# The Monte Carlo wave-function method: a robust adaptive algorithm and a study in convergence

## TL;DR

This paper introduces an adaptive-timestep Quantum Jump algorithm that enhances robustness and convergence in solving quantum master equations, addressing numerical challenges in traditional implementations.

## Contribution

It develops a stepwise adaptive-timestep version of the Quantum Jump method with proven robustness and analyzes its convergence properties based on a key parameter.

## Key findings

- The adaptive algorithm remains stable where standard methods fail.
- Convergence depends on the maximal jump probability parameter.
- The method is broadly applicable to various Quantum Jump implementations.

## Abstract

We present a stepwise adaptive-timestep version of the Quantum Jump (Monte Carlo wave-function) algorithm. Our method has proved to remain robust even for problems where the integrating implementation of the Quantum Jump method is numerically problematic. The only specific parameter of our algorithm is the single a priori parameter of the Quantum Jump method, the maximal allowed total jump probability per timestep. We study the convergence of ensembles of trajectories to the solution of the full master equation as a function of this parameter. This study is expected to pertain to any possible implementation of the Quantum Jump method.