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

Abstract

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 to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localisations of the momentums 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 nonstochastic Schrödinger equation only in the mean field approximation, whereas the central limit yelds Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.1

Mixing quantum and classical mechanics and uniqueness of Planck's constant

“…Peres and Temo [15] develop a hybrid formalism by taking recourse to the Koopman operator representation of classical Hamiltonians and conclude that the correspondence principle is violated due to the interaction. Belavkin and his collaborators [16] develop a stochastic Hamiltonian theory for coupling of a quantum system with an apparatus. In this approach, attention is focussed on providing purely dynamical arguments to derive entanglement, decoherence and collapse of the coupled system consisting of a quantum system and a (semi-classical) apparatus.…”

Quantum causality, stochastics, trajectories and information

“…where ψ = ψ|ψ 1/2 (see also [85] for the infinite-dimensional case). It corresponds to the nonlinear stochastic Master-equation…”

“…Hence the quantum jumps, decoherence and spontaneous localization are simply derived from this dynamical model as the results of inference, or quantum filtering without the projection postulate by simple conditioning. The statistical equivalence of the nondemolition countings N for an unstable system corresponding to R = 0 and C † C ≤ I was proved in the interaction representation picture in [97,85].…”

“…Thus, the quantum jumps problem has been solved as the time-continuous quantum boundary-value problem formulated as Given a reduction family Note that despite strong continuity of the unitary group evolutions T t and U t , the interaction evolution U (t, ω) is time-discontinuous for each ω. It is defined as U (t) = I 0] ⊗ W (t) for any positive t by the stochastic evolution W (t) = W t] 0 ⊗ I t resolving the Schrödinger unitary jump equation [97,85] …”

“…are equal to the square root of the flow intensity ν. The operators G i + = ϕ i Q i and G − k = Q k ϕ k corresponding to the different couplings λ k can also be obtained from the purely jump model in the central limit ν → ∞ as it was done in [85]. In this case…”