Quantum interference in the time-of-flight distribution for atomic Bose-Einstein condensates

We investigate the matter-wave interference in time domain, specifically in the time-of-flight (TOF) signal or arrival time distribution for an atomic Bose-Einstein condensate (BEC). This is in contrast to interference in space at a fixed time observed in the reported BEC experiments. We predict and quantify the interference in the centre-of-mass motion by calculating the time distribution of matter-wave arrival probability at some fixed spatial point. Specifically, we consider the free fall of an atomic BEC in a non-classical Schrödinger-cat state (prepared by coherent splitting) which is the linear superposition of two mesoscopically distinguishable Gaussian wave packets peaked around different heights, namely, z = 0 and z = -d, along the vertical free fall axis. During the free fall, the distinct superposed wave packets of the Schrödinger cat overlap or interfere (spatially as well as temporally) which leads to an interference in the quantum TOF distribution. The interference in the TOF distribution (or signal) can then be observed by taking a note or record of the particle counts over various tiny time windows at a fixed detector location. Under this purely quantum scenario, the classical or semi-classical analyses for calculating TOF distribution are not adequate and a quantum analysis for the TOF distribution is necessary. We have used the probability current density approach to calculate the quantum TOF distribution which is logically consistent and also physically motivating. Furthermore, our proposal of measuring matter-wave interference in TOF distribution has the potential to empirically resolve ambiguities inherent in the various theoretical formulations of the quantum TOF distribution.