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An optimal design problem for a charge qubit

Dario Mazzoleni, Cyrill B. Muratov, Berardo Ruffini

TL;DR

The paper studies a shape optimization problem for the ground-state Hartree energy $E_q(\Omega)$, balancing a Dirichlet (kinetic) term against a nonlocal Coulomb-like repulsion for a fixed-volume superconducting island. Using a multi-layer variational framework, it establishes existence and $C^{2,\alpha}$-regularity of minimizers for small charge $q$, showing they are $C^{2,\alpha}$-nearly spherical; a free boundary approach yields Lipschitz regularity and sharp boundary structure. A surgery technique removes the equiboundedness restriction, enabling a global existence result and a detailed qualitative picture, including a large-$q$ regime where the ball ceases to be optimal and minimizers, if they exist, must have diameter at least on the order of $q^{1/2}$. The work combines nonlinear nonlocal PDE techniques with quantitative isoperimetric tools (Fraenkel asymmetry and quantitative Faber–Krahn) to derive regularity, shape rigidity, and asymptotic behavior of minimizing domains, with implications for the physics of charge qubits. These results illuminate how island geometry can influence the spectrum and stability of Cooper-pair states in nanoscale superconducting devices.

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 $C^{2,α}$-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.

An optimal design problem for a charge qubit

TL;DR

The paper studies a shape optimization problem for the ground-state Hartree energy , balancing a Dirichlet (kinetic) term against a nonlocal Coulomb-like repulsion for a fixed-volume superconducting island. Using a multi-layer variational framework, it establishes existence and -regularity of minimizers for small charge , showing they are -nearly spherical; a free boundary approach yields Lipschitz regularity and sharp boundary structure. A surgery technique removes the equiboundedness restriction, enabling a global existence result and a detailed qualitative picture, including a large- regime where the ball ceases to be optimal and minimizers, if they exist, must have diameter at least on the order of . The work combines nonlinear nonlocal PDE techniques with quantitative isoperimetric tools (Fraenkel asymmetry and quantitative Faber–Krahn) to derive regularity, shape rigidity, and asymptotic behavior of minimizing domains, with implications for the physics of charge qubits. These results illuminate how island geometry can influence the spectrum and stability of Cooper-pair states in nanoscale superconducting devices.

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 -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.
Paper Structure (16 sections, 31 theorems, 237 equations)

This paper contains 16 sections, 31 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Main results and detailed strategy of their proof
  3. Preliminaries
  4. Minimization of $E_q(u, \Omega)$ in $u$.
  5. Fraenkel asymmetry and quantitative Faber-Krahn inequality
  6. Some facts about quasi-open sets
  7. An existence result for an auxiliary problem
  8. Removing the $L^2$ norm constraint in $E_q$
  9. Another auxiliary problem: removing the volume constraint
  10. Free boundary formulation
  11. Equivalence between the minimizations of $E_q$ and $E_{q,M,\eta}$
  12. Optimality conditions and improvement of flatness
  13. The structure of $\mu_u$: blow up limits
  14. The surgery result and the proof of Theorem \ref{['thm:main']}
  15. The case of $q$ large: proof of Theorem \ref{['thm:nomin']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

For all $\varepsilon>0$ there exists $q^*=q^*(\varepsilon)>0$ and $\alpha\in (0,1)$ such that, for all $0 < q \leq q^*$, there exists an optimal set for problem eq:minmin1. Furthermore, every optimal set $\Omega$ is $C^{2,\alpha}$-nearly spherical, namely, there is a function $\varphi_\varepsilon\co

Theorems & Definitions (75)

  • Remark 1.1: The shape optimization viewpoint
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 65 more