First Detection Probability In Quantum Resetting Via Random Projective Measurements

Abstract

We provide a general framework to compute the probability distribution $F_r(t)$ of the first detection time of a 'state of interest' in a generic quantum system subjected to random projective measurements. In our 'quantum resetting' protocol, resetting of a state is not implemented by an additional classical stochastic move, but rather by the random projective measurement. We then apply this general framework to Poissoinian measurement protocol with a constant rate $r$ and demonstrate that exact results for $F_r(t)$ can be obtained for a generic two level system. Interestingly, the result depends crucially on the detection schemes involved and we have studied two complementary schemes, where the state of interest either coincides or differs from the initial state. We show that $F_r(t)$ at short times vanishes universally as $F_r(t)\sim t^2$ as $t\to 0$ in the first scheme, while it approaches a constant as $t\to 0$ in the second scheme. The mean first detection time, as a