Resonator-induced dissipation of transverse nuclear-spin signals in cold nanoscale samples

The back action of typical macroscopic resonators used for detecting nuclear magnetic resonance can cause a reversible decay of the signal, known as radiation damping. A mechanical resonator that is strongly coupled to a microscopic sample can in addition induce an irreversible dissipation of the nuclear-spin signal, distinct from radiation damping. We provide a theoretical description of resonator-induced transverse relaxation that is valid for samples of a few nuclear spins in the low-temperature regime, where quantum fluctuations play a significant role in the relaxation process, as well as for larger samples and at higher temperatures. Transverse relaxation during free evolution and during spin locking are analyzed, and simulations of relaxation in example systems are presented. In the case where an isolated spin $\frac{1}{2}$ interacts with the resonator, transverse relaxation is exponential during free evolution, and the time constant for the relaxation is $T_2=2/R_h$, where $R_h$ is the rate constant governing the exchange of quanta between the resonator and the spin. For a system of multiple spins, the time scale of transverse relaxation during free evolution depends on the spin Hamiltonian, which can modify the relaxation process through the following effects: (1) changes in the structure of the spin-spin correlations present in the energy eigenstates, which affect the rates at which these states emit and absorb energy, (2) frequency shifts that modify emission and absorption rates within a degenerate manifold by splitting the energy degeneracy and thus suppressing the development of resonator-induced correlations within the manifold, and (3) frequency shifts that introduce a difference between the oscillation frequencies of single-quantum coherences ${\rho }_{ab}$ and ${\rho }_{cd}$ and average to zero the transfers between them. This averaging guarantees that the spin transitions responsible for the coupling between ${\rho }_{ab}$ and ${\rho }_{cd}$ cause irreversible loss of order rather than a reversible interconversion of the coherences. In systems of a few spins, transverse relaxation is accelerated by a dipolar Hamiltonian that is either the dominant term in the internal spin Hamiltonian or a weak perturbation to the chemical-shift Hamiltonian. A pure chemical-shift Hamiltonian yields exponential relaxation with $T_2=2/R_h$ in the case where the Larmor frequencies of the spins are distinct and sufficiently widely spaced. During spin locking with a nutation frequency fast enough to average the evolution under the internal spin Hamiltonian but not the interactions occurring during the correlation time of the resonator, relaxation of the spin-locked component is exponential with time constant $T_{1\rho }=2/R_h$.

Figure 1

Prototypical resonator design proposed in Ref. 12 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

Transverse relaxation of a system of three dipolar-coupled H nuclei (CSD entry: celbaw) from an initial state aligned along the $x$ axis. Two of the dipolar couplings $|{\omega }_{ij}|/2\pi$ are $\sim$$4\mathrm{k}\mathrm{Hz}$, while the third is $\sim$$1\mathrm{k}\mathrm{Hz}$. The dash-dotted curve (red) shows relaxation during free evolution, while the dashed curve (blue) shows relaxation during spin locking along the $x$ axis, with ${\omega }_1/2\pi =50\mathrm{k}\mathrm{Hz}$. For purposes of comparison, the relaxation corresponding to $H_{\mathrm{spin}}=0$ is shown as a dotted black curve, and the ideal case of exponential relaxation with time constant $2/R_0=2\mathrm{s}$ is shown as a solid black curve.

Figure 3

Transverse relaxation of a system of four dipolar-coupled H nuclei (CSD entry: sedtuq01). The two largest dipolar couplings $|{\omega }_{ij}|/2\pi$ are $\sim$17 kHz and $\sim$$17\mathrm{k}\mathrm{Hz}$. The curves are defined in the same way as in Fig. 2.

Figure 4

Transverse relaxation of a system of five dipolar-coupled H nuclei (CSD entry: wincur). The two largest dipolar couplings $|{\omega }_{ij}|/2\pi$ are $\sim$$8\mathrm{k}\mathrm{Hz}$. The curves are defined in the same way as in Fig. 2.

Figure 5

Longitudinal and transverse relaxation of five isochronous spins that interact only with the resonator. The spins 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 $\langle I_x\rangle$. For purposes of comparison, the solid curve that starts at zero shows exponential longitudinal relaxation with time constant $T_1=1/R_0$, while the solid curve that decays toward zero shows exponential transverse relaxation with time constant $T_2=2/R_0$.

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 couplings in the interaction-frame master equation to be modulated. Resonator-induced correlations begin to develop within degenerate manifolds, but the slow modulation due to $H_{\mathrm{CS}}$ prevents these correlations from determining the time scale of the relaxation.

Figure 7

Adding chemical-shift offsets to the simulation of Fig. 5, with the Larmor frequencies of the five spins spaced in steps of $3\mathrm{Hz}$, causes only small oscillations about the exponential curves.

Figure 8

Longitudinal and transverse relaxation of four dipolar-coupled H nuclei (CSD entry: pabboj) from an initial state aligned along the $x$ axis. The calculated dipolar couplings $|{\omega }_{ij}|/2\pi$ fell roughly between 50 and $100\mathrm{Hz}$, aside from a single coupling $\sim$$250\mathrm{Hz}$. To illustrate the relaxation occurring in the regime $H_D\ll H_{\mathrm{CS}}$, chemical-shift offsets in steps of $1.2\mathrm{k}\mathrm{Hz}$ were assigned to the four spins, that is, the offset on spin $j$ was $j\times 1.2\mathrm{k}\mathrm{Hz}$, where the spins were numbered according to the order in which they were listed in the database entry. (Note that these offsets correspond to steps of 2 ppm at a Larmor frequency of $600\mathrm{M}\mathrm{Hz}$.) The dashed (blue) and dash-dotted (red) curves show the respective evolution of $\langle I_z\rangle$ and $\langle I_x\rangle$. For purposes of comparison, the solid curve that starts at zero shows longitudinal relaxation with time constant $T_1=1/R_0$, while the solid curve that decays toward zero shows transverse relaxation with time constant $T_2=2/R_0$.