Running measurement protocol for the quantum first-detection problem

Abstract

The problem of the detection statistics of a quantum walker has received increasing interest, connected as it is with to the problem of quantum search. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time τ , with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of τ , from a state where detection decreases exponentially in time and the total detection is very small, to a state with power-law decay and a significantly higher probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical τ is 10/3, as opposed to 3 for every larger τ . In addition, the moving detector strongly modifies the Zeno effect.

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

“…The TBM (and related systems) when subject to projective measurements has in recent years been extensively studied in the context of detection problem corresponding to a quantum particle evolving under the dynamics to arrive at a chosen set of sites [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. In the context of the present work, we briefly summarize the contribution of [59] that also considers projective measurements at random times.…”

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

“…The TBM and related systems when subject to projective measurements with time evolution following scheme 2 of dynamics have in recent years been studied to address the issue of when does a quantum particle evolving under the dynamics arrive at a chosen set of sites [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. A crucial difference is that, barring [27], these contributions almost exclusively implement measurements at regular intervals of times, unlike the scheme considered in this work in which sequence of measurements at random times is implemented.…”

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

“…The TBM (and related systems) when subject to projective measurements has in recent years been extensively studied in the context of detection problem corresponding to a quantum particle evolving under the dynamics to arrive at a chosen set of sites [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. However, all these papers focus on the first time the quantum particle arrives at a chosen set of detection sites, unlike our treatment in which repeated detections are taken into account.…”