Operational Recoverability Bounds for Proper-Time Interferometry with Open Quantum Clocks

Abstract

Clock interferometry probes general-relativistic proper-time differences through the entanglement between a particle’s center-of-mass path and its internal degrees of freedom. Existing analyses treat the internal clock as either a closed system or study how environmental noise modifies the interferometric visibility. Here we address a different question: how much of the proper-time-induced visibility loss remains recoverable when the experimentalist is restricted to local unitary control of the internal clock. We model the clock as a two-level system undergoing spontaneous decay while the two arms of a Mach-Zehnder interferometer accumulate different proper times. For a single refocusing pulse that exchanges the internal ground and excited states midway through the evolution, we derive the exact echo visibility by keeping the complete environmental jump structure, including the two-jump history made possible by this pulse. This yields a closed analytic expression for the echo-recovered visibility and a tight phase-maximized envelope for the single-echo protocol. In the absence of environmental coupling, the pulse restores full visibility. With environmental decay, generic proper-time differences leave an echo-recovery deficit, whereas equal proper times retain unit visibility exactly. We also establish the conditions under which a proper-time-parametrized Gorini-Kossakowski-Sudarshan-Lindblad equation is valid by appealing to the local comoving frame and the equivalence principle, distinguishing comoving-bath and laboratory-bath parametrizations. Numerical evaluations of the composed Kraus channel, implemented with QuTiP, agree with the analytic expressions. Experimental estimates for cold-atom and cavity-QED platforms indicate that the irreversible visibility deficit is modest for natural optical linewidths but can be enhanced by longer interrogation times or Purcell-enhanced decay. After calibration of technical contrast losses, the echo-recovery deficit therefore serves as an operational indicator of irreversible environmental decoherence, distinguishing it from the reversible path-clock entanglement of the closed-system proper-time mechanism.