  We study the propagation of a Bose-Einstein condensate (BEC) of long-lifetime
exciton polaritons in a high-quality microcavity with an embedded semiconductor
quantum well or a graphene layer using the Gross-Pitaevskii equation. It is
shown that in these heterostructures the BEC of the long-lifetime polaritons
can propagate over the distance up to 0.5 mm. The obtained results are
consistent with the recent experimental observations for GaAs/AlGaAs
microcavity. It is demonstrated that the BEC density in a polariton trace in a
microcavity with embedded graphene at large distances from the excitation spot
is higher for the microcavity with higher dielectric constant. It is also
predicted that the propagation of a polariton BEC in a microcavity with
graphene is dynamically tunable by changing the gap energy, that makes it
potentially useful for applications in integrated optical circuits.
