We introduce an equivalence transformation to study the oscillation behavior of solutions for linear neutral differential equations of canonical and noncanonical types. The new approach leads to several novel oscillation criteria. Moreover, we show that the same arguments can be applied to nonlinear neutral equations under suitable monotonicity conditions. The importance of the results is also supported by examples.

In this paper, we investigate the oscillatory behavior of certain second-order delay differential equations in both canonical and noncanonical forms. We establish new oscillation criteria that extend and enhance existing results addressing cases where previously known theorems do not apply. A key contribution is the introduction of an oscillation-preserving transformation, which generates novel results and expands the applicability of oscillation criteria. The results are demonstrated through examples, emphasizing their effectiveness and generality in analyzing a wide range of delay differential equations.

The nonlinear phenomenon known as optical bistability, which enables optical systems to maintain two stable output states for a single input, is thoroughly examined in this review article. Through a detailed analysis of two important devices—the Simple Fiber Ring (SFR) resonator and the Nonlinear Optical Loop Mirror (NOLM)—it investigates the basic concepts of optical bistability, namely nonlinearity and feedback. Numerical simulations demonstrating dynamic dynamics and stability conditions bolster a strong theoretical framework. Recent advancements in bistable optical technologies, including developments in epsilon-near-zero materials, silicon photonics, hybrid plasmonic structures, neuromorphic photonics, and AI-driven photonic processors are also discussed. Furthermore, this review highlights emerging applications of optical bistability in optical logic circuits, high-speed communication, and quantum information processing. This review work is a useful tool for researchers and engineers creating next-generation optical systems because it connects theoretical insights with cutting-edge technological developments.

In this study, we investigate the oscillation behavior of second-order self-adjoint q - di ff erence equations, focusing on the renowned Leighton oscillation theorem. Through an example, we demonstrate that the q -version of Leighton's classical oscillation theorem does not hold and requires refinement. To address this, we introduce an oscillation-preserving transformation and establish alternative theorems to the ones existing in the literature. The strength of our work lies in the absence of any sign condition on the potential function. We also provide illustrative examples to support our findings and mention directions for future research.

In this study, we investigate the oscillation behavior of second-order self-adjoint $ q $-difference equations, focusing on the renowned Leighton oscillation theorem. Through an example, we demonstrate that the $ q $-version of Leighton's classical oscillation theorem does not hold and requires refinement. To address this, we introduce an oscillation-preserving transformation and establish alternative theorems to the ones existing in the literature. The strength of our work lies in the absence of any sign condition on the potential function. We also provide illustrative examples to support our findings and mention directions for future research.

The performance of intensity modulation (IM) with direct detection (DD) transmission systems is enhanced through a novel combination of multidimensional coding, Nyquist pulse shaping, and electronic dispersion compensation (EDC) at the transmitter using a finite impulse response (FIR) filter.A 24dimensional (24D) extended Golay binary code effectively transforms each incoming 12-bit message into a 24-bit codeword, achieving a coding efficiency of 0.5 bits per symbol for a 56 Gb/s on-off keying (OOK) transmission over 80 km of single mode fiber.While this encoding process introduces a 50% overhead, the required bandwidth is maintained at 56 GHz through doubling the symbol rate and the application of Nyquist pulse shaping with a raised cosine (RC) profile and a roll-off factor of zero, resulting in a flat power spectral density.This flat distribution contrasts with standard OOK transmission at 56 Gb/s with a roll-off factor of 1.0, where signal power is predominantly concentrated in the lower frequency range.One of the key advantages of the 24D Golay code is its substantial error correction capability.However, the benefits of this multidimensional coding and Nyquist pulse shaping extend beyond error correction.It is shown that, while both the proposed and standard OOK methods exhibit comparable performance in a white Gaussian noise channel at back-to-back, they differ significantly under frequency selective power fading conditions caused by the interplay of chromatic dispersion (CD) and direct detection.The misalignment between the frequency notches introduced by the FIR pre-EDC and those inherent in the channel response, especially severe at lower frequencies, favors transmission schemes with a flat power spectral density, like the 24D Golay-coded Nyquist pulses.

