Figure 1

Phase-space representation of the standard quantum limit for rotations. Coherent states have phase-space size of the order $\epsilon \simeq \sqrt{\hslash }$. A coherent state of complex amplitude $\sqrt{\hslash }\alpha$ is rotated around the origin by a small angle $\theta$. The initial and final coherent states are distinguishable (approximately orthogonal) when the linear displacement $\epsilon \approx s\simeq \sqrt{\hslash }\mid \alpha \mid \theta$, which leads to SQL sensitivity for rotations $\theta \simeq 1∕\mid \alpha \mid =1∕\sqrt{\overline{n}}$.Reuse & Permissions

Figure 2

(Color) The Wigner functions in the $\alpha$ plane for: (a) the displaced cat state $\mid \overline{{\text{cat}}_2}⟩\equiv \widehat{D}\left(\alpha \right)\mid {\text{cat}}_2⟩$, and (b) the compass state $\mid \text{compass}⟩$ for $\alpha =4i$. The displaced cat state is quasiorthogonal to the rotated state $\widehat{R}\left(\theta \right)\mid \overline{{\text{cat}}_2}⟩(\theta =\pi ∕4{\mid \alpha \mid }^2)$ in (c) at the Heisenberg limit scale $\theta \sim 1∕{\mid \alpha \mid }^2$. The compass state is quasiorthogonal to the translated compass state $\widehat{D}\left(\beta \right)\mid \text{compass}⟩(\beta =e^{i\pi ∕4}\pi ∕2\sqrt{2}\mid \alpha \mid )$ in (d) at the Heisenberg limit scale $\mid \beta \mid \sim 1∕\mid \alpha \mid$. The insets enlarge the central interference pattern of the displayed Wigner functions. In (e) and (f) we display the respective products of the unperturbed and perturbed Wigner functions. When performing the integration over the $\alpha$ plane, the negative contributions (in blue) cancel the positive ones (in red), leading to quasiorthogonality.Reuse & Permissions