arXiv subjects

Mohammad Umar

Publications and source records attributed to Mohammad Umar.

16 recordsLinked to original sources

arXiv2026

Quantum transport and unified scaling law in graphene with polyadic Cantor electrostatic barriers

We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the $n$--$n$--$n$, $n$--$p$--$n$, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.

cond-mat.mes-hall↗

arXiv2026

L\'{e}vy-index control of spectral singularities and coherent perfect absorption in non-Hermitian space-fractional quantum mechanics

We investigate the scattering features of a non-Hermitian rectangular potential within the framework of space-fractional quantum mechanics. Using the Riesz fractional derivative, we analytically derive locus equations for spectral singularities (SSs) and their time-reversed counterparts, coherent perfect absorption (CPA), in a dimensionless complex-potential parameter space. This geometric locus formulation provides a transparent representation of the SS and CPA conditions and enables direct visualization of how fractional quantum dynamics modifies non-Hermitian scattering. We show that reducing the L\'{e}vy index $\alpha$, which enhances nonlocal transport associated with L\'{e}vy-flight dynamics, systematically lowers the gain-loss strength required for the emergence of SSs and CPAs, while increasing the mode index further suppresses this threshold. In addition, for fixed potential parameters, we demonstrate that decreasing $\alpha$ induces a blue shift of the SS energy, in direct agreement with earlier studies. From this perspective, the L\'{e}vy index $\alpha$ emerges as a tunable control knob for SS-CPA settings in fractional non-Hermitian quantum systems. Beyond its quantum-mechanical setting, this study may find applications in fractional waveguides and metamaterials governed by fractional wave equations. This work also bridges the gap between non-Hermitian quantum mechanics and space-fractional quantum mechanics.

quant-ph↗

arXiv2026

Experimental Analysis of Server-Side Caching for Web Performance

Performance in web applications is a key aspect of user experience and system scalability. Among the different techniques used to improve web application performance, caching has been widely used. While caching has been widely explored in web performance optimization literature, there is a lack of experimental work that explores the effect of simple inmemory caching in small-scale web applications. This paper fills this research gap by experimentally comparing the performance of two server-side web application configurations: one without caching and another with in-memory caching and a fixed time-tolive. The performance evaluation was conducted using a lightweight web server framework, and response times were measured using repeated HTTP requests under identical environmental conditions. The results show a significant reduction in response time for cached requests, and the findings of this paper provide valuable insights into the effectiveness of simple server-side caching in improving web application performance making it suitable for educational environments and small-scale web applications where simplicity and reproducibility are critical.

cs.DC↗

arXiv2025

Effective SU(2) gadget: holonomic walk on higher-order Poincar\'{e} sphere

In a manner commensurate to the SU(2) gadget for the Poincar\'{e} sphere, which involves a combination of two quarter-wave plates and one half-wave plate regardless of their sequential order, an analogous construct for the higher-order Poincar\'e sphere had long remained elusive. To address this, we recently demonstrated, by modifying Euler-angle parameterization, that an optical gadget consisting of two quarter-wave $q$-plates and one half-wave $q$-plate, each endowed with the same topological charge $q$, operates as a viable SU(2) gadget for the higher-order Poincar\'{e} sphere, contingent upon the fulfillment of the holonomy condition. This work presents the controlled navigation on the higher-order Poincar\'e sphere through the proposed architecture, formulated under the concept of an effective waveplate and thus can be referred to as an effective SU(2) gadget. The notion of an effective waveplate refers to a coaxial arrangement of multiple waveplates that, under specific constraints, can act as a single waveplate. In this gadget, the relative alignment of the offset angles of the constituent $q$-plates emerges as the decisive parameter governing systematic navigation on the higher-order Poincar\'e sphere. This study bear direct relevance to the deterministic control and engineering of structured light, encompassing polarization singularities, vector vortex beams and topological optical fields. Moreover, these results holds potential applications in the contemporary research frontiers in singular optics, spin-orbit photonics and quantum communication.

physics.optics↗

arXiv2025

SU(2) gadget for higher-order Poincar\'{e} sphere

The combination of two quarter-wave plates and one half-wave plate, regardless of their sequential arrangement, constitutes a well-established universal SU(2) gadget capable of implementing all polarization transformations on the standard Poincar\'{e} sphere. However, there is no analogous system for realizing all polarization transformations on the higher-order Poincar\'{e} sphere, a member of a higher topological index space. This work demonstrates that an optical gadget, comprising two quarter-wave $q$-plates and one half-wave $q$-plate, arranged in any order, is an SU(2) gadget to realize arbitrary polarization evolution on the higher-order Poincar\'{e} sphere.

