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Connectivity of two-qubit logic gates plays a crucial and indispensable role in quantum computation research. For the cold atom qubit platform, while the two-qubit Rydberg blockade gate has recently made rapid experimental progress, a pressing challenge is to improve connectivity in pursuit of genuine scalability without sacrificing speed or fidelity. A significant advancement in this direction can be achieved by introducing an extra buffer atom to extend the two-qubit gate beyond purely nearest-neighbor two-body interactions. The buffer atom couples with the two qubit atoms through nearest-neighbor interactions, even though the qubit atoms do not directly exert any physical influence on each other. The established method of off-resonant modulated driving (ORMD) is not only convenient but also lays the groundwork for this latest development. Although the atomic linkage structure here exhibits more complex interactions compared to previous two-body systems, the population can satisfactorily return to the ground state after the ground-Rydberg transition with a properly designed modulation waveform. This can be achieved through one-photon and two-photon ground-Rydberg transitions in common practices. Furthermore, with buffer atom relay or similar structures, it is possible to realize a two-qubit entangling gate between two distant qubit atoms. In addition to demonstrating that such solutions are feasible, the representative modulation patterns are analyzed, showcasing the versatility of buffer-atom-mediated two-qubit gates. From a broader perspective, these efforts enhance the resemblance between the cold atom qubit platform and the superconducting qubit system, with the buffer atom functioning like wires and junctions. [ABSTRACT FROM AUTHOR]

Polarization imaging, which can reveal indiscernible information that cannot easily be detected with traditional intensity cameras, plays a crucial role in many research fields. There are two main types of polarization imaging methods: time‐division multiplexed polarization imaging and space‐division multiplexed polarization imaging. These two methods confront the problems of temporal and spatial resolution reduction. In this work, a rapid linear‐polarization arbitrary manipulation technique based on Pancharatnam–Berry Optical Elements is proposed and achieved as an arbitrary polarization manipulation module with a manipulation speed of up to 899 Hz. Applying this compact modulator to optical or microscopic imaging systems can easily establish a rapid, high‐resolution polarization imaging system. This method is verified at microscopic detection, indoor and outdoor photographic experiments, visible and near‐infrared bands, exhibiting excellent imaging results. It is proven as a valid approach to solve the major dilemma of the incompatibility between temporal and spatial resolution in polarization imaging, it also highlights the potential of multidimensional imaging applications. [ABSTRACT FROM AUTHOR]

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems. [ABSTRACT FROM AUTHOR]

We studied the formalism for construction of geometric phases (Berry phases) in the neutral kaon mixing system. We explicitly calculated the geometric phases for the C P conserving and C P violating scenarios with effective meson field theory and the C P violating phase δ has been shown to be of geometrical origin. [ABSTRACT FROM AUTHOR]

Structured light refers to the light field tailored by various degrees of freedom including intensity, phase, and polarization states in both spatial and temporal domains, which may greatly vitalize the technologies in both optics, such as the next‐generation optical communication as well as subwavelength imaging and the materials science in both fabrication and characterization. The structured characteristics of the structure light need materials also with structured optical properties that can generate or manipulate structured light in a straightforward way, which can be well satisfied by liquid crystals, a soft mater that can self‐assemble into tunable ordered structures through external stimuli. This review summarizes the research progress of the liquid crystal‐based devices used in structured light generations and modulations, including the well‐established techniques in the market, like the spatial light modulator, q‐plate and the liquid crystal integrated optical metasurfaces. Especially, light‐matter interactions are discussed from the topological view of both the structured light and the liquid crystal structures. Such a perfect matching in topology makes the liquid crystal a promising star together with structured light in future optic and photonic technologies. [ABSTRACT FROM AUTHOR]

Optical encryption is an increasingly significant technique in the realm of information security. In the recent decade, there has been considerable interest in using planar optics elements for information encryption. However, information leakage possibly occurs due to limited encrytion channels available for single‐layer devices. To circumvent this problem, a novel encryption method is put forward using secret sharing cascaded liquid crystal (LC) elements with spatial dislocation, which can produce near‐field patterns and far‐field holographic images under different illumination conditions. Specifically, Malus's Law and its inherent one‐to‐four mapping of rotational degeneracy, along with the Pancharatnam‐Berry (PB) phase introduced by LC molecules, to achieve multi‐channel encryption are utilized. Therefore, each cascaded LC unit can manipulate the amplitude and phase imparted to output light independently, thus only by obtaining both LC devices can decryption be realized. To further enhance the encryption security, the author purposely divide each LC device into multiple regions and find that the encrypted patterns can only be recovered when the two LC elements align precisely with a specific dislocation. These experimental measurements agree well with the design, thus demonstrating the strong encryption capability and broad application prospects of the design approach in the field of optical encryption with high cost‐effectiveness. [ABSTRACT FROM AUTHOR]

