Nonadiabatic Effects in Ultracold Molecules via Anomalous Linear and Quadratic Zeeman Shifts

Anomalously large linear and quadratic Zeeman shifts are measured for weakly bound ultracold ${}^{88}\mathrm{Sr}_2$ molecules near the intercombination-line asymptote. Nonadiabatic Coriolis coupling and the nature of long-range molecular potentials explain how this effect arises and scales roughly cubically with the size of the molecule. The linear shifts yield nonadiabatic mixing angles of the molecular states. The quadratic shifts are sensitive to nearby opposite $f$-parity states and exhibit fourth-order corrections, providing a stringent test of a state-of-the-art ab initio model.

Figure 1

${}^{88}\mathrm{Sr}_2$ potentials in the long range for (a) the excited state near the ${}^{1}S_0+{}^{3}P_1$ asymptote and (b) the ground state. In the excited state, two least-bound vibrational levels with the total angular momentum $J^{\prime }=1$ and $v^{\prime }=-1$, $-2$ are shown. As visible in the inset, the $|v^{\prime },J^{\prime }⟩=|-2,3⟩$ level is nearly degenerate with $|-1,1⟩$. In the ground state, the $v=-2$ molecules occupy the rotational levels $J=0$, 2. The allowed bound-bound transitions lead to spectral peaks and are labeled $p_1\mathrm{\text{−}}{p}_5$, while the bound-free transitions lead to “shelf” line shapes and are labeled $s_1$, $s_2$. (c) A spectroscopic trace showing the five peaks and two continuum shelves is fitted with the expected line shape.

Figure 2

Sample spectra of (a) $p_5$ and (b) $p_4$ in a small magnetic field $B$, showing $\pi$-transition peaks. The $p_4$ trace includes a shifted sublevel of the nearby $p_3$. (c),(d) Their respective frequency shifts for varying field amplitudes are fitted to parabolic shapes with the required symmetry constraints. In (d), the stretched levels with $|m^{\prime }|=3$ are separately measured via $\sigma$ transitions. The crossing point offset in (c) is likely caused by tensor light shifts from the optical lattice. The inset in (c) shows the $g(\theta )$ curve from Eq. (5), and indicates the measured Coriolis mixing angle $\theta =6.1°$ for $|v^{\prime },J^{\prime }⟩=|-2,1⟩$.

Figure 3

Measured $\overline{q}$ versus the binding energy. The values are normalized to the analogous atomic coefficient $q_a$, and show a millionfold enhancement for the weakest-bound levels. The gray line represents the magnetic susceptibility model of Eq. (6) for $J^{\prime }=1$, $m^{\prime }=0$ of $0_u^{+}$ that is in excellent agreement with the relevant data points (black squares for the $J^{\prime }=1$ levels).