Quantitative study of spin noise spectroscopy in a classical gas of ${}^{41}\mathrm{K}$ atoms

We present a general derivation of the electron spin noise power spectrum in alkali gases as measured by optical Faraday rotation, which applies to both classical gases at high temperatures as well as ultracold quantum gases. We show that the spin-noise power spectrum is determined by an electron spin-spin correlation function, and we find that measurements of the spin-noise power spectra for a classical gas of ${}^{41}\mathrm{K}$ atoms are in good agreement with the predicted values. Experimental and theoretical spin noise spectra are directly and quantitatively compared in both longitudinal and transverse magnetic fields up to the high magnetic-field regime (where Zeeman energies exceed the intrinsic hyperfine energy splitting of the ${}^{41}\mathrm{K}$ ground state).

Figure 1

(Color online) Experimental schematic showing how electron spin noise in a classical vapor of ${}^{41}\mathrm{K}$ atoms is probed via optical Faraday rotation. Spin fluctuations in the vapor impart Faraday rotation fluctuations $\delta \varphi \left(t\right)$ on a linearly polarized and detuned laser beam, which are measured in a sensitive optical bridge. The external magnetic field can be applied orthogonal to (as shown) or parallel to the direction of laser propagation.

Figure 2

Schematic of the Zeeman/hyperfine structure of ${}^{41}\mathrm{K}$. (For ${}^{41}\mathrm{K}$ atoms, $I=3∕2$ and $g_I=\mathrm{0.215.}$) The total angular momentum $F=I+J$ and its projection $M_F$ are good quantum numbers at $B=0$, and can also be used to unambiguously label the atomic levels when $B\ne 0$.

Figure 3

(Color online) Energies of the Zeeman-split ${}^{41}\mathrm{K}$ atomic levels as a function of magnetic field. The levels are labeled by their $(F,M_F)$ quantum numbers when $B=0$.

Figure 4

(Color online) Squared electron spin matrix elements for the case of transverse magnetic fields $(\theta =90°)$. The selection rule in this case is $\Delta M_F=\pm 1$. The top panel depicts the case of $\Delta F=0$. Because the nuclear moment of ${}^{41}\mathrm{K}$ is very small, the following pairs of transitions are nearly degenerate in energy (frequency): $(2,0)\to (2,-1)$ and $(1,-1)\to (1,0)$; $(2,1)\to (2,0)$ and $(1,0)\to (1,1)$. The bottom panel shows transitions for which $\Delta F=\pm 1$. The following pairs of transitions are nearly degenerate in energy (frequency): $(2,1)\to (1,0)$ and $(2,0)\to (1,1)$; $(2,0)\to (1,-1)$ and $(2,-1)\to (1,0)$.

Figure 5

(Color online) Squared electron spin matrix elements in ${}^{41}\mathrm{K}$ for the case of longitudinal magnetic fields $(\theta =0°)$. The selection rule in this case is $\Delta M_F=0$.

Figure 6

(Color online) Squared electron spin matrix elements in ${}^{41}\mathrm{K}$ for the self-excitation transitions (longitudinal magnetic fields, $\theta =0°$). These transitions all appear at zero frequency. The self-excitation amplitudes of levels (1, 1) and (2, 1) are equal, as are the amplitudes for levels (1, 0) and (2, 0), and also for $(1,-1)$ and $(2,-1)$, respectively. The self-excitation amplitudes of $(2,-2)$ and (2, 2) are both equal to 1.

Figure 7

(Color online) The experimentally measured noise power spectrum from a gas of ${}^{41}\mathrm{K}$ atoms, for longitudinal magnetic fields $B_L$ $(\theta =0°)$. Here, the measured noise power density (aW/Hz) is proportional to the square of Faraday rotation noise density (which has units of $\text{radians}∕{\mathrm{Hz}}^{1∕2}$). The three allowed spin noise resonances [$(2,-1)\to (1,-1)$, $(2,0)\to (1,0)$, and $(2,1)\to (1,1)$] are measured at $B_L=25.6$, 51.4, 75.4, and $99.8\mathrm{G}$. Electronic detector and amplifier noise is not subtracted from these data. At the right, histograms compare the theoretically calculated spin noise (black bars) with the measured integrated spin noise (red bars), in units of Faraday rotation $\left(\mu \mathrm{rad}\right)$. The relative measured spin noise agrees well with calculation at all $B_L$. There is an overall (absolute) discrepancy of approximately a factor of 2 between theory and experiment.

Figure 8

(Color online) The experimentally measured noise power spectrum in a gas of ${}^{41}\mathrm{K}$ atoms, at a transverse magnetic field of $B_T=25.6\mathrm{G}$ $(\theta =90°)$. There are 12 allowed inter- and intra-hyperfine spin noise resonances. Four pairs of resonances are nearly degenerate in frequency, giving eight resolvable spin noise peaks in the data (peaks are labeled 1–8 as shown). Background electronic noise from the detectors and amplifiers has been subtracted.

Figure 9

(Color online) Histograms comparing the calculated spin noise (black bars) with the measured integrated spin noise (red bars) for all eight resolved noise resonances in transverse magnetic fields $B_T$ $(\theta =90°)$, in units of Faraday rotation $\left(\mu \mathrm{rad}\right)$. Peaks are labeled in Fig. 8. The relative measured spin noise agrees well with calculation for all values of $B_T$ (25.6, 51.4, 75.4, and $99.8\mathrm{G}$). There is an overall (absolute) discrepancy of approximately a factor of 2 between theory and experiment.