Role of Spin Noise in the Detection of Nanoscale Ensembles of Nuclear Spins

When probing nuclear spins in materials on the nanometer scale, random fluctuations of the spin polarization will exceed the mean Boltzmann polarization for sample volumes below about $(100\mathrm{nm}{)}^3$. In this Letter, we use magnetic resonance force microscopy to observe nuclear spin fluctuations in real time. We show how reproducible measurements of the polarization variance can be obtained by controlling the spin correlation time and rapidly sampling a large number of independent spin configurations. This allows significant improvement in the signal-to-noise ratio for nanometer-scale magnetic resonance imaging.

Figure 1

(a) Experimental setup. A $(4\mu \mathrm{m}{)}^3$ ${\mathrm{CaF}}_2$ particle attached to the end of a single-crystal Si cantilever ($f_c=2.6\mathrm{kHz}$, $k_c=86\mu \mathrm{N}/\mathrm{m}$, and $Q=18500$) is positioned 80 nm above a FeCo-magnetic tip and placed in a static external field $B_0=2.85\mathrm{T}$ ($B_0||\mathbf{z}$). ${}^{19}\mathrm{F}$ spins are excited by an rf magnetic field $B_1=4\mathrm{mT}$ ($B_1||\mathbf{y}$), generated by passing current $I(t)$ through a microwire situated directly under the tip [16]. All experiments are carried out at 4.2 K and in high vacuum. (b) Protocol used for spin detection and randomization. Repetitive rf frequency sweeps with a center frequency of 114.8 MHz invert the spins adiabatically twice per cantilever cycle [17, 18]. For spin randomization, the rf field is interrupted at fixed intervals ${\tau }_r$ by a burst of $\pi /2$ pulses. Spin polarization is completely uncorrelated before and after randomization pulses. (c) Signal processing protocol as described in the text.

Figure 2

Cantilever tip oscillation amplitude as a function of time. (a) In-phase signal $X(t)$ in the presence of the ${}^{19}\mathrm{F}$ spins fluctuating naturally with a spin signal correlation time of ${\tau }_m\approx 3.5\mathrm{s}$ [13]. (b) Quadrature signal $Y(t)$ due to thermal fluctuations of the cantilever. The observed thermal noise correlation time ($\tau \approx 200\mathrm{ms}$) is set by cantilever and filter bandwidths. (c) Same as (a), but spins are periodically rerandomized every ${\tau }_r=351\mathrm{ms}$. (d) Normalized autocorrelation function of spin signals obtained by analysis of an 8 min long record and after subtracting the measurement thermal noise autocorrelation function.

Figure 3

(a) One-dimensional image scans in the $\mathbf{x}$ direction of the calcium fluoride sample. The points are the measured variances of the spin force and the solid lines are guides for the eye. Scatter of the data points in the top image is due to the uncertainty of the variance estimates. Much improved SNR is seen in the lower image where the spins are periodically randomized at ${\tau }_r=113\mathrm{ms}$. (b) Data points showing SNR for 1 min measurements as a function of the randomization interval ${\tau }_r$. The solid line represents the expected SNR based on Eq. (5) using known experimental parameters (no free fit parameters).