Magic Frequencies in Atom-Light Interaction for Precision Probing of the Density Matrix

We analyze theoretically and experimentally the existence of a magic frequency for which the absorption of a linearly polarized light beam by a vapor of alkali-metal atoms is independent of the population distribution among the Zeeman sublevels and the angle between the beam and a magnetic field. The phenomenon originates from a peculiar cancellation of the contributions of higher moments of the atomic density matrix, and is described using the Wigner-Eckart theorem and inherent properties of Clebsch-Gordan coefficients. One important application is the robust measurement of the hyperfine population.

Figure 1

Outline of the magic frequency effect. At these frequencies, the absorption from all Zeeman sublevels is equal, namely, the overall absorption depends only on the total population in the hyperfine state, and is independent of the internal population distribution. Plotted is a $\Delta {\Gamma }_F$ surface, proportional to the differences of the absorption rates of light ($D_2$ line) by ${}^{87}\mathrm{Rb}$ atoms ($T=295\mathrm{K}$) in different Zeeman sublevels of the $|F=2⟩$ hyperfine state [Eq. (7)], for a small external dc magnetic field $|\mathbf{B}|<1\mathrm{G}$. The magic frequency where $\Delta {\Gamma }_F=0$, is found at ${\Delta }_L^{\mathrm{magic}}=385\mathrm{MHz}$ (and at $-318$, not shown), where ${\Delta }_L$ is the light frequency detuning from the $|F=2⟩$ to $|F^{\prime }=0⟩$ energy difference. The light, propagating in direction $\mathbf{k}$, is linearly polarized with an angle $\varphi =0$ (i.e., polarized in the $\mathbf{B}$-$\mathbf{k}$ plane). The plot also shows that the magic frequency is independent of $\theta$, the angle between $\mathbf{B}$ and $\mathbf{k}$. The overall transition strength $S_F$ [Eq. (8)] is shown in the background. For clarity, $\Delta {\Gamma }_F$ is multiplied by 3.

Figure 2

Relative absorption profiles [normalized ${\Gamma }_{m_F}^{\mathrm{rel}}$, Eq. (3)] for Zeeman sublevels of cesium and rubidium. At least one point where all the ${\Gamma }_{m_F}^{\mathrm{rel}}$ are equal (i.e., $\Delta {\Gamma }_F=0$) always exists. These points define the magic frequencies. Shown are profiles for (a) the $6{}^{2}S_{1/2}$, $F=4\to 6{}^{2}P_{3/2}$ transitions of cesium ($D_2$ line), and (b) the $5{}^{2}S_{1/2}$, $F=2\to 5{}^{2}P_{1/2}$ transitions of ${}^{87}\mathrm{Rb}$ ($D_1$ line). ${\Delta }_L$ is the detuning of the light from the frequency difference between the ground state $|F⟩$ and the lowest $|F^{\prime }⟩$ hyperfine state. Transition frequencies to all $F^{\prime }$ levels are indicated by arrows. These absorption plots are given for linearly polarized light with $\varphi =0$ (both), $\theta =\pi /2$ in (a) and $\theta =0$ in (b) [Eq. (4)], and $T=295\mathrm{K}$.

Figure 3

Experimental demonstration of the magic frequency. (a) An example of three plots of ${\Delta }_{\mathrm{OD}}$ vs $t_R$, showing the contrast of Rabi oscillations for different frequency detunings ${\Delta }_L$ of the probe beam. Note the $\pi$ phase difference between the two outer plots. This is exactly as expected from the reversing of the relative absorption rates on the two sides of the magic frequency, as may be observed in Fig. 2. (b) Contrast of Rabi oscillations vs detuning, showing that the contrast of the oscillations between Zeeman sublevels drops to zero as the detuning of the probe frequency is nearing 339 MHz, in good agreement with the magic frequency predicted by our model (with the pressure shift).