Spin-noise correlations and spin-noise exchange driven by low-field spin-exchange collisions

The physics of spin-exchange collisions have fueled several discoveries in fundamental physics and numerous applications in medical imaging and nuclear magnetic resonance. We report on the experimental observation and theoretical justification of spin-noise exchange, the transfer of spin noise from one atomic species to another. The signature of spin-noise exchange is an increase of the total spin-noise power at low magnetic fields, on the order of 1 mG, where the two-species spin-noise resonances overlap. The underlying physical mechanism is the two-species spin-noise correlation induced by spin-exchange collisions.

Figure 1

Experimental schematic of the spin-noise measurement. For the actual measurement we used a 10-cm-long cell with Rb of natural abundance, while for a consistency check we used two 5-cm cells back-to-back, each having isotopically enriched Rb. The temperature was measured with a thermocouple placed at the oven's center, reading $112{}^{\circ }\mathrm{C}$. The temperature inferred from the collisional line width of the spin-noise resonance was $100{}^{\circ }\mathrm{C}$, and it was the corresponding Rb density that we used in the theoretical prediction. The laser power and detuning from the D2 line were 3.3 mW and 43 GHz, respectively, while the pressure-broadened optical linewidth at 100 torr of nitrogen is about 4 GHz. The magnetic field was set by a computer-controlled switch at either the desired value or a much larger value pushing spin noise out of the detector's bandwidth, enabling a fast subtraction of the background spectrum (no spin noise) from the spin-noise spectrum. The balanced polarimeter output was fed into a spectrum analyzer, and the spectra were averaged at the computer.

Figure 2

(a) Measured spin-noise spectra for three different magnetic fields. Upper graphs are from a data set with the experiment cell (C1) containing Rb of natural isotopic abundance (ratio of peak heights about 3:1), and lower graphs are from the two back-to-back cells (C2), each enriched with one of the two Rb isotopes (ratio of peak heights about 1:1). (b) Integrated spin-noise power (ISNP) for C1 (red circles) and C2 (blue squares). The former were normalized by their ISNP at $B=50$ mG, while the latter were normalized by their average value. The red solid lines are the theoretical prediction $S\left(B\right)/S\left(50\mathrm{mG}\right)$ of Eq. (7) with no free parameters, but with different (by 20%) values of the magnetic gradient as input to the theory.

Figure 3

(a) Spin-noise spectrum and background. (b) The noisy blue (gray) line is the subtraction of the two spectra shown in panel (a) and constitutes a run, while the black line is the average of 50 runs and constitutes a set. (c) The offset of the measured power spectra scales linearly with laser power, demonstrating a photon shot-noise limited measurement. (d) Total integrated spin-noise power at different temperatures. For the integral we used the ${}^{85}\mathrm{Rb}$ spin-noise resonance at high enough magnetic fields so that there is no overlap with the ${}^{87}\mathrm{Rb}$ resonance. The resonance linewidth is proportional to the atom number probed by the laser. We have corrected for the other small contributions to line broadening and for the different average laser powers in the cell at different temperatures. Spin-noise signals scale as the square root of the atom number, hence spin-noise power scales linearly with the linewidth.

Figure 4

Spin-noise spectrum with Lorentzian fit for (a) a high magnetic field and (b,c) the lowest two magnetic fields. (d) The $1/f$ noise tail falls to the photon-shot-noise level at 1.5 kHz. (e) To avoid contamination of the $B=3\mathrm{mG}$ data by the 1/$f$ noise tail we start fitting the data from the fit cutoff frequency of 1.9 kHz and on (short-dashed line). The long-dashed line at 1.5 kHz shows the line center frequency. To estimate the fit error due to the fit cutoff we generate numerical data with the same signal-to-noise ratio as the real data, fit them, and compare them with the known ISNP. For the $B=4$ mG data this error is negligible, since the peak of the resonance, being just higher than the cutoff of 1.9 kHz as shown in panel (c), is included in the fit.

Figure 5

Example of a spin-exchange collision generating spin noise. The spin states $|F,m_F\rangle$ of the colliding atoms are written in the $x$ basis. Two ${}^{85}\mathrm{Rb}$ atoms in the $F=3$ state and opposite $m_F$ collide, and after the collision one jumps to the $F=2$ manifold. The total $m_F$ is conserved; however the spin in the $F=2$ state precesses in the opposite sense. Under the action of the magnetic field $B\widehat{\mathbf{z}}$ the two atoms momentarily generate a nonzero contribution to the spin-noise signal along the $y$ axis (balanced of course by the nuclear spins). Subsequent spin-exchange collisions will damp the transverse spin and so on.