Inter national J our nal of Adv ances in A pplied Sciences (IJ AAS) V ol. 15, No. 1, March 2026, pp. 355 ∼ 371 ISSN: 2252-8814, DOI: 10.11591/ijaas.v15.i1.pp355-371 ❒ 355 State e v olution appr oach f or the axion con v ersion pr obability in magnetospher e of a neutr on star Bilal Ahmad 1 , Shehr eyar Ali 2 1 Institute of Theoretical Ph ysics, School of Ph ysics and Optoelectronic Engineering, Beijing Uni v ersity of T echnology , Beijing, China 2 Department of Ph ysics, Go vt De gree Colle ge No 2 Mardan, af liated to Abdul W ali Khan Uni v ersity Mardan, Mardan, P akistan Article Inf o Article history: Recei v ed Jun 12, 2025 Re vised No v . 24, 2025 Accepted Jan 1, 2026 K eyw ords: Axion dark matter Axion-photon con v ersion Magnetospheres Neutron star Primak of f ef fect Radiati v e po wer State e v olution ABSTRA CT Neutron stars (NS), with their e xtreme gra vitational and magnetic elds, pro vide an e xceptional astroph ysical laboratory for studying axion dark matter (DM). Through the Primak of f ef fect, axions can con v ert into photons within the magnetospheres of NS, a process that may produce observ able radio and X-ray signals. In this w ork, we in v estig ate axion-photon con v ersion using a no v el, time-dependent state e v olut ion formalism, mo ving be yond the commonly used stationary-path approximations. W e deri v e a generic analytical e xpression for the con v ersion probability and calculate the associated radiated po wer . Our analysis demonstrates that this approach allo ws NS to strongly constrain the axion-photon coupling constant, re aching sensiti vi ties of g a γ γ ≃ 10 − 14 − 10 − 15 GeV − 1 for axion masses of m a ≃ 10 − 3 − 10 − 10 eV . These results establish a ne w pathw ay to constrain g a γ via NS observ ations. Future campaigns using po werful observ atories lik e the J ames W ebb Space T elescope (JWST), Green Bank T elescope (GBT), and More Karoo Array T elescope (MeerKA T) array will be ideally suited to probe the distinct spectral signatures predicted by our model across multiple frequenc y domains. This is an open access article under the CC BY -SA license . Corresponding A uthor: Bilal Ahmad Institute of Theoretical Ph ysics, School of Ph ysics and Optoelectronic Engineering Beijing Uni v ersity of T echnology Beijing, China Email: bilalahmad@emails.bjut.edu.cn 1. INTR ODUCTION The standard model (SM) stands as one of the greatest accomplishments in modern particle p h ysi cs [1]. Despite its success in predicting the outcomes of terrestrial e xperiments, it is consi d e red incomplete due to certain problems, including the strong char ge parity (CP) problem and dark matter (DM). In the late 1970s, to resolv e the strong CP problem in quantum chromodynamics (QCD) by an additional term w as introduced in the QCD Lagrangian. It arises from the Peccei-Quinn mechanism [1], which dynamically restores CP symmetry in strong interactions; later , W ilczek and W einber g [2], [3] assigned the axion as a ph ysical outcome of the spontaneous breaking of U (1) P Q symmetry from Noether’ s theorem [2]. Hooft [4] studied one of the most important breaks do wn of U PQ (1) symmetry is possible due to the Instanton ef fects, it means that the axion eld couples to the gluons eld and acts as a shift symmetry i.e. a ( x, t ) → a ( x, t ) + ϵ ( x, t ) . Extremely l ight and weakly interacting, axions are also compelling candidates for DM [5]. Theoretical models be yond the SM, lik e string theory at lo w ener gies, often feature generic pseudoscalars in ab undance [4]. While there will e xist signicant dif ferences between the tw o en vironments, the emer gent properties of the axion will also be broadly J ournal homepage: http://ijaas.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
356 ❒ ISSN: 2252-8814 uni v ersal, and an easily characterizable theoretical benchmark will follo w [6]. This ne w particle is a possible candidate to e xplain the DM content of the uni v erse [7]. Under U PQ (1) symmetry (axion eld), we can write the ef fecti v e eld Lagrangian for QCD theory . L a = L QCD + g 2 s 32 π ¯ θ G b µν G b µν + g 2 s 32 π 2 a f a / N G b µν ˜ G b µν (1) Here, f a is the axion decay constant, b stands for the color char ge on gluons, and dots represent another possible term in the gi v en Lagrangian, and the second and third sho w the axion coupling with the gluon eld. In the original model for the axion N in Peccei-Quinn–W einber g–W ilc zek model (PQWW), it becomes N = 6 , we can write as lik e N = P f X f and represent color anomaly in U PQ (1) symmetry , the sum of PQ char ges X f o v er the fermions in the theory . The remnants of superno v a e xplosions represent one of the most e xtreme en vironments in the uni v erse [8]. These compact objects possess po werful gra vitational elds, ultra-dense cores, and some of the