Non-spatial symmetries in quantum nonlinear spectroscopy

Nonlinear spectroscopy is a powerful approach to extracting information, in particular higher-order correlations of physical quantities. Quantum nonlinear spectroscopy (QNS) can access exponentially more types of correlations than its classical counterpart, since the responses to ``classical forces'' correspond to contour-time-ordered correlations (CTOCs) that involve only commutators, while in QNS ``quantum forces'' from a quantum sensor induce responses corresponding to CTOCs that involve both commutators and anti-commutators. Symmetries are important for constraining and analyzing nonlinear spectroscopy. Quantum and classical nonlinear spectroscopy have similar spatial symmetry properties. QNS, however, is expected to have its characteristic non-spatial symmetry properties since commutators and anti-commutators of physical quantities 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), and their combination, i.e., chiral (S) 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). 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.