Pontryagin-optimal control of a non-Hermitian qubit

Abstract

Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a workaround for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity even in the case of inefficient detection.

Shortcuts to adiabaticity: theoretical framework, relations between different methods, and versatile approximations

“…Continuously monitored systems can also be viewed as open quantum systems. Enhancement of the quantum Zeno dragging was proposed by utilizing the idea of counterdiabatic driving [135].…”

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

“…An extension of the PMP was recently applied to derive time-optimal controls of two-level quantum systems with piecewise constant pulses [71]. The PMP was used to controlling a single non-Hermitian qubit similar to a system with an open spontaneous emission channel to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function balancing the desired dynamics against the controller energy [72]. It was applied to systematic optimal control study of lossy systems, to design alternative more efficient routes that, for a given admissible loss, feature an optimal transfer with respect to the cost defined as the pulse energy (energy minimization) or the pulse duration (time minimization) [73].…”