Quantum-jump vs stochastic Schr\"{o}dinger dynamics for Gaussian states with quadratic Hamiltonians and linear Lindbladians

TL;DR

This paper compares quantum-jump and stochastic Schr"odinger dynamics for Gaussian states under quadratic Hamiltonians and linear Lindbladians, highlighting differences in their individual trajectories despite identical averaged Lindblad evolution.

Contribution

It introduces a method to generate quantum-jump trajectories using Gaussian state evolution and compares these with stochastic Schr"odinger dynamics for specific quantum harmonic oscillator examples.

Findings

Gaussian states remain Gaussian under stochastic Schr"odinger dynamics.

Quantum-jump trajectories generally do not remain Gaussian.

Both unravellings converge to the same Lindblad dynamics on average.

Abstract

The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schr\"odinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schr\"odinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schr\"odinger and the quantum-jump dynamics.