Control of Optical Transitions with Magnetic Fields in Weakly Bound Molecules

In weakly bound diatomic molecules, energy levels are closely spaced and thus more susceptible to mixing by magnetic fields than in the constituent atoms. We use this effect to control the strengths of forbidden optical transitions in ${{}^{88}\mathrm{Sr}}_2$ over 5 orders of magnitude with modest fields by taking advantage of the intercombination-line threshold. The physics behind this remarkable tunability is accurately explained with both a simple model and quantum chemistry calculations, and suggests new possibilities for molecular clocks. We show how mixed quantization in an optical lattice can simplify molecular spectroscopy. Furthermore, our observation of formerly inaccessible $f$-parity excited states offers an avenue for improving theoretical models of divalent-atom dimers.

Figure 1

A magnetically enabled forbidden molecular transition. (a) Admixing of excited states by an applied static magnetic field $B$ (slanted arrows). Near the ${{}^{1}S}_0+{{}^{3}P}_1$ asymptote, two molecular potentials, $1_u$ and $0_u^{+}$, are optically accessible from the ground state, $X^1{\mathrm{\Sigma }}_g^{+}$. States with odd $J^{\prime }$ are of both $1_u$ and $0_u^{+}$ character because of nonadiabatic Coriolis coupling [7]. This coupling is not essential to this work, and only admixing of $1_u$ states is shown for clarity. (b) An optical transition from $J=0$ to $J^{\prime }=2$ is forbidden (dashed vertical arrow), while to $J^{\prime }=1$ is allowed (solid vertical arrow). With an applied field, the forbidden transition becomes allowed because of admixing with the $J^{\prime }=1$ state.

Figure 2

Magnetic control of molecular transitions in ${{}^{88}\mathrm{Sr}}_2$ near the intercombination line. Points are experimental values and curves are theoretical calculations. The $\pi$ transitions are between $X^1{\mathrm{\Sigma }}_g^{+}(v=-2,J,m)$ ground states and $1_u(v^{\prime }=-1,J^{\prime },m^{\prime })$ excited states for $J^{\prime }=1,2,4$, or the $0_u^{+}(v^{\prime }=-3,J^{\prime },m^{\prime })$ excited state for $J^{\prime }=3$ [9]. (a) An allowed transition with $\mathrm{\Delta }J=0$ has an “accidentally” forbidden $m^{\prime }=0$ component that becomes allowed with field, and $m^{\prime }=\pm 1$ components that show field-induced interference from admixing. (b) An allowed transition with $\mathrm{\Delta }J=1$ is mostly field insensitive. Its average value is used to normalize the data. (c),(d) Forbidden transitions with $\mathrm{\Delta }J=2$ and strengths that vary over 5 orders of magnitude to become comparable to allowed transitions. (e) A highly forbidden transition with $\mathrm{\Delta }J=3$ enabled by second-order admixing.

Figure 3

Highly nonlinear Zeeman shifts for the three $1_u(v^{\prime }=-1,J^{\prime })$ states listed in Table 1. The $\pi$ transitions used in the measurements are indicated in Figs. 2, 2, 2, respectively. The lines are polynomial fits using Eq. (3) with appropriate symmetry constraints.

Figure 4

Demonstration of mixed quantization of $J=2$ ground states by an optical lattice linearly polarized orthogonally to the applied magnetic field. (a) Spectrum measuring the populations of ground-state sublevels $m$. (b) A $\pi$ transition to the $m^{\prime }=-2$ sublevel of a Zeeman-resolved excited state depletes not only the population with $m=-2$, but also with $m=0$ and $m=2$. The additional population loss with $|\mathrm{\Delta }m|=2,4$ is highly forbidden by selection rules, but occurs because the optical lattice mixes the sublevels. (c) Likewise, a $\pi$ transition to $m^{\prime }=-1$ removes the populations with $m=\pm 1$. (d) A $\pi$ transition to $m^{\prime }=0$ has the same effect as that to $m^{\prime }=-2$ shown in (b).