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arXiv:2207.12706 · Physical Review A

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

Naeem Akhtar, Barry C. Sanders, Gao Xianlong

Published Oct 6, 2022• 27 citations
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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 $1/k$ compared with $1/\sqrt{k}$ 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.

DOI
10.1103/physreva.106.043704

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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 kkk. 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 1/k1/k1/k compared with 1/k1/\sqrt{k}1/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.
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 (anisotropic features), resulting in nonuniform sensitivity enhancement to phase-space displacements. Here, we construct NNN-component compass states, which are obtained by superposing N≥6N \geq 6N≥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 2πN\frac{2π}{N}N2π​. We observe that these generalized SU(1,1) compass states exhibit isotropic sub-Planck structures, leading to an isotropic enhancement in sensitivity to phase-space displacements that progressively increases with larger NNN. These states are directly relevant to quantum platforms supporting Kerr-type interactions between two bosonic modes, where the underlying SU(1,1) dynamical symmetry enables the generation of multicomponent SU(1,1) compass states. Specifically, the compact evolution of SU(1,1) coherent states under the considered dynamics enables the generation of multicomponent SU(1,1) compass states at specific times. We also investigate the effects of thermal decoherence on these multicomponent compass states within a frequency-resolved Lindblad framework, demonstrating the evolution and degradation of their nonclassical signatures.
Coherent-state superpositions are of great importance for many quantum subjects, ranging from foundational to technological, e.g., from tests of collapse models to quantum metrology. Here we explore various aspects of these states, related to the connection between sub-Planck structures present in their Wigner function and their sensitivity to displacements (ultimately determining their metrological potential). We review this for the usual Heisenberg-Weyl algebra associated with a harmonic oscillator and extend it to find analogous results for the su(2)\mathrm{su}(2)su(2) algebra, typically associated with angular momentum. In particular, in the Heisenberg-Weyl case, we identify phase-space structures with support smaller than the Planck action in both Schr\"odinger-cat-state mixtures and superpositions, the latter known as compass states. However, as compared to coherent states, compass states are shown to have N\sqrt{N}N​-enhanced sensitivity against displacements in all phase-space directions (NNN is the average number of quanta), whereas cat states and cat mixtures show such enhanced sensitivity only for displacements in specific directions. We then show that these same properties apply for analogous SU(2) states provided (i) coherent states are restricted to the equator of the sphere that plays the role of phase space for this group, (ii) we associate the role of the Planck action with the size of SU(2) coherent states in such a sphere, and (iii) we associate the role of NNN with the total angular momentum.
The superposition of coherent states has attracted extensive and intensive attention, exemplified by intriguing instances such as Schr\"odinger cat states and compass states. In this study, we derive analytical expressions of the Wigner functions and phase-space displacement sensitivity of circular states, which are superpositions of coherent states distributed equidistantly on a circle. The visualization of these phase-space functions exhibits fascinating quantum kaleidoscopic patterns, and intuitively reveals the internal coherence and symmetric structure in these states. Moreover, we unveil the inherent sub-Planck structure of circular states in terms of the zeros of sensitivity functions.
The Wigner function of the compass state (a superposition of four coherent states) develops phase-space structures of dimension much less than the Planck scale $\ensuremath{\hbar}$, which are crucial in determining the sensitivity of these states to phase-space displacements. In the present work we introduce compasslike states that may have connection to the contemporary experiments, which are obtained by either adding photons to or subtracting photons from the superposition of two squeezed-vacuum states. We show that when a significant quantity of photons is added (or subtracted), the Wigner functions of these states are shown to have phase-space structures of an area that is substantially smaller than the Planck scale. In addition, these states exhibit sensitivity to displacements that is much higher than the standard quantum limit. Finally, we show that both the size of the sub-Planck structures and the sensitivity of our states are strongly influenced by the average photon number, with the photon-addition case having a higher average photon number leading to the smaller sub-Planck structures and, consequently, being more sensitive to displacement than the photon-subtraction case. Our states offer unprecedented resolution to the external perturbations, making them suitable for quantum sensing applications.
Environmental decoherence occurs when a quantum system interacts with its surroundings, progressively reducing quantum interference and coherence, complicating the preservation of critical quantum features over time, especially during experimental implementation. The quantum features of a state can be represented in phase space via the Wigner function, which manifests across multiple scales, with decoherence potentially influencing each scale differently, as examined in this work. We consider the compass state and its photon-added and photon-subtracted variants (optimized compass states) as our representative examples, each of which exhibits phase-space features with dimensions beyond the Planck scale, making them suitable for quantum sensing applications. We investigate the interaction of these states with a heat reservoir by employing a range of well-established theoretical tools. We observe that compass states with finer-scale phase-space features are more fragile to decoherence, with parameters favoring greater sub-Planckness in phase space concomitantly increasing the fragility of these compass states to decoherence. Our findings are then validated for generic quantum states interacting with the heat reservoir, for which we provide analytical and numerical investigations, exploring the relationship between quantum state robustness to decoherence and the sizes of their phase-space features; that is, phase-space features at smaller scales decay faster under decoherence, and vice versa.
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 nnn 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.