Abstract
In this thesis, we study the one particle entanglement spectrum (OPES) after a quantum sudden quench in the one dimensional and quadratic systems. Using the topological maximal entangled state (tMES) as a signature of non-trivial topology. We find the topology of the steady state at the long time limit can be captured by a pseudo-magnetic field $\mathbf{S}_{\mbox{eff}}$, where the non-trivial or trivial Berry phase of $\mathbf{S}_{\mbox{eff}}$ determines the steady state to be topological or not. Using the dimerized chain and p-wave superconductor as practical examples, we find the topological steady states can only exist for the quenches between the same topological phase. On the other hand, the convergences to the steady states are found to be power-law like. However, we find the steady states in the dimerized chain are more universal than the steady states in the p-wave superconductor. In dimerized chain, we find the exponents of the power-law convergences are always 3/2 and the topological steady states are always found for the quenches between the same topological phase. While, in the p-wave superconductor, the exponent of the power-law convergence can be changed by tuning the parameters and one can have trivial steady states even for the quenches between the same topological phase.