“Doubly Magic” Conditions in Magic-Wavelength Trapping of Ultracold Alkali-Metal Atoms

In experiments with trapped atoms, atomic energy levels are shifted by the trapping optical and magnetic fields. Regardless of this strong perturbation, precision spectroscopy may be still carried out using specially crafted, “magic” trapping fields. Finding these conditions for particularly valuable microwave transitions in alkali-metal atoms has so far remained an open challenge. Here I demonstrate that the microwave transitions in alkali-metal atoms may be indeed made impervious to both trapping laser intensity and fluctuations of magnetic fields. I consider driving multiphoton transitions between the clock levels and show that these “doubly magic” conditions are realized at special values of trapping laser wavelengths and fixed values of relatively weak magnetic fields. This finding has implications for precision measurements and quantum information processing with qubits stored in hyperfine manifolds.

Figure 1

Left panel: Zeeman effect for the hyperfine manifold in the ground state of $I=3/2$ isotopes of alkali-metal atoms. Two clock or qubit levels $|F^{\prime }=2,M_F^{\prime }=+1⟩$ and $|F=1,M_F=-1⟩$ are shown in black. The right panel illustrates the geometry of laser-atom interaction: the degree of circular polarization, angle $\theta$, and laser wavelength may be varied.

Figure 2

Magic conditions for ${}^{133}\mathrm{Cs}$. A dependence of the product $M_{F^{\prime }}A\cos \theta$ on trapping laser frequency (in atomic units) is plotted. The shaded regions are bound by $-|M_{F^{\prime }}|$ and $+|M_{F^{\prime }}|$ lines. Magic trapping for the $|F^{\prime }=4,M_F^{\prime }⟩\to |F=3,-M_F^{\prime }⟩$ clock-qubit transition is only possible when the computed curve lies inside the corresponding shaded region.

Figure 3

Same as in Fig. 2 for ${}^{87}\mathrm{Rb}$ transition $|F^{\prime }=2,1⟩\to |F=1,-1⟩$ and for ${}^{85}\mathrm{Rb}$ transitions $|F^{\prime }=3,1⟩\to |F=2,-1⟩$ and $|F^{\prime }=3,2⟩\to |F=2,-2⟩$.