Anomalous noise spectra in a spin-exchange-relaxation-free alkali-metal vapor

We perform spin-noise spectroscopy on an unpolarized ${}^{87}\mathrm{Rb}$ vapor in the spin-exchange-relaxation-free (SERF) regime. We observe noise spectral distributions that deviate strongly from Lorentzian models that accurately describe lower-density regimes. For example, at magnetic fields of $\sim 1 \mu \mathrm{T}$ and ${}^{87}\mathrm{Rb}$ densities of $\gtrsim 1\times {10}^{14}$ atoms/${\mathrm{cm}}^3$ we observe an asymmetric spin-noise distribution in which the resonance line is depleted by about half its power, with the diverted power becoming a broad spectral component that could be mistaken for optical shot noise. The results are in good agreement with recent models accounting for correlations between the ground hyperfine states. We discuss implications for quantum sensing and absolute noise calibration in spin squeezing and entanglement detection. The results suggest similarly anomalous spectra for other noise spectroscopies, when noise mechanisms are not aligned with system dynamics.

Figure 1

Experimental setup and representative spectra. (a) Schematic representation of the experimental setup (see text). (b) Predicted non-Lorentzian spin-noise contributions (i) $S_{{\widehat{F}}_z^a,{\widehat{F}}_z^a}\left(\nu \right)$, (ii) $S_{{\widehat{F}}_z^b,{\widehat{F}}_z^b}\left(\nu \right)$, (iii) $S_{{\widehat{F}}_z^a,{\widehat{F}}_z^b}\left(\nu \right)$, and $S_{{\widehat{F}}_z^b,{\widehat{F}}_z^a}\left(\nu \right)$, computed using Eq. (6) and experimentally relevant parameters: $R_{\mathrm{se}}\approx 3.02\times {10}^5 {\mathrm{s}}^{-1}$ and $R_{\mathrm{sd}}\approx 0.03\times {10}^5 {\mathrm{s}}^{-1}$, corresponding to $3.4\times {10}^{14}$ atoms/${\mathrm{cm}}^3$ and temperature $T=169{}^{\circ }\mathrm{C}$. The magnetic field is $B=385$ nT along the $\widehat{\mathbf{x}}$ direction. (c) Example of a non-Lorentzian spectrum at a magnetic field of $B\approx 1292$ nT fitted to a Lorentzian plus dispersive curve (see Ref. [39] for details). (d) Spin-noise spectra acquired at a magnetic field of $B=385$ nT and a number density of $n\approx 3.4\times {10}^{14}$ atoms/${\mathrm{cm}}^3$. The mean PSN level is depicted by the green dashed line and has been subtracted from the spectrum. Data are fitted by a Lorentzian model (black solid line) and Eq. (1) (red solid lines) with and without “1/f-noise”. The departure from the Lorentzian spectrum is demonstrated.

Figure 2

Single-sided power spectral density (PSD) of the polarimeter signal (in volts, conversion to rotation angle 35 mrad/V) for transverse magnetic fields ranging from 280 nT to 12 $\mu \mathrm{T}$ while the vapor cell is maintained at approximately $169{}^{\circ }\mathrm{C}$. Each spectrum shows the linear average [39] of 150 spectra, each computed on a 0.5-s acquisition with a sampling rate of 200 kSa ${\mathrm{s}}^{-1}$. A 20-Hz (ten-bin) boxcar smoothing has also been applied [8]. Black solid lines: Fit of Eq. (1) (excluding 1/f and electronic noise) to the observed spectra (see text). Inset: The left axis shows spin-noise precession frequency ${\omega }_q$ normalized to ${\omega }_0=g_s{\mu }_{B}B/\left[\hslash (2I+1)\right]$ versus ${\omega }_0$ known by calibration of the coils at low density [39]. The right axis shows the spin-noise linewidth (half width at half maximum) versus ${\omega }_0$. Data are obtained by fitting the spectra with a distorted Lorentzian (see text). Error bars show $\pm 1$ standard deviation in the fit estimation parameters over 150 acquisitions. The blue (purple) solid line shows $\mathrm{Im}\left[\lambda \right]$ ($\mathrm{Re}\left[\lambda \right]$) of the eigenvalues of the drift matrix $A$, as given by Eq. (7) of Ref. [39]. The parameters are discussed in the main text.

Figure 3

Non-Lorentzian spectra and apparent (but not real) violation of the area conservation law. (a) Spin-noise spectra (single-sided PSD) as a function of the ${}^{87}\mathrm{Rb}$ number density for a fixed magnetic field of $B=918$ nT. Each spectrum shows the linear average of 100 spectra. Long high-frequency tails are apparent. Inset: Resonant noise power fraction as a function of number density as calculated using Eq. (8). The cutoff frequency ${\nu }_{\mathrm{br}}$ at 20 kHz is indicated by the red dashed line. Error bars show $\pm 1$ standard deviation in the numerical integration over 100 acquisitions. (b) For visualizing the power redistribution, the spin-noise resonances of Fig. 3 are plotted with the frequency axis shifted so that each is centered at 0. Curves are plotted for a constant field $B=918$ nT and varying number density. The values for the number densities are reported in Ref. [39]. The inset shows spin-noise resonances for the lower ($5.1\times {10}^{12} /\mathrm{c}{\mathrm{m}}^3$) and higher ($4.4\times {10}^{14} /\mathrm{c}{\mathrm{m}}^3$) atomic densities acquired.