Tune-Out and Magic Wavelengths for Ground-State ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ Molecules

We demonstrate a versatile, state-dependent trapping scheme for the ground and first excited rotational states of ${}^{23}\mathrm{Na}{}^{40}\mathrm{K}$ molecules. Close to the rotational manifold of a narrow electronic transition, we determine tune-out frequencies where the polarizability of one state vanishes while the other remains finite, and a magic frequency where both states experience equal polarizability. The proximity of these frequencies of only 10 GHz allows for dynamic switching between different trap configurations in a single experiment, while still maintaining sufficiently low scattering rates.

Figure 1

Overview of the rotational-state dependent trapping scheme near the $X\leftrightarrow b$ transition. (a) Level diagram of the NaK molecule containing the $X\leftrightarrow b$ transition and the two nearest transitions from $|1⟩$. (b) Schematic depiction of the potential experienced by $|0⟩$ (dark blue) and $|1⟩$ (bright blue) molecules in a dipole trap at the tune-out detuning for $|0⟩$ (left panel), a tune-out detuning for $|1⟩$ (center panel) and the magic detuning (right panel). (c) Frequency-dependent polarizability for $|0⟩$ (dark blue) and $|1⟩$ (bright blue), assuming light polarization parallel to the quantization axis. Each pole corresponds to one of the transitions shown in (a).

Figure 2

Polarizability and loss rate of molecules in state $|0⟩$. (a) Experimental data for polarizability ${\alpha }_0(\mathrm{\Delta })$ (orange circles) and theoretical curve determined using parameters from intensity-independent measurements (orange line.) The black dashed line indicates zero. Inset: Determination of the tune-out detuning for $|0⟩$ by measuring cloud size after resonant heating. Grey circles are root-mean-square cloud sizes and the solid line is a fit used to find the minimum, see [49]. (b) Observed loss rate ${\gamma }_L$ of molecules in $|0⟩$ subjected to 866-nm light at an intensity of $1150\mathrm{W}/{\mathrm{cm}}^2$ (blue circles). The loss rate at $I=0$ was subtracted from these data points. The blue solid line is a fit of Eq. (5) with ${\omega }_0$ and ${\mathrm{\Gamma }}_e$ as the fit parameters, where data points between $\mathrm{\Delta }=2\pi \times 1.5\mathrm{GHz}$ and $\mathrm{\Delta }=2\pi \times 2.5\mathrm{GHz}$ were excluded to avoid biasing the fit. The grey dashed line shows the prediction of the photon scattering rate ${\gamma }_{\mathrm{sc}}$ assuming ${\mathrm{\Gamma }}_e=2\pi \times 11.1\mathrm{kHz}$. Error bars denote the standard error of the fit. Inset: Intensity dependence of the loss rate at $\mathrm{\Delta }=2\pi \times 1\mathrm{GHz}$. The solid line is a linear fit to the data.

Figure 3

Differential polarizability and magic detuning. (a) Experimental data for differential polarizability ${\alpha }_{0\leftrightarrow 1}$ from microwave spectroscopy (orange circles) and fit to the data (orange line). The fit function is a combination of three resonances as described by Eq. (1), with a constant offset as well as the linewidths and positions for each resonance as fit parameters. (b) Determination of the magic detuning ${\mathrm{\Delta }}_m$ via Ramsey spectroscopy. Bright (dark) blue circles are experimentally measured contrast after free evolution time $t=0.4\mathrm{ms}(1.0\mathrm{ms})$ in the presence of 866-nm light. Lines are Lorentzian fits to the respective data sets, used to determine the center. Error bars denote one standard deviation and are determined from the covariance matrix of the fits.

Figure 4

Dependence of (a) differential polarizability ${\alpha }_{0\leftrightarrow 1}$ and (b) hyperpolarizability $\beta$ in $|1⟩$ on light polarization angle $\theta$ at the magic detuning with a dc electric field of $86\mathrm{V}/\mathrm{cm}$. $\beta$ represents the second-order dependence of the light shift on the light intensity [49].