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

In the nonequilibrium steady state of electronic transport across a spin‐resolved quantronic junction, the role of cotunneling on the emergent statistics under phase‐different adiabatic modulation of the reservoirs' chemical potentials is investigated. By explicitly identifying the sequential and inelastic cotunneling rates, the geometric or Pancharatnam–Berry contribution to the spin exchange flux between the spin system and the right reservoir is numerically evaluated. The relevant conditions wherein the sequential and cotunneling processes compete and selectively influence the total geometric flux upshot are identified. The Fock space coherences are found to suppress the cotunneling effects when the system reservoir couplings are comparable. The cotunneling contribution to the total geometric flux can be made comparable to the sequential contribution by creating a right‐sided asymmetrically stronger system‐reservoir coupling strength. Using a recently proposed geometric thermodynamic uncertainty relationship, the total rate of minimal entropy production is estimated. The geometric flux and the minimum entropy is found to be nonlinear as a function of the interaction energy of the junctions' spin orbitals. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the geometric phase (GP) of two-mode entangled squeezed-coherent states (ESCSs), undergoing unitary cyclic evolution. Results show that increasing the squeezing parameter of either mode of the balanced ESCS compresses the GP elliptically with respect to the coherence parameter of the corresponding mode. While in the case of unbalanced ESCS, the GP is compressed hyperbolically by increasing the squeezing parameters of the either mode. By generalizing the approach to higher constituting-state dimensions, it is found that the GPs of both balanced and unbalanced ESCSs, increase for a specific value of the coherence parameter. By analyzing the states through the Schmidt decomposition method, we find that, locally, the balanced and unbalanced ESCSs are unitarily equivalent. Finally, based on the interferometry technique, we suggest a theoretical scheme for the physical generation of ESCSs. [ABSTRACT FROM AUTHOR]

Phase and amplitude modes, also called polariton modes, are emergent phenomena that manifest across diverse physical systems, from condensed matter and particle physics to quantum optics. We study their behavior in an anisotropic Dicke model that includes collective matter interactions. We study the low-lying spectrum in the thermodynamic limit via the Holstein–Primakoff transformation and contrast the results with the semi-classical energy surface obtained via coherent states. We also explore the geometric phase for both boson and spin contours in the parameter space as a function of the phases in the system. We unveil novel phenomena due to the unique critical features provided by the interplay between the anisotropy and matter interactions. We expect our results to serve the observation of phase and amplitude modes in current quantum information platforms. [ABSTRACT FROM AUTHOR]

Abstract Propagating waves and surface waves are two distinct types of light-transporting modes, the free control of which are both highly desired in integration photonics. However, previously realized devices are bulky in sizes, inefficient, and/or can only achieve one type of light-manipulation functionality with a single device. Here, we propose a generic approach to design bi-channel meta-devices, constructed by carefully selected meta-atoms possessing reflection phases of both structural-resonance and geometric origins, which can exhibit two distinct light-manipulation functionalities in near-field (NF) and far-field (FF) channels, respectively. After characterizing the scattering properties of basic meta-atoms and briefly stating the theoretical strategy, we design/fabricate three different meta-devices and experimentally characterize their bi-channel wave-control functionalities in the telecom regime. Our experiments show that the first two devices can multiplex the generations of NF and FF optical vortices with different topological charges, while the third one exhibits anomalous surface plasmon polariton focusing in the NF and hologram formation in the FF simultaneously. Our results expand the wave-control functionalities of metasurfaces to all wave-transporting channels, which may inspire many exciting applications in integration optics.

