  Dynamically varying system parameters along a path enclosing an exceptional
point is known to lead to chiral mode conversion. But is it necessary to
include this non-Hermitian degeneracy inside the contour for this process to
take place? We show that a slow enough variation of parameters, even away from
the system's exceptional point, can also lead to a robust asymmetric state
exchange. To study this process, we consider a prototypical two-level
non-Hermitian Hamiltonian with a constant coupling between elements. Closed
form solutions are obtained when the amplification/attenuation coefficients in
this arrangement are varied in conjunction with the resonance detuning along a
circular contour. Using asymptotic expansions, this input-independent mode
conversion is theoretically proven to take place irrespective of whether the
exceptional point is enclosed or not upon encirclement. Our results
significantly broaden the range of parameter space required for the
experimental realization of such chiral mode conversion processes.
