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We give an explicit stochastic Hamiltonian model of discontinuous unitaryevolution for quantum spontaneous jumps like in a system of atoms in quantumoptics, or in a system of quantum particles that interacts singularly with"bubbles" which admit a continual counting observation. This model allows towatch a quantum trajectory in a photodetector or in a cloud chamber byspontaneous localisations of the momentums of the scattered photons or bubbles.Thus, the continuous reduction and spontaneous localization theory is obtainedfrom a Hamiltonian singular interaction as a result of quantum filtering, i.e.,a sequential time continuous conditioning of an input quantum process by theoutput measurement data. We show that in the case of indistinguishableparticles or atoms the a posteriori dynamics is mixing, giving rise to anirreversible Boltzmann-type reduction equation. The latter coincides with thenonstochastic Schroedinger equation only in the mean field approximation,whereas the central limit yelds Gaussian mixing fluctuations described by aquantum state reduction equation of diffusive type.

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