Koopman Operator and Path Integral of Quantum Free-Electron Laser Model

A quantum model of a free-electron laser (FEL) is considered. Two different approaches for the exploration of the the FEL system are considered. In the first case, the Heisenberg equations of motion are mapped on the basis of the initial wave functions, which consists of the photon coherent states and many-dimensional electron coherent states. This mapping is an exact procedure, which makes it possible to obtain an exact equation of motion for the intensity of the laser field in a closed form. The obtained equation is controlled by a Koopman operator. The analytical expression for the evolution of the FEL intensity is obtained in the framework of a perturbation theory, which is constructed for a small time scale. The second way of the consideration is based on the construction of the many-dimensional path integrals for the evolution of the wave function. This method also makes it possible to estimate the time evolution and the gain of the FEL intensity.