Stable Atomic Magnetometer in Parity-Time Symmetry Broken Phase

Random motion of spins is usually detrimental in magnetic resonance experiments. The spin diffusion in nonuniform magnetic fields causes broadening of the resonance and limits the sensitivity and the spectral resolution in applications like magnetic resonance spectroscopy. Here, by observation of the parity-time (PT) phase transition of diffusive spins in gradient magnetic fields, we show that the spatial degrees of freedom of atoms could become a resource, rather than harmful, for high-precision measurement of weak signals. In the normal phase with zero or low gradient fields, the diffusion results in dissipation of spin precession. However, by increasing the field gradient, the spin system undergoes a PT transition, and enters the PT symmetry broken phase. In this novel phase, the spin precession frequency splits due to spatial localization of the eigenmodes. We demonstrate that, using these spatial-motion-induced split frequencies, the spin system can serve as a stable magnetometer, whose output is insensitive to the inevitable long-term drift of control parameters. This opens a door to detect extremely weak signals in imperfectly controlled environments.

Figure 1

Experimental setup. A cubic glass cell containing Xe gas and Rb metal is placed inside a magnetic shielding and heated. A parametric magnetometer [37, 38, 39, 40, 41] is used to detect the nuclear spin signals. See Sec. I of the Supplemental Material [42] for more details.

Figure 2

Spatial distribution of the eigenmodes of the Torrey equation. (a) The distribution of $M_{+}(z)$ in PT-symmetric phase with $G=80\mathrm{nT}/\mathrm{cm}$. The eigenmode $M_{-}(z)$ is nearly uniform along the $z$ direction. (b),(c) Similar to (a) but for $M_{\pm }(z)$ in PT-broken phase with $G=250\mathrm{nT}/\mathrm{cm}$. (d),(e) The amplitude ratio $\eta (G;z_{\text{p}})$ and the phase shift $\delta \theta (G;z_{\text{p}})$ at different probe beam positions. $\delta \theta \equiv {\theta }_{+}-{\theta }_{-}$ for the PT-symmetric phase and the PT-broken phase with $G<0$; $\delta \theta \equiv {\theta }_{-}-{\theta }_{+}$ for the PT-broken phase with $G>0$. The symbols are measured data points extracted from the FID spectrum. The displacement of the probe beam in our experiment is $z_{\text{p},\mathrm{red}}-z_{\text{p},\mathrm{blue}}=1.18\pm 0.02\mathrm{mm}$. The dashed lines are theoretical results using the same parameters as in Fig. 3 and with probe beam positions $z_{\text{p},\mathrm{red}}=-0.56\mathrm{mm}$ and $z_{\text{p},\mathrm{blue}}=-1.74\mathrm{mm}$. The red (blue) data points are the mean value of five (three) repeated measurements. The error bars [50] in the PT-symmetric phase may be underestimated; see Sec. III D of the Supplemental Material [42].

Figure 3

Eigenvalue spectrum of diffusive spins. (a) Fourier spectrum of measured FID signals of ${}^{129}\mathrm{Xe}$ as a function of $G$ at pump power 250 mW. (b),(c) The peak linewidth ${\mathrm{\Gamma }}_{\pm }$ and precession frequency ${\omega }_{\pm }/(2\pi )$ of ${}^{129}\mathrm{Xe}$ spins. The symbols are measured data extracted from the FID spectrum, and the dashed curves are the calculated eigenvalues of Eq. (1) with $|\gamma B_0|/(2\pi )=257.45\mathrm{Hz}$, ${\mathrm{\Gamma }}_{2\mathrm{c}}=0.306{\mathrm{s}}^{-1}$, and $D=0.211{\mathrm{cm}}^2/\mathrm{s}$. The blue data points in the $|G|<70\mathrm{nT}/\mathrm{cm}$ region are less reliable due to the fast decay of the excited mode $M_{+}(z)$ (see Sec. III D of the Supplemental Material [42]). All data points are the mean value of five repeated measurements. See Sec. III of the Supplemental Material [42] for details of FID spectrum fitting.

Figure 4

Stable comagnetometer. (a)–(c) Magnetic field dependence of three measured frequencies. $[f_{129},\mathrm{\Delta }{f}_{129},f_{131}]\equiv [{\omega }_{129},\mathrm{\Delta }{\omega }_{129},{\omega }_{131}]/(2\pi )$. These frequencies are shifted by 257.16, 1.17, and 76.24 Hz respectively for reasons of clarity. The $B_z$ shift is relative to $21.92\mathrm{\mu }\mathrm{T}$. The fitted slopes of (a)–(c) are ($11.74\pm 0.64$), ($0.000\pm 0.011$), and $(3.49\pm 0.19)\mathrm{mHz}/\mathrm{nT}$. (d)–(f) Pump power dependence of the three measured frequencies with the fitted slopes $(-1.57\pm 0.19)$, ($0.075\pm 0.021$), and $(-0.452\pm 0.055)\mathrm{mHz}/\mathrm{mW}$ respectively. (g) Traces of 1000 successive FID measurements at zero gradient (PT-symmetric phase). The pump power changes among [40, 45, 50, 55, 60] mW periodically, and the color of points represents the value of pump power. The histogram on the right shows the distribution and average value of ${\mathrm{\Omega }}_{\mathrm{rot}}^{(2\omega )}$ at 40, 50, and 60 mW. (h) The same as (g), but for 2000 successive FID measurements at $|G|=250\mathrm{nT}/\mathrm{cm}$ (PT-broken phase). (i) The average values in (g) and (h). Solid and dashed lines are linear fitting of the data points. The slope of the blue solid line is $(16.7\pm 2.0)\mathrm{\mu }\mathrm{Hz}/\mathrm{mW}$, while that of the red dashed line is $(0.0\pm 2.2)\mathrm{\mu }\mathrm{Hz}/\mathrm{mW}$. Data points in (a)–(f) are the mean values of 400 repeated measurements.