Intrinsic radio-frequency gradiometer

Two optically pumped atomic magnetometer geometries, with sub-$\mathrm{fT}/\sqrt{\mathrm{Hz}}$ sensitivity for radio frequencies, are contrasted: one with a probe beam of high power and few passes through the atoms, the other with low power and many passes. Theoretical noise sensitivity for these conditions is modeled and compared to experimental values, with the latter geometry achieving a sensitivity of 0.19 $\pm$ 0.02 $\mathrm{fT}/\sqrt{\mathrm{Hz}}$. Further, a general approach for choosing probe parameters to optimize sensitivity is given. With the use of a $\lambda$/2 wave plate, both systems are also configured to act as an intrinsic gradiometer, enabling the rejection of common-mode interference. Common-mode interference is reduced by over 50 dB in both geometries.

Figure 1

The intrinsic gradiometer setup for two experimental configurations. Left: High probe power, few-pass system, using ${}^{39}\mathrm{K}$ vapor cells. The beam traverses the left and then right cell, with a beam width $\sim 1.3$ cm. Right: Low probe power, many-pass system, using a ${}^{87}\mathrm{Rb}$ vapor cell. In the many-pass configuration, the fiber-coupled probe beam enters and the beam traverses the first voxel 24 times and the second voxel 22 times, alternating between the two voxels. It walks across the mirror after each pass before exiting the geometry. The effective beam width in the cell is $\sim$1 cm, as shown by the fluorescence image in the inset.

Figure 2

The DC optical rotation in the M-geometry is measured as a function of probe wavelength and incident power, $I_0$, and has a wavelength dependence which deviates from the theoretical model. The model incorporates measured polarization and number density. The number density employed was the average from two separate measurement techniques, differing by $\pm$15%; one technique used the spin-exchange rate, the other the transmission of the probe light. The latter is shown in the inset, when the probe exit power, $I_{\text{pr}}$, is shown as a function of the optical cross section, $\sigma$; the slopes in the graph are the number density.

Figure 3

Shown are different cases of optical subtraction for the M-geometry (a), (b) and V-geometry (c), (d) for oscillating 1 MHz fields; spectra are centered around this same frequency. The plots shown on the left are for the individual voxels of each system. Three cases were examined: interference only, signal only, and interference$+$signal. For the M-geometry, the fields applied are a 5.2 pT common-mode interference and a 32 fT/cm gradient signal. For the V-geometry, the fields applied are 22 pT common-mode interference and a 2.3 pT/cm gradient signal. The interference is reduced by a factor of 390 in the M-geometry and 630 in the V-geometry. As the V-geometry is pulsed, the slightly different relaxation rates in each voxel give different spectral widths, resulting in nonuniform cancellation with respect to frequency, as seen in (d). The data shown in each geometry are normalized to the gradient signal so the relative field sizes are more easily seen.

Figure 4

Due to the larger angle between the arms of the probe beam, an out-of-phase component is picked up in the V-geometry system. The M-geometry doesn't have such an angle and the out-of-phase component can be well canceled.

Figure 5

In the M-geometry, the sensitivity in each operational mode, using rf magnetic fields at 1.05 MHz and an acquisition window of 1.5 ms, is shown. The sensitivity in the additive mode is 0.9 $\mathrm{fT}/\sqrt{\mathrm{Hz}}$, using a 2.4 pT common-mode magnetic field. The sensitivity operating as in intrinsic gradiometer is 0.5 $\mathrm{fT}/\mathrm{cm}/\sqrt{\mathrm{Hz}}$, using a 32 fT/cm gradient magnetic field.

Figure 6

On the left, the sensitivity from the V-intrinsic gradiometer; the noise is normalized to a 0.7 pT, 1 MHz reference signal. A typical time-domain signal is given in the inset of Fig. 6, showing significant decay over 12 ms. The raw noise is higher when the full cell is illuminated (magenta or darker) compared to illumination of individual voxels (cyan and orange or lightest and lighter). However, the signal for full illumination would presumably be doubled so the sensitivity is improved. The sensitivity is measured to be $0.19\pm 0.02\mathrm{fT}/\sqrt{\mathrm{Hz}}$, an improvement over previous results, while using a $10\times$ smaller cell volume. The corresponding gradient sensitivity is 0.20 $\pm$ 0.02 $\mathrm{fT}/\mathrm{cm}/\sqrt{\mathrm{Hz}}$. With the short acquisition time, the spectral resolution is poor. Therefore, the noise power is calculated from the Fourier transform of 12 ms of data and plotted in the graph to the right. Fits to a Lorentzian function are shown as thick solid lines and the results analyzed within the text.