Self-propelling organisms locomote via generation of patterns of self-deformation. Despite the diversity of body plans, internal actuation schemes and environments in limbless vertebrates and invertebrates, such organisms often use similar traveling waves of axial body bending for movement. Delineating how self-deformation parameters lead to locomotor performance (e.g. speed, energy, turning capabilities) remains challenging. We show that a geometric framework, replacing laborious calculation with a diagrammatic scheme, is well-suited to discovery and comparison of effective patterns of wave dynamics in diverse living systems. We focus on a regime of undulatory locomotion, that of highly damped environments, which is applicable not only to small organisms in viscous fluids, but also larger animals in frictional fluids (sand) and on frictional ground. We find that the traveling wave dynamics used by mm-scale nematode worms and cm-scale desert dwelling snakes and lizards can be described by time series of weights associated with two principal modes. The approximately circular closed path trajectories of mode weights in a self-deformation space enclose near-maximal surface integral (geometric phase) for organisms spanning two decades in body length. We hypothesize that such trajectories are targets of control (which we refer to as "serpenoid templates"). Further, the geometric approach reveals how seemingly complex behaviors such as turning in worms and sidewinding snakes can be described as modulations of templates. Thus, the use of differential geometry in the locomotion of living systems generates a common description of locomotion across taxa and provides hypotheses for neuromechanical control schemes at lower levels of organization. [ABSTRACT FROM AUTHOR]

Metasurface, a forefront in emerging optical devices, has demonstrated remarkable potential for complex amplitude manipulation of light beams. However, prevailing approaches face challenges in spatial resolution and complexities associated with integrating dynamic phases, impeding the simplified design and reproducible fabrication of metasurfaces. Here, we introduce an innovative approach for complex amplitude modulation within 3D nano-printed geometric phase metasurfaces. Our approach enables the generation of self-accelerating beams by encoding amplitude through phase-only manipulation, achieving high spatial resolution. Notably, this method circumvents the conventional need to adjust the geometric parameters of metasurface unit structures for amplitude manipulation, offering a streamlined and efficient route for design and fabrication complexity. This novel methodology holds promise for expedited and low-cost manufacturing of complex amplitude manipulation metasurfaces. [ABSTRACT FROM AUTHOR]

In this work, we propose the use of geometric phases obtained for both polarization and Hermite–Gaussian first-order transverse modes to create controlled phase gates. We present optical circuits that provide an arbitrarily controlled phase gate, including π / 8 , a universal quantum gate. Both polarization and HG mode can be used as control and target qubits, alternately, which is a great advantage in implementing quantum computing protocols. The phase gates are obtained by using a robust and precise geometric phase scheme, that is fault-tolerant and can provide fine adjustment. [ABSTRACT FROM AUTHOR]

A general strategy for designing bi‐layer reflection‐transmission integrated dual‐mode metasurfaces is presented with flexible bandwidth control. The design of such metasurfaces is enlightened by introducing a cutting hole in the ground plane, which not only acts as a filter for reflection and transmission but also has an eloquent role for bandwidth control. For proof of concept, a bi‐layer metasurface consisting of split ring resonators (SRRs) is designed on the top side and I‐shaped resonators on both sides. The proper adjustment of ground cut/hole shifts the two co‐polarized resonances (initial resonances of the SRRs) close to each other, and in addition, converts wideband cross‐polarized reflection (in between the two initial co‐polarized resonances of SRR) to co‐polarized reflection. This results in a broadband response by shifting two co‐polarized resonances close to each other and converting cross‐polarized reflection into co‐polarized reflection. It is demonstrated that a simple SRR can be responsible for broadband (7.9–12 GHz) reflection while an I‐shaped structure is used for transmission with central frequency at 17.58 GHz with independent phase controls. To endorse the proposed strategy, a metasurface prototype is fabricated and tested for the generation of vortex beams and focusing. [ABSTRACT FROM AUTHOR]

It is aimed to study a hybrid quantum‐classical system in which a qubit as a quantum system coupled to a semi‐infinite 1D photonic crystal (PC) waveguide, and is controlled by a classical driving field. PC waveguides with engineering capability serve as engineered reservoirs where various manifestations of quantum mechanics can be studied. Here, the main purpose of the study is to evaluate the effect of coherent control on the dynamics of quantum Fisher information (QFI) and geometric phase (GP). It is revealed that the intensity (the Rabi frequency) of the classical field, memory time, and geometric length of the waveguide are key variables whose judicious choice aids in optimizing the estimation of the quantum parameters that are initially encoded in the qubit state. It is observed that the optical length of the waveguide and the Rabi frequency of the classical field have an interplay effect on the dynamic of the QFI and geometric phase. As an interesting finding, it is disclosed that as the Rabi frequency increases, the effect of the optical length of the waveguide fades away, and the Rabi frequency solely controls the geometric phase. Moreover, it is demonstrated that the estimation accuracy increases as the memory time decreases. On the other hand, lengthening the memory time lessens the dependency of the geometric phase on the length of the waveguide. [ABSTRACT FROM AUTHOR]

