Proposal for suppressing ac Stark shift in the He($2\,^3S_1\rightarrow 3\,^3S_1$) two-photon transition using magic wavelengths

Motivated by recent direct measurement of the forbidden $2\,^3S_1\rightarrow3\,^3S_1$ transition in helium [K. F. Thomas et al., Phys. Rev. Lett. 125, 013002(2020)], where the ac Stark shift is one of the main systematic uncertainties, we propose a dichroic two-photon transition measurement for $2\,^3S_1\rightarrow3\,^3S_1$ which could effectively suppress the ac Stark shift by utilizing magic wavelengths: one magic wavelength is used to realize state-insensitive optical trapping, the other magic wavelength is used as one of the two lasers driving the two-photon transition. Carrying out calculations based on the no-pair Dirac-Coulomb-Breit Hamiltonian with mass shift operator included, we report the magic wavelength of 1265.615 9(4) nm for $^4$He [or 1265.683 9(2) nm for $^3$He] can be used to design an optical dipole trap; the magic wavelength of 934.234 5(2) nm for $^4$He [or 934.255 4(4) nm for $^3$He] can be as one excitation laser in the two-photon process, and the ac Stark shift can be reduced to less than 100 kHz, as long as the intensity of the other excitation laser does not exceed $1\times10^4~W/cm^2$. Alternatively, by selecting detuning frequencies relative to the $2\,^3P$ state in the region of 82$\sim$103 THz, as well as adjusting the intensity ratios of the two lasers, the ac Stark shift in the $2\,^3S_1\rightarrow3\,^3S_1$ two-photon transition can be cancelled.