Finer sub-Planck structures and displacement sensitivity of SU(1,1) circular states

Abstract

Quantum states with sub-Planck features exhibit sensitivity to phase-space displacements beyond the standard quantum limit, making them useful for quantum metrology. In the context of the SU(1,1) group, sub-Planck features have been constructed through the superposition of four Perelomov coherent states on the hyperbolic plane (the SU(1,1) compass state). However, these structures differ in scale along different phase-space directions (anisotropic features), resulting in nonuniform sensitivity enhancement to phase-space displacements. Here, we construct $N$-component compass states, which are obtained by superposing $N \geq 6$ SU(1,1) coherent states with an even total number, evenly arranged along a circular path on the hyperbolic plane; that is, all components lie at the same distance from the origin and have equal angular spacing of $\frac{2π}{N}$. We observe that these generalized SU(1,1) compass states exhibit isotropic sub-Planck structures, leading to an isotropic enhancement in sensitivity to phase-space displacements that progressively increases with larger $N$. These states are directly relevant to quantum platforms supporting Kerr-type interactions between two bosonic modes, where the underlying SU(1,1) dynamical symmetry enables the generation of multicomponent SU(1,1) compass states. Specifically, the compact evolution of SU(1,1) coherent states under the considered dynamics enables the generation of multicomponent SU(1,1) compass states at specific times. We also investigate the effects of thermal decoherence on these multicomponent compass states within a frequency-resolved Lindblad framework, demonstrating the evolution and degradation of their nonclassical signatures.