Instantaneous Stability and Robustness Investigations in Quantum Optimal Control: Harmonic Oscillator Under Linear Dipole and Quadratic Control Agents

Abstract

Summary: We have investigated the stability and robustness of the optimal control solutions to a quantum system when the control duration goes to zero. The solutions at this limit are called ``instantaneous solutions''. These investigations are based on the second variation of the cost functional evaluated at control solution values when the first variations of wave and costate functions are related to the first variation of external field amplitude via control equations. This form of cost functional's second variation is purely quadratic in the first variation of the external field amplitude. Investigations are conducted for an illustrative model system, one dimensional quantum harmonic oscillator under linear dipole interaction, purely quadratic objective operator in position, and purely quadratic penalty operator in momentum. We have not found the stability operator's spectrum explicitly. Instead we have employed a bound analysis to understand the system's stability and robustness.