Quantum Optimal Control of a Lambda System in the Density Matrix Formulation

TL;DR

This paper develops a geometric control framework for optimizing state transfer in a quantum Lambda system, balancing control energy and population in the excited state, with theoretical properties and numerical case studies.

Contribution

It introduces a novel geometric approach combining Pontryagin's maximum principle and symmetry reduction for quantum control of Lambda systems, with proven properties and differential equations for optimal solutions.

Findings

Optimal controls are normal and smooth.

Derived differential equations characterize optimal trajectories.

Numerical simulations demonstrate the approach on a Hadamard-like transformation.

Abstract

In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.