arXiv subjects

Rahul Kumar

Publications and source records attributed to Rahul Kumar.

At least 73 records · Page 4Linked to original sources

arXiv2022

Near-field Probing of Optical Superchirality for Enhanced Bio-detection

Nanophotonic platforms in theory uniquely enable < femtomoles of chiral biological and pharmaceutical molecules to be detected, through the highly localised changes in the chiral asymmetries of the near-fields that they induce. However, current chiral nanophotonic based strategies are intrinsically limited because they rely on far-field optical measurements that are sensitive to a much larger near-field volume, than that influenced by the chiral molecules. Consequently, they depend on detecting small changes in far-field optical response restricting detection sensitivities. Here we exploit an intriguing phenomenon, plasmonic circularly polarised luminescence (PCPL), which is an incisive local probe of near-field chirality. This allows chiral detection of monolayer quantities of a de novo designed peptide, which is not achieved with a far-field response. Our work demonstrates that by leveraging the capabilities of nanophotonic platforms with the near-field sensitivity of PCPL, optimal biomolecular detection performance can be achieved, opening new avenues for nanometrology.

physics.optics↗

arXiv2022

Explicit transformations of certain Lambert series

An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}σ_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-zero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for $ζ(2m+1)$. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. When $a$ is an odd integer, the well-known modular transformations of the Eisenstein series on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's formula for $ζ(2m+1)$ are derived from the master identity. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function $K_ν(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function $r_k(n)$. Some results on functions self-reciprocal in the Watson kernel are also obtained.

math.NT↗

arXiv2022

Extended Higher Herglotz function \textup{II}

Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for $ζ(2m+1)$. We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.

math.NT↗

arXiv2021

Equivalent criterion for the grand Riemann hypothesis associated to Maass cusp forms

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $Γ_0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated to the primitive Maass cusp form over $Γ_0(N)$.

math.NT↗

arXiv2021

Unravelling the coupling between excitonic quasiparticles-electron-phonon and role of interlayer coupling in vertically and horizontally aligned layered MoS2

Excitonic quasi-particles, excitons/trions/bi-excitons, and their coupling with phonons and charge carriers play a crucial role in controlling the optical properties of atomically thin semiconducting 2D materials. In this work, we unravelled the dynamics of excitons/trions and their coupling with phonons and charge carriers in a few layers vertically and horizontally aligned MoS2. We observed trion signature up to the highest recorded temperature (330 K) in both systems and have shown that the dynamics of excitons/trions and their coupling with phonons and electrons are more affected in vertically aligned MoS2. A homogeneous linewidth broadening is observed with an increase in temperature. The linewidth broadening is attributed mainly to acoustic phonons in a low-temperature regime (<100 K). In contrast, acoustic and longitudinal optical phonons contributions to the linewidth broadening are observed at high temperature. We also observed the significant effects of interlayer coupling in both systems via understanding the temperature-dependent valence band splitting and trion binding energy. A decrease of 22 and 12% in valence band splitting with temperature rise is observed for the vertically and horizontally aligned MoS2, respectively, suggesting that the valence band splitting is affected more in the case of vertically than horizontally aligned. Furthermore, we also notice a significant thermal quenching in the intensity of the trion band than that of exciton bands, attributed to the small binding energy of the trion.

cond-mat.mtrl-sci↗

arXiv2021

Shadows of black hole surrounded by anisotropic fluid in Rastall theory

Due to the gravitational lensing effect, a black hole casts a shadow larger than its horizon over a bright background, and the shape and size can be calculated. The Event Horizon Telescope collaboration has produced the first direct image (shadow) of the black hole and it is in accordance with the shadow of a Kerr black hole of general relativity. However, deviations from the Kerr black hole arising from modified theories of gravity are not ruled out and they are important as they offer an arena to test these theories through astrophysical observation. This stimulates us to investigate rotating black holes surrounded by anisotropic fluid in Rastall theory namely a rotating Rastall black hole, which is characterized by mass $M$, spin $a$, field structure parameter $N_s$ and the Rastall parameter $ψ$. It encompasses, as special cases, Kerr ($N_s \to 0$) and Kerr-Newman ($s=0$ and $N_s = -Q^2 $) black holes. The rotating Rastall black hole is characterized by an additional cosmological-like horizon apart from Cauchy and event horizons. We derive an analytical formula for the shadow of a rotating Rastall black hole and go on to visualize the shadow of black holes for various values of the parameters for an observer at a given coordinate ($r_O, θ_O$) in the domain $[r_+,r_q]$.