strongest magnetic elds kno wn, often e xceeding 10 8 G to 10 15 G (from 10 4 T to 10 12 T) [9], [10], in the case of magnetars , from 10 15 G reaching up to 10 19 G (from 10 11 T to 10 15 T) for more information visit [11]. Such e xtreme conditions mak e neutron stars (NS) unique astroph ysical laboratories for probing fundamental ph ysics, particularly the beha vior of e xotic particles lik e axions. NS are natural sources for axions and en vironments where these particles can lea v e observ able i mprints. Axions can be produced in the dense interiors of NS through v arious mechanisms, such as nucleon-nucleon bremsstrahlung. F or more details, visit [12], [13], and pionic processes are studied by [14]. Once produced, these axions can escape the star due to their weak interactions, carrying a w ay ener gy and contrib uting to the cooling of the NS. This ener gy loss mechanism has been e xtensi v ely studied and pro vides stringent constraints on axion properties, such as their coupling strength to matter and their mass. Moreo v er , the dense magnetized plasma surrounding NS of fers a unique setting for axion-photon con v ersion, a process that could lead to detectable electromagnetic signals. The electrodynamics of the axion, in terms of Chern-Simons coupling L ⊃ α / 8 π f a F µν ˜ F µν [15], creates a rich electrodynamical structure in which mixing between axions and photons in e xternal ⃗ E · ⃗ B enables resonant con v ersion processes. That such an interaction, in terms of axion-impro v ed Maxwell’ s equations, yields magnetized birefringence, photon-axion spectral splitting in plasmas, and stimulated decays in astroph ysical settings, is predicted. In the vicinity of a NS, the intense magnetic elds f acilitate this con v ersion via the Primak of f ef fect and are e xplored further by [16], enabling axions to transform into observ able photons. Axion-photon con v ersion is not limited to the immediate vicinity of NS. Axions produced in the core can form dense clouds around the star due to gr a vi tational attra ction, and the inuence of the magnetic eld is e xplored [17], [18]. These axion clouds, which gro w o v er time, pro vide an additional reserv oir for photon production. The density and spatial distrib ution of the axion cloud depend on f actors such as the NS’ s magnetic eld strength, rotation rate, and age. Observ ations of anomalous X-ray or radio emissions from NS could thus serv e as indirect e vidence for axion clouds and their con v e rsion into photons. Diagonalizat ion in an anisotropic en vironment for coupled axion-photon motion re v eals a mixture of polarization states with eigen v alues sensiti v e to plasma frequenc y ω p and coupling between axions and photons g a γ γ . Non-perturbati v e computations in lattices s pecify , in addition, photon-emitting topological defects in ALP elds, with ne w signatures in NS magnetospheres and NS binaries. Axion DM with its µeV mass and weak couplings to SM elds is a v oided in con v entional models [19]. Be yond their role in axion-photon con v ersion, NS also serv e as potential sources of axion DM. In the early uni v erse, axions could ha v e been produced non-thermally through mechanisms such as the misalignment mechanism. F or more information, see [20], and for string decay , visit [21], leading to a cold and dif fuse background of axion DM. NS, with their strong gra vitational elds, can capture and accumulate these ambient axions, further enhancing their local density . This accumulation not only amplies the potential for axion-photon con v ersion b ut also pro vides a unique opportunity to probe the properties of axion DM through astroph ysical observ ations. It is e xclusi v ely produced through non-thermal processes such as v acuum misalignment and decay of cosmological topological defects. The misalignment mechanism, with its timescale determined by θ i and of the axion eld during an e xpanding uni v erse, creates coherent oscillation that redshifts as cold DM. In a k e y feature, the dynamically link ed axion mass m a to QCD topological susceptibility m a ∼ Λ 2 QCD /f a , has its origin in f a the decay constant of the axion, a relation xing the axions parameter space to both cosmology and high-ener gy ph ysics. Ringw ald and Saika w a [22] studied the axion eld dynamics after ination and this study is also present [23], Peccei-Quinn (PQ) symmetry breaking creates Int J Adv Appl Sci, V ol. 15, No. 1, March 2026: 355–371 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Adv Appl Sci ISSN: 2252-8814 ❒ 357 axionic strings study are discussed [24], [25] and domain w alls [26], whose decay puts entrop y in the density eld of axions, and such brea kings during [27] ination remo v e inhomogeneities, lea ving θ i a stochastic v ariable under anthropi c selection. Cosmic ination and such