Distinguishing Coherent States from Phase-Mixed Coherent States with Only a Variable Beam Splitter and Single-Photon Detector, In Principle

It is generally assumed that on-off detectors with single-photon sensitivity cannot distinguish coherent states from phase-mixed coherent states without some form of quadrature-based tomography. Here, we show that it is theoretically possible to distinguish these states without quadrature-based methods by comparing the vacuum probability of one output port of a variable beam splitter (VBS) that has two different phase-mixed coherent states as its inputs to the well-known case of dual coherent-state inputs. As an application, a method is proposed to test whether a laser field is in a coherent state over a given time by simply measuring the VBS output populations with an on-off single-photon detector. Unfortunately, timing limitations of present technology prohibit such simple tests, but the technique may become practical in the future.