In this article, the first experimental demonstration of a non-iterative electronic dispersion compensation (EDC) solution implemented at the transmitter using a finite impulse response (FIR) filter optimized with the Gerchberg-Saxton (GS) algorithm, is presented, for intensity-modulation and direct-detection (IM/DD) systems. The theoretical framework and preliminary simulations have been presented in Part I of this work. Here, the performance is compared between the GS FIR filter and the iterative GS algorithm in 56-Gb/s on-off keying (OOK) transmissions over 80-km single mode fiber (SMF) with a post feed-forward equalizer (FFE) for combating residual inter-symbol interference. Furthermore, the influence of the pulse shape (raised cosine or rectangular) and modulation format (return-to-zero (RZ) or non-return-to-zero (NRZ)) on the measured bit error ratio (BER) is investigated, while changing the number of FIR taps, post-FFE taps, and post-FFE samples-per-symbol. It is shown that within the range of the target digital extinction ratios (DERs) for which the original iterative GS algorithm offers benefit, both analytical and numerical methods for calculating the optimum FIR taps, outlined in Part I, produce similar BER performance as predicted. Hence, the former method is extended, here, through a non-recursive frequency response formula, which offers insight into the action of the GS filter with different pulse shaping and enables the derivation of the explicit GS impulse response. It is also shown that rectangular RZ exploits the full benefit of GS filtering through a uniform spectrum, achieving BER < 3.8×10−3, with a 641-tap T/2-spaced pre-EDC FIR filter and a 3-tap adaptive T-spaced post-FFE.

Short-reach transmission systems traditionally utilize intensity modulation (IM) at the transmitter and direct detection (DD) at the receiver due to their cost-effectiveness, small footprint, and low power consumption. However, with the exponential increase in bandwidth demand, coherent optical communication systems have become necessary for long-haul distances, requiring application-specific integrated circuits (ASIC) and advanced digital signal processing (DSP) algorithms coupled with high-speed digital-to-analog and analog-to-digital converters to achieve Tbit/s speeds. As coherent technology matures, it will eventually become feasible for short-reach transmission. In this context, self-coherent systems have emerged as an intermediary solution, offering advantages over traditional IM/DD systems. While comprehensive review studies exist on self-coherent transceivers, they do not cover recent advances in phase retrieval methods for short-reach optical communications. This review article highlights recent developments in cost-effective self-coherent detection for short-reach systems through comparing the benefits of single sideband (SSB) transmission and Kramers-Kronig detection to carrier-assisted phase retrieval, the Gerchberg-Saxton (GS) algorithm, and the transport of intensity equation (TIE) method.

Short-reach fiber optical links employing intensity modulation (IM) at the transmitter (Tx) and direct detection (DD) at the receiver (Rx), suffer from linear and nonlinear sources of impairments due to the interaction of chromatic dispersion (CD) with DD. In this article, joint electronic dispersion compensation (EDC) at the Tx, using two distinct Gerchberg-Saxton(GS) based approaches, and at the Rx, using a functional link neural network (FLNN) equalizer is demonstrated for IM/DD transmission. The first Tx approach utilizes the modified iterative GS algorithm, which partially mitigate linear and nonlinear sources of inter-symbol interference (ISI). The second Tx pre-EDC approach only pre-compensates for the linear power fading effect through implementing a GS based finite impulse response (FIR) filter. At the Rx, a T/2-spaced or T-spaced adaptive post-feed forward equalizer (FFE) is employed to fully compensate residual chromatic dispersion prior to attempting nonlinear equalization. Furthermore, a Volterra nonlinear equalizer (VNLE) is introduced to benchmark the performance and complexity of the FLNN, Subsequently, either a FLNN or a VNLE are utilized for nonlinear system identification and subsequent post-equalization mitigating uncompensated nonlinear sources of ISI. The FLNN nonlinear taps resulting from the functional expansion block are optimized using the recursive least square (RLS) algorithm. The Tx-FIR and Rx-FLNN enable 112 Gbit/s non-return to zero (NRZ) on-off keying (OOK) transmission over 20 km of single mode fiber (SMF) and 112 Gbit/s 4-level pulse-amplitude modulation (PAM-4) transmission over 10 km of SMF. It is shown that the use of Tx pre-EDC reduces the complexity of the required equalization at the Rx. In addition, the FLNN was found to offer a 74% reduction in computational complexity relative to the third-order VNLE.

