Imaging of relaxation times and microwave field strength in a microfabricated vapor cell

We present a characterization technique for atomic vapor cells, combining time-domain measurements with absorption imaging to obtain spatially resolved information on decay times, atomic diffusion, and coherent dynamics. The technique is used to characterize a 5-mm-diameter, 2-mm-thick microfabricated Rb vapor cell, with N${}_2$ buffer gas, placed inside a microwave cavity. Time-domain Franzen and Ramsey measurements are used to produce high-resolution images of the population ($T_1$) and coherence ($T_2$) lifetimes in the cell, while Rabi measurements yield images of the ${\sigma }_{-}$, $\pi$, and ${\sigma }_{+}$ components of the applied microwave magnetic field. For a cell temperature of $90{}^{\circ }$C, the $T_1$ times across the cell center are found to be a roughly uniform $265\mu$s, while the $T_2$ times peak at around $350\mu$s. We observe a “skin” of reduced $T_1$ and $T_2$ times around the edge of the cell due to the depolarization of Rb after collisions with the silicon cell walls. Our observations suggest that these collisions are far from being 100$\%$ depolarizing, consistent with earlier observations made with Na and glass walls. Images of the microwave magnetic field reveal regions of optimal field homogeneity, and thus coherence. Our technique is useful for vapor cell characterization in atomic clocks, atomic sensors, and quantum information experiments.

Figure 1

(a) The microfabricated vapor cell used in this paper, with glass windows and a silicon frame. (b) The ${}^{87}$Rb $D_2$ line. Due to Doppler and collisional broadening on the optical transitions, the excited state hyperfine levels $F^{\prime }$ are not resolved. Transitions between the Zeeman-split $m_F$ levels of the ground-state hyperfine structure can be individually addressed by the microwave field. The three hyperfine transitions used in this work ($i=1,4,7$) are shown by dashed blue arrows. (c) A double-resonance spectrum, showing laser transmission through the cell as the microwave frequency is scanned. Transmission is reduced whenever the microwave comes on resonance with a hyperfine transition. (d) The experimental setup.

Figure 2

Optical density of the cell as a function of temperature. The theory curve has been produced using the model of Ref. [28], modified to include pressure and Rb dipole-dipole broadening. The theory has no free parameters. An arrow marks $90{}^{\circ }$C, where all other presented data were taken.

Figure 3

Cell OD response to (a) Franzen, (b) Ramsey, and (c) Rabi sequences, recorded using a photodiode. Data is shown as blue dots, while the fitting curves (described in the text) are in red. Note the different scale in (c). The insets show the laser and microwave sequences used. The OD increases with laser dark time, as the hyperfine population difference relaxes.

Figure 4

$T_1$ times as a function of temperature. Error bars are 95$\%$ confidence bounds from the fitting. The theory curve shows a calculation of $T_1$ using Eq. (6) with no free parameters.

Figure 5

Measured $T_1$ and $T_2$ times across the cell. The top panels show (a) $T_1$ times obtained from the $1/e$ decay time of a Franzen sequence (see text); (b) $T_1$ times obtained from fitting a Ramsey sequence, fitting uncertainty $\pm 1\%$; and (c) $T_2$ times obtained from the same Ramsey sequence, fitting uncertainty $\pm 4\%$. The bottom panels show radial profiles of each image in the form of a two-dimensional histogram. The radial distance from the cell center is binned into $27.5-\mu$m-wide bins for the Franzen data, and $38.8-\mu$m-wide bins for the Ramsey data. Franzen $T_1$ and Ramsey $T_1$ and $T_2$ times are binned into $0.99$-, $1.4$-, and $2.1-\mu$s-wide bins, respectively. The $T_1$ profiles are compared to theory as described in Sec. 4b. Close to the walls, there is a significant decrease in $T_1$ and $T_2$ due to Rb-wall collisions.

Figure 6

Image and radial profile of $u_0$, the hyperfine population difference in the optically pumped steady state, obtained from Franzen data. The red data points in the lower panel show the mean $u_0$ for each radial position, binned in $27.5\mu \mathrm{m}$ bins. The error of the mean is smaller than the symbols. Note the change in scaling of the bottom axis at $r=2$ mm to magnify the region near the cell wall. The data is compared to theory as described in Sec. 4b. The fit of Eq. (10) to the data near the cell wall is shown in solid blue. The “theory” and “analytic theory” curves, respectively, model $u_0$ with [Eq. (15)] and without [Eq. (14)] the inclusion of the central dip in optical pumping efficiency, which was caused by a Rb deposit on the front cell window.

Figure 7

Top: Rabi sequences have been used to obtain images of the (a) ${\sigma }_{-}$, (b) $\pi$, and (c) ${\sigma }_{+}$ components of the microwave magnetic field. Bottom: Images of the corresponding Rabi oscillation lifetimes, ${\tau }_2$. The white dots on the $\pi$ images show the approximate locations of the pixels examined in Fig. 8.

Figure 8

Representative pixels of the $\pi$ images in Fig. 7. Fitted data is shown for pixels in (a) the high ${\tau }_2$ region ($x=3.64\mathrm{mm}$, $y=1.19\mathrm{mm}$) and (b) the low ${\tau }_2$ region ($x=1.10\mathrm{mm}$, $y=3.98\mathrm{mm}$). Atoms in the high ${\tau }_2$ region perform an unusually large number of Rabi oscillations.