dynamics become intertwined, and isocurv ature perturbations with constraints under Planck, tie the ab undance of the axion with ination’ s Hubble scale H inf and tensor -to-scalar ratio r . Be yond their role in electromagnetic signatures, axions also impact the thermal e v olution of NS. Axion emission through processes lik e n + n → n + n + a contrib utes to cooling, particularly in young NS where the core temperature is high. Ho we v er , this cooling ef fect diminishes o v er time as neutrino and photon emissions [28] dominate in older stars. Observ ations of NS cooling, such as those of the Cassiopeia A superno v a remnant [29], ha v e been used to constrain axion properties, although uncertainties in microph ysics complicate the interpretation of data. Ov erall, the st udy of axion electrodynamics in NS backgrounds of fers profound insights into fundamental ph ysics, bridging particle ph ysics, astroph ysics, and cosmology , while pro viding a promising a v enue for indirect axion detection. The study of axion-photon con v ersion in NS en vironments is a rapidly e v olving eld, dri v en by adv ances in observ ational techniques and theoretical modeling. Noordhuis et al. [30] ha v e e xplored the implications of axion clouds for NS obs erv ables, i ncluding their cooling rates, spin-do wn beha vior , and electromagnetic emissions. F or instance, detecting anomalous X-ray or g amma-ray signals from NS could pro vide direct e vidence for axion-photon con v ersion. Similarly , radio observ ations of NS magnetospheres of fer a complementary approach to probing axion properties. This w ork distinguishes itself through its core methodological approach. While man y studies of axion-photon con v ersion in NS magnetospheres rely on the W entzel–Kramers–Brillouin (WKB) approximation or s tationary-phase inte gration along a path [31]–[33], we emplo y a time-dependent state e v olution formalism. This technique, inspired by quantum mechanical tw o-le v el systems, solv es the coupled equations of motion by diagonalizing the m ixing matrix in the time domain. This pro vides a direct and transparent frame w ork for deri ving the con v ersion probability , which is particularly suited for analyzing coherent e v olution o v er time. The primary no v elty of this paper is the deri v ation of a ne w analytical e xpression for the axion-photon con v ersion probability from this state e v olution perspecti v e, and the subsequent demonstration that this approach predicts a radiated po wer approximately 10 23 orders of magnitude lar ger than that estimated from static or propag ating-state formalisms [9], from this study , we are sure the cooling rate of NS will be much f aster , as already discussed in [34], [35]. The state e v olution approach sheds light on fundamental ph ysics and links axion DM with the NS cooling rate, pro viding a shred of strong e vidence for the kilono v a signal . W e need to consider a multi-directional approach to enhance and g ain e xtra sensiti vi ty in ongoing e xperiments. In the future, the Green Bank T elescope (GBT) [36], More Karoo Array T elescope (MeerKA T) [37], and James W ebb Space T elescope (JWST) [38], [39] projects will e xplore this re gion g a γ γ ≃ 10 − 14 − 10 − 15 GeV − 1 . The roadmap of this w ork is as follo ws : in section 2, we pro vide a basic o v ervie w of ongoing e xperimental and theoret ical limits on the axion-photon coupling constant. In section 3, we discuss axion-photon mixing, state e v olution probability , and ux analysis. W e discuss the radiati v e po wer of axion-photon con v ersion for state e v olution in section 4. In section 5, the result and discussion of this research is presented. Finally , section 6 and 7 present limitation and conclude this w ork. 2. LITERA TURE REVIEW Noordhuis et al. [32] demonstrated that NS can accumulate dense “axion clouds” through non-stationary pair plasma dischar ges in their polar cap re gions, particularly for axion masses in the range 10 − 9 ≤ m a ≤ 10 − 4 eV . These axions remain gra vitationally bound and accumulate o v er astroph ysical timescales, reaching densities that can e xceed O (10 22 ) GeV cm − 3 , e v en for v ery small axion-photon couplings. The authors sho w that such clouds dissipate ener gy primarily via resonant axion-photon con v ersion in the magnetosphere, producing distincti v e radio signatures such as narro w spectral lines and transient b ursts. Their w ork highlights NS as promising l aboratories for probing axion-lik e particles, with potential detectability using current radio telescopes lik e lo w-frequenc y array (LOF AR) [40] and GBT [41]. It underscores the importance of time-dependent and plasma-a w are modeling in predicting observ able signals. In a si gnicant adv ancement of magnetospheric modeling, Miguel [10] de v eloped a com prehensi v e frame w ork for axion-photon con v ersion that i ncorporates both pair multiplicity f actors and relati vistic plasma ef fects, mo ving be yond the traditional Goldreich-Julian density prole [42]. This w ork demonstrated that State e volution appr oac h for the axion con ver sion pr obability in ... (Bilal Ahmad) Evaluation Warning : The document was created with Spire.PDF for Python.
