[Paper Review] Quantum Optimal Control Theory of Harmonic Generation

Ido Schaefer|arXiv (Cornell University)|Feb 29, 2012

Laser-Matter Interactions and Applications12 references3 citations

TL;DR

This master's thesis introduces a novel quantum optimal control theory (QOCT) framework for controlling harmonic generation by formulating a frequency-domain maximization functional with a frequency-dependent penalty factor. Using the relaxation method for optimization, the approach enables precise spectral shaping of the dipole moment response, achieving enhanced harmonic generation efficiency in model systems like two-level, three-level, and anharmonic oscillators, with key results showing improved control over target harmonic frequencies while suppressing unwanted spectral components.

ABSTRACT

A new method for controlling harmonic generation, in the framework of quantum optimal control theory (QOCT), is developed. The problem is formulated in the frequency domain using a new maximization functional. The relaxation method is used as the optimization procedure. The new formulation is generalized to other control problems with requirements in the frequency domain. The method is applied to several simple problems. The results are analysed and discussed. General conclusions on harmonic generation mechanisms are obtained.

Motivation & Objective

• To develop a new quantum optimal control theory (QOCT) framework for controlling harmonic generation in the frequency domain.
• To address the challenge of selectively enhancing specific harmonic frequencies while suppressing others in quantum systems.
• To generalize the method to control problems requiring spectral constraints, extending beyond standard time-domain QOCT.
• To analyze and understand the underlying mechanisms of harmonic generation in model quantum systems using the new control approach.
• To evaluate the performance and limitations of the method in realistic quantum systems, including anharmonic oscillators and multi-level systems.

Proposed method

• Formulates a new maximization functional in the frequency domain using a frequency-dependent penalty factor to constrain the forcing field spectrum.
• Applies the relaxation method as the primary optimization procedure, enabling efficient convergence in the frequency domain.
• Introduces a modified functional that directly controls the spectral content of the dipole moment expectation value ⟨μ^(t)⟩.
• Employs a discrete cosine transform (DCT-I) for accurate spectral representation and integration of time-dependent functions via Chebyshev collocation.
• Uses a Krotov-like iterative scheme and BFGS method as alternative optimization strategies, with convergence monitored via relative field difference tolerance.
• Implements a numerical propagator based on Chebyshev time-spectral methods with adaptive convergence tolerance to ensure accuracy.

Experimental results

Research questions

• RQ1 Can a frequency-domain formulation of QOCT effectively control harmonic generation by shaping the dipole moment spectrum?
• RQ2 How does the inclusion of a frequency-dependent penalty factor improve spectral selectivity in harmonic generation?
• RQ3 What are the performance and limitations of the relaxation method in optimizing fields for harmonic generation in multi-level and anharmonic systems?
• RQ4 How do system anharmonicity and level structure influence the efficiency and feasibility of targeted harmonic generation?
• RQ5 To what extent can the new method suppress unwanted spectral components while enhancing specific harmonic orders?

Key findings

• The new QOCT formulation successfully enhances target harmonic frequencies in two-level and three-level systems, with spectral selectivity improved by the frequency-dependent penalty factor.
• In the eleven-level system, the method achieves significant suppression of non-target harmonics, demonstrating scalability to larger systems.
• For the HCl molecule and Toda anharmonic oscillator, the method enables effective control over harmonic emission, with measurable enhancement in the target frequency components.
• The relaxation method converges reliably, with convergence tolerance set at τ = 10−3, and propagator tolerance at ζ = 10−3τ for numerical stability.
• Boundary effects and weak response regimes were identified as challenges, particularly when the target harmonic signal is small relative to the linear response.
• The method generalizes well to other frequency-domain control problems, offering a robust framework for spectral shaping in quantum control.