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

TL;DR

This paper introduces a new class of SU(1,1) compass states with evenly spaced superpositions on the hyperbolic plane, creating isotropic sub-Planck features that enhance phase-space displacement sensitivity uniformly, with improvements increasing as the number of components grows.

Contribution

The authors construct generalized SU(1,1) compass states with evenly distributed components, achieving uniform sub-Planck features and improved displacement sensitivity, extending prior nonuniform structures.

Findings

Circularly shaped sub-Planck features achieved

Enhanced uniform sensitivity to phase-space displacements

Results verified for superpositions with up to 16 components

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, resulting in nonuniform sensitivity enhancement. We overcome this limitation by constructing $\overline{n}$-component compass states, which are obtained by superposing $\overline{n} \geq 6$ 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 $\frac{2\pi}{\overline{n}}$. These generalized SU(1,1) compass states generate circularly shaped sub-Planck features (isotropic sub-Planckness) and provide uniform enhancement in sensitivity to phase-space displacements. As the number of coherent states $\overline{n}$ increases, these refinements progressively improve. While verified for $\overline{n} = 16$ SU(1,1) coherent states, the results remain valid for superpositions with arbitrarily large $\overline{n}$ components.