  Open quantum systems are often represented by non-Hermitian effective
Hamiltonians that have complex eigenvalues associated with resonances. In
previous work we showed that the evolution of tight-binding open systems can be
represented by an explicitly time-reversal symmetric expansion involving all
the discrete eigenstates of the effective Hamiltonian. These eigenstates
include complex-conjugate pairs of resonant and anti-resonant states. An
initially time-reversal-symmetric state contains equal contributions from the
resonant and anti-resonant states. Here we show that as the state evolves in
time, the symmetry between the resonant and anti-resonant states is
automatically broken, with resonant states becoming dominant for $t>0$ and
anti-resonant states becoming dominant for $t<0$. Further, we show that there
is a time-scale for this symmetry-breaking, which we associate with the "Zeno
time." We also compare the time-reversal symmetric expansion with an asymmetric
expansion used previously by several researchers. We show how the present
time-reversal symmetric expansion bypasses the non-Hilbert nature of the
resonant and anti-resonant states, which previously introduced exponential
divergences into the asymmetric expansion.
