Symmetry-suppressed two-photon transitions induced by hyperfine interactions and magnetic fields

Two-photon transitions between atomic states of total electronic angular-momentum $J_a=0$ and $J_b=1$ are forbidden when the photons are of the same energy. This selection rule is analogous to the Landau-Yang theorem in particle physics that forbids decays of vector particle into two photons. It arises because it is impossible to construct a total angular-momentum $J_{2\gamma }=1$ quantum-mechanical state of two photons that is permutation symmetric, as required by Bose-Einstein statistics. In atoms with nonzero nuclear spin, the selection rule can be violated due to hyperfine interactions. Two distinct mechanisms responsible for the hyperfine-induced two-photon transitions are identified, and the hyperfine structure of the induced transitions is evaluated. The selection rule is also relaxed, even for zero-nuclear-spin atoms, by application of an external magnetic field. Once again, there are two similar mechanisms at play: Zeeman splitting of the intermediate-state sublevels, and off-diagonal mixing of states with different total electronic angular momentum in the final state. The present theoretical treatment is relevant to the ongoing experimental search for a possible Bose-Einstein-statistics violation using two-photon transitions in barium, where the hyperfine-induced transitions have been recently observed, and the magnetic-field-induced transitions are being considered both as a possible systematic effect, and as a way to calibrate the measurement.

Figure 1

(Color online). A two-photon transition between $J_a=0$ and $J_b=1$ states must proceed via a virtual intermediate state with $J_n=1$. There are two quantum paths between the initial and the final state that differ by the order of absorption of the photons, and which cancel each other in the case of degenerate photons. The transition to one particular upper-state Zeeman component $\left(M_J=0\right)$ is shown as an example.

Figure 2

(Color online). With the addition of nuclear-spin $I=3/2$, hyperfine-structure splitting in the intermediate-states $n$ may lift the cancellation between the two quantum paths of the two-photon transition. Zeeman splitting of the intermediate state has a similar effect (see text).

Figure 3

(Color online). Off-diagonal hyperfine interaction mixes the upper state with nearby states with $J_c\ne 1$, which leads to a nonvanishing degenerate two-photon amplitude. Upper-state mixing can also be induced by an external magnetic field.

Figure 4

(Color online). Fluorescence intensity of Ba $5d6d{{}^{3}D}_1$ during two-photon excitation. Upper trace shows nondegenerate excitation: The two photons are separated in frequency by 60 GHz. Lower trace shows degenerate excitation: The two photons have the same frequency. The horizontal frequency variable is ${\Omega }_1+{\Omega }_2-E_b$ of ${}^{138}\text{B}\text{a}$. “Sticks” indicate the line positions of the isotopic and hyperfine components. Stick heights indicate the calculated relative intensities of the peaks. The two traces are plotted on the same frequency scale (with a possible mismatch of no more than 50 MHz). The vertical scale of the lower plot is expanded $\sim 100\times$ the upper. The excitation intensity is $\sim {10}^3$ to ${10}^4$ times higher in the case of the lower spectrum than in the upper one. The broad asymmetric line shapes in the lower trace are due to the ac Stark effect (see Sec. 4A). Comparison of upper and lower traces shows that the nuclear-spin zero isotopes disappear during degenerate excitation, and that the relative intensities of the hyperfine components change, in qualitative agreement with calculations.