Controlled symmetry breaking of the Fermi surface in ultracold polar molecules

Abstract

Long-range anisotropic dipole-dipole interactions between ultracold polar molecules are predicted to drive exotic quantum phases, yet direct many-body signatures of these interactions in degenerate Fermi gases have remained elusive. Here, we report the observation of an interaction-induced controlled deformation of the Fermi surface, providing a clear many-body signature in a deeply degenerate Fermi gas of $^{23}\text{Na}^{40}\text{K}$ molecules. Using double microwave (MW) shielding, we prepare $8 \times 10^3$ molecules at $0.23(1)$ times the Fermi temperature, achieving a three-fold suppression of inelastic losses compared to single MW shielding while preserving strong elastic dipolar scattering. We observe Fermi surface deformations of up to $7\,\%$, more than two times larger than those observed in magnetic atoms, despite operating at two orders of magnitude lower densities. Crucially, we demonstrate continuous tuning of the interaction potential from axial U(1) to biaxial C$_{2}$ symmetry, directly imprinting this geometry onto the Fermi surface. We find excellent agreement between our experimental results and parameter-free Hartree-Fock theory. These results establish MW-shielded polar molecules as a highly tunable platform for exploring strongly correlated dipolar Fermi matter and offer a promising path towards topological superfluidity.

Figures (10)

