Measuring the complex geometric phase in the dynamics of a non-Hermitian system

When the Hamiltonian of a linear system is slowly tuned around a closed circuit in parameter space, the adiabatic theorem guarantees that an eigenstate of the system returns to itself up to an overall phase. This phase has two dominant components: a dynamical component that depends on the time taken to traverse the circuit, and a geometric component that depends only on the shape of the circuit. For non-Hermitian systems it has been predicted that, for circuits in which the adiabatic theorem is applicable, the geometric phase may be complex, i.e. it may contain both a real phase and a gain/loss component. We measure the fully complex geometric phase accumulated by a set of coupled oscillators, whose non-Hermitian Hamiltonian is parametrically tuned in real time via optomechanical dynamical back-action. We show that this phase is well predicted by only the circuit geometry for a variety of circuits.