Vp. Belavkin et O. Melsheimer, A STOCHASTIC HAMILTONIAN APPROACH FOR QUANTUM JUMPS, SPONTANEOUS LOCALIZATIONS, AND CONTINUOUS TRAJECTORIES, Quantum and semiclassical optics, 8(1), 1996, pp. 167-187
We give an explicit stochastic Hamiltonian model of discontinuous unit
ary 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 photodetect
or or in a cloud chamber by spontaneous localizations of the momenta o
f the scattered photons or bubbles. Thus, the continuous reduction and
spontaneous localization theory is obtained from a Hamiltonian singul
ar interaction as a result of quantum filtering, i.e. a sequential tim
e continuous conditioning of an input quantum process by the output me
asurement data. We show that in the case of indistinguishable particle
s or atoms, the a posteriori dynamics is mixing, giving rise to an irr
eversible Boltzmann-type reduction equation. The latter coincides with
the non-stochastic Schrodinger equation only in the mean-field approx
imation, whereas the central limit yields Gaussian mixing fluctuations
described by a quantum state reduction equation of diffusive type.