  The link of two concepts, indistinguishability and entanglement, with the
energy-time uncertainty principle is demonstrated in a system composed of two
strongly coupled bosonic modes. Working in the limit of a short interaction
time, we find that the inclusion of the antiresonant terms to the coupling
Hamiltonian leads the system to relax to a state which is not the ground state
of the system. This effect occurs passively by just presence of the
antiresonant terms and is explained in terms of the time-energy uncertainty
principle for the simple reason that at a very short interaction time, the
uncertainty in the energy is of order of the energy of a single excitation,
thereby leading to a distribution of the population among the zero, singly and
doubly excited states. The population distribution, correlations and
entanglement are shown to be substantially depend on whether the modes decay
independently or collectively to an exterior reservoir. In particular, when the
modes decay independently with equal rates, entanglement with the complete
distinguishability of the modes is observed. The modes can be made mutually
coherent if they decay with unequal rates. However, the visibility in the
single-photon interference cannot exceed $50\%$. When the modes experience
collective damping, they are indistinguishable even if decay with equal rates
and the visibility can, in principle, be as large as unity. We find that this
feature derives from the decay of the system to a pure entangled state rather
than the expected mixed state. When the modes decay with equal rates, the
steady-state values of the density matrix elements are found dependent on their
initial values.