358 ❒ ISSN: 2252-8814 accounting for electromagnetic cascades and char ge acceleration in pulsar and magnetar magnetospheres signicantly shifts the resonant con v ersion to higher frequencies, potentially e xtending detectable signals into the millimeter band for axion masses up to approximately 1 meV . The study identied SGR 1745–2900 as a particularly promising tar get due to its strong magnetic eld and location in the Galactic Center re gion with enhanced DM density . While this model pro vides crucial insights into magnetos pheric comple xities, our w ork complements it by emplo ying a fundamental ly dif ferent, ti me-dependent state e v olution approach rather than the stationary-path approximations common in the literature. Miguel [10] focus on ho w plasma properties af fect resonance conditions, we deri v e a ne w analytical e xpression for con v ersion probability that captures coherent quantum e v olution o v er time, re v ealing a dramatically enhanced radiated po wer that could e xplain rapid NS cooling and pro vide stronger constraints on axion-photon coupling. T erc ¸ as et al. [31] in v estig ated impact of resonant axion-plasmon con v ersion in NS magnetospheres, re v ealing a signicant suppression of detectable radio signals from axion-photon interactions. The authors demonstrate that in dense plasma en vironments, axions can resonantly con v ert into longitudinal plasmon modes at a smaller radius r c , p than the st andard axion-photon con v ersion radius r c ,γ , ef fecti v ely reducing the photon-production v olume. This non-radiati v e ener gy loss diminishes the e xpected ux density reaching Earth, shifting e xperimental sensiti vity curv es into re gions already e xcluded by e xisti ng constraints. Their ndings emphasize the critical need to incorporate plasma collecti v e ef fects into axion search strate gies, as ne glecting axion-plasmon interactions may lead to o v erly optimistic projections for radio-telescope-based detection ef forts. In a complementary approach to magnetospheric con v ersion, Ro y et al. [38] in v estig ated the potential of the JWST [43] to detect eV -scale axion DM via its decay into photons within the Milk y W ay halo. Their w ork forecasts that JWST’ s end-of-mission blank-sk y observ ations will pro vide leading sensiti vity t o axion-photon couplings g a γ γ ≥ 5 . 5 × 10 − 12 GeV − 1 for axion masses between 0.18 and 2 . 6 eV , potentially ruling out nucleophobic QCD axions with masses abo v e approximately 0 . 2 eV . While their study focuses on the decay of ambient Galactic DM, our w ork e xplores a fundamental ly dif ferent production mechanism: the con v ersion of axions into photons within the e xtreme en vironment of a NS magnetosphere via the Primak of f ef fect. The tw o approaches are highly complementary; JWST probes the decay of dif fuse axions, whereas our state e v olution formalism appli ed to NS signals is sensiti v e to the local con v ersion of axions, potentially from both the ambient halo and those produced or accumulated by the NS itself. T ogether , these methods co v er distinct yet o v erlapping re gions of the axion parameter space, with our predicted sensiti vity of g a γ γ ≃ 10 − 15 GeV − 1 for m a ≃ 10 − 6 eV e xploring a dif ferent, lo wer -mass and weak er -coupl ing re gime that is be yond the scope of JWST decay searches b ut potentially accessible through tar geted radio and X-ray observ ations of NS. In a direct search for axion DM, F oster et al. [36] used the GBT and the Ef felsber g 100-m radio telescope to look for t he predicted con v ersion of axions into radio photons wit h i n the strong magnetic elds of NS. The search tar geted nearby isolated NS and the dense Galactic Center re gion, scanning the highly moti v ated axion mass range of approximately (5 − 11) µ eV for (1 . 1 to 2 . 