Differential Light-Shift Cancellation in a Magnetic-Field-Insensitive Transition of ${}^{87}\mathrm{Rb}$

The precise measurement of transition frequencies of trapped atomic samples is susceptible to inaccuracy arising from the inhomogeneous differential shift of the relevant energy levels in the presence of the trapping fields. We demonstrate near-complete cancellation of the differential ac Stark shift (“light shift”) of a two-photon magnetic-field-insensitive microwave hyperfine (clock) transition in ${}^{87}\mathrm{Rb}$ atoms trapped in an optical lattice. Up to 95(2)% of the differential light shift is cancelled while maintaining magnetic-field insensitivity. This technique should have applications in quantum information and frequency metrology.

Figure 1

Schematic of the ${}^{87}\mathrm{Rb}$ $\mu \mathrm{w}$ transitions studied here: (a) the two-photon transition $|F=1,m_F=-1⟩\leftrightarrow |F^{\prime }=2,m_{F^{\prime }}=+1⟩$, used to demonstrate the DLS reduction, and (b) the $|F=1,m_F=0⟩\leftrightarrow |F^{\prime }=2,m_{F^{\prime }}=0⟩$ clock transition, used for comparison. Atoms are trapped in an optical lattice formed at the intersection of four laser beams (${\mathbf{k}}_1$ to ${\mathbf{k}}_4$), obtained by a folded, retroreflected single laser beam (c). The intensity pattern used for DLS measurements is shown in (d), along with sections through the unit cell for three different $m_F$ states, and was experimentally optimized in order to maximize the amount of circular polarization on the right ($R$) sites. Atoms are trapped only in the $R$ sites (see text).

Figure 2

DLS dependence on intensity, expressed as the lattice depth in the corresponding $\lambda /2$ configuration. We observe a significant reduction of the total DLS of the two-photon clock transition (circles), compared to the single-photon clock transition (squares). The full lines represent linear fits to the data. Their extrapolations to zero intensity agree within 5%, and represent the (uncompensated) DLS from the vertical lattice, which was kept at a constant depth of $\approx 30$ $E_R$. The $1\sigma$ uncertainty of each data point is $\approx 1\mathrm{Hz}$.

Figure 3

Ratio between the DLS sensitivities ($\partial \nu /\partial I$) for the two- and single-photon transitions as a function of the lattice wavelength, with a minimum value of 4.5% at 806 nm. The magnetic field was kept near [0.323(3) mT] for the two-photon transition, whereas a small, $20\mu \mathrm{T}$, bias was used for the single-photon transition. The solid line represents a calculation with no adjustable parameters, following 8. Error bars represent the $1\sigma$ uncertainty, dominated by statistical errors.

Figure 4

Sensitivity of the transition frequency to the lattice laser intensity ($\partial \nu /\partial I$), as a function of the bias magnetic field, at $\lambda =806\mathrm{nm}$. At $B=0.343(3)\mathrm{mT}$ (i.e., $20\mu \mathrm{T}$ away from the magic field), $\partial \nu /\partial I=0$, and the transition becomes insensitive, at first order, to fluctuations of the light intensity, with a residual field sensitivity $\partial \nu /\partial B$ of $1.7\mathrm{Hz}/\mu \mathrm{T}$.