Generating high-order quantum exceptional points in synthetic dimensions

Abstract

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-PT -symmetry. Compared to their PT -symmetric counterparts, such anti-PT -symmetric systems could be easier to scale and, thus, can serve as a promising platform for engineering quantum systems with high-order EPs.

$\mathcal{PT}$ phase transition in open quantum systems with Lindblad dynamics

“…Therefore, it is very important to investigate this question and we plan to do so in near future. Note that the studies on Liouvillian exceptional points have been actively conducted recently [16,17,42,45,46]. Also, as a natural extension, we may consider application of the third quantization to an open 2N -spin model which has alternately pumping spins along opposite directions.…”

“…Similar discussions have been conducted and it has been shown that there are infinite order Liouvillian exceptional points for other quadratic bosonic models in Refs. [42,45,46]. Also, it is known that at an exceptional point, the eigenspace cannot be spanned only by the eigenstates, but generalized eigenstates are required [40].…”

“…Also, in Refs. [42,45,46] it has been found that a matrix corresponding i H − N + M appears in the time derivative of one and two point correlation functions and it has conventional anti-PT symmetry [47] for the case where K = L = 0. Those results, however, have been obtained based on rather heuristic arguments and concrete calculations for specific models.…”