Quantum decay and amplification in a non-Hermitian unstable continuum

The decay of a bound state weakly-coupled to a non-Hermitian tight-binding unstable continuum, i.e. a continuum of states comprising energies with positive imaginary part, is theoretically investigated. As compared to quantum decay in an Hermitian continuum, in the non-Hermitian case a richer scenario can be found as a result of non-unitary dynamics. Different behaviors are observed depending on the kind of instability of the continuum. These include complete or fractional decay in convectively-unstable continua, the absence of quantum decay for a bound state with energy embedded in the continuum loop, and unstable (secular) growth with pseudo exponential amplification in the absolutely-unstable regime. Analytical results are presented for a nearest-neighboring tight-binding continuum with asymmetric hopping rates $\kappa_1$ and $\kappa_2$, which shows a transition from convective to absolute instability when the sign $\kappa_1 \kappa_2$ changes from positive to negative. In the convectively unstable regime the model describes the decay of a bound state coupled to a tight-binding lattice with an imaginary gauge field, which shows a pseudo-Hermitian dynamics. In the absolutely-unstable regime, pseudo-Hermitian dynamics is broken and a pseudo exponential secular growth is observed.