We explore a single degenerate optical cavity supporting a synthetic two-dimensional space, which includes the frequency and the orbital angular momentum (OAM) axes of light. We create the effective gauge potential inside this synthetic space and show that the system exhibits topologically protected one-way edge states along the OAM axis at the boundaries of the frequency dimension. In this synthetic space, we present a robust generation and manipulation of entanglement between the frequency and OAM of photons. Our Letter shows that a higher-dimensional synthetic space involving multiple degrees of freedom of light can be achieved in a “zero-dimensional” spatial structure, pointing towards a unique platform to explore topological photonics and to realize potential applications in optical communications and quantum information processing.

Chern number is usually characterized by Berry curvature. Here, by investigating the Dirac model of even-dimensional Chern insulator, we give the general relation between Berry curvature and quantum metric, which indicates that the Chern number can be encoded in quantum metric as well as the surface area of the Brillouin zone on the hypersphere embedded in Euclidean parameter space. We find that there is a corresponding relationship between the quantum metric and the metric on such a hypersphere. We give the geometrical property of quantum metric. Besides, we give a protocol to measure the quantum metric in the degenerate system.

Efficient manipulation of quantum states is a key step towards applications in quantum information, quantum metrology, and nonlinear optics. Recently, atomic arrays have been shown to be a promising system for exploring topological quantum optics and robust control of quantum states, where the inherent nonlinearity is included through long-range hoppings. Here we show that a one-dimensional atomic array in a periodic magnetic field exhibits characteristic properties associated with an effective two-dimensional Hofstadter-butterfly-like model. Our work points out super- and sub-radiant topological edge states localized at the boundaries of the atomic array despite featuring long-range interactions, and opens an avenue of exploring an interacting quantum optical platform with synthetic dimensions.

The quantum coherence (QC) of two comoving atoms on a stationary trajectory is investigated. We develop a formalism to characterize the properties of atoms on a stationary trajectory. We give a criterion under which QC is frozen to a nonzero value. The frozen condition that vanishing super- or subradiant decay rate is not so sensitive to the initial condition of state. We show that enhanced QC and a subradiant state can be gained from the initial state.

We investigate the thermal Casimir effect of a massless scalar field for two parallel plates moving in the equatorial orbit in Kerr space–time. Under the assumption that the typical cavity size is much smaller than the orbital radius, proposed by Sorge, we deduce the analytical expression of the renormalized free energy in this curved space–time. We also get the analytical representation for the renormalized internal energy, and find that there is a thermal correction to the Casimir energy, which depends on the proper temperature and the proper geometrical parameters of the plates. The asymptotic behavior of the Casimir free energy, entropy and internal energy at low temperature is also investigated

We investigate Casimir effect as well as thermal Casimir effect for a pair of parallel perfectly plates placed in general stationary space-time background. It is found that the Casimir energy is influenced by the 00-component of metric and the corresponding quantity in dragging frame. We give a scheme to renormalize thermal correction to free energy in curved space-time. It is shown that the thermal corrections to Casimir thermodynamic quantities not only depend on the proper temperature and proper geometrical parameters of the plates, but also on the determinant of space-time metric.Comment: 4 pages, 2 figure