Observing random walks of atoms in buffer gas through resonant light absorption

Using resonant light absorption, random-walk motions of rubidium atoms in nitrogen buffer gas are observed directly. The transmitted light intensity through atomic vapor is measured, and its spectrum is obtained, down to orders of magnitude below the shot-noise level to detect fluctuations caused by atomic motions. To understand the measured spectra, the spectrum for atoms performing random walks in a Gaussian light beam is computed, and its analytical form is obtained. The spectrum has $1/f^2$ ($f$ is frequency) behavior at higher frequencies, crossing over to a different, but well-defined, behavior at lower frequencies. The properties of this theoretical spectrum agree excellently with the measured spectrum. This understanding also enables us to obtain the diffusion constant, the photon cross section of atoms in buffer gas, and the atomic number density from a single spectral measurement. We further discuss other possible applications of our experimental method and analysis.

Figure 1

Configuration of the experiment: a resonant laser light beam is shone through a rubidium cell, its transmitted power is measured by the photodetectors (PD), and the power fluctuation spectrum is computed. (a) The conceptual design of the experiment. (b) The setup used in the measurements. Differential measurement from the two beams is used to remove the correlated noise ($+-$), and the correlations of the transmitted power from the same beam ${\mathcal{D}}_1,{\mathcal{D}}_2$ are used to extract the signal (see text). The photodetector outputs are converted to a digital signal by the analog to digital converters (ADC), Fourier transforms are performed (FFT), and then correlations are computed and averaged. FFT and averagings are performed on a computer.

Figure 2

An observed spectrum with and without correlation measurements (solid red and dashed blue lines, respectively). Without the correlation measurement, the spectrum reduces to the shot-noise level at higher frequencies. Shot-noise level is also shown (grey dots). Some extraneous noise still remains at higher frequencies. $\mathcal{P}=58\mu \mathrm{W}, w=0.34\mathrm{mm}$, and temperature was $46.4{}^{\circ }\mathrm{C}$ in this measurement. For comparison, the transit-noise spectrum of the rubidium gas without buffer gas with the same $w$ and similar $\mathcal{P}$ is also shown for $f>1.6\mathrm{kHz}$ ($\mathcal{P}=68\mu \mathrm{W}$, short-dashed magenta line).

Figure 3

Power spectra of rubidium atoms in nitrogen gas (200 Torr) and the corresponding theoretical spectra (thin gray lines), Eq. (13). The spectra have larger magnitudes for larger light power $\mathcal{P}$. The theoretical and experimental spectra agree well. Inset: the experimental spectra rescaled by their overall coefficient, which show that the shapes of the spectra are the same to a high degree. The vertical scale is denoted as “$S\left(f\right)$ Resc.,” and the units on the vertical scale are arbitrary. $\mathcal{P}=58.0$ (red), 92.8 (green), 197 (blue), 429 (magenta), and 912 (cyan)  $\mu \mathrm{W}$. $w=0.96$ mm, and the temperature of the gas is $46.4{}^{\circ }\mathrm{C}$. Larger $\mathcal{P}$ leads to a larger signal.

Figure 4

Power $\mathcal{P}$ dependence of $C(w,n,\mathcal{P})$: $C(w,n,\mathcal{P})$ and fits to them proportional to ${\mathcal{P}}^2$ (gray dashed lines) are plotted for measurements at the following temperatures and $w$ values: $44.6{}^{\circ }\mathrm{C}$, 0.96 mm ($\square$), $46.4{}^{\circ }\mathrm{C}$, 0.34 mm ($\circ$), and $50.0{}^{\circ }\mathrm{C}$, 0.34 mm ($▵$). Inset: $\mathcal{P}$ dependence of $C(w,n,\mathcal{P})w^4/n_{\mathrm{theory}}$ for the same spectra and its fit to ${\mathcal{P}}^2$ (gray dashed line). The data can be seen to fall on a single line. The vertical scale is labeled as “$C(w,n,\mathcal{P})$ Resc.,” and arbitrary units are used.

Figure 5

Cross section $\sigma$ and the rubidium number density $n$ extracted from the spectra used in Fig. 4 (the same symbols are used). Top: measured $\sigma$. The theoretical cross section, Eq. (5), is shown (gray solid line). Bottom: measured rubidium number density $n$ relative to the number density in the literature $n_{\mathrm{theory}}$.