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arXiv:2406.11055 · Optics Express

Ultra-broadband, ultra-narrowband, and ultra-passband composite polarization half-wave plates, ultra-broadband composite polarization -rotators, and quantum-classical analogy

Hayk L. Gevorgyan

Published Jan 24, 2025• 2 citations
arXivHTMLPDF

Abstract

Composite pulses, which produce ultra-broadband, ultra-narrowband and ultra-passband x-, y-rotations R x (θ)=exp⁡{−iθσ x /2}, R y (θ)=exp⁡{−iθσ y /2} by the rotational angle θ = π on the Bloch-Poincaré sphere, are presented. In polarization optics, the first class plays a role in the design of achromatic polarization retarders, while the second class corresponds to chromatic polarization filters. The third class is an assortment of the above two classes. Besides, composite pulses, which produce ultra-broadband z-rotations R z (ζ)=exp⁡−iζσ z /2 by the phasal angle ζ = π on the same sphere, are presented. These phasal pulses coincide with achromatic polarization π rotators. The evolution matrix and the Jones matrix have similar forms in quantum optics and polarization optics. In the context of quantum optics, quantum computing, and quantum information processing, following the quantum-classical analogy, we achieve ultra-robust, ultrasensitive, and ultrasquare quantum control of an X (NOT) gate, and ultra-robust control of a Z phase gate, with respect to pulse area error. Our proposed UBBN (BatN), UNBN (SnakeN), UPBN (OctopusN), and UBBPhN (BatPhN) sequences are novel due to the novel methods of derivation. They surpass and outperform the superior candidates of the similar subclass of altering variable rotations in literature — BB2, NB2, and PB2, in precision parameters such as broadness, narrowness, FWHM, or execution time. We suggest that, in general, the presented composite pulse sequences are interdisciplinary and can have other possible applications, such as nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) and spectroscopy, quantum sensing, local addressing, optical switching, cold-atom interferometry, quantum communication, quantum cryptography, quantum image processing, and quantum steganography.

