Spread complexity and dynamical transition in two-mode Bose-Einstein condensations

We study the spread complexity in two-mode Bose-Einstein condensations and unveil that the long-time average of the spread complexity C̄K can probe the dynamical transition from self-trapping to Josephson oscillation. When the parameter ω increases over a critical value ωc, we reveal that the spread complexity exhibits a sharp transition from lower to higher value, with the corresponding phase space trajectory changing from self-trapping to Josephson oscillation. Moreover, we scrutinize the eigen-spectrum and uncover the relation between the dynamical transition and the excited state quantum phase transition, which is characterized by the emergence of singularity in the density of states at critical energy Ec. In the thermodynamical limit, the cross point of Ec(ω) and the initial energy E₀(ω) determines the dynamical transition point ωc. Finally, we show that the different dynamical behavior for the initial state at a fixed point can be distinguished by the long-time average of the spread complexity, when the fixed point changes from unstable to stable.