Abstract
We point out that in the first order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function $\sinc x \equiv \sin x / x$. These physical effects appear in many systems with approximately equally spaced spectra, and is also
 robust for larger-amplitude coupling beyond the domain of perturbation theory.
 We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.
Highlights
Coarse graining is a common trick in physics
We can replace the real density of states, which consists of delta functions, by a coarse-grained one, which is a continuous and smooth function of energy
The slope of the line changes periodically with the period being inversely proportional to the level spacing of the spectrum
Summary
Coarse graining is a common trick in physics. In principle, it is invoked whenever one replaces a summation by an integral. P(t) denotes the probability of the system remaining in the initial state The transition dynamics can be nonsmooth and have a single level resolution beyond some critical time These effects can be demonstrated by using the one-dimensional tight-binding model as we shall do below. III, we demonstrate the effect by taking two examples of tight-binding lattices and driving a local parameter (potential of a single site) In these realistic models, the two assumptions are only approximately satisfied, but we still see sharp kinks (nonsmooth behavior) and sudden bifurcations (level resolution). The two assumptions are only approximately satisfied, but we still see sharp kinks (nonsmooth behavior) and sudden bifurcations (level resolution) All these effects persist even beyond the perturbative regime, i.e., for large-amplitude driving
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