Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states

TL;DR

This paper explores sub-Planck structures in SU(1,1) quantum states, revealing how superpositions enhance phase-space sensitivity and scale inversely with the Bargmann index, with implications for quantum measurement precision.

Contribution

It introduces a detailed analysis of sub-Planck structures in SU(1,1) states and demonstrates quadratic scaling improvements in sensitivity for superpositions compared to individual states.

Findings

Sub-Planck structures scale as 1/k for SU(1,1) superpositions.

Superpositions of four SU(1,1) coherent states exhibit nearly isotropic sub-Planck features.

Displacement sensitivity scales quadratically with the inverse of the Bargmann index.

Abstract

We investigate the sub-Planck-scale structures associated with the SU(1,1) group by establishing that the Planck scale on the hyperbolic plane can be considered as the inverse of the Bargmann index $k$. Our discussion involves SU(1,1) versions of Wigner functions, and the quantum-interference effect is easily visualized through plots of these Wigner functions. Specifically, the superpositions of four Perelomov SU(1,1) coherent states (compass state) yield nearly isotropic sub-Planck structures in phase space scaling as $\frac1{k}$ compared with $\frac1{\sqrt{k}}$ scaling for individual SU(1,1) coherent states and anisotropic quadratically improved scaling for superpositions of two SU(1,1) coherent states (cat state). We show that displacement sensitivity exhibits the same quadratic improvement to scaling.