Quantum random walk and tight-binding model subject to projective measurements at random times

Abstract

What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N. For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m. By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ &gt; 1. These scales are such that (i) for m large and satisfying <?CDATA $m m &lt; m 1 ⋆ ( N ) , the decay of the survival probability is as m −2, (ii) for m satisfying <?CDATA ${m}_{1}^{\star }(N)\ll m m 1 ⋆ ( N ) ≪ m &lt; m 2 ⋆ ( N ) , the decay is as m −3/2, while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our results for a wide range of distributions including exponential and power-law distributions as well as for the case of measurements at regular intervals. This fact hints at robustness and ubiquity of our derived results.

Continuously monitored quantum systems beyond Lindblad dynamics

“…However, there are ways to prevent or slow down the propagation as, for instance, adding a disorder potential which induces Anderson localization [63,64]. Here, we show that the quantum Zeno effect due to the coupling of the hopping particle to a measurement apparatus can also results into a slowdown of the particle propagation [65,66]. Related protocols have been studied in the context of quantum stochastic resetting, in which the hopping particle is reset to the initial state with a certain probability [67].…”

First detection probability in quantum resetting via random projective measurements

“…This question concerning the first detection time in projective measurements has been addressed in a number of recent papers [38-42, 49, 56, 57]. Most of these papers addressed a stroboscopic measurement protocol where a measurement is attempted periodically with a fixed period T, with the exception of [49,56,57] where more general measurement protocols were discussed. [42,56] developed a renewal formalism for general random measurement protocols to compute the first detection probability directly and used it to study a quantum random walk model [42,49].…”

“…Most of these papers addressed a stroboscopic measurement protocol where a measurement is attempted periodically with a fixed period T, with the exception of [49,56,57] where more general measurement protocols were discussed. [42,56] developed a renewal formalism for general random measurement protocols to compute the first detection probability directly and used it to study a quantum random walk model [42,49]. Below we present an alternative general formalism that is also adaptable to any measurement protocol.…”

Non-Hermitian description of sharp quantum resetting

“…Our work is concerned with the first detected time of the particles under the quantum evolution as explored in the Ref [17,18,19,20,21]. In a parallel study, the random resetting of the quantum state to its initial state is been studied as a process to generate non-equilibrium steady states, and the dynamics can be modeled as an open quantum system dynamics [22,23,24,25,26,27,28,29,30,31].…”