## (a) Variation of splitting between quasi-degenerate energy levels (in units of *J*) with number of particles *N* for α = 1.7

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**Figure 4.** (a) Variation of splitting between quasi-degenerate energy levels (in units of *J*) with number of particles *N* for α = 1.7. The solid line represents numerical results and the dashed line indicates the analytical result of energy splitting obtained from semiclassical approximation (given in equation (24)). (b) Time evolution of number imbalance Δ*N*/*N* for α = 1.25 (time in units of 1/*J*). The initial coherent state is localized at a fixed point corresponding to the self-trapped state (MST) with cos θ = 0.6 and = π. The solid line shows the oscillation of the BJJ with *N* = 50 between the self-trapped states with equal and opposite number imbalance due to MQT. The dashed line indicates the self-trapping of a BJJ with *N* = 100.

**Abstract**

We study the quantum dynamics of a Bose–Josephson junction made up of two coupled Bose–Einstein condensates. We analyse different dynamical branches of Josephson oscillations within an 'effective potential' approach. At a critical coupling strength, a transition takes place between the dynamical branches of Josephson oscillations, which is also manifested in the energy spectrum, and pairs of (quasi)-degenerate excited states appear above the critical coupling strength. This phenomenon can be understood in terms of change in shape of the 'effective potential'. Possible novel quantum phenomena like decay of metastable 'π-oscillation' by 'macroscopic quantum tunnelling' (MQT) and MQT between the 'self-trapped' states with equal and opposite number imbalance become evident from the simple picture of 'effective potential'.