Calculation of transition probabilities and ac Stark shifts in two-photon laser transitions of antiprotonic helium

Numerical ab initio variational calculations of the transition probabilities and ac Stark shifts in two-photon transitions of antiprotonic helium atoms driven by two counter-propagating laser beams are presented. We found that sub-Doppler spectroscopy is, in principle, possible by exciting transitions of the type (n,L)→(n-2,L-2) between antiprotonic states of principal and angular momentum quantum numbers n~L-1~35, first by using highly monochromatic, nanosecond laser beams of intensities 10⁴–10⁵ W/cm², and then by tuning the virtual intermediate state close (e.g., within 10–20 GHz) to the real state (n-1,L-1) to enhance the nonlinear transition probability. We expect that ac Stark shifts of a few MHz or more will become an important source of systematic error at fractional precisions of better than a few parts in 10⁹. These shifts can, in principle, be minimized and even canceled by selecting an optimum combination of laser intensities and frequencies. We simulated the resonance profiles of some two-photon transitions in the regions n=30–40 of the p̅ ⁴He⁺ and p̅ ³He⁺ isotopes to find the best conditions that would allow this.