7) GHz . Their analysis, which emplo yed a rob ust lik elihood-based frame w ork to identify ultra-narro w spectral lines, found no signicant e vidence for an axion signal. Consequently , the study placed some of the most stringent constraints to date on the axion-photon coupling constant g aγ γ for this mass range, e xcluding ne w parameter space be yond pre vious laboratory e xperiments and demonstrating the po werful potential of radio telescopes in hunt for particle DM. In their 2021 study , W itte et al. [44] address k e y theoretical uncertainties in axion DM searches via NS radio signals by de v eloping an end-to-end ray-tracing simulation that incorporates plasma ef fects within the Goldreich-Julian magnetosphere model [42]. Their analysis re v eals se v eral critical phenomena pre viously o v erlook ed: strong anisotrop y in the radio ux, signicant spectral line broadening due to photon-plasma interactions, premature axion-photon dephasing from refraction, and time-dependent signal v ariations inuenced by vie wing angle and magnetospheric geometry . The authors also highlight that e xceptionally strong magnetic elds—such as those of magnetars—can lead to c yclotron absorption, reducing detectability . This w ork highlights the i mportance of incorporating plasma dynamics into future axion search strate gies and of fers a e xible computational frame w ork to mitig ate theoretical uncertainties in indirect detection ef forts. This w ork presents a no v el time-domain search for axion DM using radio observ ations of the pulsar PSR J2144-3933 with the MeerKA T telescope [37]. Unlik e pre vious frequenc y-domain approaches, the authors emplo y a matched-lter technique to le v erage the predicted time-v arying signature of axion-photon con v ersion in the pulsar’ s magnetosphere, which arises from its rotating, non-axisymmetric plasma structure. Analyzing Int J Adv Appl Sci, V ol. 15, No. 1, March 2026: 355–371 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Adv Appl Sci ISSN: 2252-8814 ❒ 359 4,416 seconds of data, the y nd no signicant signal and place an upper limit on the axion-photon coupling of g aγ < 5 . 5 × 10 − 11 GeV − 1 o v er the mass range (3 . 9 − 4 . 7) µ eV (assuming a pulsar distance of 0 . 165 Kp c ) and constraints are sho wn in Figure 1. The study demonstrates that time-domain information can enhance sensiti vity compared to time-a v eraged ux measurements, particularly for NS with lar ge magnetic elds. It also discusses prospects for tar gets lik e the Galactic Center Magnetar and ne xt-generation telescopes, such as the square kilometre array (SKA) [45]. The Conseil Europ ´ een pour la Recherche Nucl ´ eaire (CERN) axion solar telescope (CAST)-center for axion and precision ph ysics research (CAPP) e xperiment [46], represent s a signicant adv ancement in the direct search for g alactic DM axions within the mass range of (19 . 74 − 22 . 47) µ eV , emplo ying a haloscope composed of four phase-matched resonant ca vities operating inside CERN’ s 8 . 8 T dipole magnet. By utilizing a f ast frequenc y tuning mechanism (10 MHz / min) and coherent signal combination across multiple ca vities, the collaboration achie v ed enhanced sensiti vity , collecting data o v er 4124 hours from 2019 to 202 1. The analysis e xcluded axion-photon couplings do wn to g aγ γ = 8 × 10 − 14 GeV − 1 at 90% condence le v el, probing pre viously une xplored parameter space, as sho wn in Figure 2. The e xperiment also demonstrated no v el techniques, such as phase-matching and rapid scanning, that pa v e the w ay for future lar ge-scale axion searches, including sensiti vity to transient signals from axion streams or mini-clusters. Figure 1. The axion-photon coupling from 1 hour observ ations of PSR J2144-3933 from the SKA (red) [37] Figure 2. CAST -CAPP , sensiti vity on g a γ γ as a function of axion mass with 95% condence le v el [46] 3. METHODS This w ork in v estig ates a x i on-photon con v ersion in NS magnetospheres using a no v el, ti me-dependent state e v olution formalism. This approach pro vides a distinct and po werful alternati v e to the commonly emplo yed WKB or stationary-phase approximations [31]–[33]. By framing the problem in the time domain and dra wing an analogy to a quantum mechanical tw o-le v el system, we directly solv e the coupled equations of motion for the axion and photon elds. The core of our methodology in v olv es diagonalizing the axion-photon State e volution appr oac h for the axion con ver sion pr obability in ... (Bilal Ahmad) Evaluation Warning : The document was created with Spire.PDF for Python.
360 ❒ ISSN: 2252-8814 mixing matrix to deri v e a ne w , generic analytical e xpression for the con v ersion probability . This technique is particularly well-suited for capturing t he coherent e v olution of the system o v er time, of fering a transparent frame w ork that re v eals a signicantly enhanced radi ated po wer compared to static or propag ating-state formalisms, with profound implications for NS cooling rates and observ able signals. 