  While non-Hermitian Hamiltonians have been experimentally realized in cold
atom systems, it remains an outstanding open question of how to experimentally
measure their complex energy spectra in momentum space for a realistic system
with boundaries. The existence of non-Hermitian skin effects may make the
question even more difficult to address given the fact that energy spectra for
a system with open boundaries are dramatically different from those in momentum
space; the fact may even lead to the notion that momentum-space band structures
are not experimentally accessible for a system with open boundaries. Here, we
generalize the widely used radio-frequency spectroscopy to measure both real
and imaginary parts of complex energy spectra of a non-Hermitian quantum system
for either bosonic or fermionic atoms. By weakly coupling the energy levels of
a non-Hermitian system to auxiliary energy levels, we theoretically derive a
formula showing that the decay of atoms on the auxiliary energy levels reflects
the real and imaginary parts of energy spectra in momentum space. We further
prove that measurement outcomes are independent of boundary conditions in the
thermodynamic limit, providing strong evidence that the energy spectrum in
momentum space is experimentally measurable. We finally apply our non-Hermitian
absorption spectroscopy protocol to the Hatano-Nelson model and non-Hermitian
Weyl semimetals to demonstrate its feasibility.
