We develop a generic method to compute the dynamics induced by quenches in completely connected quantum systems. These models are expected to provide a mean-field description at least of the short time dynamics of finite dimensional system. We apply our method to the Bose-Hubbard model, to a generalized Jaynes-Cummings model, and to the Ising model in a transverse field. We find that the quantum evolution can be mapped onto a classical effective dynamics, which involves only a few intensive observables. For some special parameters of the quench, peculiar dynamical transitions occur. They result from singularities of the classical effective dynamics and are reminiscent of the transition recently found in the fermionic Hubbard model. Finally, we discuss the generality of our results and possible extensions. ‡ This consists in a technique similar to the ones developed in [24]. § Thermalization is very likely to occur in finite dimensions for the Bose-Hubbard model, because the system is not integrable. Instead for completely connected models we do not expect thermalization * This symmetry may be spontaneously broken in systems with attractive interactions, which we do not consider here. ♯ The minus sign in this definition is for later convenience.

We study the off-equilibrium dynamics of the infinite dimensional Bose Hubbard Model after a quantum quench. The dynamics can be analyzed exactly by mapping it to an effective Newtonian evolution. For integer filling, we find a dynamical transition separating regimes of small and large quantum quenches starting from the superfluid state. This transition is very similar to the one found for the fermionic Hubbard model by mean field approximations.Significant advances in the field of ultra cold atoms have allowed one to engineer quantum many-body systems in almost perfect isolation from the environment. Thanks to the ability to rapidly tune different parameters, e.g. the interaction strength between the atoms or the creation of controlled excitations, the realm of non-equilibrium many body physics of (almost) isolated quantum systems has thus been accessed and can now be studied experimentally. For example, Greiner et al. [1] studied the dynamics of interacting bosons loaded on an optical lattice. The physics of this system is well captured by the Bose-Hubbard model. By changing the intensity of the lasers one can effectively tune the parameters in the corresponding Bose-Hubbard model. Rapid changes induce interesting non-equilibrium dynamics [1]. The activity in this field is booming: several new experiments have been performed, including on fermionic systems [2,3]; questions about thermalization [4,5], its absence [6][7][8], quantum dynamical phase transitions out of equilibrium [9,10] are currently addressed. A protocol inducing an off-equilibrium dynamics, which has received a lot of attention recently, is the so called quantum quench. It corresponds to preparing the system in the ground state of the HamiltonianĤ i , to changing suddenly at time t = 0 a parameter of the Hamiltonian, for example the interaction strength, and then letting the system evolve with the new HamiltonianĤ f . Several studies have been performed for the Bose Hubbard model, which as discussed above is relevant for experiments. There have been numerical analysis of one dimensional systems by exact diagonalization and t-DMRG [4,5,7,8]. Saddle point approximation [11], GrossPitaevskii equations [12] and Gutzwiller approximation [14,15] have been used to analyze higher dimensional and realistic cases. The fermionic Hubbard model has been also studied by mean field theories recently [9,10]. In this work we present a complete analysis of quantum quenches in the Bose Hubbard model (BHM) in the limit of infinite dimensions. The advantage of this limit is that the model can then be analyzed exactly even out of equilibrium. Its solution at equilibrium played an important role in determining the phase diagram and the properties of the Mott-superfluid quantum phase transition of the three dimensional BHM [13]. Studying its off-equilibrium dynamics is therefore a natural route to follow. We will discuss in the conclusion the limitations of this approach and possible extensions. To obtain a well defined infinite dimensional limit one has to scale the ...