Multichannel modeling and two-photon coherent transfer paths in NaK

We explore possible pathways for the creation of ultracold polar $\mathrm{NaK}$ molecules in their absolute electronic and rovibrational ground state starting from ultracold Feshbach molecules. In particular, we present a multichannel analysis of the electronic ground and $\text{K}\left(4p\right)+\text{Na}\left(3s\right)$ excited-state manifold of NaK, analyze the spin character of both the Feshbach molecular state and the electronically excited intermediate states and discuss possible coherent two-photon transfer paths from Feshbach molecules to rovibronic ground-state molecules. The theoretical study is complemented by the demonstration of stimulated Raman adiabatic passage from the $X{}^1{\Sigma }^{+}$($v=0$) state to the $a{}^3{\Sigma }^{+}$ manifold on a molecular beam experiment.

Figure 1

Scheme of the coherent transfer sequence from a Feshbach level to the absolute ground state of the molecule. In this example, the intermediate level is of the resonant $A{}^1\Sigma \sim b{}^3\Pi$ type. The wave-function amplitudes reflect the singlet or triplet components.

Figure 2

Binding energies of ${}^{23}$Na${}^{40}$K for the total quantum number $\mathcal{M}=-{}^3/{}_2$ as function of magnetic field leading to $s$-wave resonances for the entrance channel (1,1)${}_{\text{Na}}$ $+$ (9/2,$-$5/2)${}_{\text{K}}$. For various magnetic fields the expectation value $\langle S\rangle$ of the electronic spin is given. (Inset) Singlet character as a function of $B$ for the resonance situated at 136 G. For the sharp resonances at lower fields, only a subtle change in spin character is observed.

Figure 3

Multicomponent wave function for the bound level at 125 G, projected onto the Hund's case (e) basis. The wave function displays a distinct open-channel fraction by the large amplitude at large internuclear distance $R>40\text{Å}$ (violet line). This component has the quantum numbers of the open channel with lowest energy and these are given in the figure.

Figure 4

Multicomponent wave function of the bound level at 125 G (top graph) and 110 G (bottom graph), projected onto the Hund's case (b) basis. For clarity, all but the strongest triplet and singlet channels were removed. As dashed lines the vibrational wave functions of the pure singlet and triplet states are given for the least bound level, scaled to the amplitude of the appropriate inner parts of the multichannel wave function.

Figure 5

Diabatic PECs of the $\text{Na}\left(3s\right)+\text{K}\left(4p\right)$ manifold used in the study. The energy zero is chosen at the atomic ground-state asymptote. Shown are the diabatic PECs applied in this section (dashed lines) as well as RKR curves (solid lines) for comparison. (Inset) Potential difference between the $B{}^1\Pi$ RKR potential, the raw ab initio curve (dashed line), and the refined curve (solid line), respectively.

Figure 6

Integrated square of the individual $\Omega =1$ channel wave functions at each eigenenergy, together with a $J=1$ perturbative model for the ${}^1\Pi {(}^3{\Sigma }^{+})$ case (open stars). A label of the form ${}^3{\Sigma }^{+}{(}^3\Pi )$ marks the ${}^3{\Sigma }^{+}$ fraction within a dominant ${}^3\Pi$ state. The vertical line at $E_0$ is positioned at the first state with dominant ${}^1\Pi$ character.

Figure 7

Absolute values of the Stokes ($\left|i\right.〉\to \left|X\right.〉$ top graph) and pump [$\left|F\right.〉\to \left|i\right.〉$ bottom graph, decomposed into the singlet (red squares) and triplet (blue circles) parts] laser transition dipole matrix elements for all intermediate states. The energy is given with respect to the $P_{{}^3/{}_2}$ dissociation limit.

Figure 8

Absolute value of the two-photon dipole matrix element. The circles mark double dots, indicating degeneracy-induced resonances.

Figure 9

Part of the hyperfine spectrum of the $v=5,N=6$ level of the $a{}^3{\Sigma }^{+}$ state by STIRAP transfer ($X{}^1{\Sigma }^{+},v=0\to$ intermediate state $\to a{}^3{\Sigma }^{+}$) on a molecular beam of NaK.