Spin-noise spectrum of hot vapor atoms in an anti-relaxation-coated cell

We study the spin-noise spectrum (SNS) of unpolarized Rb vapor in an evacuated anti-relaxation-coated cell using parametric Faraday rotation of a flat-top probe beam with variable diameter. We identify a distinct structure in the spectrum, called the Ramsey peak. We find that the power of the Ramsey peak is inversely proportional to the total number of atoms in the cell, and the power of the total spin noise is inversely proportional to the number of atoms within the probe beam. We further present a full theoretical analysis of the SNS by comparing three models of spin diffusion: Fickian diffusion, the ballistic flight, and Langevin's diffusion. We find that Fickian diffusion fails, while the ballistic flight is acceptable. However, Langevin's diffusion yields a remarkable agreement with the experimental spectrum.

Figure 1

Experimental setup. PBS, polarizing beam splitter; Laser lock, laser frequency locking module; λ/2, half-wave plate; $\pi$ shaper, optical system converting a Gaussian beam into a flat-top beam; LP, linear polarizer; BE, beam expander; Iris, diaphragm for adjusting beam area; WP, Wollaston prism; CL, converging lens; BPD, balanced photon detector; SA, spectrum analyzer. Inset: probe frequency and the energy diagram of the ${}^{87}\mathrm{Rb}D1$ transition.

Figure 2

Probe beam area dependence of the spin-noise spectrum taken at 0.024 G. The cell's tip and body temperatures are $104.4{}^{\circ }\mathrm{C}$ and $107.1{}^{\circ }\mathrm{C}$, respectively. All data are taken with a constant probe power of 60 μW. (a) The SNS for different probe beam diameters. The main plot shows the SNS from a full frequency scan (100 kHz) of the SA. The resonance amplitude decreases with increasing probe beam diameter from 2 (black), 3 (red), 4 (blue), 5 (pink), and 6 (green) to 7 (purple) mm. The inset shows the SNSs of a 3.125-kHz frequency scan around the Ramsey peak with their transit bases subtracted.

Figure 3

The probe beam area dependence of the power of SPN. The black square represents the power of the total SPN, and the solid line is a linear fit. The red triangle represents the power of the Ramsey peak.

Figure 4

The SNS fitted by the Fickian diffusion model. The inset is a zoom-in around the Ramsey peak. Empty shapes of different color represent experiment data for different probe diameters from 2 (black), 3 (red), 4 (blue), 5 (pink), and 6 (green), to 7 (purple) mm. Each solid line of the same color is the corresponding theoretical spectrum given by $2S\left(f\right)$. Each experimental curve is obtained from the spectrum of a 100-kHz scan with its data points around the resonance center being replaced by the spectrum of a 3.125-kHz scan around the resonance. The theoretical curves are generated using $D=0.15{\mathrm{m}}^2/\phantom{{\mathrm{m}}^2\mathrm{s}}\mathrm{s}$.

Figure 5

Schematics of an atom's flying path in the cell. $R_{\mathrm{C}}$: radius of the cell; $R_{\mathrm{P}}$: radius of the probe beam; $\mathbf{r}$: initial position of an atom; $\mathbf{v}$: atom's initial velocity; $\varphi$: the relative angle between $\mathbf{v}$ and $\mathbf{r}$; $L_{\mathrm{P}}$: distance the atom travels to exit the probe; $L_{\mathrm{C}}$: distance the atom travels to reach the cell wall.

Figure 6

The SNS fitted by the ballistic flight model. The inset shows a zoom-in around the Ramsey peak. The color coding is the same as that of Fig. 4.

Figure 7

The SNS fitted by Langevin's diffusion model. The inset shows a zoom-in at the Ramsey peak. The color coding is the same as that of Fig. 4.

Figure 8

Calculated diffusion correlation functions of different models for the 2-mm beam radius. They belong to Fickian diffusion (blue dash), ballistic flight with $R_{\mathrm{f}}=R_{\mathrm{C}}$ (red dash-dot) and $R_{\mathrm{f}}=2\mathrm{mm}$ (green dot), and Langevin's diffusion (black solid).