Moving Detector Quantum Walk with Random Relocation

We study a discrete-time quantum walk in presence of a detector at $x_D$ initially. The detector here is repeatedly removed after a span of $t_R$, the removal time, and reinserted at random locations. Two relocation rules are considered here: In Model~1, the detector is reinserted at any site beyond $x_D$, while in Model~2, reinsertion is done within a restricted window around the position of the detector at that time. Both variants behave like Semi Infinite Walk (SIW) for large $t_R$, where the detector behaves effectively as a fixed boundary. However, in the rapid-relocation regime, i.e., when $t_R$ is small, the behaviours are different. Model~1 permits greater spreading due to unrestricted reinsertion, which is different from Model~2. The time evolution of occupation probability ratio of our walker to that of an infinite walker at $x_D$, i.e., $f(x_D,t)/f_\infty(x_D,t)$, initially show the feature of a SIW upto $t=t_R$, then show some oscillatory behaviour and finally reach a saturation value for both the models. The ratio enhancing under certain conditions of $x_D$ and $t_R$, is a purely quantum mechanical effect. The saturation ratio shows a crossover behavior below and above a removal time $t_R^*$. At sites $x \neq x_D$ the occupation probablity ratios at a certain time reveals that for small $t_R$, the behaviours of the two models are drastically different from each other, as well as from Semi Infinite Walk (SIW), Quenched Quantum Walk (QQW) and Moving Detector Quantum Walk (MDQW). The correlation ratios of the two models with that of Infinite Walk (IW) show interesting time dependence for sites to the left or right of the initial detector position $x_D$.