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Finite-time stabilization of ladder multi-level quantum systems

Zeping Su, Sen Kuang, Daoyi Dong

TL;DR

The paper tackles the problem of finite-time stabilization of $n$-level ladder quantum systems to an eigenstate of the intrinsic Hamiltonian under $n-1$ external controls. It designs a universal fractional-order, continuous non-smooth controller based on the Lyapunov function $V = 1 - |\langle \psi_f | \psi \rangle|^2$ and analyzes the non-Lipschitz closed-loop dynamics via Filippov differential inclusions and LaSalle's invariance principle. The authors prove global asymptotic stability and, crucially, finite-time convergence to the target eigenstate, deriving an explicit bound on the convergence time $T_f$. Numerical validation on a three-level rubidium ladder demonstrates finite-time convergence and highlights reduced chattering and faster convergence compared with standard Lyapunov and bang-bang strategies, underscoring practical applicability to high-dimensional quantum control.

Abstract

In this paper, a novel continuous non-smooth control strategy is proposed to achieve finite-time stabilization of ladder quantum systems. We first design a universal fractional-order control law for a ladder n-level quantum system using a distance-based Lyapunov function, and then apply the Filippov solution in the sense of differential inclusions and the LaSalle's invariance principle to prove the existence and uniqueness of the solution of the ladder system under the continuous non-smooth control law. Both asymptotic stability and finite-time stability for the ladder system is rigorously established by applying Lyapunov stability theory and finite-time stability criteria. We also derive an upper bound of the time required for convergence to an eigenstate of the intrinsic Hamiltonian. Numerical simulations on a rubidium ladder three-level atomic system validate the effectiveness of the proposed method.

Finite-time stabilization of ladder multi-level quantum systems

TL;DR

The paper tackles the problem of finite-time stabilization of -level ladder quantum systems to an eigenstate of the intrinsic Hamiltonian under external controls. It designs a universal fractional-order, continuous non-smooth controller based on the Lyapunov function and analyzes the non-Lipschitz closed-loop dynamics via Filippov differential inclusions and LaSalle's invariance principle. The authors prove global asymptotic stability and, crucially, finite-time convergence to the target eigenstate, deriving an explicit bound on the convergence time . Numerical validation on a three-level rubidium ladder demonstrates finite-time convergence and highlights reduced chattering and faster convergence compared with standard Lyapunov and bang-bang strategies, underscoring practical applicability to high-dimensional quantum control.

Abstract

In this paper, a novel continuous non-smooth control strategy is proposed to achieve finite-time stabilization of ladder quantum systems. We first design a universal fractional-order control law for a ladder n-level quantum system using a distance-based Lyapunov function, and then apply the Filippov solution in the sense of differential inclusions and the LaSalle's invariance principle to prove the existence and uniqueness of the solution of the ladder system under the continuous non-smooth control law. Both asymptotic stability and finite-time stability for the ladder system is rigorously established by applying Lyapunov stability theory and finite-time stability criteria. We also derive an upper bound of the time required for convergence to an eigenstate of the intrinsic Hamiltonian. Numerical simulations on a rubidium ladder three-level atomic system validate the effectiveness of the proposed method.
Paper Structure (8 sections, 4 theorems, 34 equations, 4 figures)

This paper contains 8 sections, 4 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model and Control Problem
  3. Finite-Time Controller Design
  4. Finite-Time Stability Analysis
  5. Asymptotic Stability
  6. Finite-time Stability
  7. Numerical Simulations
  8. Conclusion

Key Result

Lemma 1

Suppose that $\sum_{j=1}^{n}{r_{j}^{2}}=1$, where ${{r}_{j}}\ge 0$. Then, for $\forall \alpha \in (0,1)$, the following inequality holds

Figures (4)

  • Figure 1: Configuration of a there-level ladder rubidium atom, where $5S_{1/2}$, $5P_{3/2}$, and $5D_{5/2}$ correspond to $|0\rangle$, $|1\rangle$, and $|2\rangle$, respectively, and the control fields $u_1$ and $u_2$ are exerted to the system shown in the figure.
  • Figure 2: For the initial state $|\psi(0)\rangle = |0\rangle$ and the target state $|\psi_f\rangle = |2\rangle$, the population evolution curves of the ladder quantum system under the continuous non-smooth controls, the standard Lyapunov controls, and the standard bang-bang Lyapunov controls.
  • Figure 3: For $|\psi(0)\rangle = |0\rangle$, the continuous non-smooth controls $u_1$ and $u_2$, the standard Lyapunov controls $u_1^s$ and $u_2^s$, and the standard bang–bang Lyapunov controls $u_1^b$ and $u_2^b$.
  • Figure 4: For the initial state $| \psi (0) \rangle = [ \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2} ]^T$ and the target state $| \psi_f \rangle = | 2 \rangle$, the population evolution curves and the continuous non-smooth controls $u_1$ and $u_2$ in the ladder quantum system.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4