Fig. 1: Composite parametric oscillator (PO) as a D-dimensional hyperspin.
a The spin consists of D degenerate POs (colored dots) saturating the same pump field (gray arrows and area) with equal intrinsic loss (purple arrows). b Compact representation of the scheme in a. The colored dots denote the D POs, described by dynamical variables x1, …xD, and the gray circle represents the common pump. c, d, e, f Fixed points of the composite PO system. The fixed points lie on the surface of a D-dimensional hypersphere. For D = 1, there are two fixed points on the x-axis representing the two states of an Ising spin; For D = 2, the fixed points lie on a circumference in the xy-plane, encoding the continuous phase of a XY spin; For D = 3, the fixed points lie on the surface of a sphere in the xyz-space, encoding the two angles of an Heisenberg spin; For D = 4, the fixed points are represented by encoding three of the four coordinates into a point within the volume of a sphere in the xyz-space, and the fourth coordinate w is encoded as a color with extremal values wmin and wmax in the colormap. g, h, i, j Left panels are the composite PO representation of the D-dimensional spin as in panel b, while right panels are a three-dimensional representation of the composite PO as a spin \(\overrightarrow{\sigma }\) in standard hyperspherical coordinates (blue arrow). For D = 4, borrowing the terminology from QCD, the arrow represents the “vector” xyz-component of the QCD spin, and the color assigned to the outer sphere encodes the “scalar” w-component.
