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

TL;DR

This work addresses the anisotropy of sub-Planck features in SU(1,1) compass states by constructing $\overline{n}$-component, evenly spaced circular superpositions on the Poincaré disk to realize isotropic sub-Planck structures. The method generalizes SU(1,1) coherence into $\overline{n}$-component circular superpositions, demonstrating isotropic enhancement in phase-space displacement sensitivity that grows with $\overline{n}$, with explicit analysis up to $\overline{n}=16$. The paper connects displacement sensitivity to overlaps $S(\delta)$ and Wigner functions, showing a transition from anisotropic to isotropic metrological performance as $\overline{n}$ increases. The results offer robust, isotropic resources for quantum metrology that surpass the standard quantum limit, and they draw connections to two-mode squeezed states in the large-$k$ limit.

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 $\frac{2π}{\overline{n}}$. 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.

Figures (4)

• Figure 1: SU(1,1) circular states are composed of $\overline{n}$ SU(1,1) coherent states (red circles) evenly distributed along a circular path in the Poincaré disk (the SU(1,1) phase space). We consider superpositions with $\overline{n} > 4$, restricted to even integers, where each adjacent component is separated by an equal angular spacing of $\frac{2\pi}{\overline{n}}$. In the limit $\overline{n} \to \infty$, the superposition approaches a continuous circular form.
• Figure 2: The SU(1,1) Wigner distributions of our $\overline{n}$-component compass states are mapped onto the Poincaré disk for different values of $\overline{n}$, with each case containing a distinct central phase-space region highlighted by a black dot. (a)-(c) depict the SU(1,1) cat state: (a) displays the Wigner distribution, (b) highlights the central feature, and (c) displays the zeros associated with this central feature, with red and purple curves corresponding to $k=12$ and $k=16$, respectively. (d)-(f) represent the case when $\overline{n}=4$, it is the SU(1,1) compass state, where its Wigner distribution is shown in (d), its central feature in (e), and the zeros associated with this feature in (f), again with red and purple curves corresponding to $k=12$ and $k=16$, respectively. The case $\overline{n}=6$ is presented in (g)–(i), where (g) shows the Wigner distribution, (h) emphasizes the central feature, and (i) depicts the zeros corresponding to this feature (red curve for $k=12$, purple curve for $k=16$). Finally, for $\overline{n}=8$ (j)–(l), (j) shows the Wigner distribution, (k) the central feature, and (l) the zeros associated with it for $k=12$ (red curve) and $k=16$ (purple curve). In all cases, $\overline{\tau}=1.5$.
• Figure 3: The left panels show the overlap $S_{\ket{\bigcirc_{\overline{n}}}}(\delta)$ with $\delta = \delta x + \text{i} \delta p$, where the corresponding zeros are illustrated in the right panels of each image. (a) $\overline{n}=2$; (b) $\overline{n}=4$; (c) $\overline{n}=6$; (d) $\overline{n}=8$. In all exhibited scenarios, $\overline{\tau}=1.5$.
• Figure 4: The SU(1,1) Wigner distribution and their associated overlaps for $\overline{n}$ component SU(1,1) superpositions of the present work: (a) Wigner distribution for $\overline{n} = 10$, with its central close-up in (d) and associated overlap in (g); (b) Wigner distribution for $\overline{n} = 12$, with its central close-up in (e) and associated overlap in (h); and (c) Wigner distribution for $\overline{n} = 16$, with its central close-up in (f) and associated overlap in (i). In all cases, $\overline{\tau}=1.5$.