Decoherence of Bell states by local interactions with a suddenly quenched spin environment

We study the dynamics of disentanglement of two qubits initially prepared in a Bell state and coupled at different sites to an Ising transverse field spin chain (ITF) playing the role of a dynamic spin environment. The initial state of the whole system is prepared into a tensor product state $\rho_{Bell}\otimes \rho_{chain}$ where the state of the chain is taken to be given by the ground state $|G(\lambda_i)\rangle$ of the ITF Hamiltonian $H_E(\lambda_i)$ with an initial field $\lambda_i$. At time $t=0^+$, the strength of the transverse field is suddenly quenched to a new value $\lambda_f$ and the whole system (chain $+$ qubits) undergoes a unitary dynamics generated by the total Hamiltonian $H_{Tot}=H_E(\lambda_f) + H_I$ where $H_I$ describes a local interaction between the qubits and the spin chain. The resulting dynamics leads to a disentanglement of the qubits, which is described through the Wooter's Concurrence, due to there interaction with the non-equilibrium environment. The concurrence is related to the Loschmidt echo which in turn is expressed in terms of the time-dependent covariance matrix associated to the ITF. This permits a precise numerical and analytical analysis of the disentanglement dynamics of the qubits as a function of their distance, bath properties and quench amplitude. In particular we emphasize the special role played by a critical initial environment.