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Global approximate controllability of quantum systems by form perturbations and applications

Aitor Balmaseda, Davide Lonigro, Juan Manuel Pérez-Pardo

TL;DR

This work develops a rigorous framework for the global approximate controllability of infinite-dimensional quantum systems driven by form perturbations of a drift Hamiltonian, including singular perturbations such as point interactions. By extending Lie–Galerkin methods to form-bilinear settings and establishing explicit $L^1$ control bounds, the authors show that approximate controllability holds for initial and target states in broad subspaces and, under compact perturbations, for arbitrary states as well. The core strategy combines well-posedness and stability results for time-dependent forms with regular approximations to transfer controllability from regular (operator) systems to form-based perturbations. A key application demonstrates approximate controllability of a quantum particle in a one-dimensional box with a tunable center delta interaction, highlighting the practical impact for modeling impurities and localized perturbations in quantum devices.

Abstract

We provide sufficient conditions for the approximate controllability of infinite-dimensional quantum control systems corresponding to form perturbations of the drift Hamiltonian modulated by a control function. We rely on previous results on controllability of quantum bilinear control systems and obtain a priori $L^1$-bounds of the controls for generic initial and target states. We apply a stability result for the non-autonomous Schrödinger equation to extend the results to systems defined by form perturbations, including singular perturbations. As an application of our results, we prove approximate controllability of a quantum particle in a one-dimensional box with a point-interaction with tuneable strength at the centre of the box.

Global approximate controllability of quantum systems by form perturbations and applications

TL;DR

This work develops a rigorous framework for the global approximate controllability of infinite-dimensional quantum systems driven by form perturbations of a drift Hamiltonian, including singular perturbations such as point interactions. By extending Lie–Galerkin methods to form-bilinear settings and establishing explicit control bounds, the authors show that approximate controllability holds for initial and target states in broad subspaces and, under compact perturbations, for arbitrary states as well. The core strategy combines well-posedness and stability results for time-dependent forms with regular approximations to transfer controllability from regular (operator) systems to form-based perturbations. A key application demonstrates approximate controllability of a quantum particle in a one-dimensional box with a tunable center delta interaction, highlighting the practical impact for modeling impurities and localized perturbations in quantum devices.

Abstract

We provide sufficient conditions for the approximate controllability of infinite-dimensional quantum control systems corresponding to form perturbations of the drift Hamiltonian modulated by a control function. We rely on previous results on controllability of quantum bilinear control systems and obtain a priori -bounds of the controls for generic initial and target states. We apply a stability result for the non-autonomous Schrödinger equation to extend the results to systems defined by form perturbations, including singular perturbations. As an application of our results, we prove approximate controllability of a quantum particle in a one-dimensional box with a point-interaction with tuneable strength at the centre of the box.
Paper Structure (16 sections, 19 theorems, 101 equations)

This paper contains 16 sections, 19 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Basic notions
  4. Form bilinear control systems
  5. Main results
  6. Dynamics and stability of form bilinear control systems
  7. Time-dependent Hamiltonians with constant form domain
  8. Dynamics of form bilinear control systems
  9. Approximations of form bilinear systems
  10. Compact form perturbations
  11. Controllability of the bilinear Schrödinger equation with L1-bounded controls
  12. A theorem on the bilinear Schrödinger equation
  13. Preliminary results
  14. Proof of Theorem \ref{['thm:bounded_control_span']}
  15. Proof of Theorems \ref{['thm:main1']}--\ref{['thm:compact']}
  16. ...and 1 more sections

Key Result

Theorem 2.12

Let $(H_0, h_1,r)$ be a form bilinear control system satisfying Assumption assump:form and admitting a regular approximating family (cf. Definition def:approx_family). For every $m\in\mathbb{N}$, $\varepsilon > 0$, $r > 0$, and $\Psi_0, \Psi_1 \in \mathrm{span}\left\{\Phi_j \mid 1 \leq j \leq m\righ and such that where $U_u(t,s)$, $t,s\in[0,T_u]$, is the piecewise weak solution of the Schrödinger

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Unitary propagator
  • Definition 2.4: Weak and piecewise weak solution of the Schrödinger equation
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.9
  • Definition 2.10
  • Remark 2.11
  • ...and 39 more