Superposing compass states for asymptotic isotropic sub-Planck phase-space sensitivity
Atharva Shukla, Barry C. Sanders
Abstract
Compass states deliver a sub-Planck phase-space structure in the sense that sensitivity to phase-space displacement is superior to the sensitivity of displacing the vacuum state in any direction, but this sensitivity is anisotropic: better sensitivity for some directions of phase-space displacement versus others. Here we introduce generalized compass states as superpositions of $n$ compass states, with each oriented by $\ensuremath{\pi}/2n$ with respect to its predecessor. Specifically, we derive Wigner functions for these generalized compass states and approximate closed-form expressions for overlaps between generalized compass states and their displaced counterparts. Furthermore, we show that generalized compass states, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, display isotropic sensitivity to phase-space displacement in any direction.