The role of the geometric phases in adiabatic populations tracking for   non-hermitian hamiltonians
  We show that the definition of instantaneous eigenstate populations for a dynamical non-self-adjoint system is not obvious. The naive direct extension of the definition used for the self-adjoint case leads to inconsistencies; the resulting artifacts can induce a false inversion of population or a false adiabaticity. We show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations. An example is given which demonstrates both the anomalous effects and their removal by our approach. 
Arnaud Leclerc, David Viennot and Georges Jolicard
quant-ph