Stochastic energy spreading

Returning now to the FGR expression (2.45), one might wonder how it is possible to get dissipation (irreversible growth in the mean energy) at all, given that the up and down transition rates are always equal! The answer will be that heating is possible because the density of states $g(E)$ may differ slightly at the two energies $E = E_m \pm \hbar\omega$ being `pumped' into, and therefore more states may fall under the upper delta function than the lower, giving an increase in mean energy. However to understand this in more detail, the up and down transitions (2.45) must be interpreted as giving diffusion in energy-space. A diffusion equation for time-evolution of a continuous energy distribution function $\rho(E)$ can then be written--from this follows the rate of increase of mean energy.

The linear growth of $p_n(t) \equiv \vert a_n(t)\vert^2$ resulting from the FGR means a linear growth in the second moment (variance) of the distribution about the initial energy $E_m$,

$$\delta E^2 \; \equiv \; \langle ({\mathcal{H}}(t) - E_m)^2 ... ..._{{\mbox{\tiny E}}_m} \; = \; \sum_{n} (E_n - E_m)^2 p_n(t) .$$ (2.50)

This can be identified with diffusive growth $\delta E^2 = 2 D_{{\mbox{\tiny E}}}\cdot t$ with diffusion constant given by

$$D_{{\mbox{\tiny E}}}(\omega) \; = \; \frac{1}{2} \sum_n (E_n - E_m)^2 \, \Gamma_{nm}(\omega)$$

$$= \frac{1}{2} \cdot \mbox{\small$\frac{1}{2}$}(\omega A)^2 \cdot \m... ...\tiny E}}}(\omega) + \tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(-\omega)]$$

$$= \frac{1}{2} \cdot \mbox{\small$\frac{1}{2}$}(\omega A)^2 \cdot \tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$$ (2.51)

Use was made of (2.49), which is appropriate since we are considering a continuum limit. Also, as in the previous section, an averaging over a classically-small range of adjacent states was performed. Thus we see that the band profile at the driving frequency $\omega$ determines the resulting diffusion rate. The term $\mbox{\small$\frac{1}{2}$}(\omega A)^2$ is simply the mean square parametric driving velocity $\langle \dot{x}^2 \rangle$.

So far the evolution of the coefficients $a_n(t)$ has been entirely coherent (this is unaffected by the final averaging over initial state $m$). Therefore an important issue arises: does the energy diffusion eventually stop at $t \sim t_{{\mbox{\tiny H}}}$, when quantum recurrences due to the discreteness of the spectrum will surely occur? (For instance, in the kicked rotator system [75], periodic driving results in saturation of energy spreading, giving localization). The answer (discussed in detail in Section 8 and Appendix B of [46]) is that preservation of coherent effects is actually very fragile. Either $x(t)$ can have some slight stochasticity (frequency jitter), or there is some slight dephasing process (interaction with the environment, universal in real systems). Correlations due to coherence are then lost on timescales longer than that associated with jitter or dephasing. So if diffusive spreading and loss of coherence can be established before $t_{{\mbox{\tiny prt}}}$, the `breaktime' (i.e. time limit of applicability) of the FGR, then stochastic spreading will continue for ever. In other words the `rate equation' (2.49) will remain valid, and we will not be surprised by recurrence (interference) effects at long times. The FGR breaktime will be discussed more in Section 2.2.6.

Figure 2.4: Fermi occupation function $f(E)$ evolving under energy diffusion, causing growth of mean energy. The initial ensemble occupation function is shown by the dotted line.