Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit

TL;DR

This paper explores quantum system dynamics under continuous measurements, showing how non-Hermitian Hamiltonians describe the evolution, and provides analytical results for a particle on a lattice with implications for the continuum limit.

Contribution

It introduces a non-Hermitian effective Hamiltonian framework for continuous quantum measurements and derives analytical results for particle survival and arrival times.

Findings

Avoidance of the Zeno effect with specific system-detector coupling

Analytic expressions for survival probability and first arrival time

Continuum limit yields free Schrödinger evolution with complex boundary conditions

Abstract

The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schrodinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented.