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Approximation of diffeomorphisms for quantum state transfers

Eugenio Pozzoli, Alessandro Scagliotti

TL;DR

The paper addresses the problem of achieving approximate state transfers in bilinear Schrödinger PDEs on the torus by bridging infinite-dimensional diffeomorphism control with ensemble optimal control. It shows that small-time diffeomorphism reachability implies the ability to approximate $|DP|^{1/2}(\psi_0\circ P)$, and it recasts the task as a finite ensemble control problem via Gamma-convergence, enabling explicit control laws. Numerically, explicit controls are computed for transfers from the ground state to targeted eigenstates with low $L^2$ error, using a discretized ODE on $\mathbb{T}$ and a damped-BFGS solver. The approach provides a constructive path to quantum state transfer in molecular rotation and optical-lattice settings and demonstrates the practicality of turning existence proofs into implementable controls.

Abstract

In this paper, we seek to combine two emerging standpoints in control theory. On the one hand, recent advances in infinite-dimensional geometric control have unlocked a method for controlling (with arbitrary precision and in arbitrarily small times) state transfers for bilinear Schrödinger PDEs posed on a Riemannian manifold $M$. In particular, these arguments rely on controllability results in the group of the diffeomorphisms of $M$. On the other hand, using tools of $Γ$-convergence, it has been proved that we can phrase the retrieval of a diffeomorphism of $M$ as an ensemble optimal control problem. More precisely, this is done by employing a control-affine system for \emph{simultaneously} steering a finite swarm of points towards the respective targets. Here we blend these two theoretical approaches and numerically find control laws driving state transitions (such as eigenstate transfers) in small time in a bilinear Schrödinger PDE posed on the torus. Such systems have experimental relevance and are currently used to model rotational dynamics of molecules, and cold atoms trapped in periodic optical lattices.

Approximation of diffeomorphisms for quantum state transfers

TL;DR

The paper addresses the problem of achieving approximate state transfers in bilinear Schrödinger PDEs on the torus by bridging infinite-dimensional diffeomorphism control with ensemble optimal control. It shows that small-time diffeomorphism reachability implies the ability to approximate , and it recasts the task as a finite ensemble control problem via Gamma-convergence, enabling explicit control laws. Numerically, explicit controls are computed for transfers from the ground state to targeted eigenstates with low error, using a discretized ODE on and a damped-BFGS solver. The approach provides a constructive path to quantum state transfer in molecular rotation and optical-lattice settings and demonstrates the practicality of turning existence proofs into implementable controls.

Abstract

In this paper, we seek to combine two emerging standpoints in control theory. On the one hand, recent advances in infinite-dimensional geometric control have unlocked a method for controlling (with arbitrary precision and in arbitrarily small times) state transfers for bilinear Schrödinger PDEs posed on a Riemannian manifold . In particular, these arguments rely on controllability results in the group of the diffeomorphisms of . On the other hand, using tools of -convergence, it has been proved that we can phrase the retrieval of a diffeomorphism of as an ensemble optimal control problem. More precisely, this is done by employing a control-affine system for \emph{simultaneously} steering a finite swarm of points towards the respective targets. Here we blend these two theoretical approaches and numerically find control laws driving state transitions (such as eigenstate transfers) in small time in a bilinear Schrödinger PDE posed on the torus. Such systems have experimental relevance and are currently used to model rotational dynamics of molecules, and cold atoms trapped in periodic optical lattices.

Paper Structure

This paper contains 10 sections, 3 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Schrödinger equation
  3. An equation posed on $M={\mathbb T}={\mathbb R}/2\pi{\mathbb Z}$.
  4. Our contribution
  5. Structure of the paper
  6. From controlled Schrödinger equation to diffeomorphisms control
  7. Ensemble optimal control formulation
  8. Numerical experiments
  9. Evolution approximation
  10. Numerical computation of state transfers

Key Result

Theorem 2.1

For any $P\in {\rm Diff}^0({\mathbb T}),\psi_0\in L^2({\mathbb T},{\mathbb C})$, the state $|DP|^{1/2}(\psi_0\circ P)$ is small-time approximately reachable for eq:rotor from $\psi_0$.

Figures (2)

  • Figure 1: State transfer from the ground state $\psi_0 \equiv \frac{1}{\sqrt{2\pi}}$ to the target state $\psi_1'$ (see \ref{['eq:psi_cos_2x']}). We used a random initialization for the control $u\in {\mathcal{U}}^M$, and we performed $\sim 2.2\cdot 10^3$ iterations of the Damped-BFGS scheme ($\sim15$ mins of computational time). The resulting relative mismatch in the $L^2$-norm amounts at $0.0225$.
  • Figure 2: State transfer from the ground state $\psi_0 \equiv \frac{1}{\sqrt{2\pi}}$ to the target state $\psi_1"$ (see \ref{['eq:psi_cos_3x']}). We used a random initialization for the control $u\in {\mathcal{U}}^M$, and we performed $2.8\cdot 10^3$ iterations of the the Damped-BFGS scheme ($\sim20$ mins of computational time). The resulting relative mismatch in the $L^2$-norm amounts at $0.0227$.

Theorems & Definitions (13)

  • Definition 1
  • Theorem 2.1
  • proof
  • Remark 1
  • Proposition 3.1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 3.2: $\Gamma$-convergence
  • ...and 3 more