Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation
Consider a system governed by the time-dependent Schrödinger equation in its ground state. When subjected to weak (size $ε$) parametric forcing by an "ionizing field" (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay-rate of this metastable state is governed by {\it Fermi's Golden Rule}, $Γ[V]$, which depends on the potential $V$ and the details of the forcing. We pose the potential design problem: find $V_{opt}$ which minimizes $Γ[V]$ (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior points of the constraint set and to be smooth. The multi-scale structures that emerge incorporate the physical mechanisms of energy confinement via material contrast and interference effects. An analysis of locally optimal potentials reveals local optimality is attained via two mechanisms: (i) decreasing the density of states near a resonant frequency in the continuum and (ii) tuning the oscillations of extended states to make $Γ[V]$, an oscillatory integral, small. Our approach achieves lifetimes, $\sim (ε^2Γ[V])^{-1}$, for locally optimal potentials with $Γ^{-1}\sim\mathcal{O}(10^{9})$ as compared with $Γ^{-1}\sim \mathcal{O}(10^{2})$ for a typical potential. Finally, we explore the performance of optimal potentials via simulations of the time-evolution.