Berlin 2014 – scientific programme
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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik
MP 3: Quanten-Information III
MP 3.2: Talk
Tuesday, March 18, 2014, 10:50–11:10, SPA SR125
Controlling atoms in a cavity - applications of infinite dimensional Lie-algebras — •Michael Keyl1, Robert Zeier2, and Thomas Schulte-Herbrüggen2 — 1Zentrum Mathematik, M5, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany — 2Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching, Germany
We consider control theory for a number of two-level atoms interacting with one mode of the electromagnetic field in a cavity. In the rotating wave approximation this provides a very useful toy-model to study several aspects of quantum control theory in infinite dimensions, in particular the emergence of infinite dimensional system algebras. In this context we provide a short discussion of problems arising with infinite dimensional Lie-algebras and Lie-algebras consisting of unbounded operators. For the models under consideration these problems can be solved by splitting the set of control Hamiltonians up into two groups: The first obeys an Abelian symmetry and can be treated in terms of infinite dimensional Lie-algebras and strongly closed subgroups of the unitary group of the system Hilbert space. The second breaks this symmetry and their discussion needs new arguments. At the end full controlability can be achieved in a certain strong sense. As an example we study a time dependent version of the Jaynes Cummings model and show that with an appropriate tuning of the coupling constants every unitary of the coupled system (atom and cavity) can be approximated with arbitrary small error.