Generalization of Wigner Time Delay to Sub-Unitary Scattering Systems

Wigner time delay has been of great interest to the physics community because of its importance in describing the scattering properties of open quantum systems. Defined as energy derivative of the total scattering phase shift, the Wigner time delay measures how long a particle/wave lingers in an interaction region before leaving the system through scattering channels. The statistics of Wigner time delay in the lossless limit has been well studied by theorists, but relatively little research has been done for the case when loss or decoherence is present. Here we introduce a complex generalization of Wigner time delay by carefully considering the effects of introducing loss into the system. Through a series of experiments on microwave analogs of quantum graphs and billiards in which we have precision control of the loss strength inside the system, we demonstrate the evolution of the complex Wigner time delay spectrum along with the migration of the scattering matrix poles. We demonstrate experimentally and numerically that the complex Wigner time delay can be used as a practical counter of the imaginary parts of the scattering matrix poles.