Quantum Uncertainty Considerations for Gravitational Lens Interferometry

The measurement of the gravitational lens delay time between light paths has relied, to date, on the source having sufficient variability to allow photometric variations from each path to be compared. However, the delay times of many gravitational lenses cannot be measured because the intrinsic source amplitude variations are too small to be detectable. At the fundamental quantum mechanical level, such photometric “time stamps” allow which-path knowledge, removing the ability to obtain an interference pattern. However, if the two paths can be made effectively equal (zero time delay) then interference can occur. We describe an interferometric approach to measuring gravitational lens delay times using a “quantum-eraser/restorer” approach, whereby the light travel time along the two paths may be rendered unmeasurable. Energy and time being non-commuting observables, constraints on the photon energy in the energy-time uncertainty principle—via adjustments of the width of the radio bandpass —dictate the uncertainty of the time delay and therefore whether the “path taken” along one or the other gravitational lens geodesic is “knowable.” If one starts with interference, for example, which-path information returns when the bandpass is broadened (constraints on the energy are relaxed) to the point where the uncertainty principle allows a knowledge of the arrival time to better than the gravitational lens delay time itself, at which point the interference will disappear. We discuss the near-term feasibility of such measurements in light of current narrow-band radio detectors and known short time-delay gravitational lenses. PACS: 95.75Kk (interferometry), 98.62Sb (gravitational lenses), 95.85Bh (radio observations).