The ability of liquid crystal to self‐organize into complicated structures can be harnessed in applications such as thin film optical devices, diffraction gratings or smart windows. In this work we focus on liquid crystal layers in contact with one or two substrates that have been patterned by photoalignment. This leads to geometric phase gratings, in which the phase of transmitted or reflected light depends on the location on the substrate. It has been shown that geometric phase gratings with uniform rotation can achieve very high diffraction efficiencies (>90%) for incident light that is circularly polarized. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the Berry phase of a spinor accumulated along a path corresponding to an adiabatically varying magnetic field and the conformal deformation along the path in the parameter space. The first part demonstrates that the accumulated phase can be controlled by continuous deformation characterized by a single conformal parameter. The source of the corresponding field strength responsible for the deformed phase path is shown to be a conformally deformed monopole located at the origin in the parameter space with the fundamental strength g=±1/2. In the second part, we model a rotating magnetic field coupled to a charged spinor with a time-varying amplitude subjected to the conformal deformation, realizing a single-variable deformation of the evolution process of the phase path.

We investigated the mixed state geometric phase of a two-level system which is coupled to a bosonic reservoir in both weak and strong coupling regimes, and driven by a classical field. Besides, we extended the direct-drive model to the indirect-drive model by incorporating an additional qubit. In the direct-drive model, the geometric phase undergoes corresponding alterations through the amplification of the classical field strength. Moreover, the combination of the classical driving and the strong coupling between the two-level system and the bosonic reservoir significantly changes the geometric phase. Furthermore, we elucidate the asymmetric effect of the detuning between the two-level system and the Lorentzian spectrum on the geometric phase, while achieving stability in the geometric phase regardless of the detuning value by appropriately adjusting the classical driving strength. In the indirect-drive model, intriguing phenomena arise, such as different behaviors of the geometric phase and the restoration of the symmetric effects of the detuning. The explicit dynamics of the geometric phase has been numerically analyzed by investigating the time-dependent factor of the reduced density matrix. Finally, the energy flow of the entire system was clarified by utilizing quasimode theory and visualizing the temporal evolution of the Bloch vector. [ABSTRACT FROM AUTHOR]

Optical information and communication technologies are essentially based on the ability to shape the light field, either at the transmission or at the reception of the optical signal. Nowadays, wavelength and polarization state are already implemented in current telecom architectures and adding the spatial structuring of the light field as a novel degree of freedom to enhance data rate is desirable. In this context, the development of optical devices to shape optical modes belonging to orthogonal basis remains challenging. Here, in the context of singular optics, this work reports on the design, fabrication, and characterization of modal optical vortex beam shaping of paraxial light in the visible domain, in the framework of the Laguerre–Gauss basis. Both the azimuthal and radial degrees of freedom are univocally shaped via optical spin‐orbit interaction mediated by dielectric metasurfaces by combining the spatial modulation of amplitude, geometric phase, and dynamic phase. These results not only demonstrate the realization of new optical components, but also highlight how metasurface‐based technologies can contribute to optical information processing in the classical or quantum regime. [ABSTRACT FROM AUTHOR]

The nonadiabatic holonomic quantum computation based on the geometric phase is robust against the built-in noise and decoherence. In this work, we theoretically propose a scheme to realize nonadiabatic holonomic quantum gates in a surface electron system, which is a promising two-dimensional platform for quantum computation. The holonomic gate is realized by a three-level structure that combines the Rydberg states and spin states via an inhomogeneous magnetic field. After a cyclic evolution, the computation bases pick up different geometric phases and thus perform a holonomic gate. Only the electron with spin up experiences the holonomic gate, while the electron with spin down is decoupled from the state-selective driving fields. The arbitrary controlled-U gate encoded on the Rydberg states and spin states can then be realized. The fidelity of the output state exceeds 0.99 with experimentally achievable parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a rotating fullerene molecule with Ih symmetry within the framework of non-inertial space-times. We use a low-energy geometric theory to describe the molecule as a two-dimensional spherical space of Gödel type. Using the well-known fictitious 't Hooft–Polyakov monopole to decouple the doublet associated with the lattice, we employ the Dirac factor method to derive a geometric phase for the fermions in the system. We also discuss the chronology protection feature of the phase through specific geometric considerations for the system. [ABSTRACT FROM AUTHOR]

