  In a two-qubit system the coupling with an environment affects considerably
the entanglement dynamics, and usually leads to the loss of entanglement within
a finite time. Since entanglement is a key feature in the application of such
systems to quantum information processing, it is highly desirable to find a way
to prolonging its lifetime. We present a simple model of an interacting
two-qubit system in the presence of a thermal Markovian environment. The qubits
are modeled as interacting spin-$\half$ particles in a magnetic field and the
environment is limited to inducing single spin-flip events. A simple scheme
allows us to calculate the relaxation rates for all processes. We show that the
relaxation dynamics of the most entangled state exhibit critical slowing down
as a function of the magnetic field, where the relaxation rate changes from
exponentially small values to finite values in the zero-temperature limit. We
study the effect of temperature and magnetic field on all the other relaxation
rates and find that they exhibit unusual properties, such as non-monotonic
dependence on temperature and a discontinuity as a function of magnetic field.
In addition, a simple scheme to include non-Markovian effects is presented and
applied to the two-qubit model. We find that the relaxation rates exhibit a
sharp, cusp-like resonant structure as a function of the environment
memory-time, and that for long memory-times all the different relaxation rates
merge into a single one.