3.1. Axion electr odynamics Axion electrodynamics in the NS background represents a f ascinating interplay of particle ph ysics and astroph ysics. In this conte xt, axions are h ypothetical light pseudo-scalar particles interact with electromagnetic elds in the e xtreme en vironments surrounding NS. NS, characterized by its immense gra vitational elds, ultra-strong magnetic elds (ranging from 10 15 G to 10 19 G in magnetars) [9]–[11], and dense plasma go v erned by Maxwell’ s equations as in (2) and (3) [42]. Pro vide unique conditions for studying axions. ⃗ ∇ · ⃗ E = ρ , ⃗ ∇ × ⃗ E = − ∂ t ⃗ B (2) ⃗ ∇ · ⃗ B = 0 , ⃗ ∇ × ⃗ B = J + ∂ t ⃗ E (3) 3.2. Modied Maxwell’ s equations f or axion dark matter A set of Maxwell’ s equations acquired from this approximation e xactly describes the reacted elds generated from the axion-photon interaction. This interaction leads to the con v ersion of axions into photons through the in v erse Primak of f ef fect in the presence of a NS magnetic eld [16]. Man y of the successful e xperiments searching for axions rely on this axion-photon coupling, along with the assumption that axions constitute halo DM [15], [47], [48], and are therefore referred to as axion haloscope searches. T o account for the axion interaction with electromagnetic elds, classical Maxwell’ s equations must be modied accordingly . The ef fecti v e Lagrangian that describes the axion-photon interaction, including an axion-lik e term, can be deri v ed in system international (SI) units as (4). L a = 1 2 ∂ ν a∂ ν a − 1 2 m 2 a 2 + g aγ 4 aF µν ˜ F µν + · · · } (4) Modied Gauss la w: ∂ µ F µν = g aγ ∂ µ a ˜ F µν + ∂ µ F µν = 1 2 g aγ ϵ µν ρσ ∂ µ aF ρσ + J ν Here, 1 2 ϵ µν ρσ F ρσ , if ν = 0 and µ = 0 , k , then (5). ∂ i E i = 1 2 g aγ ⃗ ∇ a · ⃗ B + ρ a (5) Modied ampere la w: after simplication as in (6). ⃗ ∇ × ⃗ B − ∂ E ∂ t = g aγ ⃗ ∇ a × E − ∂ a ∂ t ⃗ B + J (6) Modied F araday la w: as we kno w the duality transformation of the ⃗ E and ⃗ B , hold equally with asymmetric nature, after doing transformation in (6), in result we get (7). ⃗ ∇ × ⃗ E + ∂ ⃗ B ∂ t = − g aγ ⃗ ∇ a × ⃗ B + ∂ a ∂ t ⃗ E + J (7) Equation of motion for ⃗ E and ⃗ B elds: let’ s use (8). ⃗ ∇ × ⃗ B = ∂ t ⃗ E − g aγ ⃗ E × ⃗ ∇ a − ⃗ B ∂ t a (8) No w taking curl of ⃗ ∇ × ⃗ B , then we get (9). ⃗ ∇ × ⃗ ∇ × ⃗ B = ⃗ ∇ × ∂ t ⃗ E − g aγ ⃗ ∇ × ⃗ E × ⃗ ∇ a − ⃗ B ∂ t a (9) Int J Adv Appl Sci, V ol. 15, No. 1, March 2026: 355–371 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Adv Appl Sci ISSN: 2252-8814 ❒ 361 After using v ector identities, ∇ 2 ⃗ B + ∂ 2 t ⃗ B = − g aγ ⃗ E ⃗ ∇ · ⃗ ∇ a − g aγ ⃗ ∇ a ⃗ ∇ · ⃗ E + g aγ ⃗ ∇ a · ⃗ ∇ ⃗ E − g aγ ⃗ E · ∇ ⃗ ∇ a + g aγ h ⃗ ∇ ( ∂ t a ) × ⃗ B + ∂ t a ⃗ ∇ × ⃗ B i Simplied form for ⃗ B -eld, using Coulomb g auge ∂ µ ∇ µ = 0 as in (10). ∇ 2 ⃗ B − ∂ 2 t ⃗ B = − g aγ ∂ t ⃗ E ∂ t a (10) Similarly equation of motion for ⃗ E eld will be (11). ⃗ ∇ 2 ⃗ E − ∂ 2 t ⃗ E = g aγ ∂ t ⃗ B ∂ t a (11) Equation of motion for axion and photon elds: from classical eld theory , we can write the action for the axion eld as (12). S = Z d 4 x − 1 2 ∂ µ a∂ µ a + m 2 a a 2 − 1 4 g aγ γ aF µν ˜ F µν (12) Let’ s equate the interaction terms set to zero. Using the KG-w a v e equation for the axion eld as in (13). − 1 2 ∂ µ a∂ µ a + m 2 a a 2 = □ a + m 2 a a (13) After simplications, the result equation of motion will be (14). □ a + m 2 a a = − g a γ γ ⃗ E · ⃗ B n (14) The term ⃗ E · ⃗ B n represents the EM component and comes from the axion-photon interaction. In astroph ysical en vironments lik e NS magnetospheres, re gions with non-zero ⃗ E · ⃗ B n (e.g., due to dynamical screening in v acuum g aps) can ef ciently produce axions where axion mix ed with ⃗ E · ⃗ B n and photon equation of motion as in (15). □ A = g a γ γ ∂ t a ⃗ B n (15) The dispersion relation of a photon in a plasma is (16). ω 2 − k 2 ≃ ω p , ω p = r 4 π α n e m e (16) Here, ω p is plasma frequenc y , n e electron number density , and α is the ne-structure constant. In general, ω p = 1 . 31 × 10 18 p n e / 10 26 cm − 3 R − 1 ⊙ , here R ⊙ is solar radius. 3.3. The mixing of axion-photon elds Axion is a v ery elusi v e particle that only interacts via gra vitational interaction, the detection of the axion is trick y so the detection of axion can be probed via the con v ersion of the photon into axion and vice v ersa, as occurs in the sun which is kno wn as Primak of f con v ersion [49], this process is the k e y phenomenon through which an y neutral particle can be con v erted into tw o photons in the presence of columbic eld of the nucleus. Similarly , the axion can be con v erted into tw o photons in the presence of an e xternal electromagnetic eld, which inuences the con v ersion phenomenon. In an y e xperiment, the con v ersion of the axion into a photon can happen statistically , which can be predicted through the probability of the con v ersion of the axion into a photon. In this w ork, we will e xplore the con v ersion probability of axions into photons in the presence of a NS magnetic eld, utilizing the axion-photon mixing mechanism. In astroph ysical en vironments such as NS, axions can be produced through processes lik e nucleon-nucleon bremsstrahlung in dense nuclear matter and can couple to photons via the axion-photon interaction term. This coupling f acilitates processes such as the Primak of f ef fect, where axions con v ert into photons in the presence of strong magnetic elds. As a result, the magnetospheres of NS pro vide ideal en vironments for detecting axion-induced signals. Axion con v erts into photons in the presence of a NS magnetic eld ⃗ B n , which could be dubbed as an oscillation of axion into photons. W e did our analysis in the time domain oscillation instead of the spatial State e volution appr oac h for the axion con ver sion pr obability in ... (Bilal Ahmad) Evaluation Warning : The document was created with Spire.PDF for Python.
