We show that a two-level non-Hermitian Hamiltonian with constant off-diagonalexchange elements can be analyzed exactly when the underlying exceptional pointis perfectly encircled in the complex plane. The state evolution of this systemis explicitly obtained in terms of an ensuing transfer matrix, even for largeencirclements, regardless of adiabatic conditions. Our results clearly explainthe direction-dependent nature of this process and why in the adiabatic limitits outcome is dominated by a specific eigenstate irrespective of initialconditions. Moreover, numerical simulations suggest that this mechanism canstill persist in the presence of nonlinear effects. We further show that thisrobust process can be harnessed to realize an optical omni-polarizer: aconfiguration that generates a desired polarization output regardless of theinput polarization state, while from the opposite direction it always producesthe counterpart eigenstate.
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