Using the technique of quantum transport of the interfering pair we examine the Hanbury-Brown-Twiss (HBT) interferometry signatures for the particle-emitting sources of pions and kaons produced in the heavy ion collisions at GSI-FAIR energies. The evolution of the sources is described by relativistic hydrodynamics with the system equation of state of the first-order phase transition from quark-gluon plasma (QGP) to hadronic matter. We use quantum probability amplitudes in a path-integral formalism to calculate the two-particle correlation functions, where the effects of particle decay and multiple scattering are taken into consideration. We find that the HBT radii of kaons are smaller than those of pions for the same initial conditions. Both the HBT radii of pions and kaons increase with the system initial energy density. The HBT lifetimes of the pion and kaon sources are sensitive to the initial energy density. They are significantly prolonged when the initial energy density is tuned to the phase boundary between the QGP and mixed phase. This prolongations of the HBT lifetimes of pions and kaons may likely be observed in the heavy ion collisions with an incident energy in the GSI-FAIR energy range.
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