# Short-time expansion of Heisenberg operators in open collective quantum spin systems

## TL;DR

This paper introduces a short-time expansion method for Heisenberg operators in large open collective spin systems, enabling accurate calculations of quantum properties under decoherence relevant for quantum metrology and information scrambling.

## Contribution

The authors develop a Taylor expansion approach for Heisenberg operators in collective spins, allowing efficient short-time analysis in large, decohering quantum systems, including extensions to spin-boson models.

## Key findings

- Accurately computed spin squeezing and correlation functions for 10^4 spins.
- Demonstrated effectiveness in strong-decoherence regimes inaccessible to previous methods.
- Applicable to systems relevant for trapped ion and cavity QED experiments.

## Abstract

We present a new method to compute short-time expectation values in large collective spin systems with generic Markovian decoherence. Our method is based on a Taylor expansion of a formal solution to the equations of motion for Heisenberg operators. This expansion can be truncated at finite order to obtain virtually exact results at short times that are relevant for metrological applications such as spin squeezing. In order to evaluate the expansion for Heisenberg operators, we compute the relevant structure constants of a collective spin operator algebra. We demonstrate the utility of our method by computing spin squeezing, two-time correlation functions, and out-of-time-ordered correlators for $10^4$ spins in strong-decoherence regimes that are otherwise inaccessible via existing numerical methods. Our method can be straightforwardly generalized to the case of a collective spin coupled to bosonic modes, relevant for trapped ion and cavity QED experiments, and may be used to investigate short-time signatures of quantum chaos and information scrambling.