362 ❒ ISSN: 2252-8814 component because, in NS magnetospheres, the magnetic eld and plasma density can change o v er time due to processes lik e magneto-rotational spin-do wn or glitches. These changes af fect the axion-photon con v ersion dynamics, making a time-dependent analysis more appropriate for axion con v ersion into photons. The plane w a v e solution for the axion and photon elds is (17) and (18). a ( r , t ) = a 0 e i ⃗ k r − iω t (17) A ( r , t ) = A 0 e i ⃗ k r − iω t (18) Here, both plane w a v es satisfy both equation of motion instead of xing an y specic direction we are dealing with in the time domain, because if we x the axion eld oscillation or the photon eld propag ation in one direction, we might be we lose the friction of data, then as a result we get (19) and (20). ∂ 2 t − ⃗ k 2 − m 2 a a = − g aγ γ ω A ⃗ B n (19) ( ∂ 2 t − ⃗ k 2 ) A = − g aγ γ ω a ⃗ B n (20) No w (19) and (20) we can write in matrix form and also use (16), so we get a v ery simplied form as in (21). ∂ t A ( t ) a ( t ) = " − ω 2 p 2 ω − 1 2 g aγ γ ⃗ B n − 1 2 g aγ γ ⃗ B n − m 2 a 2 ω # A ( t ′ ) a ( t ′ ) (21) Where, ω is photon frequenc y and m a is axion mass. Here, ∆ aγ = − 1 2 g aγ γ ⃗ B n , ∆ p = − ω 2 p 2 ω , and ∆ a = − m 2 a 2 ω , let’ s substituted back then we get (22). M = ∆ p ∆ a γ ∆ a γ ∆ a (22) 3.4. Mathematical modeling The deri v ation of the Schr ¨ odinger -lik e e v olution in (22) relies on a set of specic ph ysical approximations which we no w clarify . – Linearly polarized photons and constant magnetic eld: we consider the con v ersion of axions into a single dominant polarization mode of the photon, parallel to the e xternal magnetic eld ⃗ B n , which is assumed to be constant and homogeneous o v er the con v ersion re gion for this initial deri v ation. This allo ws us to treat the photon eld as a scalar , A . – High-ener gy approximation ( ω ≫ m a , ω p ): we assume the particle ener gy ω is much lar ger than both the axion mass m a and the plasma frequenc y ω p . This justies the use of the relati vistic dispersion relation and allo ws us to approximate the d’Alembert operator as ( ∂ 2 t − ∇ 2 ) ≈ 2 iω ( ∂ t + ∂ z ) for a plane w a v e e − iω t + ik z [50]. – T ime-domain focus and forw ard propag ation: we ne glect spatial deri v ati v es perpendicular to the propag ation direction and focus on the time e v olution, ef fecti v ely considering a localized re gion of the magnetosphere. This simplies the problem to a rst-order dif ferential equation in time, ∂ t ≈ − i H , where H i s the Hamiltonian. – Ne glect of back-reaction: we assume the photon eld generated by axion con v ersion is small and does not signicantly back-react on the axion eld or the e xternal magnetic eld. W e no w assume that the mass matrix in (21) is independent of an y x ed direction. This implies that the magnetic eld remains constant, with a x ed magnitude and direction, and that the con v ersion occurs in a homogeneous plasma at a constant frequenc y . Under these conditions, the mass matrix can be diagonalized with R M R † , allo wing us to transform it unmi x edl y . Let’ s solv e eigen v alue, since ( M − λI ) v = 0 , then we get eigen v alues, for our case as in (23). λ ± = − m 2 a + ω 2 p ± q 4 ⃗ B 2 n g 2 a γ γ ω 2 + m 2 a − ω 2 p 2 4 ω (23) Int J Adv Appl Sci, V ol. 15, No. 1, March 2026: 355–371 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Adv Appl Sci ISSN: 2252-8814 ❒ 363 From neutrino oscillation using a mixing angle approach ˜ M = R T M R (24). R = cos θ sin θ − sin θ cos θ (24) And R T is transpose of rotational metrics R . Then, as a result, we get an unmix ed matrix for the axion con v ersion probability as in (25). D = − m 2 a + ω 2 p + p 4 B 2 g 2 aγ γ ω 2 + ( m 2 a − ω 2 p ) 2 4 ω 0 0 − m 2 a + ω 2 p − p 4 B 2 g 2 aγ γ ω 2 + ( m 2 a − ω 2 p ) 2 4 ω (25) The matrix D encapsulates the e v olution of the axion-photon system from the initial point to an arbitrary point. This solution w as deri v ed using the equation of motion gi v en in (25). 