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 $\gg 100/R_0$, 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.

Figure 1

Prototypical resonator design proposed in Ref. [21] for NMR spectroscopy of nanoscale samples. The sample is “sandwiched” between magnetic cylinders and rotates with the sandwich about the torsional beams. The nuclear-spin dipole couples to the oscillating transverse field generated by the cylinders.

Figure 2

The dashed (blue) and dash-dotted (red) curves show the evolution of $\langle I_z\rangle$ in two systems of isochronous spins that interact only with the resonator and relax from an initially disordered state, with the respective systems having $N=144$ and $N=36$ spins. The curves are normalized to take the value 1 when the system has relaxed to its trapped state, and the value of $\langle I_z\rangle$ in this state is denoted by ${\langle I_z\rangle }_{\text{trap}}$. For both curves, the time $t={\left(R_0\sqrt{N/2}\right)}^{-1}$ corresponds to a point at which $\langle I_z\rangle$ has relaxed to about 70$\%$ of its value in the trapped state. For $N=144$ and $N=36$, these respective times are $t=0.12\mathrm{s}$ and $t=0.24\mathrm{s}$. As in all simulations presented in this paper, the resonator temperature is $T_h=0\mathrm{K}$, and the rate constant for resonator-enhanced spontaneous emission is $R_0=1{\mathrm{s}}^{-1}$.

Figure 3

Simulated relaxation of five isochronous spins that interact only with the resonator and are initially completely disordered. The dashed (blue) and dash-dotted (red) curves show the respective evolution of $\langle I_z\rangle$ and $\langle I_x^2+I_y^2-5/2\rangle$. The solid curve shows ideal exponential relaxation with rate constant $R_0=1{\mathrm{s}}^{-1}$ toward full polarization, for purposes of comparison with $\langle I_z\rangle$. As in all simulations presented in this paper, the resonator temperature is $T_h=0\mathrm{K}$.

Figure 4

Adding chemical shift offsets to the simulation of Fig. 3, such that the Larmor frequencies of the five spins are spaced in steps of $1\mathrm{Hz}$, causes the indirect spin-spin torques to be modulated quickly enough that their contribution to relaxation is suppressed.

Figure 5

Simulated relaxation of five isochronous spins that interact only with the resonator and are initially aligned along the $x$ axis. The dashed (blue) and dash-dotted (red) curves show the respective evolution of $\langle I_z\rangle$ and $\langle I_x^2+I_y^2-5/2\rangle$. The solid curve shows ideal exponential relaxation with rate constant $R_0=1{\mathrm{s}}^{-1}$ toward full polarization, for purposes of comparison with $\langle I_z\rangle$.

Figure 6

Adding chemical shift offsets to the simulation of Fig. 5, with the Larmor frequencies of the five spins spaced in steps of $1\mathrm{Hz}$, causes the indirect spin-spin torques to be modulated. When the transverse spin components are initially aligned, the indirect torques add coherently, and so a faster modulation would be needed to completely suppress their effect on longitudinal relaxation.

Figure 7

Cooling of a system of three H nuclei (CSD entry: nimfoe) from an initially disordered state. The dash-dotted curves (red) show relaxation under the dipolar Hamiltonian $H_D$, while the dashed curves (blue) show relaxation under $H_D+H_{\text{CS}}$. For each simulation, the upper curve shows the relaxation of $\langle I_z\rangle$, and the lower shows $\langle I_x^2+I_y^2-3/2\rangle$. The solid curve shows ideal exponential relaxation with rate constant $R_0=1{\mathrm{s}}^{-1}$ toward full polarization, for purposes of comparison with $\langle I_z\rangle$.

Figure 8

Cooling of a system of four H nuclei (CSD entry: cukcam04) from an initially disordered state. The dash-dotted curves (red) show relaxation under the dipolar Hamiltonian $H_D$, while the dashed curves (blue) show relaxation under $H_D+H_{\text{CS}}$. For each simulation, the upper curve shows the relaxation of $\langle I_z\rangle$, and the lower shows $\langle I_x^2+I_y^2-4/2\rangle$. The solid curve shows ideal exponential relaxation with rate constant $R_0=1{\mathrm{s}}^{-1}$ toward full polarization, for purposes of comparison with $\langle I_z\rangle$.

Figure 9

Cooling of a system of four H nuclei (CSD entry: vazmep) from an initially disordered state. The curves are defined in the same way as in Fig. 8.

Figure 10

Cooling of a system of five H nuclei (CSD entry: cejmaf) from an initially disordered state. The dash-dotted curves (red) show relaxation under the dipolar Hamiltonian $H_D$, while the dashed curves (blue) show relaxation under $H_D+H_{\text{CS}}$. For each simulation, the upper curve shows the relaxation of $\langle I_z\rangle$, and the lower shows $\langle I_x^2+I_y^2-5/2\rangle$. The solid curve shows ideal exponential relaxation with rate constant $R_0=1{\mathrm{s}}^{-1}$ toward full polarization, for purposes of comparison with $\langle I_z\rangle$.

Figure 11

Cooling of a system of five H nuclei (CSD entry: wijzoe) from an initially disordered state. The curves are defined in the same way as in Fig. 10.