Analysis of Magnetoacoustic Quadrupole Resonance and Application to Probe Quadrupole Degrees of Freedom in Quantum Magnets

Abstract

Motivated by the recent progress of high-frequency ultrasonic measurements, we propose a theory of magnetoacoustic resonance as a microscopic probe for quadrupole degrees of freedom hidden in magnetic materials. A local strain driven by an acoustic wave couples to electronic states of a magnetic ion through various quadrupole-strain couplings, and this provides a periodically time-dependent oscillating field. As a typical two-level system with the quadrupole, we consider a non-Kramers doublet and investigate single-and multiphonon-mediated transition processes on the basis of the Floquet theory. An analytic form of the transition probability is derived within the weak coupling theory, which helps us analyze the magnetoacoustic quadrupole resonance. We apply the theory to realistic non-Kramers doublet systems for the f 2 configuration in O h and D 4h symmetries, and discuss how to identify the relevant quadrupole by controlling the quadrupole-strain coupling with an applied magnetic field in ultrasonic measurements.

Magnetoacoustic Resonance to Probe Quadrupole–Strain Coupling in a Diamond Nitrogen-Vacancy Center as a Spin-Triplet System

“…Very recently, an evaluation of spin-strain coupling with O zx has been performed by measurements of an acoustically driven single-quantum spin transition. 20) As reported in our recent studies, 21,22) it is also important that the transition via quadrupole couplings can be changed by rotating an applied magnetic field. This is useful for probing such a spin-strain coupling with O zx that is difficult to measure.…”

“…( 9). 22) Conversely, the maximum broadening shifts toward φ = 0 as A zx /A v increases in Fig. 1 (b), owing to cos φ for the coupling with the zx quadrupole.…”

“…leads to simple analytic forms for the transition probability at fixed ε 0 = nω (n = 1, 2) as 22) P( 1)…”

“…In this case, the infinite-dimensional matrix form of H F is reduced to an effective 2 × 2 matrix using the Van Vleck perturbation theory. 29,30) By analogy with previous studies, 21,22) the effective Hamiltonian in the subspace of |α0 and |β, −n is given bỹ…”

“…( 7) has been frequently studied by the Floquet theory. 21,22,[27][28][29][30] Following Shirley,27) the problem of solving the time-dependent Schrödinger equation is transformed to a time-independent eigenvalue problem using an infinitedimensional matrix form of the Floquet Hamiltonian. The matrix element αn|H F |βm = H [n−m] αβ + nωδ nm δ αβ is constructed using the Floquet states |αn = |α ⊗ |n .…”