The generation of moiré lattices by superimposing two identical sublattices at a specific twist angle has garnered significant attention owing to its potential applications, ranging from two-dimensional materials to manipulating light propagation. While macroscale moiré lattices have been widely studied, further developments in manipulating moiré lattices at the subwavelength scale would be crucial for miniaturizing and integrating platforms. Here, we propose a plasmonic metasurface design consisting of rotated nanoslits arranged within N + N′ round apertures for generating focused moiré lattices. By introducing a spin-dependent geometric phase through the rotated nanoslits, an overall lens and spiral phase can be achieved, allowing each individual set of round apertures to generate a periodic lattice in the focal plane. Superimposing two sets of N and N′ apertures at specific twist angles and varying phase differences allows for the superposition of two sublattices with different periods, leading to the formation of diverse moiré patterns. Our simulations and theoretical results demonstrate the feasibility of our proposed metasurface design. Due to their compactness and tunability, the utilization of metasurfaces in creating nanoscale photonic moiré lattices is anticipated to find extensive applications in integrated and on-chip optical systems. [ABSTRACT FROM AUTHOR]

Proposed paper presents the latest results in the framework of polarization-sensitive low-coherence interferometry related to new approaches for using the geometric phase to reproduce the polarization structure of a biological transparent anisotropic micro (nano) object. The polarization parameters of an anisotropic object are measured in real time on the basis of a modified Mach-Zehnder interferometer. The advantage of using the geometric phase is the diagnostic of polarization anisotropic surface (subsurface) nanosized layers in a non-contact, non-invasive manner.

Control over symmetry breaking in three-dimensional electromagnetic systems offers a pathway to tailoring their optical activity. We introduce fractured Pancharatnam–Berry-phase metasurface systems, in which a full-waveplate geometric phase metasurface is fractured into two half-waveplate-based metasurfaces and actively configured using shear displacement. Local relative rotations between stacked half-nanowaveplates within the metasurface system are transduced by shear displacement, leading to dynamic modulation of their collective geometric phase properties. We apply this concept to pairs of periodic Pancharatnam–Berry-phase metasurfaces and experimentally show that these systems support arbitrary and reconfigurable broadband circular birefringence response. High-speed circular birefringence modulation is demonstrated with modest shearing speeds, indicating the potential for these concepts to dynamically control polarization states with fast temporal responses. We anticipate that fractured geometric phase metasurface systems will serve as a nanophotonic platform that leverages systems-level symmetry breaking to enable active electromagnetic wave control.

Phase and amplitude modes, also called polariton modes, are emergent phenomena that manifest across diverse physical systems, from condensed matter and particle physics to quantum optics. We study their behavior in an anisotropic Dicke model that includes collective matter interactions. We study the low-lying spectrum in the thermodynamic limit via the Holstein–Primakoff transformation and contrast the results with the semi-classical energy surface obtained via coherent states. We also explore the geometric phase for both boson and spin contours in the parameter space as a function of the phases in the system. We unveil novel phenomena due to the unique critical features provided by the interplay between the anisotropy and matter interactions. We expect our results to serve the observation of phase and amplitude modes in current quantum information platforms.

The nonadiabatic holonomic quantum computation based on the geometric phase is robust against the built-in noise and decoherence. In this work, we theoretically propose a scheme to realize nonadiabatic holonomic quantum gates in a surface electron system, which is a promising two-dimensional platform for quantum computation. The holonomic gate is realized by a three-level structure that combines the Rydberg states and spin states via an inhomogeneous magnetic field. After a cyclic evolution, the computation bases pick up different geometric phases and thus perform a holonomic gate. Only the electron with spin up experiences the holonomic gate, while the electron with spin down is decoupled from the state-selective driving fields. The arbitrary controlled-U gate encoded on the Rydberg states and spin states can then be realized. The fidelity of the output state exceeds 0.99 with experimentally achievable parameters.