physics.optics↗

arXiv2025

Mathematics of effective $q$-plate in polarization optics

The $q$-plate is a spatially inhomogeneous SU(2) birefringent optical element that has garnered significant interest due to its ability to mediate the spin-orbit interaction of light and facilitate the generation of optical vortices. The $q$-plate features a spatially varying fast axis orientation defined by two parameters: the topological charge $q$ and the offset angle $\alpha_0$. The notion of an effective waveplate arises when multiple waveplates, whether homogeneous or inhomogeneous, are aligned coaxially such that, under specific constraints, the composite system emulates the behavior of a single effective waveplate. This work presents a comprehensive mathematical formalism for realizing an effective waveplate through a cascaded configuration of three $q$-plates, each chosen as either a quarter-wave $q$-plate, a half-wave $q$-plate, or a combination thereof. This yields a total of eight distinct configurations. Some configurations result in an effective waveplate exhibiting a constant retardance, whereas others allow continuous modulation of the effective retardance over the full range from $0$ to $2\pi$ through the systematic variation of the relative offset angles between the constituent $q$-plates. This feature enables holonomic polarization transformations on the higher-order Poincar\'e sphere, making the concept of the effective waveplate applicable to topological index spaces. Moreover, tunable effective retardance holds significant potential for applications involving structured light corresponding to the higher-order Poincar\'{e} sphere, particularly in scenarios demanding controlled spatial modulation of polarization states or dynamic tailoring of polarization topologies.

physics.optics↗

arXiv2025

Holonomically constrained polarization transformation

In polarization optics, various topological constructs, namely Poincar\'e spheres of different orders, are used to represent uniform and structured polarization distributions. Similarly, there are also structured polarization optical elements. Consequently, various topological indices are defined for structured beams and elements. These topological aspects naturally allow us to holonomy-based categorization of polarization transformations. In this paper, we introduce holonomically constrained polarization transformations on topological constructs. The conditions on the topological parameters of the beams, elements and spheres to achieve holonomically constrained polarization transformations are discussed in detail. A topological treatment of holonomic systems is needed, since abundant polarization transformations reported in the literature on beams with structured polarization are nonholonomic. It is prudent to carry out this study since it can not be assumed that switching between holonomic and nonholonomic transformations does not affect the system output. It is shown here how these concepts enable us to introduce topological index spaces for polarization optics.

physics.optics↗

arXiv2025

SU(2) polarization evolution on higher-order Poincar\'e sphere by using general $q$-plate

This paper investigates the rotational dynamics on the higher-order Poincar\'e sphere with the use of $q$-plate by exploring three key aspects: the topological condition, the global-local rotation, and the SU(2) polarization evolution on the sphere. The polarized light beam corresponding to this sphere and $q$-plates shares analogous topological features, characterized by azimuthal variation. We have formulated the topological condition that establishes a connection between the $q$-plate and the higher-order Poincar\'e sphere, enabling the SU(2) polarization evolution on the same higher-order Poincar\'e sphere. Leveraging this correspondence, we have shown that a single \textit{global} SO(3) rotation on the higher-order Poincar\'e sphere is a collection of multiple \textit{local} SO(3) rotations on the standard Poincar\'e sphere. SO(3) is related to SU(2) through a two-to-one surjective homomorphism, with SU(2) serving as its double cover. Moreover, we demonstrate that a general $q$-plate, defined by a continuously tunable retardance ranging from $0$ to $2\pi$ and an offset angle ranging from $0$ to $\pi/2$, provides the complete coverage on the higher-order Poincar\'e sphere.

physics.optics↗

arXiv2025

Tunneling time in non-Hermitian space fractional quantum mechanics

We investigate the tunneling time of a wave packet propagating through a non-Hermitian potential $V_{r} - iV_{i}$ in space-fractional quantum mechanics. By applying the stationary phase method, we derive a closed-form expression for the tunneling time for this system. This study presents the first investigation of tunneling time at the interplay of non-Hermitian quantum mechanics and space-fractional quantum mechanics. The variation in tunneling time as the system transitions from a real to a complex potential is analyzed. We demonstrate that the tunneling time exhibits a dependence on the barrier width $d$ in the limit $d\rightarrow \infty$, showing the absence of the Hartman effect. A particularly striking feature of our findings is the potential manifestation of the Hartman effect for a specific combination of the absorption component $V_{i}$ and the Levy index $\alpha$. This behavior arises from the fact that the presence of the absorption component $V_{i}$ leads to a monotonic increase in tunneling time with barrier thickness, whereas the Levy index $\alpha$ reduces the tunneling time. The interplay of these contrasting influences facilitates the emergence of the Hartman effect under a specific combination of $V_{i}$ and the fractional parameter $\alpha$.

quant-ph↗

arXiv2024

Transmission through Cantor structured Dirac comb potential

In this study, we introduce the Cantor-structured Dirac comb potential, referred to as the Cantor Dirac comb (CDC-$\rho_{N}$) potential system, and investigate non-relativistic quantum tunneling through this novel potential configuration. This system is engineered by positioning delta potentials at the boundaries of each rectangular potential segment of Cantor potential. This study is the first to investigate quantum tunneling through a fractal geometric Dirac comb potential. This potential system exemplifies a particular instance of the super periodic potential (SPP), a broader class of potentials that generalize locally periodic potentials. Utilizing the theoretical framework of SPP, we derived a closed-form expression for the transmission probability for this potential architecture. We report various transmission characteristics, including the appearance of band-like features and the scaling behavior of the reflection coefficient with wave vector $k$, which is governed by a scaling function expressed as a finite product of the Laue function. A particularly striking feature of the system is the occurrence of sharp transmission resonances, which may prove useful in applications such as highly sharp transmission filters.

