Controlling Ensemble States of a Quantum System

We develop a method for controlling the ensemble states of a two-level quantum system on the Bloch sphere. A spectrum of resonance frequencies parametrizes the ensemble. The evolution of the ensemble states under externally applied controls is uniquely determined by their respective resonance frequencies. Therefore, the control fields must be designed to simultaneously evolve the ensemble states across the domain of resonance frequencies. The objective is to evolve the discretized quantum ensemble states from an arbitrary initial dispersion to a targeted final configuration on the Bloch sphere. The proposed method holds potential for broad applicability in areas such as nuclear magnetic resonance (NMR), electron spin resonance (ESR), quantum computing and information processing, sampling-based learning control (SLC), quantum sensing and metrology, and quantum error correction.