Perfect vortex beams (PVBs) possessing orbital angular momentum (OAM) and constant intensity profile enable practical applications in information encoding and transmission due to an unbounded number of orthogonal OAM channels and fixed annular intensity distributions. Geometric metasurfaces, which are 2D counterparts of metamaterials, have provided an ultra‐compact platform to flexibly design perfect vortex beams in a single flat device. However, the previous reported PVBs based on geometric metasurfaces are limited to ring‐shaped intensity profiles and intrinsic spin‐coupling between two orthogonal spin‐components. Here, spin‐decoupled geometric metasurfaces encoding with two‐step coordinate transformations are proposed to generate helicity‐independent PVBs with transmutable intensity profiles. By tailoring local phase gradient along the azimuthal direction, spin‐independent and polarization‐rotated terahertz (THz) PVBs with CN‐fold rotationally symmetric intensity profiles have been theoretically designed and experimentally demonstrated. Furthermore, THz PVBs with arbitrary intensity profiles have also been realized. The unique approach for simultaneously manipulating the spiral phase, focusing phase, as well as intensity profiles will open a new avenue to develop multifunctional integrated devices and systems, which enables potential applications in information processing and optical communication. [ABSTRACT FROM AUTHOR]

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Geometric phase enabled by spin‐orbit coupling has attracted enormous interest in optics over the past few decades. However, it is only applicable to circularly‐polarized light and encounters substantial challenges when applied to wave fields lacking the intrinsic spin degree of freedom. Here, a new paradigm is presented for achieving geometric phase by elucidating the concept of topological complementary pair (TCP), which arises from the combination of two compact phase elements possessing opposite intrinsic topological charge. Twisting the TCP leads to the generation of a linearly‐varying geometric phase of arbitrary order, which is quantified by the intrinsic topological charge. Notably distinct from the conventional spin‐orbit coupling‐based theories, the proposed geometric phase is the direct result of the cyclic evolution of orbital‐angular‐momentum transformation in mode space, thereby exhibiting universality across classical wave systems. As a proof of concept, the existence of this geometric phase is experimentally demonstrated using scalar acoustic waves, showcasing the remarkable ability in the precise manipulation of acoustic waves at subwavelength scales. These findings engender a fresh understanding of wave‐matter interaction in compact structures and establish a promising platform for exploring geometric phase, offering significant opportunities for diverse applications in wave systems. [ABSTRACT FROM AUTHOR]

A quantum system, when subjected to an adiabatic cyclic evolution of parameters governing its Hamiltonian, gains a phase factor known as the geometric phase. One of its most celebrated formulations in the regime of quantum optics is the Pancharatnam-Berry phase. It is said to be acquired by a beam of light undergoing an evolution of its polarization state. This article builds on the background and notion of the Pancharatnam-Berry phase in quantum optics. A convenient tool for visualizing the evolution of polarization states is the Poincaré sphere, which has been briefly discussed here. The close relation of the phase to a few other important concepts, such as Bargmann invariants and geodesics, has also been discussed in this article. [ABSTRACT FROM AUTHOR]