DOI
10.1364/oe.545605

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•Oct 1, 2026•0 citations
Composite pulses, which produce ultra-broadband, ultra-narrowband and ultra-passband x-, y-rotations R x (θ)=exp⁡{−iθσ x /2}, R y (θ)=exp⁡{−iθσ y /2} by the rotational angle θ = π on the Bloch-Poincaré sphere, are presented. In polarization optics, the first class plays a role in the design of achromatic polarization retarders, while the second class corresponds to chromatic polarization filters. The third class is an assortment of the above two classes. Besides, composite pulses, which produce ultra-broadband z-rotations R z (ζ)=exp⁡−iζσ z /2 by the phasal angle ζ = π on the same sphere, are presented. These phasal pulses coincide with achromatic polarization π rotators. The evolution matrix and the Jones matrix have similar forms in quantum optics and polarization optics. In the context of quantum optics, quantum computing, and quantum information processing, following the quantum-classical analogy, we achieve ultra-robust, ultrasensitive, and ultrasquare quantum control of an X (NOT) gate, and ultra-robust control of a Z phase gate, with respect to pulse area error. Our proposed UBBN (BatN), UNBN (SnakeN), UPBN (OctopusN), and UBBPhN (BatPhN) sequences are novel due to the novel methods of derivation. They surpass and outperform the superior candidates of the similar subclass of altering variable rotations in literature — BB2, NB2, and PB2, in precision parameters such as broadness, narrowness, FWHM, or execution time. We suggest that, in general, the presented composite pulse sequences are interdisciplinary and can have other possible applications, such as nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) and spectroscopy, quantum sensing, local addressing, optical switching, cold-atom interferometry, quantum communication, quantum cryptography, quantum image processing, and quantum steganography.
High-precision, robust quantum gates are essential components in quantum computation and information processing. In this study, we present an alternative perspective, exploring the potential applicability of quantum gates that exhibit heightened sensitivity to errors. We investigate such sensitive quantum gates, which, beyond their established use in nuclear magnetic resonance spectroscopy and polarization optics, may offer significant utility in the other areas where selectivity, filtering, sensing, localization, or addressing properties are of interest. Utilizing the composite pulses technique, we derive three fundamental quantum gates with narrowband and passband characteristics—the () gate, the gate, and gates enabling arbitrary rotations. To systematically design these composite pulse sequences, we introduce the SU(2), modified-SU(2), and regularization random search methodologies. These approaches demonstrate superior performance compared to established sequences in the literature, including NB1, SK1, and PB1.
Broadband and ultra-broadband half-wave plates with composite design were previously experimentally demonstrated by E. Dimova et. al. , in Opt. Commun. 366 , 382 (2016). To achieve broadband and ultra-broadband performance, the authors utilized the relative rotation between the individual identical half-wave plates as control parameters, thus simulating the well-known control technique of composite pulses from quantum optics. In this paper, we extend the broadband and ultra-broadband designs by relaxing the condition of having identical constituent half-wave plates and allow for their thicknesses to also be variable control parameters. We further relax the condition of having symmetric rotation angles and solve numerically for both control parameters - rotation angles and thicknesses of the birefringent plates. We show that the presented design is broadband over a much wider range of the wavelengths, compared to the original design, when using the same number of constituent half-wave plates.
We develop a systematic approach to derive narrowband (NB) and passband (PB) composite sequences which can produce any preselected transition probability with any desired accuracy. The NB composite pulses are derived by successive cancellation of derivative terms in the propagator. The PB pulses are built by combining two $\ensuremath{\pi}/2$ NB pulses. Both the NB and the PB pulses feature vanishing wings on both sides of the central region around the target transition probability, and the PB pulses offer also a flat (broadband) top around the target probability. The NB sequences are particularly suitable for sensing and metrology applications, whereas the PB sequences can be very useful for qubit control in the case of tightly spaced laser-driven qubits for they ensure both selectivity (suppressed cross talk to neighbors) and robustness (suppressed laser pointing instability). Other possible applications include selective (NB) or both selective and robust (PB) control of close transitions in frequency space.
States can be mapped between the Poincaré space (representing photon polarizations) and the Bloch space (representing states of a quantum spin- particle). Using this mapping and analogous mappings for unitary operators, techniques and formalisms developed in one space can be applied in the other. As an example, error-correction strategies originally developed in the context of pulsed nuclear magnetic resonance experiments are applied to linear optics to propose the design of a new class of composite high-fidelity broadband linear retarder.
Composite pulse sequences, which produce arbitrary predefined rotations of a qubit on the Bloch sphere, are presented. The composite sequences contain up to 17 pulses and can compensate up to 8 orders of experimental errors in the pulse amplitude and the pulse duration. Composite sequences for three basic quantum gates, X (not), Hadamard and arbitrary rotation, are derived. Three classes of composite sequences are presented: one symmetric and two asymmetric. They contain as their lowest members two well-known composite sequences: the three-pulse symmetric SCROFULOUS pulse and the four-pulse asymmetric BB1 pulse, which compensate first- and second-order errors, respectively. The shorter sequences are derived analytically, and the longer ones numerically (instead by nesting and concatenation, as usually done hitherto). Consequently, the composite sequences derived here match or outperform the existing ones in terms of either speed or accuracy, or both. For example, we derive a second-order composite sequence, which is faster (by about 13%) than the famous BB1 sequence. For higher-order sequences, the speedup becomes much more pronounced. This is important for quantum information processing as the sequences derived here provide more options for finding the sweet spot between ultrahigh fidelity and high speed.
FINDING: Composite pulse sequences on the Bloch sphere achieve arbitrarily accurate SU(2) rotations by nesting error-compensating rotations, with the key structural insight being that any single-qubit gate is a 3D rotation in SU(2) ≅ SO(3), parameterized by Euler angles on the sphere. MATH: - Qubit state: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, with θ ∈ [0,π], φ ∈ [0,2π) — the Bloch sphere coordinates. - SU(2) generators: Pauli matrices σ_x, σ_y, σ_z, satisfying [σ_i, σ_j] = 2iε_{ijk}σ_k. - Rotation operator: R_n̂(θ) = exp(−iθ n̂·σ/2) = cos(θ/2)I − i sin(θ/2)(n̂·σ). - Composite pulse: U_total = Π_k R_{n̂_k}(θ_k), designed so that first-order (and higher) error terms in pulse amplitude/duration vanish: ∂U_total/∂ε |_{ε=0} = 0, up to desired order N. - The arXiv paper (1810.03833) explicitly constructs sequences with arbitrary N, using nested commutators of rotation axes to cancel error terms — a Lie-algebraic approach. CONNECTION: - The Bloch sphere is the unit sphere S², and SU(2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com