gr-qc↗

arXiv2021

Point Absorber Limits to Future Gravitational-Wave Detectors

High-quality optical resonant cavities require low optical loss, typically on the scale of parts per million. However, unintended micron-scale contaminants on the resonator mirrors that absorb the light circulating in the cavity can deform the surface thermoelastically, and thus increase losses by scattering light out of the resonant mode. The point absorber effect is a limiting factor in some high-power cavity experiments, for example, the Advanced LIGO gravitational wave detector. In this Letter, we present a general approach to the point absorber effect from first principles and simulate its contribution to the increased scattering. The achievable circulating power in current and future gravitational-wave detectors is calculated statistically given different point absorber configurations. Our formulation is further confirmed experimentally in comparison with the scattered power in the arm cavity of Advanced LIGO measured by in-situ photodiodes. The understanding presented here provides an important tool in the global effort to design future gravitational wave detectors that support high optical power, and thus reduce quantum noise.

physics.ins-det↗

arXiv2021

Electron-Phonon Coupling, Thermal Expansion Coefficient, Resonance Effect and Phonon Dynamics in High Quality CVD Grown Mono and Bilayer MoSe2

Probing phonons, quasi-particle excitations and their coupling has enriched our understanding of these 2D materials and proved to be crucial for developing their potential applications. Here, we report comprehensive temperature, 4-330 K, and polarization-dependent Raman measurements on mono and bilayer MoSe2. Phonon's modes up to fourth-order are observed including forbidden Raman and IR modes, understood considering Frohlich mechanism of exciton-phonon coupling. Most notably, anomalous variations in the phonon linewidths with temperature pointed at the significant role of electron-phonon coupling in these systems, especially for the out-of-plane (A1g) and shear mode (E22g), which is found to be more prominent in the narrow-gaped bilayer than the large gapped monolayer. Via polarization-dependent measurements, we deciphered the ambiguity in symmetry assignments, especially to the peaks around ~ 170 cm-1 and ~ 350 cm-1. Temperature-dependent thermal expansion coefficient, an important parameter for the device performance, is carefully extracted for both mono and bilayer by monitoring the temperature-dependence of the real-part of the phonon self-energy parameter. Our temperature-dependent in-depth Raman studies provide a pave for uncovering the deeper role of phonons in these 2D layered materials from a fundamental as well as application point of view.

cond-mat.mtrl-sci↗

arXiv2021

Extended higher Herglotz functions I. Functional equations

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F(x)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $\mathscr{F}_{k,N}(x)$, an extension of the Herglotz function which also subsumes \emph{higher Herglotz function} of Vlasenko and Zagier. We call it the \emph{extended higher Herglotz function}. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by $\mathscr{F}_{k,N}(x)$. Radchenko and Zagier gave a beautiful relation between the integral $\displaystyle\int_{0}^{1}\frac{\log(1+t^x)}{1+t}\, dt$ and $F(x)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $\mathscr{F}_{k,N}(x)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $\mathscr{F}_{k, N}(x)$ and some generalized Lambert series are also obtained along with other supplementary results.

math.NT↗

arXiv2021

Number field analogue of divisor function à la Koshliakov

In this article, we study a divisor function in an arbitrary number field akin to Koshliakov's work on Vorono\"{\dotlessi} summation formula. More precisely, we generalize Koshliakov's kernel and Koshliakov's transform over any number field to obtain identities for the Lambert series associated to the divisor function in an arbitrary number field.

math.NT↗

arXiv2021

Assessing the Effectiveness of Syntactic Structure to Learn Code Edit Representations

In recent times, it has been shown that one can use code as data to aid various applications such as automatic commit message generation, automatic generation of pull request descriptions and automatic program repair. Take for instance the problem of commit message generation. Treating source code as a sequence of tokens, state of the art techniques generate commit messages using neural machine translation models. However, they tend to ignore the syntactic structure of programming languages. Previous work, i.e., code2seq has used structural information from Abstract Syntax Tree (AST) to represent source code and they use it to automatically generate method names. In this paper, we elaborate upon this state of the art approach and modify it to represent source code edits. We determine the effect of using such syntactic structure for the problem of classifying code edits. Inspired by the code2seq approach, we evaluate how using structural information from AST, i.e., paths between AST leaf nodes can help with the task of code edit classification on two datasets of fine-grained syntactic edits. Our experiments shows that attempts of adding syntactic structure does not result in any improvements over less sophisticated methods. The results suggest that techniques such as code2seq, while promising, have a long way to go before they can be generically applied to learning code edit representations. We hope that these results will benefit other researchers and inspire them to work further on this problem.