Conventional metasurfaces based on geometric phase are constrained to spin‐locked phase profile, resulting in mirrored functionalities for different spins. A single flat device that enables independent manipulation of wavefronts in two orthogonal circularly polarized channels is of paramount importance in wireless and optical communications. In this work, by tuning the dimension and rotation angle of H‐shaped meta‐atoms to synthesize propagating phase and geometric phase, spin‐dependent plasmonic metasurfaces are presented to manipulate circularly polarized waves in the visible band. To verify the capability of spin‐dependent wavefront manipulation, three metasurfaces are implemented. The first metasurface generates vortex beams with orbital angular momentum (OAM) l = 1 under left‐handed circularly polarized (LCP) incidence and l = 2 under right‐handed circularly polarized (RCP) incidence. By introducing convolution operation, the second metasurface is capable of producing vortex beams with different OAMs and different directions for two spins. The third metasurface produces dual‐beam and quad‐beam with different OAMs for different circular polarizations. This scheme can provide a new pathway in ultracompact nanophotonic devices and systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the interaction of Nitrogen Vacancies in Diamond (NVD) with quantized cavity field. The system is explored analytically and the effect of the system parameters is analyzed. The stability of a quantum system with influencing factors is investigated using the Mandal Parameter. The generated correlation between the NVD and the quantized cavity field is quantified using the negativity. This study also investigates geometric phase and its dependence on the system parameters. The results show that this system holds great potential applications in quantum computation and quantum memory. Additionally, the features of the system can be controlled by the system parameters. [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) information is vital for providing detailed features of the physical world, which is used in numerous applications such as industrial inspection, automatic navigation and identity authentication. However, the implementations of 3D imagers always rely on bulky optics. Metasurfaces, as the next-generation optics, shows flexible modulation abilities and excellent performance combined with computer vision algorithm. Here, we demonstrate an active 3D positioning and imaging method with large field of view (FOV) by single fringe projection based on metasurface and solve the accurate and robust calibration problem with the depth uncertainty of 4 μm. With a compact metasurface projector, the demonstrated method can achieve submillimeter positioning accuracy under the FOV of 88°, offering robust and fast 3D reconstruction of the texture-less scene due to the modulation characteristic of the fringe. Such scheme may accelerate prosperous engineering applications with the continued growth of flat-optics manufacturing process by using metadevices.

Abstract Geometric phase enabled by spin‐orbit coupling has attracted enormous interest in optics over the past few decades. However, it is only applicable to circularly‐polarized light and encounters substantial challenges when applied to wave fields lacking the intrinsic spin degree of freedom. Here, a new paradigm is presented for achieving geometric phase by elucidating the concept of topological complementary pair (TCP), which arises from the combination of two compact phase elements possessing opposite intrinsic topological charge. Twisting the TCP leads to the generation of a linearly‐varying geometric phase of arbitrary order, which is quantified by the intrinsic topological charge. Notably distinct from the conventional spin‐orbit coupling‐based theories, the proposed geometric phase is the direct result of the cyclic evolution of orbital‐angular‐momentum transformation in mode space, thereby exhibiting universality across classical wave systems. As a proof of concept, the existence of this geometric phase is experimentally demonstrated using scalar acoustic waves, showcasing the remarkable ability in the precise manipulation of acoustic waves at subwavelength scales. These findings engender a fresh understanding of wave‐matter interaction in compact structures and establish a promising platform for exploring geometric phase, offering significant opportunities for diverse applications in wave systems.

The concept of geometric phase has attracted enormous interest in the metasurface community for achieving phase variation with simple geometrical rotation. Here, this work demonstrates for the first time a highly efficient dual‐band metasurface with independent phase responses for structures with different‐fold rotational symmetries. For proof of concept, a hybrid unit cell structure with nested resonators of different‐fold (single‐fold and threefold) is proposed, where geometric rotation independent phase shifts of two times and six times the rotation angle are achieved in lower and higher bands, respectively. By rotating the resonators within the unit cell, highly efficient independent full phase controls can be achieved with circularly polarized incident waves. The co‐polarized reflection efficiency exceeds 90% at the frequency domain from 10.95 to 11.75 GHz (corresponding to the inner resonator) and exceeds 80% between 14.10 and 14.43 GHz (corresponding to the outer resonator), with peak values of 94.7% at 11.55 GHz and 85.5% at 14.23 GHz, respectively. To validate the versatility of the design paradigm, a metasurface porotype is fabricated and experimentally tested by generating independent holographic images. [ABSTRACT FROM AUTHOR]

We obtained the geometric phase for states of a particle in a spherical infinite potential well with moving walls in two different cases: First, when the radius of the well increases (or decreases) monotonically. Secondly, when the radius changes are oscillatory. In the latter case, we have solved the Schrödinger equation and found its solutions approximately. We obtained the transition rate for the possible real situation in an acousto-optic case which can reveal the effect of the geometric phase. We show that the absorption or radiation peaks will appear if the energy gap between the incident photon and the modified energy difference of two levels by the geometric phase is equal to the integer multiple of oscillation frequency. [ABSTRACT FROM AUTHOR]

