# Quantum dynamics of long-range interacting systems using the positive-P and gauge-P representations

## TL;DR

This paper develops and tests stochastic positive-P and gauge-P simulation methods for long-range interacting bosonic systems, showing they can effectively extend simulation times and handle large systems without truncating the quantum state.

## Contribution

It introduces optimized stochastic gauges and analytical estimates to improve the efficiency and applicability of positive-P and gauge-P methods for long-range quantum systems.

## Key findings

- Long-range interactions do not significantly limit positive-P and gauge-P methods.
- Stochastic gauges can extend simulation times effectively.
- Different gauges are optimal depending on system size.

## Abstract

We provide the necessary framework for carrying out stochastic positive-P and gauge-P simulations of bosonic systems with long range interactions. In these approaches, the quantum evolution is sampled by trajectories in phase space, allowing calculation of correlations without truncation of the Hilbert space or other approximations to the quantum state. The main drawback is that the simulation time is limited by noise arising from interactions. We show that the long-range character of these interactions does not further increase the limitations of these methods, in contrast to the situation for alternatives such as the density matrix renormalisation group. Furthermore, stochastic gauge techniques can also successfully extend simulation times in the long-range-interaction case, by making using of parameters that affect the noise properties of trajectories, without affecting physical observables. We derive essential results that significantly aid the use of these methods: estimates of the available simulation time, optimized stochastic gauges, a general form of the characteristic stochastic variance and adaptations for very large systems. Testing the performance of particular drift and diffusion gauges for nonlocal interactions, we find that, for small to medium systems, drift gauges are beneficial, whereas for sufficiently large systems, it is optimal to use only a diffusion gauge. The methods are illustrated with direct numerical simulations of interaction quenches in extended Bose-Hubbard lattice systems and the excitation of Rydberg states in a Bose-Einstein condensate, also without the need for the typical frozen gas approximation. We demonstrate that gauges can indeed lengthen the useful simulation time.