Regime of applicability of linear response

In the adiabatic limit $V\rightarrow0$ an initial state $\left\vert m(0)\right\rangle$ will remain in the local eigenstate $\left\vert m(x)\right\rangle$ for all $x$. Hence for extremely small $V$, LRT will not be valid because Landau-Zener transitions (avoided crossings) become the dominant heating mechanism, as explained by Wilkinson [200]. This is called the quantum adiabatic regime. For a given system and given parameter $x$, there will be a typical change in $x$ which results in each energy level encountering about one avoided crossing. More strictly this is expressed by the criterion that eigenstate overlaps (see Section 6.4) should mix about one nearest neighbour. This parameter change is called $\delta x^{{\mbox{\tiny qm}}}_{{\mbox{\tiny c}}}$. If this change takes longer than $t_{{\mbox{\tiny H}}}$ to occur, then the system remains `localized' on the initial state, and we have adiabaticity. The above also requires that no dephasing processes occur before $t_{{\mbox{\tiny H}}}$.

LRT also has an upper velocity limit, beyond which non-perturbative effects dominate. The key criterion is whether LRT is valid for a single correlation time $\tau_{{\mbox{\tiny cl}}}$ (the timescale required to establish diffusive spreading [46]). Once this is true, stochastic energy spreading is expected to continue forever. FOPT will break down when there is non-perturbative mixing between levels. If the band profile is flat near $\omega=0$, this happens first between neighbouring levels, when $x$ reaches $\delta x^{{\mbox{\tiny qm}}}_{{\mbox{\tiny c}}}$. Therefore the limit of applicability of FOPT is $V \ll \delta x^{{\mbox{\tiny qm}}}_{{\mbox{\tiny c}}}/ \tau_{{\mbox{\tiny cl}}}$. This was once thought [202] to be the point at which the heating rate departs from LRT. Beyond this, a `core' region will be created which is non-perturbatively mixed, before diffusive growth is established. However, Cohen[46] has realised that the existence of this core need not invalidate the LRT result, because it is the `tail' region (everything outside the core) which dominates the energy spreading. His modification extends the applicability of LRT up to $V \sim \delta x^{{\mbox{\tiny qm}}}_{{\mbox{\tiny c}}}/ t_{{\mbox{\tiny prt}}}$, where $t_{{\mbox{\tiny prt}}}$ is the timescale for much of the probability to have left the initial level (beyond which perturbation theory is past rescue).

Beyond this, the LRT picture breaks down. However, at the highest $V$, it is expected that there is semiclassical correspondence, so that the classical dissipation rate applies. The above sequence of spreading profiles, FOPT $\rightarrow$ core-tail $\rightarrow$ semiclassical, is demonstrated in our work [48] (the calculations for which appear in Section 6.4 of this thesis).