The tensor of interaction of a two-level system with an arbitrary strain field

TL;DR

This paper develops a general tensor model for the interaction between two-level systems and strain fields in solids, predicting phonon interactions and TLS polarization effects consistent with experimental observations.

Contribution

It introduces a tensor-based framework for TLS-strain interaction, generalizing previous models and deriving specific predictions for isotropic solids.

Findings

Longitudinal phonons interact more strongly with TLSs than transverse phonons.

Transversal waves leave certain TLSs unperturbed depending on their orientation.

Longitudinal strain polarizes the TLS ensemble.

Abstract

The interaction between two-level systems (TLS) and strain fields in a solid is contained in the diagonal matrix element of the interaction hamiltonian, $\delta$, which, in general, has the expression $\delta=2[\gamma]:[S]$, with the tensor $[\gamma]$ describing the TLS ``deformability'' and $[S]$ being the symmetric strain tensor. We construct $[\gamma]$ on very general grounds, by associating to the TLS two objects: a direction, $\hat\bt$, and a forth rank tensor of coupling constants, $[[R]]$. Based on the method of construction and on the invariance of the expression of $\delta$ with respect to the symmetry transformation of the solid, we conclude that $[[R]]$ has the same structure as the tensor of stiffness constants, $[[c]]$, from elasticity theory. In particular, if the solid is isotropic, $[[R]]$ has only two independent parameters, which are the equivalent of the Lam\'e constants. Employing this model we calculate the absorption and emission rates of phonons on TLSs and show that in isotropic solids, on average, the longitudinal phonons interact stronger with the TLSs than the transversal ones, as it is observed in experiments. We also show that in isotropic solids, a transversal wave leaves unperturbed all the TLSs with the direction contained in one of the two planes that are perpendicular either to the wave propagation direction or to the polarization direction and that a longitudinal strain applied to the solid polarises the TLS ensemble.