Schemes of Engineering Nodal Surfaces with an Open Four-Level Quantum System

Quantum systems coupled to the environment with gain and loss can often be described by non-Hermitian Hamiltonians. Such open systems can exhibit phenomena difficult to observe in closed systems, such as the emergence of novel topological matter induced by non-Hermiticity. For example, in two-dimensional momentum space, a Hermitian Weyl node can become a Weyl exceptional ring when subject to non-Hermitian perturbations. Whereas such nodal points and lines have already led to extensive studies of topological (semi)metals, nodal surfaces may further enrich topological physics but are not well-understood. Here, we propose realizing various nodal surfaces by utilizing an open four-level quantum system. Take atomic systems for example, one can choose four pseudospin states and couple them with microwaves or radio-frequency waves, whose coupling strengths, phases, and detunings constitute a parameter space. With engineered gain and loss, we show that nodal surfaces such as spheres, hyperboloids, cones, and cylinders can emerge in the parameter space. Our results may provide insights into exploring high-dimensional topological defects in synthetic quantum matter.