Measuring the Complex Berry Phase in a Non-Hermitian Superconducting Circuit

The Berry phase is a geometric phase acquired during closed-loop adiabatic evolution. It plays an essential role in geometric quantum gates and other phase-based protocols. In non-Hermitian systems, the Berry phase is complex, introducing fundamentally new geometric effects. In this work, we experimentally measure both the real and imaginary parts of the Berry phase in a superconducting transmon circuit with engineered dissipation. Our results show that the imaginary part does not require an enclosed path in parameter space while the real part of the Berry phase does. These findings establish a clear geometric distinction between the real and imaginary components of the Berry phase and experimentally confirm the unique adiabatic behavior of non-Hermitian quantum systems.