A stochastic Hamiltonian approach for quantum jumps, spontaneous localizations, and continuous trajectories

We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with 'bubbles' which admit a continual counting observation. This model allows one to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localizations of the momenta of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e. a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms, the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the non-stochastic Schrodinger equation only in the mean-field approximation, whereas the central limit yields Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.