cs.LG↗

arXiv2021

Parameter estimation of hairy Kerr black holes from its shadow and constraints from M87*

The recently obtained hairy Kerr black holes, due to additional sources or surrounding fluid, like dark matter, with conserved energy-momentum tensor, have a deviation $α$ and primary hair $l_0$, apart from rotation parameter $a$ and mass $M$. In the wake of the \textit{Event Horizon Telescope} (\textit{EHT}) observations of the supermassive black hole M87*, a recent surge in interest in black hole shadows suggests comparing the black holes in general relativity (GR) and modified theories of gravity (MoG) to assess these models' differences. Motivated by this, we take on an extensive study of the rotating hairy Kerr black holes, which encompasses, in particular cases, the Kerr black hole ($α=0$). We investigate ergosphere and shadows of the black holes to infer that their size and shape are affected due to the $l_0$ and are found to harbour a richer chaotic structure. In particular, the hairy Kerr black holes possess smaller size but more distorted shadows when compared with Kerr black holes. We also estimate the parameters $l_0$ and $a$ associated with hairy Kerr black holes using the shadow observables. The inferred circularity deviation $ΔC \leq 0.1$ for the M87* black hole is satisfied, whereas shadow angular diameter $θ_{d}=42 \pm 3 μas$, within $1 σ$ region, for a given choice of $α$, places bounds on the parameters $a$ and $l_0$. Interestingly, the shadow axial ratio obeying $1< D_x \lesssim 4/3$ is in agreement with the \textit{EHT} results and thus eventuates in the hairy Kerr black holes being suitable candidates for astrophysical black holes.

gr-qc↗

arXiv2021

LIGOs Quantum Response to Squeezed States

Gravitational Wave interferometers achieve their profound sensitivity by combining a Michelson interferometer with optical cavities, suspended masses, and now, squeezed quantum states of light. These states modify the measurement process of the LIGO, VIRGO and GEO600 interferometers to reduce the quantum noise that masks astrophysical signals; thus, improvements to squeezing are essential to further expand our gravitational view of the universe. Further reducing quantum noise will require both lowering decoherence from losses as well more sophisticated manipulations to counter the quantum back-action from radiation pressure. Both tasks require fully understanding the physical interactions between squeezed light and the many components of km-scale interferometers. To this end, data from both LIGO observatories in observing run three are expressed using frequency-dependent metrics to analyze each detector's quantum response to squeezed states. The response metrics are derived and used to concisely describe physical mechanisms behind squeezing's simultaneous interaction with transverse-mode selective optical cavities and the quantum radiation pressure noise of suspended mirrors. These metrics and related analysis are broadly applicable for cavity-enhanced optomechanics experiments that incorporate external squeezing, and -- for the first time -- give physical descriptions of every feature so far observed in the quantum noise of the LIGO detectors.

physics.ins-det↗

arXiv2021

Explicit identities on zeta values over imaginary quadratic field

In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even and odd zeta values over $\Q$. Moreover, any complex zeta values over imaginary quadratic field may also be evaluated from our identities.

math.NT↗

arXiv2021

Normality from one family of meromorphic functions to another through sharing of values

Let F and G be two families of meromorphic functions on a domain D, and let a, b and c be three distinct points in the extended complex plane. Let G be a normal family in D such that all limit functions of G are non-constant. If for each f in F, there exists g in G such that f and g share a, b and c partially, then F is normal in D. This gives a sharp improvement of a result due to X. J. Liu, S. H. Li and X. C. Pang. We also prove some interesting related sharp results.

