Resonant Dipolar Collisions of Ultracold Molecules Induced by Microwave Dressing

We demonstrate microwave dressing on ultracold, fermionic ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ ground-state molecules and observe resonant dipolar collisions with cross sections exceeding 3 times the $s$-wave unitarity limit. The origin of these interactions is the resonant alignment of the approaching molecules’ dipoles along the intermolecular axis, which leads to strong attraction. We explain our observations with a conceptually simple two-state picture based on the Condon approximation. Furthermore, we perform coupled-channel calculations that agree well with the experimentally observed collision rates. The resonant microwave-induced collisions found here enable controlled, strong interactions between molecules, of immediate use for experiments in optical lattices.

Figure 1

Microwave dressing in ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$. (a) Schematic energy level diagram, labeled by rotational quantum numbers $J,m_J$ and microwave photon number $N$. Hyperfine structure is omitted for simplicity. The rotational ground state $|J=0,m_J=0⟩$ is coupled by a ${\sigma }^{+}$-polarized microwave field to the lowest energy state in the $J=1$ manifold, $|1,1⟩$, resulting in dressed states $|-⟩,|+⟩$. Higher-lying “spectator” states (i.e., $|1,0⟩$ and $|1,-1⟩$) are not coupled by the microwaves. Molecular wave functions are depicted with color encoding the wave function’s phase. (b) Level scheme with the relevant molecular states for Autler-Townes spectroscopy. A microwave field with Rabi frequency ${\mathrm{\Omega }}_R/2\pi =7\mathrm{kHz}$ is used to address the $|g_2⟩\to |f⟩$ transition. The weaker probe microwave field has a frequency detuning ${\delta }_P$ that is scanned around the $|g_1⟩\to |f⟩$ transition. (c) An Autler-Townes doublet is observed when we scan the probe microwave. The solid line shows a double Lorentzian fitted to the line shape.

Figure 2

Observation of resonant dipolar collisions between dressed molecules. (a) Dressed energies as a function of the microwave frequency detuning $\delta$ from the $|g_1⟩\to |f⟩$ transition at 129 G. $\delta$ is swept from far off-resonance (typically 12 kHz below ${\delta }_{\text{final}}$) to a final detuning at a rate of $3\mathrm{kHz}{\mathrm{ms}}^{-1}$ and held for a varying hold time $t$. (b) Evolution of the molecule number under microwave dressing with ${\mathrm{\Omega }}_R/2\pi =1.7\mathrm{kHz}$ for ${\delta }_{\text{final}}=-20,-5$, and 0 kHz (in blue triangles, red circles, and gray diamonds). A lifetime curve of $|g_1⟩$ taken without microwaves is shown with open circles. Dashed lines are two-body decay fits. (c) Collision rates (left axis) obtained from loss coefficients (right axis) of dressed molecules as a function of ${\delta }_{\text{final}}$ for ${\mathrm{\Omega }}_R/2\pi =1.7\mathrm{kHz}$ (squares) and 2.4 kHz (diamonds). The black dot-dashed line shows the universal $p$-wave loss at 560 nK [54], and the green dotted line includes the additional loss from first-order dipole-dipole interactions. The unitarity limit for the loss rate from a single partial wave is shown as the black dashed line. The blue dashed line shows the coupled-channel prediction, assuming pure ${\sigma }^{+}$ microwave polarization, and the red line depicts the rate given by the Condon approximation. Including all microwave (MW) polarizations, the Condon approximation increases to the red dashed line.

Figure 3

Two-state picture of dipolar interactions between microwave-dressed molecules, valid for detunings larger than the Rabi coupling and shown here for $M_L=1$. (a) Interaction potentials for molecules in the $|--⟩$ and $|-+⟩$ states. An effective microwave Rabi coupling between the branches causes an avoided crossing at $R_C$. Excluding spectator states, the molecules remain aligned with the microwave field. ${\mathrm{\Omega }}_{\mathrm{MW}}$ is the effective Rabi frequency between the two states, proportional to ${\mathrm{\Omega }}_R$ [48]. (b) The same as (a) but including spectator states: the relevant excited potential comes from two molecules in the $\widehat{\mathcal{R}}(|00⟩|10⟩+|10⟩|00⟩)/\sqrt{2}$ state [48], which represents the molecules aligning at short range along the intermolecular axis $\widehat{R}$. Here, molecules experience strong, resonant dipole-dipole interactions.

Figure 4

Hyperfine-state-dependent interactions, calculated at 129 G. (a) The short-range loss, which occurs when the incoming molecules reach the absorptive boundary condition that models photoinduced loss, is strongest for the lowest hyperfine state $|f⟩$, whereas (b) excited hyperfine states have larger inelastic loss due to transitions to different field-dressed levels and hyperfine states. The single-channel unitarity limit is shown as the dotted line.