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 12 interacts with the resonator, transverse relaxation is exponential during free evolution, and the time constant for the relaxation is T2 = 2/Rh, where Rh 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 ρab and ρcd and average to zero the transfers between them. This averaging guarantees that the spin transitions responsible for the coupling between ρab and ρcd 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 T2 = 2/Rh 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 T1ρ = 2/Rh.