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Ensemble control of n-level quantum systems with a scalar control

Ruikang Liang, Ugo Boscain, Mario Sigalotti

TL;DR

The paper addresses ensemble controllability for a continuum of $n$-level quantum systems driven by a single scalar input, where the system Hamiltonian $H(\alpha)$ depends on uncertain parameters. It develops a rigorous control strategy that cascades rotating wave and adiabatic approximations, aided by high-order averaging, to realize population inversion between arbitrary eigenstates across parameter dispersion. An explicit control law is provided, with finite-time fidelity bounds that improve under stronger resonance separation; a concrete 4-level example demonstrates both feasibility and the sharpness of the required hypotheses. The work extends ensemble control methods to multi-level systems with scalar control under dispersion, with potential impact on robust quantum state manipulation in heterogeneous quantum platforms.

Abstract

In this paper we discuss how a general bilinear finite-dimensional closed quantum system with dispersed parameters can be steered between eigenstates. We show that, under suitable conditions on the separation of spectral gaps and the boundedness of parameter dispersion, rotating wave and adiabatic approximations can be employed in cascade to achieve population inversion between arbitrary eigenstates. We propose an explicit control law and test numerically the sharpness of the conditions on several examples.

Ensemble control of n-level quantum systems with a scalar control

TL;DR

The paper addresses ensemble controllability for a continuum of -level quantum systems driven by a single scalar input, where the system Hamiltonian depends on uncertain parameters. It develops a rigorous control strategy that cascades rotating wave and adiabatic approximations, aided by high-order averaging, to realize population inversion between arbitrary eigenstates across parameter dispersion. An explicit control law is provided, with finite-time fidelity bounds that improve under stronger resonance separation; a concrete 4-level example demonstrates both feasibility and the sharpness of the required hypotheses. The work extends ensemble control methods to multi-level systems with scalar control under dispersion, with potential impact on robust quantum state manipulation in heterogeneous quantum platforms.

Abstract

In this paper we discuss how a general bilinear finite-dimensional closed quantum system with dispersed parameters can be steered between eigenstates. We show that, under suitable conditions on the separation of spectral gaps and the boundedness of parameter dispersion, rotating wave and adiabatic approximations can be employed in cascade to achieve population inversion between arbitrary eigenstates. We propose an explicit control law and test numerically the sharpness of the conditions on several examples.
Paper Structure (7 sections, 12 theorems, 110 equations, 2 figures)

This paper contains 7 sections, 12 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Sketch of the proof
  4. Proof of Theorem \ref{['theorem:main']}
  5. Example
  6. Conclusion
  7. Appendix

Key Result

Theorem 1

Let us assume that for all $1\leq j<k\leq n$, and for all $\alpha\in \mathcal{D}$, $\lambda_{k}(\alpha)-\lambda_{j}(\alpha)>0$. Fix $1\leq p<q\leq n$. Assume that $\delta_{pq}$ belongs to a closed interval $\mathcal{I}_{pq}=\left[\delta_{pq}^{0},\delta_{pq}^{1}\right]$ such that $0\notin\mathcal{I}_ Fix $T>0$ and take $u,f\in\mathcal{C}^{2}([0,T],\mathbb{R})$ such that Denote by $\psi_{\epsilon_{

Figures (2)

  • Figure 1: Simulations for $\epsilon_1=10^{-5/3}$, $\epsilon_2=10^{-7/3}$, and $\alpha\in\{-0.6,-0.3,-0.1,0.1,0.3\}$: To better illustrate the convergence, we also show the logarithmic convergence of $1-\text{fid}(s)$, specifically $\log(1-\text{fid}(s))/\log10$. The convergence is only guaranteed when the hypotheses of Theorem \ref{['theorem:main']} are satisfied. When $\lambda_{4}(\alpha)-\lambda_{3}(\alpha)$ is not in $(v_0,v_1)$, the population inversion between the third and the fourth eigenstates does not occur at all (see the blue curve). When $\lambda_{3}(\alpha)-\lambda_{1}(\alpha)$ is in $[v_0,v_1]$, the fidelity stays quite far from 1 and a total population inversion is not accomplished (see the purple and red curves).
  • Figure 2: Simulations for $\alpha=-0.1$, $\epsilon_{1}=10^{-5/3},\epsilon_{2}=10^{-7/3}$: Population inversions between $(\textbf{e}_1,\textbf{e}_2)$, $(\textbf{e}_2,\textbf{e}_3)$, and $(\textbf{e}_3,\textbf{e}_4)$ happen successively.

Theorems & Definitions (29)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • Proposition 5: Change of variables
  • proof
  • Definition 6
  • Remark 7
  • Remark 8
  • Lemma 9
  • ...and 19 more