Quantum Optimal Control Theory in the Linear Response Formalism

TL;DR

This paper reformulates quantum optimal control theory using linear response and Keldysh formalism, enabling the use of diagrammatic techniques for complex many-electron systems.

Contribution

It introduces a straightforward derivation of gradients in QOCT via linear response theory, extending to time-dependent Hamiltonians and connecting with non-equilibrium Green's functions.

Findings

Gradients in QOCT can be derived using linear response theory.

The formalism allows application of diagrammatic techniques to many-electron problems.

Reformulation facilitates computational approaches for complex quantum systems.

Abstract

Quantum optimal control theory (QOCT) aims at finding an external field that drives a quantum system in such a way that optimally achieves some predefined target. In practice this normally means optimizing the value of some observable, a so called merit function. In consequence, a key part of the theory is a set of equations, which provides the gradient of the merit function with respect to parameters that control the shape of the driving field. We show that these equations can be straightforwardly derived using the standard linear response theory, only requiring a minor generalization -- the unperturbed Hamiltonian is allowed to be time-dependent. As a result, the aforementioned gradients are identified with certain response functions. This identification leads to a natural reformulation of QOCT in term of the Keldysh contour formalism of the quantum many-body theory. In particular, the gradients of the merit function can be calculated using the diagrammatic technique for non-equilibrium Green's functions, which should be helpful in the application of QOCT to computationally difficult many-electron problems.