1. The Rabi regime (, weakly interacting oscillators)

2. The Josephson regime ()

3. The Fock regime (, dominated by quantum effects)

The Rabi and Josephson regimes are the ones most often investigated, and their behavior is exhibited in this Demonstration. In the mean-field approximation, their dynamics are determined largely by the population imbalance between the oscillators , with the total number of particles and their relative phase. The trajectories are typically plotted in a phase space, as shown in the diagram on the lower right.

A complete and more insightful representation makes use of the Bloch sphere, on the lower left. We show here the two representations side by side, as functions of the initial conditions and of the population imbalance and relative phase at . We show both: the general solution for arbitrary interaction strength in red and the non-interacting case , the pure Rabi regime, which is represented by a circle on the sphere) in blue. We also take account of detuning between the oscillators [2].

The Demonstration shows how phenomena such as self-trapping or running phases also occur in the pure Rabi regime (no interactions). An unambiguous and general criterion to distinguish the regimes is provided by a stability analysis of the dynamical equations that identifies the Josephson regime by the presence of at least one fixed saddle point. The positions of the fixed points are also shown, as are the zeros of the steady-state equations. More details on this regime, as well as a discussion of the case with finite lifetime and interactions between the modes can be found in [3].