First-detection return statistics in quantum walks with long-range hopping

Quantum walks are paradigms for many-body dynamics and are analog realization of quantum algorithms such as the quantum search. Key characterizing concepts are quantum recurrence, which describes the ability of a quantum walker to return to its initial state, and the associated first-detection time, which is the time interval elapsed between the initial time and the recurrence. In this work, we analyze a quantum walk on a chain with long-range hopping , where the coupling between sites at distance d decays as d^-α, with α ≥ 0. The walker evolves unitarily between stroboscopic projective measurements on the initial site performed at times t_n = nτ, n ∈ Ν. Our study shows that the nature of the walk is controlled by the hopping exponent α. In the strong long-range regime α < 1, interference tends to localize the walker and the quantum walks are recurrent: the walker returns to the origin with probability one. For α > 1, instead, the first-detection probability decays algebraically, with exponent depending on α, leading to a transient quantum walk. We connect these behaviors with the spectral features of the model.