I will study out-of-equilibrium dynamics of two-dimensional Bose gases. I will create samples with spatial variations of density, temperature or internal state and investigate their relaxation towards an equilibrium or metastable state. The two main breakthroughs of this project will consist in the experimental realization of (i) quantum transport of bosons through a single-mode channel. (ii) diffusion of a few single-particle impurities in a bath of atoms in another internal state, realizing a quantum Brownian motion experiment. This project is based on an experimental setup that I have recently developed in my team and which is fully operational today. This platform allows us to confine Bose gases in a light sheet restricting the motion of the atoms to a two-dimensional plane and to limit their in-plane motion to an arbitrary-shaped potential. Such a setup, allowing high resolution tailoring of optical potentials is rather original in our field and will offer us a large flexibility to investigate out-of-equilibrium dynamics. A first line of research will be devoted to transport properties of an atomic channel. We will start our work two-dimensional channels of a few micron width, directly available in our setup and hosting many conducting modes. We will study particle and heat transport and characterize the influence of disorder in the channel, which is expected to modify much more strongly the behavior of the normal part with respect to the superfluid part. Then, we will decrease the thickness of the channel to enter the single mode regime and where dramatic effects, like the quantification of heat of conductance, are expected. A second line of research will focus on spin dynamics. The rubidium atom that we use has several internal hyperfine states in the electronic ground state. These states can be easily coupled thanks to a microwave field or a two-photon Raman transition, the latter easily allowing spatial resolution. We will focus on binary mixtures. For instance, we will realize quenches by abruptly superimposing two immiscible internal states and monitor their relaxation, a situation for which we already obtained preliminary results. Then, we will move to the study of the dynamics of an impurity in a bath of atoms in another internal state with tunable interaction. We will develop new tools to measure the motion of a few individual particles with sub-micron resolution. By doing so, we will achieve an experimental configuration where we will observe the diffusive behavior of quantum Brownian particles. A superdiffusive behavior is expected in this regime, which goes beyond the memory-less Markov regime for diffusion. The duration of the project will be of 48 months. A large part of the proposed objectives can be directly tackled in our team. In addition, we will develop new tools (optical aberration correction techniques to achieve single mode channels, spatially resolved Raman coupling with tunable momentum transfer, single-atom fluorescence imaging) to achieve original regimes that goes beyond the current state-of-the-art.