Exceptional Points revealed by the integrated imaginary scattering eigenphase
We propose and analytically demonstrate that the eigenphases of the scattering matrix provide direct, phase-sensitive signatures of EPs in open PT-symmetric systems. Using a 1D PT-symmetric quantum dimer with balanced gain and loss, coupled to continuum leads, we track the evolution of the scattering eigenphases across the PT-exact to broken transition. The imaginary parts of the eigenphases develop localized structures of opposite sign as the EP is approached, reflecting the emergence of gain/loss asymmetry in the scattering eigenmodes. The EP is located by the algebraic degeneracy condition of the S-matrix eigenvalues, and corroborated by two further, formally distinct analyses--the merging of the transmission resonances and the dynamics of the complex poles and zeros of S(E)--which place the transition within the same narrow parameter window, providing strong evidence for its location and physical reality. We then introduce the integrated imaginary scattering eigenphase G(gamma), which condenses the eigenphase information into a single experimentally accessible scalar and constitutes a fourth, resonance-free route to the same transition. G(gamma) is negligibly small in the PT-exact phase and undergoes a sharp transition--a pronounced inflection followed by saturation into a plateau--near the EP. The inflection point of G(gamma) precedes gamma_EP and coincides with the maximum amplitude of the transmission resonances, revealing that peak gain/loss asymmetry and eigenstate coalescence are governed by independent conditions in open systems--a direct fingerprint of their finite coupling to the continuum. Because the analysis is formulated entirely at the level of the S-matrix--without reference to any specific physical realization--the results are universally applicable across wave systems, like photonic, acoustic and microwave platforms where phase-resolved measurements are available.