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. Previous proposals to realize non-Hermitian Hamiltonians (NHHs) with high-order EPs have been mainly based on either direct construction of spatial networks of coupled modes or utilization of synthetic dimensions, e.g., mapping of spatial lattices to time or photon-number space. Both methods rely on the construction of effective NHHs describing classical or postselected quantum fields, which neglect the effects of quantum jumps and which, thus, suffer from a scalability problem in the quantum regime, when the probability of quantum jumps increases with the number of excitations and dissipation rate. 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 operator 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 alternative NHHs describing, e.g., a spatial lattice of coupled resonators, where spatial sites are represented by high-order field moments in the synthetic space of field moments. Notably, such a mapping allows correct reproduction of the results of the Liouvillian dynamics, including quantum jumps. As an example, we consider a $U\left(1\right)$-symmetric quadratic Liouvillian describing a bimodal cavity with incoherent mode coupling, which can also possess $\mathrm{anti}\text{−}\mathcal{P}\mathcal{T}$ symmetry, whose field moment dynamics can be mapped to an NHH governing a spatial network of coupled resonators with high-order EPs.

• Received 1 March 2021
• Revised 14 June 2021
• Accepted 23 June 2021

DOI:https://doi.org/10.1103/PhysRevA.104.012205