Dynamics of two interacting dipolar two-level systems in a multi-mode electromagnetic cavity: sudden death and revival of the concurrence within the Born-Markov approximation

Abstract

Interacting dipolar two-level systems form a special class of qubits that interact with a cavity in a particular way. We first prove that the Markovian dynamics of one 1/2-spin in interaction with a quantised magnetic field from a multi-mode cavity at thermal equilibrium is equivalent to a two-level atom interacting in the dipole approximation with the electric field of the cavity. We then use the Born-Markov approximation to study the dynamics of two spins interacting through the antiferromagnetic Heisenberg coupling in the same environment. We find the exact expression of the density matrix of the system, with the off-diagonal coherence decay time and spin relaxation time. The concurrence for the stationary state is explicitly derived for any kind of initial state and the role of the singlet state is brought to light. The temporal evolution of the concurrence is numerically computed for different initial states, the phenomenon of sudden death and revival is observed even within the Born-Markov approximation, and an analysis of its behaviour is conducted for Werner states. We finally derive the equations and the stationary concurrence for the XXZ coupling.