We study magnetic traps with very high trap frequencies where the spin is coupled to the motion of the atom. This allows us to investigate how the Born–Oppenheimer approximation fails and how effective magnetic and electric fields appear as the consequence of the non-adiabatic dynamics. The results are based on exact numerical diagonalization of the full Hamiltonian describing the coupling between the internal and external degrees of freedom. The position in energy and the decay rate of the trapping states corresponding to the imaginary part of the resonances of this Hamiltonian are computed using the complex rotation method.

In this article, the first experimental demonstration of a non-iterative electronic dispersion compensation (EDC) solution implemented at the transmitter using a finite impulse response (FIR) filter optimized with the Gerchberg-Saxton (GS) algorithm, is presented, for intensity-modulation and direct-detection (IM/DD) systems. The performance of the GS-based FIR filter is compared to the performance of the standard iterative GS algorithm in the transmission of a 56-Gb/s on-off keying (OOK) signal over 80 km of single mode fiber (SMF) with a chirp-free single drive Mach-Zehnder modulator (MZM). The transmitter employed a T/2-spaced N-tap GS based FIR filter with an 8-bit finite-precision arithmetic, while the receiver employed a J-tap adaptive T-spaced or T/2-spaced feed-forward equalizer (FFE) for combating residual inter-symbol interference (ISI). Furthermore, the influence of the pulse shape and modulation format on the measured bit error ratio (BER) is experimentally investigated, while changing the number of FIR taps N, the number of post-FFE taps J and the number of post-FFE samples-per-symbol spsRx. The pulse shapes and modulation formats considered in this work are: raised-cosine (RC) non-return-to-zero (NRZ), rectangular NRZ and rectangular return-to-zero (RZ). It is shown that within the range of target digital extinction ratio (DER) for which the original iterative GS algorithm offers benefit, both methods for calculating the optimum FIR taps, outlined in Part I of this work, produce similar BER performance as predicted theoretically. Furthermore, the measured BER, using rectangular RZ with a 641-tap T/2-spaced static FIR filter at the transmitter and a 3-tap adaptive T-spaced post-FFE at the receiver, is below the 7% hard-decision (HD) forward error correction (FEC) limit of 3.8x10-3.

Holonomic quantum computation is the idea to use non-Abelian geometric phases to implement universal quantum gates that are robust to fluctuations in control parameters. Here, we propose a compact design for a holonomic quantum computer based on coupled three-level systems. The scheme does not require adiabatic evolution and can be implemented in arrays of atoms or ions trapped in tailored standing wave potentials.

The adiabatic Aharonov-Bohm (AB) effect is a manifestation of the Berry phase acquired when some slow variables take a planar spin around a loop. While the effect has been observed in molecular spectroscopy, direct measurement of the topological phase shift in a scattering experiment has been elusive in the past. Here, we demonstrate an adiabatic AB effect by explicit simulation of the dynamics of unpolarized very slow neutrons that scatter on a long straight current-carrying wire.

Motivated by the Mobius transformation for symmetric points under the generalized circle in the complex plane, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. In terms of these states, we construct the maximally entangled complete set of two-qubit coherent states, which in the limiting cases reduces to the Bell basis. A specific property of our symmetric coherent states is that they never become unentangled for any value of psi from the complex plane. Entanglement quantifications of our states are given by the reduced density matrix and the concurrence determinant, and it is shown that our basis is maximally entangled. Universal one- and two-qubit gates in these new coherent state basis are calculated. As an application, we find the Q symbol of the XYZ model Hamiltonian operator H as an average energy function in maximally entangled two- and three-qubit phase space. It shows regular finite-energy localized structure with specific local extremum points. The concurrence and fidelity of quantum evolution with dimerization of double periodic patterns are given.

concurrence determinant methods, it is shown that our basis is maximally entangled. In addition we find that the average of spin operators in these states vanish, as it must be according to another, operational definition of completely entangled states. Universal one qubit and two qubit gates in this new basis are calculated and time evolution of these states for some spin systems is derived. We find that the average energy for XYZ model in two qubit case (Q symbol of H) shows regular finite energy localized structure with characteristic extremum points, and appears as a soliton in maximally entangled two qubit phase space. Generalizations to three and higher qubit states are discussed.

