Polarization of nuclear spins by a cold nanoscale resonator

A cold nanoscale resonator coupled to a system of nuclear spins can induce spin relaxation. In the low-temperature limit where spin-lattice interactions are ``frozen out,'' spontaneous emission by nuclear spins into a resonant mechanical mode can become the dominant mechanism for cooling the spins to thermal equilibrium with their environment. We provide a theoretical framework for the study of resonator-induced cooling of nuclear spins in this low-temperature regime. Relaxation equations are derived from first principles, in the limit where energy donated by the spins to the resonator is quickly dissipated into the cold bath that damps it. A physical interpretation of the processes contributing to spin polarization is given. For a system of spins that have identical couplings to the resonator, the interaction Hamiltonian conserves spin angular momentum, and the resonator cannot relax the spins to thermal equilibrium unless this symmetry is broken by the spin Hamiltonian. The mechanism by which such a spin system becomes ``trapped'' away from thermal equilibrium can be visualized using a semiclassical model, which shows how an indirect spin-spin interaction arises from the coupling of multiple spins to one resonator. The internal spin Hamiltonian can affect the polarization process in two ways: (1) By modifying the structure of the spin-spin correlations in the energy eigenstates, and (2) by splitting the degeneracy within a manifold of energy eigenstates, so that zero-frequency off-diagonal terms in the density matrix are converted to oscillating coherences. Shifting the frequencies of these coherences sufficiently far from zero suppresses the development of resonator-induced correlations within the manifold during polarization from a totally disordered state. Modification of the spin-spin correlations by means of either mechanism affects the strength of the fluctuating spin dipole that drives the resonator. In the case where product states can be chosen as energy eigenstates, spontaneous emission from eigenstate populations into the resonant mode can be interpreted as independent emission by individual spins, and the spins relax exponentially to thermal equilibrium if the development of resonator-induced correlations is suppressed. When the spin Hamiltonian includes a significant contribution from the homonuclear dipolar coupling, the energy eigenstates entail a correlation specific to the coupling network. Simulations of dipole-dipole coupled systems of up to five spins suggest that these systems contain weakly emitting eigenstates that can trap a fraction of the population for time periods $\ensuremath{\gg}100/{R}_{0}$, where ${R}_{0}$ is the rate constant for resonator-enhanced spontaneous emission by a single spin 1/2. Much of the polarization, however, relaxes with rates comparable to ${R}_{0}$. A distribution of characteristic high-field chemical shifts tends to increase the relaxation rates of weakly emitting states, enabling transitions to states that can quickly relax to thermal equilibrium. The theoretical framework presented in this paper is illustrated with discussions of spin polarization in the contexts of force-detected nuclear-magnetic-resonance spectroscopy and magnetic-resonance force microscopy.