Generating high-order quantum exceptional points

Recently, there has been intense research in proposing and developing various methods for constructing high-order exceptional points (EPs) in dissipative systems. These EPs can possess a number of intriguing properties related to, e.g., chiral transport and enhanced sensitivity. Proposals to realize high-order EPs have been based on the use of non-Hermitian Hamiltonians (NHHs) of composite systems, i.e., the operators describing the evolution of coupled post-selected systems or coupled intense light fields. In both cases, quantum jumps play no role. Here, by considering the full quantum dynamics of a quadratic Liouvillian superoperator, we introduce a simple and effective method for engineering NHHs with high-order quantum EPs, derived from evolution matrices of system operators moments. That is, by quantizing higher-order moments of system operators, e.g., of a quadratic two-mode system, the resulting evolution matrices can be interpreted as the new NHHs describing, e.g., networks of coupled resonators. Notably, such a mapping allows to correctly reproduce the results of the Liouvillian dynamics, including quantum jumps. By applying this mapping, we demonstrate that quantum EPs of any order can be engineered in dissipative systems and can, thus, be probed by the coherence and spectral functions. As an example, we consider a $U(1)$-symmetric quadratic Liouvillian describing an optical cavity with incoherent mode coupling, which can also possess anti-$\cal PT$-symmetry. Compared to their $\cal PT$-symmetric counterparts, such anti-$\cal PT$-symmetric systems could be easier to scale and, thus, can serve as a promising platform for engineering quantum systems with high-order EPs.