math.CV↗

arXiv2021

Point absorbers in Advanced LIGO

Small, highly absorbing points are randomly present on the surfaces of the main interferometer optics in Advanced LIGO. The resulting nano-meter scale thermo-elastic deformations and substrate lenses from these micron-scale absorbers significantly reduces the sensitivity of the interferometer directly though a reduction in the power-recycling gain and indirect interactions with the feedback control system. We review the expected surface deformation from point absorbers and provide a pedagogical description of the impact on power build-up in second generation gravitational wave detectors (dual-recycled Fabry-Perot Michelson interferometers). This analysis predicts that the power-dependent reduction in interferometer performance will significantly degrade maximum stored power by up to 50% and hence, limit GW sensitivity, but suggests system wide corrections that can be implemented in current and future GW detectors. This is particularly pressing given that future GW detectors call for an order of magnitude more stored power than currently used in Advanced LIGO in Observing Run 3. We briefly review strategies to mitigate the effects of point absorbers in current and future GW wave detectors to maximize the success of these enterprises.

physics.ins-det↗

arXiv2021

Photon ring structure of rotating regular black holes and no-horizon spacetimes

The Kerr black holes possess a photon region with prograde and retrograde orbits radii, respectively, $M\leq r_p^-\leq 3M$ and $3M\leq r_p^+\leq 4M$, and thereby always cast a closed photon ring or a shadow silhouette for $a\leq M$. For $a>M$, it is a no-horizon spacetime (naked singularity) wherein prograde orbits spiral into the central singularity, and retrograde orbits produce an arc-like shadow with a dark spot at the center. We compare Kerr black holes' photon ring structure with those produced by three rotating regular spacetimes, viz. Bardeen, Hayward, and nonsingular. These are non-Kerr black hole metrics with an additional deviation parameter of $g$ related to the nonlinear electrodynamics charge. It turns out that for a given $ a $, there exists a critical value of $ g $, $g_E$ such that $Δ=0$ has no zeros for $ g > g_E$, one double zero at $ r = r_E $ for $ g = g_E $, respectively, corresponding to a no-horizon regular spacetime and extremal black hole with degenerate horizon. We demonstrate that, unlike the Kerr naked singularity, no-horizon regular spacetimes can possess closed photon ring when $g_E< g \leq g_c$, e.g., for $a=0.10M$, Bardeen ($g_E=0.763332M<g\leq g_c= 0.816792M$), Hayward ($g_E=1.05297M < g\leq g_c = 1.164846M$) and nonsingular ($g_E=1.2020M < g \leq g_c= 1.222461M$) no-horizon spacetimes have closed photon ring. These results confirm that the mere existence of a closed photon ring does not prove that the compact object is necessarily a black hole. The ring circularity deviation observable $ΔC$ for the three no-horizon rotating spacetimes satisfy $ΔC\leq 0.10$ as per the M87* black hole shadow observations. We have also appended the case of Kerr-Newman no-horizon spacetimes (naked singularities) with similar features.

gr-qc↗

arXiv2021

Parameters estimation and strong gravitational lensing of nonsingular Kerr-Sen black holes

The recent time witnessed a surge of interest in strong gravitational lensing by black holes is due to the Event Horizon Telescope (EHT) results, which suggest comparing the black hole lensing in both general relativity and heterotic string theory. That may help us to assess the phenomenological differences between these models. Motivated by this, we consider gravitational lensing by the nonsingular Kerr-Sen black holes, which encompass Kerr black holes as a particular case, to calculate the light deflection coefficients $p$ and $q$ in strong-field limits, while the former increases with increasing parameters $k$ and charge $b$, later decrease. We also find a decrease in the light deflection angle $α_D$, angular position $θ_{\infty}$ decreases more slowly and impact parameter for photon orbits $u_{m}$ more quickly, but angular separation $s$ increases more rapidly with parameters $b$ and $k$. We compare our results with those for Kerr black holes, and also the formalism is applied to discuss the astrophysical consequences in the case of the supermassive black holes NGC 4649, NGC 1332, Sgr A* and M87*. In turn, we also investigate the shadows of the nonsingular Kerr-Sen black holes and show that they are smaller and more distorted than the corresponding Kerr black holes and nonsingular Kerr black holes shadows. The inferred circularity deviation $ΔC\leq 0.10$, for the M87* black hole shadow, put constraints on the nonsingular Kerr-Sen black hole parameters ($a, k$) and ($a, b$). The maximum shadow angular diameter for $b=0.30M$ and $k=0.30M$ are, respectively, $θ_d=35.3461\,μ$as and $θ_d=35.3355\,μ$as. We also estimate the parameters associated with nonsingular Kerr-Sen black holes using the shadow observables.

gr-qc↗