Decoherence of tripartite states—a trapped ion coupled to an optical cavity

Abstract

We investigate the decoherence process of a three-qubit system obtained by manipulating the state of a trapped two-level ion coupled to an optical cavity. Interaction of the ion with a resonant laser and the cavity field tuned to red sideband of ionic vibrational motion generates tripartite entanglement of the internal state of the ion, vibrational state of ionic centre of mass motion and the cavity-field state. Non-dissipative decoherence occurs due to entanglement of the system with the environment, modelled as a set of non-interacting harmonic oscillators. Analytic expressions for the state operator of tripartite composite system, the probability of generating maximally entangled GHZ state and the population inversion have been obtained. Coupling to environment results in exponential decay of off-diagonal matrix elements of the state operator with time as well as a phase decoherence of the component states. Numerical calculations to examine the time evolution of GHZ state generation probability and population inversion for different values of system environment coupling strengths are performed. Using negativity as an entanglement measure and linear entropy as a measure of mixedness, the entanglement dynamics of the tripartite system in the presence of decoherence sources has been analysed. The maximum tripartite entanglement is found to decrease with increase in the strength of system–environment coupling. The negativity as well as the linear entropy as entanglement measures gives qualitatively similar results, uniquely identifying maximally entangled and separable states of the system. For large values of system–environment coupling strength, the mixed states of the composite system lying at the boundary of an entangled–separable region are reached. For these states, whereas the negativity measures only quantum correlations, the linear entropy measures classical correlations as well.