Accelerating the analysis of optical quantum systems using the Koopman operator

The prediction of photon echoes is an important technique for gaining an understanding of optical quantum systems. However, this requires a large number of simulations with varying parameters and/or input pulses, which renders numerical studies expensive. This article investigates how we can use data-driven surrogate models based on the Koopman operator to accelerate this process. In order to be successful, we require a model that is accurate over a large number of time steps. To this end, we employ a bilinear Koopman model using extended dynamic mode decomposition and simulate the optical Bloch equations for an ensemble of inhomogeneously broadened two-level systems. Such systems are well suited to describe the excitation of excitonic resonances in semiconductor nanostructures, for example, ensembles of semiconductor quantum dots. We perform a detailed study on the required number of system simulations such that the resulting data-driven Koopman model is sufficiently accurate for a wide range of parameter settings. We analyze the L2 error and the relative error of the photon echo peak and investigate how the control positions relate to the stabilization. After proper training, the dynamics of the quantum ensemble can be predicted accurately and numerically very efficiently by our methods.