quant-ph↗

arXiv2024

Polyadic Cantor potential of minimum lacunarity: Special case of super periodic generalized unified Cantor potential

To bridge the fractal and non-fractal potentials we introduce the concept of generalized unified Cantor potential (GUCP) with the key parameter $N$ which represents the potential count at the stage $S=1$. This system is characterized by total span $L$, stages $S$, scaling parameter $ρ$ and two real numbers $μ$ and $ν$. Notably, the polyadic Cantor potential (PCP) system with minimal lacunarity is a specific instance within the GUCP paradigm. Employing the super periodic potential (SPP) formalism, we formulated a closed-form expression for transmission probability $T_{S}(k, N)$ using the $q$-Pochhammer symbol and investigated the features of non-relativistic quantum tunneling through this potential configuration. We show that GUCP system exhibits sharp transmission resonances, differing from traditional quantum systems. Our analysis reveals saturation in the transmission profile with evolving stages $S$ and establishes a significant scaling relationship between reflection probability and wave vector $k$ through analytical derivations.

quant-ph↗

arXiv2023

Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential

In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power $ n $, denoted as SVC$\left(ρ, n\right)$. Bridging the gap between the general Cantor and SVC systems, this novel potential offers a fresh perspective on Cantor-like potential systems within quantum mechanics that unify fractal and non-fractal potentials. Utilizing the Super Periodic Potential (SPP) formalism, we derive the close form expression of the transmission probability $ T_{G}(k) $. Notably, the system exhibits exceptionally sharp transmission resonances, a characteristic that distinguishes it from other quantum systems. Furthermore, the multifaceted transmission attributes of the SVC$\left(ρ, n\right)$ are found to be critically dependent on both parameters, $ ρ$ and $ n $, offering an intricate interplay that warrants deeper exploration. Our findings highlight a pronounced scaling behavior of reflection probability with $ k $, which is underpinned by analytical derivations.

quant-ph↗

arXiv2023

Quantum tunneling from a new type of Unified Cantor Potential

We introduce a new type of potential system that combines the families of general Cantor (fractal system) and general Smith-Volterra-Cantor (non-fractal system) potentials. We call this system as Unified Cantor Potential (UCP) system. The UCP system of total span $L$ is characterized by scaling parameter $ρ>1$, stage $G$ and two real numbers $α$ and $β$. For $α=1$, $β=0$, the UCP system represents general Cantor potential while for $α=0$, $β=1$, this system represent general Smith-Volterra-Cantor (SVC) potential. We provide close-form expression of transmission probability from UCP system for arbitrary $α$ and $β$ by using $q$-Pochhammer symbol. Several new features of scattering are reported for this system. The transmission probability $T_{G}(k)$ shows a scaling behavior with $k$ which is derived analytically for this potential. The proposed system also opens up the possibility for further generalization of new potential systems that encompass a large class of fractal and non-fractal systems. The analytical formulation of tunneling from this system would help to study the transmission feature at breaking threshold when a system transit from fractal to non-fractal domain.

quant-ph↗

arXiv2023

Tunneling from general Smith-Volterra-Cantor potential

We study the tunneling problem from general Smith-Volterra-Cantor (SVC) potential of finite length $L$ characterized by the scaling parameter $ρ$ and stage $G$. We show that the SVC($ρ$) potential of stage $G$ is the special case of super periodic potential (SPP) of order $G$. By using SPP formalism developed by us earlier, we provide the close form expression of tunneling probability $T_{G}(k)$ with the help of $q$-Pochhammer symbol. The profile of $T_{G}(k)$ with wave vector $k$ is found to saturate with increasing stage $G$. Very sharp transmission resonances are found to occur from this system which may find applications in the design of sharp transmission filters.

quant-ph↗

arXiv2022

Quantum tunneling from family of Cantor potentials in fractional quantum mechanics

We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials. We consider two types of potentials: general Cantor and general Smith-Volterra-Cantor potential. The Cantor potential is an example of fractal potential while the Smith-Volterra-Cantor potential doesn't belong to the category of a fractal system. The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics. We report several new features of scattering in the domain of space fractional quantum mechanics including the emergence of energy-band like features from these systems and extremely sharp transmission features. Further the scaling relation of the scattering amplitude with wave vector $k$ is presented analytically for both types of potentials.

quant-ph↗

arXiv2021

Locally finite free space as limiting case of PT-symmetric medium

We explicitly prove that the transfer matrix of a finite layered $PT$-symmetric system of fix length $L$ consisting of $N$ units of the potential system `$+iV$' and `$-iV$' of equal thickness becomes a unit matrix in the limit $N \rightarrow \infty$. This result is true for waves of arbitrary wave vector $k$. This shows that in this limit, the transmission coefficient is always unity while the reflection amplitude is zero for all waves traversing this length $L$. Therefore, a free space of finite length $L$ can be represented as a $PT$-symmetric medium.

quant-ph↗