Direct measurement of excited-state dipole matrix elements using electromagnetically induced transparency in the hyperfine Paschen-Back regime

Applying large magnetic fields to gain access to the hyperfine Paschen-Back regime can isolate three-level systems in a hot alkali metal vapors, thereby simplifying usually complex atom-light interactions. We use this method to make the first direct measurement of the $|\langle 5P|\left|er\right||5D\rangle |$ matrix element in ${}^{87}\mathrm{Rb}$. An analytic model with only three levels accurately models the experimental electromagnetically induced transparency spectra and extracted Rabi frequencies are used to determine the dipole matrix element. We measure $|\langle 5P_{3/2}\left|\right|er\left|\right|5D_{5/2}\rangle |=(2.290\pm 0.{002}_{\mathrm{stat}}\pm 0.{04}_{\mathrm{syst}})e{a}_0$, which is in excellent agreement with the theoretical calculations of Safronova, Williams, and Clark [Phys. Rev. A 69, 022509 (2004)].

Figure 1

(a) Schematic of the experimental apparatus. PBS: Polarizing beam splitter; $\lambda /4$: quarter waveplate; L: lens with 200 mm focal length; Exp. Cell: 2 mm rubidium vapor cell; PD: high gain photodetector. (b) Axial magnetic field profile (red) of the permanent ring magnets (gray blocks). The maximum field strength is 0.6 T and the maximum variation over the region occupied by the vapor cell (shown in blue) is less than 1 mT.

Figure 2

(a) A typical weak probe transmission spectrum (red). (b) A diagram of the transitions associated with each of the spectral features in (a). On the left of the panel is an energy level diagram for the relevant states, showing the evolution into the hyperfine Paschen-Back regime at large magnetic fields. The eigenstates of the system in the $|m_J,m_I\rangle$ basis are shown to the right of the diagram for a magnetic field strength $B=0.6$ T. Even at this large field there still exists a significant admixture of states with opposite spin (blue text) in the $5S_{1/2}$ manifold, these result in the weak transitions indicated by the blue arrows. The large splittings between ground states as well as the electric dipole selection rule $\mathrm{\Delta }{m}_I=0$ ensure that only three atomic states are involved in each of the two-photon resonances. An expanded view of the three-level EIT resonance $5S_{1/2}|-\frac{1}{2},-\frac{1}{2}\rangle \to 5P_{3/2}|\frac{1}{2},-\frac{1}{2}\rangle \to 5D_{5/2}|\frac{3}{2},-\frac{1}{2}\rangle$ is shown in (c) for control-beam powers of 0, 3, 9, and 27 mW. Vertical offsets have been added for clarity. Zero detuning is given as the weighted $D_2$ line center of naturally abundant rubidium in zero-magnetic field [19].

Figure 3

Experimental transmission spectrum (blue) showing a purely three-level EIT resonance in a hot ${}^{87}\mathrm{Rb}$ vapor. The red line is a least-squares fit to the data using a three-level EIT model. The fit results in a measurement of the control-beam Rabi frequency ${\mathrm{\Omega }}_{\mathrm{c}}/2\pi =255.0\pm 0.2$ MHz, where the uncertainty is from the statistical error of the fit. The residual (R) shows the excellent agreement between the model and data, with the small amount of structure near line center being explained by electromagnetically induced absorption (see main text).

Figure 4

Experimental transmission spectra (blue) and numerical fits (red) for three control-beam detunings. The fit parameters except for control detuning are constrained to be equal in all three data sets. The extracted control-beam detunings are ${\mathrm{\Delta }}_{\mathrm{c}}/2\pi =-18.8\pm 0.6,408.0\pm 0.6,834.7\pm 0.7$ MHz. The residuals (R) show the excellent agreement between the model and data.

Figure 5

Extracted values of the control Rabi-frequency (${\mathrm{\Omega }}_c$) for increasing control-beam power ($P$). The red line is a least-squares fit to the function ${\mathrm{\Omega }}_{\mathrm{c}}=\alpha \sqrt{P}$, with $\alpha /2\pi =(46.47\pm 0.04)$ $\mathrm{MHz}/\sqrt{\mathrm{mW}}$. The residuals (R) are shown below and the fit has a reduced chi squared of 0.5. The inset shows the data on a logarithmic scale.

Figure 6

Predicted electromagnetically induced absorption (EIA) line shape for a 4% back reflection of the control beam overlapping with 15% of the interaction region. The dashed line shows the fit to this line shape using a three-level EIT line shape and the residual (R) shows the same structure as is found in the experimental data.