Strong-coupling theory for the spin-phonon model

We present a strong-coupling approach to the spin-phonon model with a cubic spectral density, which relies on a matrix representation for the reduced time evolution operator on the four-dimensional space of pseudospins. The $4\times 4$ self-energy matrix $\Sigma$ is written as an infinite series in powers of bath fluctuation operators, and the lowest-order approximation is evaluated explicitly. The diagonal element ${\Sigma }_{\mathrm{zz}}$ corresponds to the result of the noninteracting blip approximation (NIBA); retaining the full matrix is equivalent to taking into account certain blip-blip interactions. Contrary to NIBA, this approach agrees with a rigorous result for the quadratic term of the rate. As a main result, we find a crossover to incoherent motion at a temperature $T^{*}$, which is compared to previous theoretical work and discussed in view of experimental data for various two-level tunneling systems.