• Figure 1: Fermi Surface Deformation of polar molecules with U(1) and C$_2$ symmetric dipolar interactions.a) Experimental geometry: Molecules confined in an optical dipole trap are shielded by two MW fields, a circularly polarized field and a nearly orthogonally linearly polarized field. b) Mechanism of FSD: Anisotropic Fock exchange energy (green) arising from dipolar interaction elongates the momentum distribution along the direction of attractive interactions, while the isotropic kinetic pressure (red) counteracts it, yielding a squeezed, ellipsoidal FS (blue). c) Hartree–Fock prediction of the dimensionless deformation $\Delta_{xy} = \sigma_x/\sigma_y-1$ as a function of ellipticity $\xi$ and Rabi frequency $\Omega_\pi$ at $T/T_\mathrm{F}=0.2$. Here $\sigma_x$ ($\sigma_y$) denotes the cloud width along $x$ ($y$), the principal axes of the deformed FS. Fixed parameters: $(\Delta_\sigma, \; \Delta_\pi, \;\Omega_\sigma) = 2\pi\times (20,\, 30,\, 30) \,\mathrm{MHz}$. The dashed line marks the dipolar cancellation point, while shaded gray regions indicate proximity to field-linked resonances Chen2022, where the perturbative theory is no longer valid. 3D plots show the dipolar interaction energy landscape at a representative parameter, with attractive (repulsive) interactions shown in red (blue). d) Top: Difference between an elliptical and a circular Fermi–Dirac fit (see \ref{['sec:image_analysis']}) to the averaged absorption image (65-70 shots) at $\xi=\pm\ang{10}$. Black ellipses indicate the direction of the MW major axis. Scale bar: $30\,\mu\mathrm{m}$. Bottom: Difference images obtained by subtracting a $\ang{90}$-rotated copy of the cloud from the original image, residuals highlighting the quadrupolar deformation. Black solid (dashed) lines indicate the directions of attractive (repulsive) interactions. $x'-y'$ axes indicate the camera axes, whereas $x-y$ axes indicate the principal axes of the deformed FS.
• Figure 2: Radial Fermi surface deformation vs Fermi degeneracy.a) Extracted deformation $\Delta_{xy} = \sigma_x/\sigma_y -1$ versus reduced temperature $T/T_{\mathrm{F}}$. Blue, orange, and green circles correspond to $\xi = \ang{-10},\, \ang{0},$ and $+\ang{10}$, respectively. Error bars are the standard deviation from the fit to the averaged (15–20 shots) images. The shaded areas are finite-temperature Hartree-Fock predictions with no free parameter, taking into account the uncertainty in molecule numbers, $T/T_{\mathrm{F}}$, and $\xi$. Insets Representative absorption images after $14 \, \mathrm{ms}$ of ballistic expansion at $\xi = \pm \ang{10}$ (filled circles). Each panel averages 65-70 shots, each pixel corresponds to $2\times 2$ binning of raw images, with normalized optical depth shown in false colour. b) Residuals from subtracting circular from elliptical Fermi–Dirac fits for the data points marked by filled circles in (a), Scale bar: 30 $\mu\mathrm{m}$. $x-y$ axes in panels (i) and (iii) show the principal axes of clouds for $\xi = \ang{-10}$ and $+\ang{10}$, respectively. c) Corresponding self-subtracted images (see text for definition), highlighting the quadrupolar four-lobe pattern. d) Dimensionless angle dependence of long-range interaction energy, $r^3V_\mathrm{dd}$, integrated along $z'$ direction for negative, zero and positive $\bar{C}_3$ (top to bottom). In panels (b--d), dashed ellipses show the orientations of MW fields, rows (i), (ii), (iii) correspond to $\xi = -\ang{10}$, $\ang{0}$, $+\ang{10}$ from top to bottom.
• Figure 3: Axial Fermi surface deformation vs Fermi degeneracy.a) Measured deformation $\Delta_{\rho z} = \sigma_\rho/\sigma_z - 1$ versus $T/T_\mathrm{F}$ for $\Omega_\pi = 2\pi\times 15\,\mathrm{MHz}$ (blue, upper), $2\pi\times 30\,\mathrm{MHz}$ (orange, middle), and $2\pi\times 40\,\mathrm{MHz}$ (green, lower). Error bars denote uncertainties from the fits. Shaded bands indicate finite-temperature Hartree–Fock calculations that include trap anisotropy and experimental uncertainties in molecule number, temperature, and $\Omega_\pi$. Insets Representative absorption images of the filled circle data points for $\Omega_\pi \in 2\pi\times (15,\,30,\,40)$ MHz, each averaged over $60$-$70$ shots, with $2 \times 2$ binning and normalized OD of raw images. b) Residuals obtained by subtracting elliptical and circular Fermi-Dirac fits from the measured distributions for filled circles marked in (a). $\rho-z$ axes in panels (i) and (iii) denote the principal axes in the MW frame when the circular and linear MW fields dominate, respectively. c) Self-subtracted images for the same data. Scale bar: $45 \, \mu\mathrm{m}$. d) Corresponding dimensionless interaction energy $r^3V_\mathrm{dd}$ integrated along $y'$. Ellipses and lines indicate the circular and linear MW field directions, respectively; in panels (b–c), their orientations are inferred from polarization measurements assuming the fields lie in the $x'$–$z'$ plane, while panel (d) shows the ideal case of orthogonal MW fields.
• Figure 4: Control of Fermi surface deformation via tunable dipolar interactions.a)$\Delta_{\rho z}$ versus $\Omega_\pi$ at $\xi = \ang{0}$. The solid line shows a finite temperature Hartree-Fock calculation at $0.3\,T_\mathrm{F}$, while data span 0.26-0.34 $T_\mathrm{F}$. Error bars denote the fit uncertainties in deformation. Bottom inset: Angle between the cloud major and camera $z'$ axes. Dashed lines indicate the strongest attraction direction for $\ang{30},\, \ang{20}, \, \ang{10}$ (dark to light) misalignment between circular and linear MW field. Top right inset Independent polarization measurements resulting in the normal to circular MW and linear MW fields tilted by $\ang{12}$ and $\ang{14}$ from $z'$, respectively. b) Dipolar length, $a_\mathrm{dd}$ (in units of the Bohr radius $a_0$) from cross-dimensional rethermalization, exhibiting a minimum near the dipolar cancellation point. Error bars include fit uncertainties with density fluctuations of 3–4 runs. Solid lines are $a_\mathrm{dd}$ calculated from coupled channel calculation. Inset: Raw rethermalization data showing the evolution of temperature difference after directional heating of the cloud at $\Omega_\pi = 2\pi\times 10\,\mathrm{MHz}$ (dark blue) and $\Omega_\pi = 2\pi\times 30\,\mathrm{MHz}$ (light blue), dashed curves are exponential fits. Error bars are the standard error of the mean of 3-4 iterations. c)$\Delta_{xy}$ versus $\xi$ at $\Omega_\pi = 2\pi\times 22.5\,\mathrm{MHz}$. Deformation magnitude grows with $|\bar{C}_3|$ and changes sign at $\xi = \ang{0}$. Inset: Fitted cloud orientation shifts by $\sim \ang{90}$ at zero ellipticity, reflecting near-orthogonal rotation of the MW major axis. d)$a_\mathrm{dd}$ versus $\xi$ shows a minimum at $\xi = \ang{0}$, and symmetric growth with $|\xi|$ that follows deformation in (c). Inset: rethermalization traces at $\xi = \ang{0}$ (dark blue) and $\ang{13}$ (light blue).
• Figure 5: Time-of-flight dependence of the axial deformation.$\Delta_{\rho z}$ measured as a function of ballistic expansion time for two linear-field strengths: $\Omega_\pi = 2\pi\times 15\; \mathrm{MHz}$ (blue circles) and $\Omega_\pi = 2\pi\times 40\; \mathrm{MHz}$ (green circles). The orange-dashed line shows the expected evolution for a non-interacting, isotropic Fermi gas released from the same oblate trap. Blue and green dashed lines are finite-temperature Hartree–Fock predictions that include both the intrinsic momentum-space deformation and the residual real-space anisotropy imposed by the trap and dipole interactions. The dot-dashed line shows the corresponding value of only momentum-space deformation achievable at a long time of flight. Error bars are the standard fitting error of 8-10 averaged images per point. The low density of the cloud resulted in large fitting uncertainties at 18 ms data. Inset: Time-of-flight evolution of the major axis orientation of the deformed cloud. At short times, the orientation is set by the trap axes, while at long times it aligns with the direction of the strongest dipolar attraction. The figures illustrate that both the aspect ratio and the reorientation of the cloud are momentum-space dominated at 16 ms, as shown in Figs. \ref{['fig:fig_3']} and \ref{['fig:fig_4']}.