Open Quantum Systems
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Submission date: 29. Oct. 2008
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These notes give an informal introduction to open quantum systems. Starting from Schr¨odinger’s equation, we are lead to density matrices and then to states which are positive, normalized linear functionals on observable algebras. We then go through the notions of KMS states and C*-, W*-, and quantum dynamical systems and mathematically formulate the property of return to equilibrium, which we then demonstrate to hold true for a spin-boson model. Finally, we give an application to decoherence of qubits of a quantum computer.
These notes are based on articles whose precise reference is appended. (Particularly useful were Pillet’s article:
S. Attal, A. Joye, and C.-A. Pillet, editors. Open Quantum Systems I: The Hamiltonian Approach, volume 1880 of Lecture Notes in Mathematics. Springer-Verlag, 2006.
M. Mück. Thermal Relaxation for Particle Systems in Interaction with several bosonic Heat Reservoirs. PhD thesis, Johannes Gutenberg-Universität Mainz, July 2004. ISBN 3-8334-1866-4.)
It is a pleasure to thank Klaus Sibold and Manfred Salmhofer for organizing a most interesting and pleasant workshop at the Institute for Theoretical Physics of Leipzig University in February 2008 and Kay-Uwe Giering for excellent typesetting.Copyright remains with the author.