Electric-field control of atom-molecule Feshbach resonances

Abstract

Ultracold molecules provide opportunities for exploring quantum matter, chemical dynamics and information processing thanks to their rich interactions, which can be controlled by external fields. Magnetic fields tune interactions through Feshbach resonances, enabling the formation of ultracold dimers and triatomic molecules from atom-dimer collisions. Here we demonstrate electric-field control of atom-molecule Feshbach resonances. In mixtures of ground-state sodium-potassium molecules and potassium atoms, electric fields shift resonance positions systematically, revealing specific trimer bound states and their electric-field dependent energies. The response differs markedly from isolated dimers, showing hindered rotation of the molecular constituent near an atom. Electric fields therefore add an independent knob for atom-molecule resonances, open spectroscopic access to triatomic quantum states, and advance controlled polyatomic quantum matter.

Figures (5)

• Figure 1: Electric-field dependence of magnetic resonance positions. Resonance positions $B_{\textrm{res}}$ are plotted versus the applied electric field $E$ for the collision partner K prepared in (a) $\left|F=1, m_F=0\right\rangle_{\textrm{K,s}}$ and in (b) $\left|F=2, m_F=-2\right\rangle_{\textrm{K,s}}$. Panel (a) shows a single resonance; panel (b) displays two distinct resonances labeled H (orange) and L (blue). Solid curves are quadratic functions of $E$ and serve as visual guides. Resonance positions are obtained from fits to molecule loss spectra versus $B_{\textrm{target}}$. An example spectrum for $E = 0.15kV\,cm^{-1}$ (dashed vertical line) is shown in the inset. Error bars in the inset represent the standard error of the relative molecule number from averaged images over 6-8 runs. To account for losses during the magnetic field ramp over the resonance, we use a Gaussian scaled with an error function as a fit model (inset, black solid line). Error bars (smaller than plot marker if not visible) in (a) and (b) correspond to the standard error from these fits.
• Figure 2: Method to determine the bound state energy. The energy of the scattering state as a function of magnetic field is approximated by the Zeeman effect of $^{39}$K (gray solid lines). Energies are referenced to the center of gravity of the K hyperfine structure plus the Stark shift of the uncoupled NaK molecule. A measured resonance in the channel $\left|F=2, m_F=-2\right\rangle_{\textrm{K,s}}$ is shown as a green circle. By the selection rules for $\Delta m_F$ (see main text), this resonance can only originate from a bound state with $\left|F=1, m_F=-1\right\rangle_{\textrm{K,b}}$ or $\left|F=2, m_F=-1\right\rangle_{\textrm{K,b}}$. Propagating these states back to $B=0$ (green dashed lines), starting from the measured resonance, we can extract the possible energy of the bound state $U^{\textrm{b}}_{F,m_F}(E)$ (arrows) referenced to the Stark shift of the uncoupled NaK molecule for the applied electric field $E$. Knowing the energy of the scattering state, we obtain the energy of the trimer $U^{\textrm{NaK}_2}_{F,m_F}(E)=$$U^{\textrm{NaK}}_{N=0,m_N=0}(E)+U^{\textrm{b}}_{F,m_F}(E)$, including the Stark shift of NaK and referenced to zero fields and center of gravity of the hyperfine structure of K and the prepared state of NaK.
• Figure 3: Stark shifts of the bound state. Solid lines show the electric field dependence $U^{\textrm{NaK}}_{ N, m_N}(E)$ of the respective NaK states. Points indicate the extracted energy shifts of the bound trimer state at $B=0$, assuming the K contribution as specified in the legend. Error bars represent the standard errors. Dashed lines are fits of the bound-state energy as a function of electric field using Eq. \ref{['eq:Fitformular']} for $N=0$, $m_N=0$. Data and fits are offset by $U^{\textrm{b}}_{F,m_F}(0)$. All fit parameters are listed in Table \ref{['tab:FitPar']}. Panels (a) and (b) correspond to the resonance measured in $\left|1, 0\right\rangle_{\textrm{K,s}}$, with (b) providing a magnified view. Panels (c) and (d) correspond to the two resonances in $\left|2, -2\right\rangle_{\textrm{K,s}}$: H and L, respectively (see Fig. \ref{['fig:EfieldData']} (b)).
• Figure 4: Experimental sequence. After molecule preparation, sodium is removed and potassium is prepared in a selected hyperfine state. The magnetic field is rapidly ramped to $B_{\textrm{target}}$, held for a time $t_{\textrm{h}}$, and then ramped back in a controlled manner to remove potassium once the field is stable and to image the remaining molecules. The fast ramp to $B_{\textrm{target}}$ leads to overshoot, which broadens the observed resonances by up to $\Delta B=0.4G$.
• Figure 5: Stark shifts of the bound state. Solid lines show the electric field dependence $U^{\textrm{NaK}}_{ N, m_N}(E)$ of the respective NaK states. Points indicate the extracted energy shifts of the bound trimer state at $B=0$, assuming the K contribution as specified in the legend. Error bars represent the standard errors. Dashed lines are fits of the bound-state energy as a function of electric field using Eq. \ref{['eq:Fitformular']} for $N=0$, $m_N=0$. Data and fits are offset by $U^{\textrm{b}}_{F,m_F}(0)$. All fit parameters are listed in Table \ref{['tab:FitPar']}. Panels (a) and (b) correspond to the resonance measured in $\left|1, 0\right\rangle_{\textrm{K,s}}$, with (b) providing a magnified view. Panels (c) and (d) correspond to the two resonances in $\left|2, -2\right\rangle_{\textrm{K,s}}$: H and L, respectively (see Fig. \ref{['fig:EfieldData']} (b)).