Magic wavelengths for the $2{}^{3}S\to 2{}^{1}S$ transition in helium

We have calculated ac polarizabilities of the $2{}^{3}S$ and $2{}^{1}S$ states of both ${}^4\mathrm{He}$ and ${}^3\mathrm{He}$ in the range 318 nm to 2.5 $\mu \mathrm{m}$ and determined the magic wavelengths at which these polarizabilities are equal for either isotope. The calculations, only based on available ab initio tables of level energies and Einstein A coefficients, do not require advanced theoretical techniques. The polarizability contribution of the continuum is calculated using a simple extrapolation beyond the ionization limit, yet the results agree to better than $1\%$ with such advanced techniques. Several promising magic wavelengths are identified around 320 nm with sufficient accuracy to design an appropriate laser system. The extension of the calculations to ${}^3\mathrm{He}$ is complicated due to the additional hyperfine structure, but we show that the magic wavelength candidates around 320 nm are predominantly shifted by the isotope shift.

Figure 1

Calculated polarizabilities of the $2{}^3{S}_1$ (dashed, blue) and $2{}^1{S}_0$ (dotted, black) states shown together with the differential polarizability (full, red) in the wavelength range 318–327 nm. The blue and black vertical lines indicate the positions of the $2{}^3{S}_1\to 4{}^{3}P$ and the $2{}^1{S}_0\to n{}^{1}P(n=9–13)$ transitions, respectively. There are five magic wavelengths (black dots) in this range, all listed in Table 2.

Figure 2

Calculated polarizabilities of the $2{}^3{S}_1$ (dashed, blue) and $2{}^1{S}_0$ (dotted, black) states shown together with the differential polarizability (full, red) for wavelengths ranging from 327 nm to 420 nm. The blue and black vertical lines indicate the positions of the $2{}^3{S}_1\to 3{}^{3}P$ and the $2{}^1{S}_0\to n{}^{1}P(n=4–8)$ transitions, respectively. There are four magic wavelengths (black dots) in this range, all listed in Table 2. The inset shows the wavelength region 411–415 nm, displaying the magic wavelength at 411.863 nm and the tune-out wavelength of the $2{}^3{S}_1$ state at 414.197 nm.

Figure 3

Calculated polarizabilities of the $2{}^3{S}_1$ (dashed, blue) and $2{}^1{S}_0$ (dotted, black) states shown together with the differential polarizability (full, red) for wavelengths ranging from 420 nm to $2.5\mu \mathrm{m}$. The blue and black vertical lines indicate the positions of the $2{}^3{S}_1\to 2{}^{3}P$ and the $2{}^1{S}_0\to n{}^{1}P(n=2,3)$ transitions, respectively. There are no magic wavelengths in this range and the polarizabilities converge to the dc polarizabilities for $\lambda >2.5\mu \mathrm{m}$.