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Physical Review A

Optimal control of strongly driven quantum systems: Fully variational formulation and nonlinear eigenfields

K. G. Kim, Marvin D. Girardeau

Published Aug 1, 1995• 16 citations
Publisher

Abstract

The theory of optimal control of quantum systems is combined with a variational method for quantum-statistical dynamics to facilitate efficient nonperturbative calculations. The linear eigenfield approach is extended to the nonlinear regime to treat saturation and other higher-order effects. The application to a three-level \ensuremath{\Lambda} system generates, via an efficient iterative scheme, an eigenfield producing optimal constructive interference between direct and three-step excitation paths.

DOI
10.1103/physreva.52.r891

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The theory of optimal control of quantum systems is combined with a variational method for quantum-statistical dynamics to facilitate efficient nonperturbative calculations. The linear eigenfield approach is extended to the nonlinear regime to treat saturation and other higher-order effects. The application to a three-level \ensuremath{\Lambda} system generates, via an efficient iterative scheme, an eigenfield producing optimal constructive interference between direct and three-step excitation paths.
This paper explores the integration of nonlinear control theory with quantum system modeling, presenting a novel approach to designing and optimizing control strategies for quantum systems. The core claim centers on leveraging the optimization techniques within nonlinear control theory to address the inherent complexities of quantum system dynamics. The mechanism involves formulating quantum systems' nonlinear dynamics into equivalent control problems, subsequently employing nonlinear control theory's optimization methods for solution. This combined approach promises enhanced precision and efficiency in quantum system control. The work demonstrates a theoretical framework for tackling this challenge, paving the way for future research and practical applications in areas such as quantum computing and quantum sensing. The key mathematical concepts utilized are presented, including Hamiltonian formalism, Liouville-Perron equation, and nonlinear control theory formulations. The final results offer a pathway to more effective manipulation of quantum states.
A Markovian master equation describing the evolution of open quantum systems in the presence of a time-dependent external field is derived within the Bloch-Redfield formalism. It leads to a system-bath interaction which depends on the control field. Optimal control theory is used to select control fields which allow accelerated or decelerated system relaxation, or suppression of relaxation (dissipation) altogether, depending on the dynamics we impose on the quantum system. The control-dissipation correlation and the nonperturbative treatment of the control field are essential for reaching this goal. The optimal control problem is formulated within Pontryagin's minimum principle and the resulting optimal differential system is solved numerically. As an application, we study the dynamics of a spin-boson model in the strong coupling regime under the influence of an external control field. We show how trapping the system in unstable quantum states and transfer of population can be achieved by optimized control of the dissipative quantum system. We also used optimal control theory to find the driving field that generates the quantum ZZZ gate. In several cases studied, we find that the selected optimal field which reduces the purity loss significantly is a multicomponent low-frequency field including higher harmonics, all of which lie below the phonon cutoff frequency. Finally, in the undriven case we present an analytic result for the Lamb shift at zero temperature.
High-harmonic generation (HHG) is a prototypical strong-field process in which intense light drives matter to emit radiation at integer multiples of the driving frequency. Extending HHG into the quantum-optical regime offers new opportunities to probe and control strongly nonlinear light-matter interactions using nonclassical states of light. Yet describing this regime requires a fully quantum treatment of the correlated electron-photon dynamics, which becomes computationally challenging for broadband, strongly squeezed fields. Here we solve the quantum-electrodynamical dynamics of a two-level system driven by bright squeezed vacuum in a converged multimode Hilbert space. Both the driving field and emitted harmonics are fully quantized, with the light-matter interaction treated nonperturbatively. This enables direct access to the multimode quantum state and its higher-order correlations beyond semiclassical sampling or perturbative descriptions. We show that squeezed-vacuum driving produces harmonic emission with qualitatively distinct second- and third-order photon correlations compared with coherent excitation. Moreover, back-action from the driven emitter strongly reshapes the incident squeezed field, generating pronounced Wigner-function negativities that evolve on attosecond timescales. Our results establish a fully quantum framework for broadband strong-field dynamics with squeezed light and provide a route to predicting and interpreting quantum-HHG experiments and their extension to more complex emitters.
This work considers optimal control of quantum-mechanical systems within the framework of perturbation theory with respect to the controlling optical electric field. The control problem is expressed in terms of a cost functional including the physical objective, the penalties, and constraints. The resultant nonlinear variational equations are linearized by considering the lowest-order term in an expansion in powers of the optical-field strength. The optical field is found to satisfy a linear integral equation, and the solution may be expressed in terms of a generalized eigenvalue problem associated with the corresponding kernel. A full determination of the field is specified through the solution to the integral equation and the roots of an accompanying linearized spectral equation for a characteristic multiplier parameter. Each discrete value of the latter parameter corresponds to a particular solution to the variational equations. As a result, it is argued that under very general conditions there will be a denumerably infinite number of solutions to well-posed quantum-mechanical optimal-control problems.
The accurate description of dynamics in complex quantum and classical systems presents a formidable challenge, particularly when dealing with phenomena involving strong interactions, non-adiabatic transitions, or chaotic behavior. Traditional methods often struggle with either computational cost or insufficient accuracy in such regimes. This paper introduces and rigorously develops an Optimized Variational-Perturbation-WKB Integration (VPWKB) method designed to achieve high precision in the propagation of dynamics within these challenging systems. By synergistically combining the strengths of variational principles for robust global optimization, perturbation theory for handling intricate interactions and higher-order corrections, and the Wentzel-Kramers-Brillouin (WKB) approximation for efficient semiclassical trajectory generation, the VPWKB framework offers a powerful new approach. The method leverages variational techniques to determine optimal initial conditions or trial functions, subsequently refines the dynamics through a controlled perturbative expansion around a WKB-derived semiclassical path, and integrates these components into a coherent, self-consistent propagation scheme. This optimization significantly enhances the accuracy and stability of semiclassical calculations, enabling a more faithful representation of quantum effects and non-classical trajectories in complex landscapes. We discuss the theoretical underpinnings, computational advantages, and potential applications of VPWKB, demonstrating its capability to provide precision dynamics where conventional techniques falter.