Sensitivity of force-detected NMR spectroscopy with resonator-induced polarization

In the low-temperature regime where the thermal polarization $P$ is of order unity and spin-lattice relaxation is “frozen out,” resonator-induced relaxation can be used to polarize a nuclear-spin sample for optimal detection sensitivity. We characterize the potential of resonator-induced polarization for enhancing the sensitivity of nuclear-magnetic-resonance spectroscopy. The sensitivities of two detection schemes are compared, one involving detection of a polarized sample dipole and the other involving detection of spin-noise correlations in an unpolarized sample. In the case where the dominant noise source is instrument noise associated with resonator fluctuations and with detection of the mechanical motion, a simple criterion can be used to compare the two schemes. Polarizing the sample improves sensitivity when $P$ is larger than the signal-to-noise ratio for detection of a fully-polarized spin during a single transient. Even if the instrument noise is decreased to a level near the quantum-mechanical limit, it is larger than spin noise for unpolarized samples containing up to a few tens of nuclei. Under these conditions, spin polarization of order unity would enhance spectroscopic detection sensitivity by an order of magnitude or more. In the limiting case where signal decay is due to resonator-induced dissipation during ideal spin locking, and where resonator fluctuations are the noise source, the only parameter of the spin-resonator system that affects the sensitivity per spin is the ratio of frequency to temperature. A balance between the coupling strength, the noise power, and the signal lifetime causes the cancellation of other parameters from the sensitivity formula. Partial cancellation of parameters, associated with a balance between the same three quantities, occurs more generally when the resonator is both the dominant noise source and the dominant source of signal decay. An intrinsic sensitivity limit exists for resonant detection of coherent spin evolution, due to the fact that the detector causes signal decay by enhancing the spins' spontaneous emission. For a single-spin sample, the quantum-limited signal-to-noise ratio for resonant detection is $1/3$. In contrast to the sensitivity, the time required for sample polarization between transients depends strongly on resonator parameters. We discuss resonator design and show that for a torsional resonator, the coupling is optimal when the resonator's magnetization remains aligned with the applied field during the mechanical oscillations.

Figure 1

Prototypical resonator design proposed in Ref. 15 for NMR spectroscopy of nanoscale samples. The sample is “sandwiched” between magnetic cylinders and rotates with the sandwich about the torsional beam. The spin dipole couples to the oscillating transverse field generated by the cylinders.

Figure 2

Comparison of the resonator's transverse field in two limiting cases. In (a), arrows represent the magnetization $\mathbf{M}$ of the resonator's ferromagnetic cylinders after a mechanical rotation away from the equilibrium position. In (b), magnetization that remains aligned with the applied field is expressed as the sum of two perpendicular components, one of which rotates with the cylinders. The transverse fields generated by the two components add constructively at the location of the spins. If $\mathbf{M}$ rotates with the cylinders, only the first of these components is present. The spin-resonator coupling is thus stronger when $\mathbf{M}$ remains aligned with the applied field during the mechanical oscillations.