  Non-Hermitian systems generically have complex energies, which may host
topological structures, such as links or knots. While there has been great
progress in experimentally engineering non-Hermitian models in quantum
simulators, it remains a significant challenge to experimentally probe complex
energies in these systems, thereby making it difficult to directly diagnose
complex-energy topology. Here, we experimentally realize a two-band
non-Hermitian model with a single trapped ion whose complex eigenenergies
exhibit the unlink, unknot or Hopf link topological structures. Based on
non-Hermitian absorption spectroscopy, we couple one system level to an
auxiliary level through a laser beam and then experimentally measure the
population of the ion on the auxiliary level after a long period of time.
Complex eigenenergies are then extracted, illustrating the unlink, unknot or
Hopf link topological structure. Our work demonstrates that complex energies
can be experimentally measured in quantum simulators via non-Hermitian
absorption spectroscopy, thereby opening the door for exploring various
complex-energy properties in non-Hermitian quantum systems, such as trapped
ions, cold atoms, superconducting circuits or solid-state spin systems.
