Abstract
The quantum stochastic differential equation of the inverse oscillator in a heat bath of oscillators is solved by the means of a calculus of continuous and differentiable Maassen kernels. It is shown that the time development operator does not only map the Hilbert space of the problem into itself, but also vectors with finite moments into vectors with finite moments. The vacuum expectation of the occupancy numbers coincides for pyramidally ordered times with a classical Markovian birth process showing the avalanche character of the quantum process.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literature
Belavkin, V.P.: A quantum non adapted Ito formula and non stationary evolution in Fock scale. Quantum probability and applications. VI, p. 137–180. World Scientific (Singapore) (1992)
Glauber, R.J.: Amplifiers, Attenuators and the Quantum Theory of Measurement. In Frontiers in Quantum Optics, ed. by E.R. Pike and S. Sarkar, Vol. X of Malveru Physics Theories (Adam Hilger), Bristol, 1986.
Haake, F., Walls, D.F.: Overdamped and Amplifying Meters in the Quantum Theory of Measurement, Phys. Rev. A. 36 (1987), p. 730–739.
Hepp, K., Lieb, E.H.: Phase Transitions in Reservoirdriven Open Systems with Applications to Lasers and Superconductors. Helv. Phys. Acta. 46 (1973), p. 573–603.
Hudson, R.L., Parthasarathy, K.R.: Construction of Quantum Diffusions. Lecture Notes in Mathematics 1055, Springer (1984), p. 173–205.
Lindsay, J.M.: Quantum and non-causal stochastic calculus. Prob. Theory Relat. Fields 97, (1993), p. 65–80.
Lindsay, J.M., Maassen, H.: The Stochastic Calculus of Bose Noise. Preprint, Nijmwegen (1988).
Lindsay, M., Maassen, H.: An Integral Kernel Approach to Noise. Lecture Notes in Mathematics 1303, Springer (1988), p. 192–208.
Maassen, H.: Quantum Markov Processes on Fock Space Described by Integral Kernels. Lecture Notes in Mathematics 1136, Springer (1985), p. 361–374.
Meyer, P.A.: Quantum Probability for Probabilists. Lecture Notes in Mathematics 1538, Springer (1993).
Palma, G.M., Vaglica, A., Leonardi, C., De Oliveira, F.A.M., Knight, P.L.: Effects of Broadband Squeezing on the Quantum Onset of Superradiance. Optics Communications, 79 (1990), p. 377–380.
Robinson, P., Maassen, H.: Quantum Stochastic Calculus and the Dynamical Stark Effect. Reports an Math. Phys. Vol. 30 (1991).
Waldenfels, W.v.: Spontaneous Light Emission Described by a Quantum Stochastic Differential Equation. Lecture Notes in Mathematics 1136, Springer (1985), p. 515–534.
Waldenfels, W.v.: The Inverse Oscillator in a Heat Bath as a Quantum Stochastic Process. Preprint 630. 1991. SFB 123 (Heidelberg).
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to P.A. Meyer to his 60th birthday
Rights and permissions
Copyright information
© 1996 Springer-Verlag
About this chapter
Cite this chapter
von Waldenfels, W. (1996). Continuous Maassen kernels and the inverse oscillator. In: Azéma, J., Yor, M., Emery, M. (eds) Séminaire de Probabilités XXX. Lecture Notes in Mathematics, vol 1626. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094646
Download citation
DOI: https://doi.org/10.1007/BFb0094646
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61336-7
Online ISBN: 978-3-540-68463-3
eBook Packages: Springer Book Archive