  Open fermion systems with energy-independent bilinear coupling to a fermionic
environment have been shown to obey a general duality relation [Phys. Rev. B
93, 81411 (2016)] which allows for a drastic simplification of time-evolution
calculations. In the weak-coupling limit, such a system can be associated with
a unique dual physical system in which all energies are inverted, in particular
the internal interaction. This paper generalizes this fermionic duality in two
ways: we allow for weak coupling with arbitrary energy dependence and describe
both occupations and coherences coupled by a quantum master equation for the
density operator. We also show that whenever generalized detailed balance holds
(Kolmogorov criterion), the stationary probabilities for the dual system can be
expressed explicitly in terms of the stationary recurrence times of the
original system, even at large bias.
  We illustrate the generalized duality by a detailed analysis of the rate
equation for a quantum dot with strong onsite Coulomb repulsion, going beyond
the commonly assumed wideband limit. We present predictions for (i) the decay
rates for transient charge and heat currents after a gate-voltage quench and
(ii) the thermoelectric linear response coefficients in the stationary limit.
We show that even for pronouncedly energy-dependent coupling, all nontrivial
parameter dependence in these problems is entirely captured by just two
well-understood stationary variables, the average charge of the system and of
the dual system. Remarkably, it is the latter that often dictates the most
striking features of the measurable quantities (e.g., positions of resonances),
underscoring the importance of the dual system for understanding the actual
one.
