Large fluctuations of the first detected quantum return time

Abstract

How long does it take a quantum particle to return to its origin? As shown previously under repeated projective measurements aimed to detect the return, the closed cycle yields a geometrical phase which shows that the average first detected return time is quantized. For critical sampling times or when parameters of the Hamiltonian are tuned this winding number is modified. These discontinuous transitions exhibit gigantic fluctuations of the return time. While the general formalism of this problem was studied at length, the magnitude of the fluctuations, which is quantitatively essential, remains poorly characterized. Here, we derive explicit expressions for the variance of the return time, for quantum walks in finite Hilbert space. A classification scheme of the diverging variance is presented, for four different physical effects: the Zeno regime, when the overlap of an energy eigenstate and the detected state is small and when two or three phases of the problem merge. These scenarios present distinct physical effects which can be analyzed with the fluctuations of return times investigated here, leading to a topology-dependent time-energy uncertainty principle.

Dark states of quantum search cause imperfect detection

“…This approach is investigated in detail in Ref. [68]. Therefore, we find another reason why the nontrivial eigenvalues obey |ζ l | < 1.…”

“…Thus, except for the zero-measure set of resonant values of τ , P det does not depend on the detection period at all. However, at these resonant τ , P det changes dramatically [68]. This formula is obviously invariant under a change of basis within any quasienergy sector.…”

Uncertainty Relation between Detection Probability and Energy Fluctuations

“…The uncertainty relation found here, can be extended to other observables. In [ 27 ], a time-energy relation was discovered for the fluctuations of the return time, with an interesting dependence on the winding number of the problem.…”

“…The theory is valid in generality, e.g., by identifying the graph with a Fock space, one can describe the dynamics of a many-body system, see an example in Ref. [ 27 ]. Initially, the particle is in state , which could be a state localized on a node of the graph.…”

“…The relation does not depend on the sampling time , and in fact is valid for a large class of measurement protocols. Recently, a time-energy uncertainty-like relation, was found in [ 27 ], with a focus on the time to detect a particle returning to its origin.…”