The theory of quantum tunnelling diffusion in the presence of strong coupling to lattice excitations has been developed in large part by Holstein,  Andreev and Lifshitz,  Flynn and Stoneham,  and most recently by Kagan, Klinger, Maksimov and Prokof'ev. [34,35,36,40] The derivation of the principal equations which appear in the literature regarding one- and two-phonon mediated tunnelling diffusion is outlined here. Some details are omitted in the interest of keeping this discussion to a readable length but these may be found in the review articles, most recently by Kagan and Prokof'ev.  Although much of this has been published previously, the literature lacks a concise treatment of quantum diffusion that would serve experimentalists well. It is hoped that a short review of this model will serve as a guide to the uninitiated in understanding the theory, and in particular the origin of temperature dependences which one can measure.
We are concerned here with the tunnelling diffusion of a light, neutral interstitial atom in an otherwise (nearly) perfect insulating crystal. The picture one should have in mind is of the particle, such as a muonium or hydrogen atom, occupying interstitial sites between atoms of an otherwise regular lattice at temperature T. The interstitial's potential is minimized by nearby lattice atoms relaxing slightly from their usual equilibrium sites to take on a new configuration where each atom resides in a potential well not very different from the perfect lattice. In the absence of a Coulomb interaction with conduction electrons, the dependence of the interstitial's potential on the displacement of nearby lattice atoms from their equilibrium positions is the principal interaction. The interstitial then is in a potential minimum partly of its own making and is therefore said to be self-trapped. The neutral interstitial together with the associated lattice distortion is called a small polaron. For example, electrons in semiconductors can induce a small polaron which extends over many more lattice sites and results in a large increase in effective mass.
Owing to the mass of our interstitial Mu atom (much heavier than electrons) the characteristic tunnelling bandwidth is small compared to energies of typical lattice excitations (we set thoughout this chapter) and, for the experiments described here, also small compared to the temperature of the lattice. At low temperatures where the number of lattice excitations is sufficiently small, the interstitial atom can propagate almost freely in a band-like (Bloch) state. At higher temperatures scattering with lattice excitations (phonons) occurs, more so as the number of excitations increases, and in this regime the diffusion rate decreases with increasing temperature. This results from the ``dynamical destruction of the band" when there are sufficient numbers of phonons that there is a significant probability of scattering during the time the interstitial spends at a given site. Under these conditions diffusion of the interstitial can still proceed via quantum tunnelling though the barrier, though the scattering results in the loss of phase coherence of the particle wavefunction. Therefore, in this regime the particle diffusion is said to be incoherent.
One can imagine that if the interstitial atom diffused to a site where the surrounding atoms were intitially unadjusted to its presence, the lattice would eventually respond, on a time scale set by the lattice vibrational modes, by re-establishing the polaron at the new site. It is much easier for the intersitial to diffuse by tunnelling from one site to another if the lattice fluctuates into a configuration so that, effectively, the entire polaron tunnels to the new site. When a particular fluctuation of the lattice brings the potential at an adjacent site into resonance with the interstitial's present potential, the interstitial can tunnel to the new site through the barrier. Transient configurations may also lead to an increase of the tunnelling rate by shifting the atoms that define the potential barrier to positions where the height and width of the barrier are reduced. In this model thermally excited fluctuations of lattice atoms' positions in these ways will play an important role in determining the tunnelling rate. In both the low- and high-temperature regimes it is through the temperature dependence of the phonon population that one can indirectly vary the particle's diffusion rate. This provides us with the principal experimental handle on the problem.
Generally the spectrum of lattice excitations can be divided into two parts: those that follow the interstitial particle adiabatically and those that are too slow to respond before the particle has moved. In the case of a light interstitial such as muonium in a lattice of much heavier atoms, the entire phonon spectrum must be considered non-adiabatic. The interstitial can easily follow the motion of the atoms, but the lattice can only respond slowly to the motion of the muonium atom, and therefore sees the muonium at its average position at the centres of the (slightly modified) potential wells. We can therefore treat the problem by writing the wave function of the system as a product of fast intra-well states and slow environmental states in the adiabatic approximation.
We shall assume that the diffusing particle occupies the lowest energy state of the intrawell part of the wave function, so that the overall temperature dependence of the hop rate is due to the overlap of states of the lattice. Intrawell degrees of freedom will play essentially no role since they are associated with relatively high energy excitations. We are therefore interested in calculating the transition rate W12 from state , the wave function of the lattice when the particle is at , to , where m,n label the configuration of the lattice.