On the collapses and revivals in the Rabi Hamiltonian Titus Sandu National Institute for R&D in Microtechnologies, Bucharest, Romania Abstract. The Rabi Hamiltonian describes the interaction between a single two-level system and a single quantized radiation mode and models many phenomena like nuclear spins interacting with magnetic fields, atoms interacting with an electromagnetic field mode, electrons coupled to a crystal lattice phonon mode, superconducting qubits interacting with a nanomechanical resonator, a transmission line resonator, etc. Although the Hamiltonian is quite simple only recently it was exactly solved [1]. Despite its complete resolution approximate but more simple and comprehensible solutions are needed. Such approximations like the rotating wave approximation [2] or more elaborate variants of it [3] are often used. However, as ultra-strong coupling regime is attained (where the coupling strength approaches the energies of the fermionic and bosonic systems) the rotating wave approximation is no longer valid. We will use several unitary transformations that approximate the dynamics of the Rabi Hamiltonian for a coupling strength that is no greater than the detuning between the two-level system and the bosonic field. The collapses and revivals in both sub-systems are discussed and characterized. [1] D. Braak, “Integrability of the Rabi model”, Phys. Rev. Lett., vol. 107, p. 100401, (2011) [2] E. T. Jaynes and F. W.Cummings, “Comparison of quantum and semi-classical radiation theories with application to the beam maser”, Proc. IEEE, vol. 51, p. 89, (1963). [3] E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett., vol. 99, p. 173601, (2007). T. Niemczyk et al., Nat. Phys. 6, 772 2010 Y. Todorov et al. PRL 105, 196402 2010 Conclusions: For ultra-strong coupling, three main regimes can be described by three unitary transformations. Collapses and revivals of both sub-systems are encountered in all three regimes. Collapses are rather not complete due to residual terms induced by the strong coupling. ACKNOWLEDGEMENTS: This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PNII-ID-PCCE-2011 -2-0069.