Dynamical instabilities in quantum many-body systems

from the introduction: "[...] This thesis addresses the problem of the nonequilibrium time-evolution of many-body sistems realized with quantum simulators. We investigate theoretically the relation between the time-evolution and the equilibrium phase diagram in the presence of a quantum phase transition. The long-time evolution of the systems is investigated, both in the case of conservative dynamics and under the action of dissipative processes. The thesis is organized as follows. Chaps. 1-3 contain theoretical and experimental facts that are relevant to the present work. In the Chap. 1 we describe the quantum simulator realized with fermionic cold atoms in the bulk of an optical trap. In the Chap. 2 we describe the microresonators where light and matter are strongly coupled, and we focus on the implementation of a quantum simulator with defect-cavities in photonic crystals. In Chap. 3 several theoretical works concerning the nonequilibrium dynamics of many-body systems are summarized. Chaps. 4{6 contain the original contributions of this thesis. In Chap. 4 we investigate the time-evolution of an ensemble of fermionic atoms after an abrupt change of the interaction strength from a vanishing to a weak attractive value. In Chap. 5 we demonstrate the possibility to measure the quantum phase transition between a super fluid and a Mott-insulator state in an array of microresonators, in the presence of the leakage of photons out of the cavities. In Chap. 6 we study a system of cold bosonic atoms coupled to a tailored bath that dissipatively drives the system into a super fluid or into a thermal state.