The arrow of time in open quantum systems and dynamical breaking of the resonance anti resonance symmetry
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 and anti resonant states becoming dominant for 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.