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The number process as low density limit of Hamiltonian models. (English) Zbl 0738.60099
Summary: We study a nonrelativistic quantum system coupled, via a quadratic interaction, to a free Boson gas in the Fock state. We prove that, in the low density limit (\(z^ 2\)=fugacity \(\to 0\)), the matrix elements of the wave operator of the system at time \(t/z^ 2\) in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation driven by some pure number process (i.e. no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes). This proves that the number (or quantum Poisson) processes, introduced by R. L. Hudson and K. R. Parthasarathy [ibid. 93, 301-323 (1984; Zbl 0546.60058)] and studied by A. Frigerio and H. Maassen [Probab. Theory Relat. Fields 83, No. 4, 489-508 (1989; Zbl 0684.60081)] arise effectively as conjectured by the latter two authors as low density limits of Hamiltonian models.

MSC:
60K40 Other physical applications of random processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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