Spectral analysis of the Koopman operator recovers Hamiltonian parameters in open quantum systems

TL;DR

The paper addresses parameter identification for open quantum systems by introducing mHAVOK, a spectral, data-driven method tied to the Koopman operator. By leveraging delay-embedded observations and a forced linear model on a reduced subspace, it extracts Hamiltonian parameters from the spectrum of a finite-dimensional Koopman-restricted operator. Through simulations of a 2D quantum harmonic oscillator with Kerr nonlinearities, Jaynes-Cummings coupling, and time-dependent modulation, the method achieves high accuracy (often <5%) and outperforms traditional estimators under strong damping, while also revealing limitations in cross-term Kerrs and large modulation regimes. The work offers a principled, interpretable framework for quantum-dynamical parameter identification and motivates further Koopman-based approaches in open quantum systems.

Abstract

Accurate identification of Hamiltonian parameters is essential for modeling and controlling open quantum systems. In this work, we demonstrate that the multichannel Hankel alternative view of Koopman (mHAVOK) algorithm is a robust and reliable spectral data-driven method for retrieving Hamiltonian parameters from the evolution of first-moment observables in open quantum systems. The method relies on the discrete spectrum of the Koopman operator to obtain these parameters, which are computed using the mHAVOK algorithm; a theoretical connection to this affirmation is presented. The method is tested on noiseless quadratures of an open two-dimensional quantum harmonic oscillator and shown to retrieve oscillation frequencies, damping rates, nonlinear Kerr shifts, the qubit-photon coupling strength of a Jaynes-Cummings interaction, and the modulated frequency of a time-dependent Hamiltonian. The majority of the recovered parameters remained within 5\% of their actual values. Compared with Fourier and matrix-pencil estimators, our approach yields lower errors for dynamics with strong dissipation. Overall, these findings suggest that Koopman operator theory provides a practical framework for studying quantum dynamical systems.

Figures (8)

• Figure 1: Overview of the pipeline employed in this work. (1) The expected values of observables are first obtained by simulating the Lindblad master equation. (2) Such data is then analyzed through the mHAVOK framework, where two dynamical matrices are retrieved. A spectral analysis is performed on one of them (3), allowing recovery of the Koopman generator's discrete spectrum and, thus, the Hamiltonian parameters of the open quantum system (4).
• Figure 2: mHAVOK phase-space reconstruction of an open 2D QHO for the $x$ and $y$ components using a damping rate of (a) $\kappa=0.1$ and (b) $\kappa=1.0$. No nonlinearities were introduced.
• Figure 3: Percent error comparison among three different methods for determining the oscillation frequencies $\hat{\omega}_i$ of the 2D QHO. mHAVOK exhibits a slower increase in percent error compared to existing alternatives.
• Figure 4: Phase-space reconstruction considering (a-c) $\chi_x,\chi_y=2,3$ and (b-d) $\chi_x=\chi_y=5$. In both cases, the framework successfully recovered Kerr strengths with few numerical errors. No cross-coupling terms were included.
• Figure 5: Error comparison for a varying of Kerr nonlinear strengths. The percent error was below 5%, even for strong nonlinearities. To achieve this, the optimal rank $r_{opt}$ was calculated and considered for each simulation.