionic ($\Psi_2^{B}$) systems at $\xi=\pi/12$ and $\hbar\Omega/E_u=157$. Accordingly, (b1,b2) show the reduced momentum-space distributions along $k_y$, from both exact solutions and effective 1D model. The length unit is $l_u$.
• Figure 5: Born-Oppenheimer potential $V_{\rm BO}({\mathbf{R}}=R\hat{y})$ between two heavy molecules mediated by the light one. Here we take $\xi=\pi/12$ and $\hbar\Omega/E_u=157$. Red and blue triangles show exact results of $V_{\rm BO}$ for two lowest eigen-levels. Pink and purple dashed lines are predictions from effective 1D models up to the second- and fourth-order angular fluctuations. The horizontal gray line shows the tetratomic binding energy of a heavy-light pair. The length and energy units are respectively $l_u$ and $E_u$.