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

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 $\frac{1}{k}$ compared with $\frac{1}{\sqrt{k}}$ 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.

arXiv comment15 pages, 7 figures

References in corpus (47)

• Decoherence, einselection, and the quantum origins of the classical
• Quantum-enhanced measurements: beating the standard quantum limit
• Sub-Planck spots of Schroedinger cats and quantum decoherence
• Operational Quantum Mechanics, Quantum Axiomatics and Quantum Structures
• Review of Entangled Coherent States
• Pumped-up SU(1,1) interferometry
• Sub-Planck phase-space structures and Heisenberg-limited measurements
• Entangled SU(2) and SU(1,1) coherent states
• Phase sensitivity at the Heisenberg limit in an SU(1,1) interferometer via parity detection
• Decay of the Loschmidt Echo for quantum states with sub-Planck scale structures
• Wigner Function Evolution of Quantum States in Presence of Self-Kerr Interaction
• Analytic representations based on SU(1,1) coherent states and their applications
• Discrete Moyal-type Representations for a Spin
• Overview of the phase space formulation of quantum mechanics with application to quantum technologies
• Mesoscopic Superposition of States with Sub-Planck Structures in Phase Space
• Experimental Observation of Quantum Chaos in a Beam of Light
• Quantum metrology at the Heisenberg limit with ion traps
• Achieving Heisenberg-limited metrology with spin cat states via interaction-based readout
• Continuous phase-space representations for finite-dimensional quantum states and their tomography
• Short time decay of the Loschmidt echo
• Superposition of two-mode squeezed states for quantum information processing and quantum sensing
• Generation of a superposition of multiple mesoscopic states of radiation in a resonant cavity
• Sub-Planck scale structures in the P{ö}schl-Teller potential and their sensitivity to perturbations
• Time-frequency Domain Analogues of Phase Space Sub-Planck Structures
• Mesoscopic superposition and sub-Planck-scale structure in molecular wave packets
• Sub-Planck scale structures in a vibrating molecule in the presence of decoherence
• Entanglement induced Sub-Planck structures
• Deterministic generation of a four-component optical cat state
• Quantum Hypercube States
• Wigner negativity in spin-$j$ systems
• Sub-Planck structures: Analogies between the Heisenberg-Weyl and SU(2) groups
• Wigner function for SU(1,1)
• SU(1,1) symmetry of multimode squeezed states
• Teleportation fidelity as a probe of sub-Planck phase-space structure
• Unitary transformations for testing Bell inequalities
• Connection between the N00N State and a superposition of SU(2) Coherent States
• Generation of Kerr non-Gaussian motional states of trapped ions
• Quantum tetrachotomous states: Superposition of four coherent states on a line in phase space
• The SU(1,1) Perelomov number coherent states and the non-degenerate parametric amplifier
• Completeness of photon-added squeezed vacuum and one-photon-states and of photon-added coherent states on a circle
• Direct Measurement of Time-Frequency Analogues of Sub-Planck Structures
• Measuring sub-Planck structural analogues in chronocyclic phase space
• Coherent control of mesoscopic superpositions in a diatomic molecule
• $SU(1,1)$ covariant $s$-parametrized maps
• Sub-Planck structure in a mixed state
• Unified approach to the nonlinear Rabi models
• Non-Gaussian features from Excited Squeezed Vacuum State

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• Sub-Planck structures and sensitivity of the superposed photon-added or photon-subtracted squeezed-vacuum states
• Superposing compass states for asymptotic isotropic sub-Planck phase-space sensitivity
• Enhanced entanglement and controlling quantum steering in a Laguerre-Gaussian cavity optomechanical system with two rotating mirrors
• Compasslike states in a thermal reservoir and fragility of their nonclassical features
• Multicomponent cat states with sub-Planckian structures and their optomechanical analogues
• Sub-shot-noise sensitivity via superpositions of two deformed kitten states
• Quantum walk for SU(1,1)