Detection of a quantum particle on a lattice under repeated projective measurements

TL;DR

This paper investigates the detection probabilities of a quantum particle on a lattice under repeated measurements, introducing a perturbative approach and revealing power-law decay in survival probabilities with connections to non-Hermitian Hamiltonian dynamics.

Contribution

It develops a general perturbative method to analyze quantum detection on lattices and provides exact solutions for survival probabilities in specific models.

Findings

Survival probability decays as a power law over time.

Exact solutions for a mean field model with all-to-all hopping.

Strong agreement between analytical and numerical results.

Abstract

We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect it's arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals $\tau$, and we consider the evolution of the wave function till the time a detection occurs. We study the probabilities of its first detection at some time and conversely the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, consisting of unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian. For some examples, of a particle moving on one and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results. A mean field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and non-trivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.