Motivated by M\"obius transformation for symmetrical points under the generalized circle in complex plane, the system of symmetrical spin coherent states corresponding to antipodal qubit states is introduced. It implies the maximally entangled spin coherent states basis, which in the limiting cases reduces to the Bell basis. A specific property of our symmetric image coherent states is that they never become unentangled for any value of $\psi$ from complex plane. By the reduced density matrix and the concurrence determinant methods, it is shown that our basis is maximally entangled. In addition we find that the average of spin operators in these states vanish, as it must be according to another, operational definition of completely entangled states. Universal one qubit and two qubit gates in this new basis are calculated and time evolution of these states for some spin systems is derived. We find that the average energy for XYZ model in two qubit case (Q symbol of H) shows regular finite energy localized structure with characteristic extremum points, and appears as a soliton in maximally entangled two qubit phase space. Generalizations to three and higher qubit states are discussed.

In the present paper an influence of the anisotropic antisymmetric exchange interaction, the Dzialoshinskii-Moriya (DM) interaction, on entanglement of two qubits in various magnetic spin models, including the pure DM model and the most general XY Z model are studied. We find that the time evolution generated by DM interaction can implement the SWAP gate and discuss realistic quasi-one-dimensional magnets where it can be realized. It is shown that inclusion of the DM interaction to any Heisenberg model creates, when it does not exist, or strengthens, when it exists, the entanglement. We give physical explanation of these results by studying the ground state of the systems at T=0. Nonanalytic dependence of the concurrence on the DM interaction and its relation with quantum phase transition is indicated. Our results show that spin models with the DM coupling have some potential applications in quantum computations and the DM interaction could be an efficient control parameter of entanglement.

The problem of magnetic vortex dynamics in an anisotropic spin liquid model is considered. For incompressible flow the model admits reduction to saturating Bogomolny inequality analytic projections of spin variables, subject the linear holomorphic Schrodinger equation. It allows us to construct N vortex configurations in terms of the complex Hermite polynomials. Using complex Galilean boost transformations, the interaction of the vortices and the vortex chain lattices (vortex crystals) is studied. By the complexified Cole-Hopf transformation, integrable N vortex dynamics is described by the holomorphic Burgers equation. Mapping of the point vortex problem to N-particle problem, the complexified Calogero-Moser system, showing its integrability and the Hamiltonian structure, is given. (C) 2006 Elsevier Ltd. All rights reserved.

In this paper, we study the Dzyaloshinskii-Moriya (DM) anisotropic $XX$ spin-chain model in the presence of an external homogeneous magnetic field. We found that the Berry phase of the system varies interestingly with small and large amounts of DM interaction and the magnetic field. In addition, we also considered the concurrence (i.e., the amount of entanglement) of the system, and the relationship between the concurrence and the Berry phase. Finally, we calculate the Berry phase of thermal states and verify that the results are consistent with that of the pure states.

We consider the Abelian Chern-Simons gauge field theory in 2+1 dimensions and its relation to the holomorphic Burgers hierarchy. We show that the relation between the complex potential and the complex gauge field as in incompressible and irrotational hydrodynamics has the meaning of the analytic Cole-Hopf transformation, linearizing the Burgers hierarchy and transforming it into the holomorphic Schrödinger hierarchy. The motion of planar vortices in Chern-Simons theory, which appear as pole singularities of the gauge field, then corresponds to the motion of zeros of the hierarchy. We use boost transformations of the complex Galilei group of the hierarchy to construct a rich set of exact solutions describing the integrable dynamics of planar vortices and vortex lattices in terms of generalized Kampe de Feriet and Hermite polynomials. We apply the results to the holomorphic reduction of the Ishimori model and the corresponding hierarchy, describing the dynamics of magnetic vortices and the corresponding lattices in terms of complexified Calogero-Moser models. We find corrections (in terms of Airy functions) to the two-vortex dynamics from the Moyal space-time noncommutativity.