A chiral nematic is embedded between multilayer mirrors to obtain voltage-inducible polarized resonance spectra. Initially, the nematic director is uniformly oriented perpendicular to the mirrors' surfaces because the chiral nematic helix is completely untwisted due to the homeotropic boundary conditions specified by the adsorbed cations. Then, a voltage is applied to remove the layer of surface-active cations from the input mirror. The obtained twisted homeoplanar configuration has a helix pitch exceeding the layer's thickness. The twisting leads to the anomalous blue shift of the o-modes in the transmittance spectrum of the photonic structure. This blue shift can be effectively compensated by repulsion of spectral peaks as a result of mode coupling in the vicinity of the virtual avoided crossing point. The experimental results obtained are confirmed numerically using the 4 × 4 transfer matrix method and explained with the contribution of a geometric phase. [ABSTRACT FROM AUTHOR]

Manipulating angular momentum of light relies on the change of geometric phase in anisotropic nanostructures or optical interface phenomena. Spin–orbit interactions demonstrated in normal incidence at a planar isotropic interface are conditioned by Brewster reflection, limited to the light beam with a single radial frequency in the k‐space. This study demonstrates a geometric (Pancharatnam–Berry) phase that overcomes Brewster angle limitations providing high variability in the modulation of light reflected from a planar isotropic interface. The examined geometric phase stems from the reflection anisotropy imitating the structural anisotropy of photoaligned liquid crystals and photonic metasurfaces. Using a geometric‐phase modulation of tightly focused light reflected from a glass slab, three modes of spin to orbital angular momentum conversion are demonstrated. The discovered reflective spin–orbit interaction allows for design of a double‐helix focusing sensor to be used in microscopy imaging for precise depth measurements (accuracy 109 nm and precision 7 nm in the range of 8 µm). Experiments with a focused astigmatic beam prove that a glass plate can perform very complex modulation of the geometric phase using illumination structuring. The results thus provide the foundation for the design of low‐cost geometric‐phase systems relying on the space‐variant reflection anisotropy. [ABSTRACT FROM AUTHOR]

Geometric and holonomic quantum computation utilizes intrinsic geometric properties of quantum-mechanical state spaces to realize quantum logic gates. Since both geometric phases and quantum holonomies are global quantities depending only on the evolution paths of quantum systems, quantum gates based on them possess built-in resilience to certain kinds of errors. This review provides an introduction to the topic as well as gives an overview of the theoretical and experimental progress for constructing geometric and holonomic quantum gates and how to combine them with other error-resistant techniques. [ABSTRACT FROM AUTHOR]

Novel options for multiplexing, such as orbital angular momentum (OAM), are sought to satisfy the explosive growth of information capacity. Consequently, spatial phase modulation with on‐demand tailoring of working bands is increasingly investigated. In this study, a polymer‐stabilized cholesteric liquid crystal is used to address this requirement. A varying DC voltage is applied, and the working band is increased over eightfold owing to the electric‐induced gradient pitch of the polymer network. Thus, the working band of an OAM processor is reversibly switched between narrowband and broadband states. An OAM‐multiplexing hologram is designed for parallel OAM encoding and decoding, enabling a wavelength‐division‐multiplexing compatible approach for in situ and non‐destructive OAM processing. The proposed design offers a promising solution for the on‐demand tailoring of working bands in liquid crystal planar optics and can promote advancements in massive information transmission and large‐capacity data processing. [ABSTRACT FROM AUTHOR]

This work comprehensively investigates possibilities of surface states realization in one-dimensional photonic systems that are terminated with ε-negative and μ-negative bandgap materials. We begin by first fathoming the topological properties of photonic band structure and notice that its bulk properties completely characterize the surface phenomena in all the foreseeable cases. This approach is inspired from topologically non-trivial behavior in low-dimensional condensed matter systems and the ensuing emergence of topologically protected edge states in such systems. Specifically, we will be following the setup of Su-Schrieffer-Heeger model and emulate the topological phenomena in one-dimensional photonic systems with a substantial advantage of relatively less demanding fabrication. More importantly, being distributed systems, the photonic crystal realizations in question further enrich the available parameter space and provide application avenues for topological phenomena. For example, unlike the atomic chains, in the case of photonic crystals, we can achieve the band inversion and topological phase transition without altering the arrangement of constituents. Our investigations primarily focus on exploiting this very aspect of higher-order photonic bandgaps and in this process, we experimentally demonstrate that the bulk-band geometric phases offer a deterministic yet customizable route for surface state engineering. [ABSTRACT FROM AUTHOR]