Proposal for suppressing the ac Stark shift in the He(2 ${}^3{S}_1\to$ 3 ${}^3{S}_1$) two-photon transition using magic wavelengths

Motivated by recent direct measurement of the forbidden $2{}^3{S}_1\to 3{}^3{S}_1$ transition in helium [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 $2{}^3{S}_1\to 3{}^3{S}_1$, 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 ${}^4\mathrm{He}$ [or 1265.683 9(2) nm for ${}^3\mathrm{He}]$ can be used to design an optical dipole trap; the magic wavelength of 934.234 5(2) nm for ${}^4\mathrm{He}$ [or 934.255 4(4) nm for ${}^3\mathrm{He}]$ 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 $1\times {10}^4{\text{W/cm}}^2$. Alternatively, by selecting detuning frequencies relative to the $2{}^{3}P$ state in the region of 82–103 THz, as well as adjusting the intensity ratios of the two lasers, the ac Stark shift in the $2{}^3{S}_1\to 3{}^3{S}_1$ two-photon transition can be canceled.

Figure 1

Dynamic dipole polarizabilities for the $2{}^3{S}_1(M=\pm 1)$ and $3{}^3{S}_1(M=0)$ states of ${}^4\mathrm{He}$. The magic wavelengths are marked with arrows. The positions of the resonances are indicated by vertical dashed lines with small arrows on top of the graph.

Figure 2

Energy-level diagram indicating the $2{}^3{S}_1\to 3{}^3{S}_1$ two-photon transition of ${}^4\mathrm{He}$ excited by two lasers with wavelengths of ${\lambda }_1$ and ${\lambda }_2$. The detuning frequency of $\mathrm{\Delta }{\omega }_d$ represents the relative position of the virtual state to the real $2{}^{3}P$ state. The single-photon transition wavelength of $2{}^3{S}_1\to 3{}^3{S}_1$ is 427.7 nm. Other relevant transition wavelengths are indicated.

Figure 3

Transition amplitude $|M^{2E1}|$ of the $2{}^3{S}_1\to 3{}^3{S}_1$ two-photon transition of ${}^4\mathrm{He}$ as a function of the detuning frequency $\mathrm{\Delta }{\omega }_d$. The two-photon transition is excited by two different lasers with wavelengths of ${\lambda }_1$ and ${\lambda }_2$. Three positions marked by arrows from left to right indicate the cases with $\mathrm{\Delta }{\omega }_d=42.46$, 73.73, and 103.31 THz. For $\mathrm{\Delta }{\omega }_d=42.46$ and 103.31 THz, ${\lambda }_2$ are respectively the 785.3 and 934.2 nm magic wavelengths and, for $\mathrm{\Delta }{\omega }_d=73.73$ THz, two lasers have equal wavelengths.

Figure 4

ac Stark shifts in the $2{}^3{S}_1\to 3{}^3{S}_1$ two-photon transition of ${}^4\mathrm{He}$ at different intensities of the ${\lambda }_1$ laser. $I_{\max }$ indicates the maximum intensity of the ${\lambda }_1$ laser when the ac Stark shift equals 100 kHz. The two-photon transition is excited by two lasers with wavelengths of ${\lambda }_1$ and ${\lambda }_2$. The ac Stark shifts are estimated using dynamic dipole polarizabilities of $2{}^3{S}_1(M=\pm 1)$ and $3{}^3{S}_1(M=0)$.

Figure 5

Ratios of $-\mathrm{\Delta }{\alpha }_1\left({\lambda }_2\right)/\mathrm{\Delta }{\alpha }_1\left({\lambda }_1\right)$ at different detuning frequencies $\mathrm{\Delta }{\omega }_d$. $\mathrm{\Delta }{\alpha }_1\left({\lambda }_1\right)$ and $\mathrm{\Delta }{\alpha }_1\left({\lambda }_2\right)$ are the differences of polarizabilities between the $3{}^3{S}_1(M=0)$ and $2{}^3{S}_1(M=\pm 1)$ states of ${}^4\mathrm{He}$ at the wavelengths of ${\lambda }_1$ and ${\lambda }_2$, respectively. The red dashed line is a horizonal zero line.