# Structures in higher-order quantum correlations due to non-spatial symmetries

## TL;DR

This paper explores how non-spatial symmetries like particle-hole, time-reversal, and chiral symmetries constrain higher-order quantum correlations in quantum nonlinear spectroscopy, revealing deep structural insights and new observable correlations.

## Contribution

It uncovers the influence of non-spatial symmetries on higher-order quantum correlations and establishes selection rules and relations to out-of-time-order correlations in QNS.

## Key findings

- Generalized C-symmetry imposes specific selection rules.
- T- and S-symmetries relate CTOCs to OTOCs.
- Time translation symmetry leads to a generalized fluctuation-dissipation theorem.

## Abstract

Quantum nonlinear spectroscopy (QNS) via a quantum sensor can access $2^{n-1}$ types of $n$-th order contour-time-ordered correlations (CTOCs) arising from different orderings of quantum operators, while classical nonlinear spectroscopy can detect only one in each order. QNS and its classical counterpart have similar spatial symmetry properties, but they are expected to have characteristically different non-spatial symmetry properties since different orderings of operators can behave differently under non-spatial transformations (such as exchange of operators). Here, we investigate how higher-order correlations extracted by QNS are constrained by non-spatial symmetries, including particle-hole (C), time-reversal (T), chiral (S) symmetry, and time translation symmetry. We find that the generalized C-symmetry imposes special selection rules on QNS, and the generalized T- and S-symmetry relate CTOCs to out-of-time-order correlations (OTOCs). The time translation symmetry leads to a generalized fluctuation-dissipation theorem for the spectra of higher-order CTOCs and OTOCs. This work discloses deep structures in higher-order quantum correlations due to non-spatial symmetries and provides access to certain types of OTOCs that are not directly observable.