Stark effect in spherical quantum dots

The Stark problem in quantum-confined geometries is challenging in high fields. In particular, the localization of electrons near sample boundaries is hard to accurately capture in perturbation methods. We analyze the problem for spherical quantum dots using a combination of numerical diagonalization and analytical perturbation approaches. Closed-form expressions for polarizabilities and hyperpolarizabilities of arbitrary states are obtained. In addition, a hypergeometric resummation ansatz replicating the correct high-field behavior is constructed. We find that a simple resummation approach is superior to even 14th-order perturbation series.

Figure 1

Nanosphere of radius $a$ perturbed by a vertical field $\vec{F}$.

Figure 2

Numerical diagonalization energy eigenvalues for several $m$ values.

Figure 3

Comparison of numerical eigenvalues (exact, circles) with low-order perturbation series (green and blue solid lines) for different $m$ quantum numbers.

Figure 4

Energy, polarizability, and hyperpolarizability for low-lying eigenstates as a function of state index $n$.

Figure 5

Comparison of exact diagonalization results (circles) with perturbative expansions (green, blue, and cyan) and hypergeometric ansatz (red line). The inset is a zoom of the low-field regime.