A sub-Planck structure from Compass states, composited by four coherent states in superposition, is reported experimentally through the correspondence between Wigner distribution function and frequency-resolved optical gating (FROG) measurement. Demonstration on the chessboard-like patterns with a time-bandwidth product smaller than that of a transform-limited pulse is realized through simultaneously sufficient time and frequency separations among optical pulses. The sub-Planck structure in the interference pattern from Compass states not only manifests itself as a generalized Schödinger's cat state, but also provides the platform to exploit quantum metrology and quantum information processing through ultrafast optics.
A sub-Planck structure from Compass states, composited by four coherent states in superposition, is reported experimentally through the correspondence between Wigner distribution function and frequency-resolved optical gating (FROG) measurement. Demonstration on the chessboard-like patterns with a time-bandwidth product smaller than that of a transform-limited pulse is realized through simultaneously sufficient time and frequency separations among optical pulses. The sub-Planck structure in the interference pattern from Compass states not only manifests itself as a generalized Schödinger's cat state, but also provides the platform to exploit quantum metrology and quantum information processing through ultrafast optics.
Exploiting the correspondence between the Wigner distribution function and the frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate the existence of chessboardlike interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase-space representation of compass states. Using superpositions of four electric pulses as the realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exactly chronocyclic correspondence to the sub-Planck structure in the interference pattern appearing for the superposition of two Schr\"odinger-cat-like states in a position-momentum phase space.
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 N-component compass states, which are obtained by superposing N≥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 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 N. 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.
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.
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) 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-enhanced sensitivity against displacements in all phase-space directions (N 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 N with the total angular momentum.
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/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.