  Ferromagnetism is an iconic example of a first-order phase transition taking
place in spatially extended systems and is characterized by hysteresis and the
formation of domain walls. In this paper we demonstrate that an extended atomic
superfluid in the presence of a coherent coupling between two internal states
exhibits a quantum phase transition from a para- to a ferromagnetic state. The
nature of the transition is experimentally assessed by looking at the phase
diagram as a function of the control parameters, at hysteresis phenomena, at
the magnetic susceptibility and the magnetization fluctuations around the
critical point. We show that the observed features are in good agreement with
mean-field calculations. Additionally, we develop experimental protocols to
deterministically generate domain walls that separate spatial regions of
opposite magnetization in the ferromagnetic state. Thanks to the enhanced
coherence properties of our atomic superfluid system compared to standard
condensed matter systems, our results open the way towards the study of
different aspects of the relaxation dynamics in isolated coherent many-body
quantum systems.
