General approach to state-dependent optical-tweezer traps for polar molecules

State-dependent optical tweezers can be used to trap a pair of molecules with a separation much smaller than the wavelength of the trapping light, greatly enhancing the dipole-dipole interaction between them. Here we describe a general approach to producing these state-dependent potentials using the tensor part of the ac Stark shift and show how it can be used to carry out two-qubit gates between pairs of molecules. The method is applicable to broad classes of molecules including bialkali molecules produced by atom association and those amenable to direct laser cooling.

Figure 1

Level shifts of a molecule with a ${}^1\mathrm{\Sigma }$ ground state in dc and laser fields. (a) Energy of states as a function of electric field. States with $|m_J|=1$ are shown in blue. Inset: States in the $|\mathcal{J}=1,m_J=\pm 1\rangle$ manifold as a function of tensor tweezer intensity, $I_{\mathrm{tens}}$. Calculated for $E_z=6B/{\mu }_e$ with $\widehat{\epsilon }=\widehat{x}$ and ${\alpha }^{\left(2\right)}<0$. (b) Schematic of relevant electronic structure in the molecule. (c) Scalar (red dashed) and tensor polarizabilities (blue) calculated using Eqs. (1) and (2) with $d_{\mathrm{\Sigma }}=d_{\mathrm{\Pi }}=d$.

Figure 2

Schematic of the two schemes. Left column [(a)–(c)]: asymmetric scheme. Right column [(d)–(f)]: symmetric scheme. [(a) and (d)] Separated tweezers. [(b) and (e)] Merged tweezers. [(c) and (f)] Subwavelength spaced tweezers. Dots show positions of tweezers, their size shows intensity and their color shows type—red for scalar tweezer and blue (green) for tensor tweezer polarized along $x$ ($y$). Solid (dashed) lines show potentials for $|\mathcal{J},-\rangle$ ($|{\mathcal{J}}^{\prime },+\rangle$) states. Black lines (offset for clarity) show the combined potential of all tweezers and the filled (open) stars show the positions of the $|\mathcal{J},-\rangle$ ($|{\mathcal{J}}^{\prime },+\rangle$) states at each step.

Figure 3

Schematic of the two-qubit gate operations. Left side shows gate using permanent dipole moment and right side shows entanglement operation using resonant dipole-dipole interaction.

Figure 4

The $|\mathcal{J}=1,m_J=\pm 1\rangle$ manifold in real molecules. (a) ${}^{23}\mathrm{Na}{}^{133}\mathrm{Cs}$ at $E_z=4.5\mathrm{k}\mathrm{V}{\mathrm{cm}}^{-1}$, assuming ${\tilde{\alpha }}^{\left(2\right)}=-8.4\times {10}^{-3}\mathrm{Hz}{\mathrm{W}}^{-1}{\mathrm{m}}^2$. For intensities above $\sim 0.015\mathrm{G}\mathrm{W}{\mathrm{m}}^{-2}$ the levels behave as in the model molecule. (b) ${}^{40}\mathrm{Ca}{}^{19}\mathrm{F}$ at $E_z=2\mathrm{k}\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$ and ${\tilde{\alpha }}^{\left(2\right)}=6.3\times {10}^{-3}\mathrm{Hz}{\mathrm{W}}^{-1}{\mathrm{m}}^2$. There are 8 states but the pair with $|m_F|=2$ are nearly degenerate to high intensity. The two pairs with $m_G=0$ split linearly with $I_{\mathrm{tens}}$ out to intensities of about $20\mathrm{G}\mathrm{W}{\mathrm{m}}^{-2}$, corresponding to energy splittings of $50\mathrm{M}\mathrm{Hz}$. Light blue (dark blue dashed) lines show states which correspond to $|1,-\rangle$ ($|1,+\rangle$) for the intensities relevant to our scheme. Gray lines are other states in the manifold not useful for our scheme.

Figure 5

Separation and interaction energy for NaCs. Blue line, left axis: Separation of trap minima as a function of $I_{\mathrm{sc}}/I_{\mathrm{tens}}$. Dashed green line, right axis: Dipole-dipole interaction energy for point molecules positioned at trap minima. Calculations are for ${\lambda }_{\mathrm{sc}}=1064\mathrm{n}\mathrm{m}$, ${\lambda }_{\mathrm{tens}}=768\mathrm{n}\mathrm{m}$ with polarizabilities calculated for NaCs as described in the Appendix, and $E_z=4.5\mathrm{k}\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$ giving $\delta \mu =0.43\times 4.6\mathrm{D}$.

Figure 6

Calculated polarizabilities and scattering rates for ${}^{23}\mathrm{Na}{}^{133}\mathrm{Cs}$ and ${}^{40}\mathrm{Ca}{}^{19}\mathrm{F}$.