First Passage Times for Continuous Quantum Measurement Currents

TL;DR

This paper develops an efficient, analytical framework for calculating first passage times in continuously measured quantum systems, applicable to quantum jump and diffusion processes, with practical applications in quantum detection and uncertainty relations.

Contribution

It introduces a charge-resolved master equation approach with absorbing boundaries to compute FPTs analytically and efficiently in quantum measurement contexts.

Findings

Framework applies to quantum jump and diffusion processes.

Method tests kinetic uncertainty relations for quantum jumps.

Optimizes qubit-based detection of Rabi pulses.

Abstract

The First Passage Time (FPT) is the time taken for a stochastic process to reach a desired threshold. In this letter we address the FPT of the stochastic measurement current in the case of continuously measured quantum systems. Our approach is based on a charge-resolved master equation, which is related to the Full-Counting statistics of charge detection. In the quantum jump unravelling this takes the form of a coupled system of master equations, while for quantum diffusion it becomes a type of quantum Fokker-Planck equation. In both cases, we show that the FPT can be obtained by introducing absorbing boundary conditions, making their computation extremely efficient {and analytically tractable}. The versatility of our framework is demonstrated with two relevant examples. First, we show how our method can be used to study the tightness of recently proposed kinetic uncertainty relations (KURs) for quantum jumps, which place bounds on the signal-to-noise ratio of the FPT. Second, we study the usage of qubits as threshold detectors for Rabi pulses, and show how our method can be employed to maximize the detection probability while, at the same time, minimize the occurrence of false positives.