Enhancing Dipolar Interactions between Molecules Using State-Dependent Optical Tweezer Traps

We show how state-dependent optical potentials can be used to trap a pair of molecules in different internal states at a separation much smaller than the wavelength of the trapping light. This close spacing greatly enhances the dipole-dipole interaction and we show how it can be used to implement two-qubit gates between molecules that are 100 times faster than existing protocols and than rotational coherence times already demonstrated. We analyze complications due to hyperfine structure, tensor light shifts, photon scattering, and collisional loss, and conclude that none is a barrier to implementing the scheme.

Figure 1

(a) State-dependent trap formed from a two-color optical tweezer. Two molecules in different internal states are trapped in different locations due to opposite circular handedness (rotating arrows) on opposite sides of the focus. (b) Contour plot of $(I/I_{\max })\overrightarrow{C}·\widehat{z}$ in $y=0$ focal plane for a single tweezer. Calculated using the vector Debye integral [43] for a lens with $\mathrm{NA}=0.55$. The input beam is polarized along $x$ and has $1/e^2$ diameter equal to that of the lens. An approximate analytical approach to this calculation, based on Ref. [44], is given in the Supplemental Material [45].

Figure 2

(a) Schematic of relevant electronic structure. The excited state ${A\hspace{0.17em}}^2\mathrm{\Pi }$ is split by spin-orbit coupling into two states with $|\mathrm{\Omega }|=\frac{1}{2},\frac{3}{2}$. Dashed lines: energies of scalar (red) and vector (orange) tweezers. (b) Scalar (blue) and vector (green dashed) polarizabilities calculated using Eq. (2). (c) Calculated potentials of scalar (red dotted), vector (orange) and combined (blue) tweezers versus displacement along $x$ for $J=\frac{1}{2}$ states of $N=0$ and 1. Solid (dashed) line shows the potential for $m_J=-\frac{1}{2}\hspace{0.17em}\hspace{0.17em}(\frac{1}{2})$. Calculations are for ${\lambda }_{\mathrm{vec}}=0.8{\lambda }_{\mathrm{sc}}$, ${\mathrm{\Delta }}_{\mathrm{sc}}=50{\delta }_{\mathrm{fs}}$ and $I_{\mathrm{sc}}/I_{\mathrm{vec}}=50$. (d) Blue line, left axis: $\delta x$ versus $I_{\mathrm{sc}}/I_{\mathrm{vec}}$. Green dashed line, right axis: enhancement of $E_{\mathrm{dd}}$ for point particles trapped at the two minima.

Figure 3

Two-qubit gate in CaF. (a) Lines: potentials for the states used, calculated for 10 mW, 780 nm scalar tweezer with incident polarization along $z$ and $35.4\mu \mathrm{W}$, 604.966 nm vector tweezer polarized along $x$. The lens has NA of 0.55. Input beams have $1/e^2$ diameter equal to the lens diameter. Wave packets show ground motional states of each potential. Energies are relative to the $|N=0,F=0⟩$ ground state in zero field. (b) Level diagram of two-molecule states relevant to the gate, not to scale. Left (right) half shows energies of states without (with) dipole-dipole interaction. Black arrow: transition driven for two-qubit gate. (c) Dipole-dipole energy, photon scattering rate and collisional loss rate versus separation of $|2_{\pm }⟩$ states. Dotted blue line, energy for fixed point dipoles; solid line, full 1D calculation.