An optimal design problem for a charge qubit

TL;DR

This paper introduces a variational model for a superconducting charge qubit's ground state, formulating a shape optimization problem to maximize qubit states, revealing different optimal shapes depending on charge size.

Contribution

It presents a novel shape optimization framework for charge qubits, analyzing the existence and shape of optimal configurations for different charge regimes.

Findings

Small charge optimal shapes are nearly spherical and exist.

Large charge regimes likely lack optimal shapes, favoring disjoint sets.

Balls are not optimal for large charges.

Abstract

In this paper we introduce a simple variational model describing the ground state of a superconducting charge qubit. The model gives rise to a shape optimization problem that aims at maximizing the number of qubit states at a given gating voltage. We show that for small values of the charge optimal shapes exist and are $C^{2,\alpha}$-nearly spherical sets. In contrast, we prove that balls are not minimizers for large values of the charge and conjecture that optimal shapes do not exist, with the energy favoring disjoint collections of sets.