The study of many-body quantum dynamics is fundamentally hindered by the exponential growth of the Hilbert space, the mathematical space in which the physics is parametrized. As a first approximation, one can therefore make an attempt to formulate the problem in terms of free quasiparticles, fictitious particles devised to capture the relevant degrees of freedom of a quantum many-body model. In this thesis, the emergence of quasiparticles is studied in a variety of out-of-equilibrium scenarios, for which we illustrate that the formalism actually permits to gain a clear insight into the problem as well.

We start by examining how a sudden change of a parameter in a model - a quantum quench – abruptly injects a population of quasiparticles into the system. Subsequently, each quasiparticle starts propagating through the system at a well-defined velocity, which we can derive from the model. This analysis allows us to understand how distant points become correlated and how this eventually makes the whole system relax to an apparent equilibrium. In particular, we examine how a long-range interaction potential between the true particles of the system influences the spreading of information, conveyed by quasiparticles, and the final relaxation dynamics.

We then move our attention to photonic systems, which constitute an entirely different notion of out of equilibrium. Now it is investigated how quasiparticles arise inside a planar microcavity – an inherent out-of-equilibrium system due to the continuous injection and escape processes of photons. It is illustrated how a weak photon interaction, mediated by a nonlinear photonic medium, generates slightly nonclassical light that comes out of the cavity. Our work then studies how this small nonclassical feature can be exploited in an optical selection and interference scheme, placed after the microcavity, to generate an approximate train of single photons – a photon turnstile device.

Finally, we conclude the work by going one step further - to consider also interactions between the quasiparticles. We illustrate that a footprint of quasiparticle collisions can be observed in the steady state of a driven chain of nonlinear cavities. At equilibrium - as we also explicitly verify later - the very same collisions inevitably drive the system into a thermalized state. The apparent contrast between these two scenarios is worked out in detail, and we conclude that this distinction can be related back to long-standing notions of equilibrium dynamics, such as the second law of thermodynamics and detailed balance.