Pendular trapping conditions for ultracold polar molecules enforced by external electric fields

We theoretically investigate trapping conditions for ultracold polar molecules in optical lattices when external magnetic and electric fields are simultaneously applied. Our results are based on an accurate electronic-structure calculation of the polar ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ polar molecule in its absolute ground state combined with a calculation of its rovibrational-hyperfine motion. We find that an electric field strength of $5.26\left(15\right)$ kV/cm and an angle of $54.7^{\circ }$ between this field and the polarization of the optical laser lead to a trapping design for ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ molecules where decoherence due to electric field strength and laser-intensity fluctuations, as well as fluctuations in the direction of its polarization, are kept to a minimum. One-standard-deviation systematic and statistical uncertainties are given in parenthesis. Under such conditions, pairs of hyperfine-rotational states of $v=0$ molecules, used to induce tunable dipole-dipole interactions between them, experience ultrastable, matching trapping forces.

Figure 1

Dynamic parallel (top panel) and perpendicular (bottom panel) polarizabilities in atomic units, a.u.=${\left(e{a}_0\right)}^2/E_{\mathrm{h}}$, at the equilibrium separation of the ground $X^1{\mathrm{\Sigma }}^{+}$ electronic state of NaK as functions of laser frequency $\omega$. Black markers are our computed data points, while the solid red curve corresponds to a fit to this data as described in the text. The data is mainly localized near zero frequency and $\omega /\left(2\pi c\right)\approx 10000{\mathrm{cm}}^{-1}$. The latter corresponds to laser frequencies typically used for trapping of ultracold molecules.

Figure 2

Eigenenergies of the lowest rotational-hyperfine states of the $v=0$ vibrational level of the electronic ground state of ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ as a function of static electric field strength $E_x$ when no trapping laser is present and $B_z=8.57$ mT. The electric field $\mathit{E}=E_x\widehat{x}$ is directed along our $\widehat{x}$ axis. Panels (a) and (b) show the same data on two different energy and electric field scales. Approximate labels $\lambda$ and $m$, valid for large electric fields, are indicated. The zero of energy is at the hyperfine barycenter of the $N=1$ rotational state when $E_x=0$.

Figure 3

Dynamic polarizabilities of the lowest 144 eigenstates of ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ as functions of the angle $\theta$ at small electric fields, $\left|\mathit{E}\right|=0$ kV/cm, 0.06 kV/cm, and 0.09 kV/cm. Panels on the top and bottom row correspond to an electric field along the $\widehat{z}$ and $\widehat{x}$ axis, respectively. In panels (a), (b), (d), and (e) the orange line and purple markers correspond to $\lambda =0$ and $\lambda =1$ hyperfine states, respectively. In panels (c) and (f) the orange, blue, and purple lines and markers correspond to $|\lambda =0,m=0\rangle ,|\lambda =1,m=0\rangle$, and $|\lambda =1,m=\pm 1\rangle$ hyperfine states, respectively. Panels (a) and (d) for zero electric field are identical. The copy is only included for easy comparison with other panels. We use $B_z=8.57$ mT, a laser wavelength of 1064 nm, and $I_{\mathrm{trap}}=2.35$ kW/${\mathrm{cm}}^2$.

Figure 4

Dynamic polarizabilities of the lowest 144 rotational-hyperfine states of the ground vibrational level of ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ as functions of the angle $\theta$ for three strong electric fields $\mathit{E}=E_x\widehat{x}$ with $E_x=2.0$ kV/cm, 5.265 kV/cm, and 8.0 kV/cm in panels (a), (b), and (c), respectively. Orange, blue, and purple lines and markers correspond to hyperfine states in the $|\lambda =0,m=0\rangle ,|\lambda =1,m=0\rangle$, and $|\lambda =1,m=\pm 1\rangle$ manifolds, respectively. We use $B_z=8.57$ mT, a laser wavelength of 1064 nm, and $I_{\mathrm{trap}}=2.35$ kW/${\mathrm{cm}}^2$.

Figure 5

Dynamic polarizabilities of $m=0$ rotational-hyperfine states of the ground vibrational level of ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ as functions of the strength of an applied electric field for the eleven angles $\theta =0^{\circ },{10}^{\circ },\cdots ,{80}^{\circ },{90}^{\circ }$ and $35.3^{\circ }$. The electric field is directed along the $\widehat{x}$ axis. Orange dashed and blue solid lines are polarizabilities for hyperfine states of the $|\lambda =0,m=0\rangle$ and $|\lambda =1,m=0\rangle$ manifold, respectively. For $\theta =35.3^{\circ }$ the polarizability for the two manifolds is the same and independent of $E_x$. At the red circle where $\theta =35.3^{\circ }$ and $E_x=5.265$ kV/cm our double magic condition holds. We use $B_z=8.57$ mT, a laser wavelength of 1064 nm, and $I_{\mathrm{trap}}=2.35$ kW/${\mathrm{cm}}^2$.

Figure 6

Dynamic polarizabilities of the lowest 144 eigenstates of the vibrational ground state of ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ as functions of the trapping laser intensity $I_{\mathrm{trap}}$ at four representative pairs $(E_x,\theta )=(0.0,{60}^{\circ })$, $(0.09,{60}^{\circ }),(2.0,35.3^{\circ })$, and $(5.265,{60}^{\circ })$ in panels (a), (b), (c) and (d), respectively. The electric field is in units of kV/cm and points along the $\widehat{x}$ axis. In all panels the orange dashed line corresponds to ${\alpha }_{\mathrm{dyn}}$ of the $36\lambda =0,m=0$ hyperfine states. In panel (a) the purple dots correspond to the $108\lambda =1$ hyperfine states with mixed $m$ character. In panels (b), (c), and (d) the blue and purple dots are eigenstates dominated by $|\lambda =1,m=0\rangle$ and (mixed) $|\lambda =1,m=\pm 1\rangle$ pendular functions. In panels (c) and (d) the polarizibilities of the $m=0$ eigenstates are indistinguishable. Both correspond to magic conditions for ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$. We use $B_z=8.57$ mT and a laser wavelength of 1064 nm.