Quantum resonance catastrophe for conductance through a periodically driven barrier

We consider the quantum conductance in a tight-binding chain with a locally applied potential which is oscillating in time. The steady state for such a driven impurity can be calculated exactly for any energy and applied potential using the Floquet formalism. The resulting transmission has a nontrivial, nonmonotonic behavior depending on incoming momentum, driving frequency, and the strength of the applied periodic potential. Hence there is an abundance of tuning possibilities, which allows finding the resonances of total reflection for any choice of incoming momentum and periodic potential. Remarkably, this implies that even for an arbitrarily small infinitesimal impurity potential it is always possible to find a resonance frequency at which there is a catastrophic breakdown of the transmission $T=0$. The points of zero transmission are closely related to the phenomenon of Fano resonances at dynamically created bound states in the continuum. The results are relevant for a variety of one-dimensional systems where local AC driving is possible, such as quantum nanodot arrays, ultracold gases in optical lattices, photonic crystals, or molecular electronics.

Figure 1

Left: Sketch of the model mapped onto a set of static coupled chains. The $n=0$ chain is locally connected to two side-coupled systems of chains with a corresponding chemical potential $n\omega$. Right: Dispersion relation with the special frequencies $\omega =2J\pm \epsilon$ below which the side-coupled chains $n=\mp 1$ start to support unbound solutions for the case $\epsilon =0.5J$.

Figure 2

Exact results for transmission coefficient $T$ as a function of amplitude $\mu$ and frequency $\omega$ for an incoming wave of energy $\epsilon =0.5J$. Top: The solid lines (blue) indicate the exact values of the resonances ($T=0$) while the dashed lines are analytical expressions from the first zeros of Bessel functions ${\mathcal{J}}_{\pm \epsilon /\omega }$ in Eq. (13) at large frequencies and the small $\mu$ approximations in Eq. (14). Bottom: Behavior close to the critical frequency $\omega \approx {\omega }_c=2.5J$. For $\omega ={\omega }_c$ the transmission approaches $T\to (2J-|\epsilon \left|\right)/8J$ as $\mu \to 0$.

Figure 3

Transmission coefficient as a function of incoming particle energy $\epsilon$ at $\omega =3J$ (top) and $\omega =10J$ (bottom). The dashed lines in the bottom plot depict the high-frequency approximation in Eq. (13). The inset shows an enlarged region of the resonance for $\mu =25J$.