Sub-Planck structures: Analogies between the Heisenberg-Weyl and SU(2) groups

Abstract

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 to a harmonic oscillator, and extend it to find analogous results for the $\mathfrak{su}(2)$ 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 Schrödinger-cat-state mixtures and superpositions, the latter known as compass states. However, as compared to coherent states, compass states are shown to have $\sqrt{N}$-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 to the size of SU(2) coherent states in such a sphere, and (iii) we associate the role of $N$ with the total angular momentum.

References in corpus (56)

• Quantum entanglement
• Decoherence, einselection, and the quantum origins of the classical
• Advances in Quantum Metrology
• Gaussian Quantum Information
• Quantum-enhanced measurements: beating the standard quantum limit
• Quantum metrology with nonclassical states of atomic ensembles
• Decoherence, the measurement problem, and interpretations of quantum mechanics
• 14-qubit entanglement: creation and coherence
• Quantum Optical Metrology -- The Lowdown on High-N00N States
• Negativity of the Wigner function as an indicator of nonclassicality
• Dynamical Reduction Models
• Generation and manipulation of Schrödinger cat states in Rydberg atom arrays
• Experimental ten-photon entanglement
• Squeezed states of light and their applications in laser interferometers
• Sub-Planck spots of Schroedinger cats and quantum decoherence
• Sub-Planck phase-space structures and Heisenberg-limited measurements
• On-chip quantum interference of a superconducting microsphere
• Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity
• Quantifying Quantumness and the Quest for Queens of Quantum
• Experimental Preparation of High NOON States for Phonons
• Decay of the Loschmidt Echo for quantum states with sub-Planck scale structures
• Dissipative optomechanical preparation of macroscopic quantum superposition states
• Wigner Function Evolution of Quantum States in Presence of Self-Kerr Interaction
• Discrete Moyal-type Representations for a Spin
• Dissipative Quantum Metrology with Spin Cat States
• Mesoscopic Superposition of States with Sub-Planck Structures in Phase Space
• Extremal quantum states and their Majorana constellations
• Experimental Observation of Quantum Chaos in a Beam of Light
• Quantum metrology at the Heisenberg limit with ion traps
• Quantum-limited Euler angle measurements using anticoherent states
• Continuous phase-space representations for finite-dimensional quantum states and their tomography
• Achieving Heisenberg-limited metrology with spin cat states via interaction-based readout
• Short time decay of the Loschmidt echo
• Engineering Schrödinger cat states with a photonic even-parity detector
• Tuning between photon-number and quadrature measurements with weak-field homodyne detection
• 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
• Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States
• Deterministic generation of a four-component optical cat state
• Entanglement induced Sub-Planck structures
• Quantum Hypercube States
• Wigner negativity in spin-$j$ systems
• Wigner function for SU(1,1)
• Superpositions of SU(3) coherent states via a nonlinear evolution
• Connection between the N00N State and a superposition of SU(2) Coherent States
• Teleportation fidelity as a probe of sub-Planck phase-space structure
• Generation of Kerr non-Gaussian motional states of trapped ions
• Relating the Entanglement and Optical Nonclassicality of Multimode States of a Bosonic Quantum Field
• Quantum tetrachotomous states: Superposition of four coherent states on a line in phase space
• Direct Measurement of Time-Frequency Analogues of Sub-Planck Structures
• Coherent control of mesoscopic superpositions in a diatomic molecule
• Measuring sub-Planck structural analogues in chronocyclic phase space
• Sub-Planck structure in a mixed state

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• Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states
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• Superposing compass states for asymptotic isotropic sub-Planck phase-space sensitivity
• How squeezed states both maximize and minimize the same notion of quantumness
• Quantum walk on the Bloch sphere
• Seeding Gaussian boson samplers with single photons for enhanced state generation
• 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
• Nonclassicality and sub-Planck structures of photon subtracted compass states
• Phase-sensitive superposition of quantum states
• Multi-headed symmetrical superpositions of coherent states