Topology of classical molecular optimal control landscapes in phase space

Abstract

Optimal control of molecular dynamics is commonly expressed from a quantum mechanical perspective. However, in most contexts the preponderance of molecular dynamics studies utilize classical mechanical models. This paper treats laser-driven optimal control of molecular dynamics in a classical framework. We consider the objective of steering a molecular system from an initial point in phase space to a target point, subject to the dynamic constraint of Hamilton's equations. The classical control landscape corresponding to this objective is a functional of the control field, and the topology of the landscape is analyzed through its gradient and Hessian with respect to the control. Under specific assumptions on the regularity of the control fields, the classical control landscape is found to be free of traps that could hinder reaching the objective. The Hessian associated with an optimal control field is shown to have finite rank, indicating the presence of an inherent degree of robustness to control noise. Extensive numerical simulations are performed to illustrate the theoretical principles on (a) a model diatomic molecule, (b) two coupled Morse oscillators, and (c) a chaotic system with a coupled quartic oscillator, confirming the absence of traps in the classical control landscape. We compare the classical formulation with the mathematically analogous quantum state-to-state transition probability control landscape.

Quantum feedback control of atomic ensembles and spinor Bose-Einstein condensates

“…where A is infinitesimal generator of ρ c . Note that E[S(ρ 2 t )] ≥ 0, hence from (32) we conclude that E[S(ρ c )] decreases monotonically. Therefore, S(ρ c ) converges to 0 as t goes to ∞.…”

“…Therefore, S(ρ c ) converges to 0 as t goes to ∞. But the only states ρ satisfying S(ρ) = 0 are the eigenstates of S z , which implies that the state ρ governed by (32) with H f = 0 must collapse onto one of the eigenstates of S z . From a physical point of view, this is the expected result as the PCI performs a measurement in the S z basis, and hence an arbitrary initial state is driven towards S z eigenstates.…”

“…A more general task is to target an arbitrary state given an arbitrary initial state, where the deviation is not necessarily small. Such tasks are central to quantum control theory [25][26][27][28][29][30], and have been successfully applied to atomic and molecular physics as well as physical chemistry [31,32]. These have been mainly analyzed for systems with small Hilbert space dimensions such as qubits [33,34], however, in the above context it is an important task to extend this towards systems involving many particles.…”

Inherently trap-free convex landscapes for full quantum optimal control

“…The characteristic control landscapes in the remaining two frameworks (i.e., C-C and C-Q) are rooted in the fundamental differences of quantum and classical mechanics. These additional landscapes are of theoretical and practical importance, with the C-C picture only partially explored to date [35], while the C-Q picture has yet to be physically defined. Since classical dynamics can be taken as a limiting process of quantum dynamics, the present Q-Q landscape analysis may provide a foundation for future research to draw together the full tetrad of classical and quantum mechanical control in a seamless fashion [16].…”

Control landscapes for a class of non-linear dynamical systems: sufficient conditions for the absence of traps

“…Following the approach of [74], successful controls were obtained using a gradient search based on the D-MORPH method [75]. This method is based on traversing a smoothly parameterized set of control fields by smoothly updating the control so as to increase the fidelity.…”