3.5. Axion con v ersion pr obability The state e v olution tactic in axion-photon mixing of fers a po werful background for understanding the time-dependent dynamics of axion-to-photon con v ersion in systems such as dense astroph ysical en vironments or laboratory e xperiments. This technique e xplains the coupled quantum equations of motion (QEoM) for the axion and photon elds, accounting for classical mean-eld ef fects and quantum uctuations that k ernel the instability dri ving the mixing. By presenting scaled v ariables and r escaling operators to knob lar ge particle numbers ef ciently , the method captures the e v olution of photon modes from an initially pure axion state through a “quantum break” mechanism, where coherent axion-photon interactions l ead to swift ener gy transfer . Signicantly , this formalism incorpora tes multi-mode ef fects, mitig ating log arithmic f actors related to single-mode approximations, while also addressing challenges lik e redshift and inhomogeneity by simulating spatially e xtended systems. The result is a vigorous e xplanation of ho w axion clouds e v olv e into mix ed states with signicant electromagnetic components, contrib uti ng insights into observ able signatures such as radio emission or X-ray signals from NS. Let’ s dene a relation for the con v ersion of axions int o photons through state e v olution as in (26). P a → γ = D A ( t ) a ( t ) E 2 (26) Using [50], [51] and deri v e, a ( t ) = − sin θ a 1 (0) + cos θ a 2 (0) and quantum st ate for photon. After time t when axion eld e v olv es, A ( t ) = cos [ θ ] e − iλ 1 t | A 1 (0) ⟩ + sin [ θ ] e − iλ 2 t | A 2 (0) ⟩ . Then (26) will be (27). P a → γ = − e − iλ 1 t sin [ θ ] cos [ θ ] ⟨ A 1 (0) | a 1 (0) ⟩ + e − iλ 1 t cos 2 [ θ ] ⟨ A 1 (0) | a 2 (0) ⟩ − e − iλ 2 t sin 2 [ θ ] ⟨ A 2 (0) | a 1 (0) ⟩ + e − iλ 2 t sin [ θ ] cos [ θ ] ⟨ A 2 (0) | a 2 (0) ⟩ 2 (27) Using the orthogonality conditions, ⟨ ϕ i | ϕ j = δ ij and must follo w if i = j it should be 1 otherwise 0 . Then (27) as in (28). P a → γ = sin [ θ ] cos [ θ ] e − iλ 2 t − e − iλ 1 t 2 (28) Using e xponential form of sin [ θ ] , so e − iλ 2 t − e − iλ 1 t = − 2 i sin [( λ 2 − λ 1 ) t/ 2] . After some straight forw ard calculation, then our e xpression will be (29). P a → γ = sin 2 [2 θ ] sin 2 ( λ 2 − λ 1 ) t 2 (29) No w use eigen v alues e xpressions, which is λ 1 = − m 2 a + ω 2 p + q 4 ⃗ B 2 n g 2 a γ γ ω 2 + m 2 a − ω 2 p 2 / 4 ω and λ 2 = − m 2 a + ω 2 p − q 4 ⃗ B 2 n g 2 a γ γ ω 2 + m 2 a − ω 2 p 2 / 4 ω , and solv e with the basic algebraic approach then the result of eigen v alues dif ference is ∆ e = λ 2 − λ 1 = q 4 ⃗ B 2 n g 2 a γ γ ω 2 + m 2 a − ω 2 p 2 / 2 ω and from here [50], [51], we can write a relation for sin[2 θ ] as in (30). sin [2 θ ] = 2 ∆ aγ γ q 4 ⃗ B 2 n g 2 a γ γ ω 2 + ( m 2 a − ω 2 p ) 2 4 ω (30) State e volution appr oac h for the axion con ver sion pr obability in ... (Bilal Ahmad) Evaluation Warning : The document was created with Spire.PDF for Python.
364 ❒ ISSN: 2252-8814 No w use the abo v e e xpressions in (29), as a result, we get a v ery well simplied form for the axion con v ersion probability as in (31). P a → γ ≃ g a γ γ ⃗ B n t 2 F ( ω , t ) . (31) Since (32). F ( ω , t ) = sin 2 " q 4 g 2 a γ γ ⃗ B 2 n ω 2 + ( m 2 a − ω 2 p ) 2 4 ω t #   q 4 g 2 a γ γ ⃗ B 2 n ω 2 + ( m 2 a − ω 2 p ) 2 4 ω ! 2 ∼ 1 (32) The mass dim ension of ω p = ω = m a = M 1 , ⃗ B n = M 2 and g a γ γ = t = M − 1 , since, con v ersion probability P a → γ must be dimensionless, in (31) conrms its ph ysical v alidity and re v eals k e y scaling beha viors. From (31), we deri v e the scaling of the con v ersion probability with the ph ysical parameters: – Magnetic eld ⃗ B n : the probability scales as P a → γ ∝ B 2 n . In the strong-eld re gime, the pre-f actor saturates to 1, and the probability oscillates sinusoidally . – Axion mass m a : in the small-mixing re gime, P a → γ ∝ m − 4 a . This strong in v erse dependence on the axion mass means that lighter axions ha v e a signicantly higher con v ersion probability in NS magnetospheres for a x ed coupling g a γ γ . – Propag ation distance (time t ): the probability oscillates as sin 2 [ θ t ] , this oscillatory beha vior with time (which corresponds to propag ation distance for a non-relati vistic axion) is a hallm ark of coherent quantum mixing. The characteristic oscillation length L osc = π /κ determines the scale o v er which the probability c ycles from zero to its maximum v alue. F or the simplied case where t he axion mass term dominates ( κ ≃ m 2 a / (4 ω ) ), the oscillation length scales as L osc ∝ ω /m 2 a . Our state e v olution formalism pro vides a foundational frame w ork that can be inte grated with broader astroph ysical modeling to enhance its predicti v e po wer and testability . A natural e xtension of this w ork in v olv es coupling our model with NS population synthesis [52]. By applying our con v ersion probability to a synthetic population of NS with v arying magnetic elds, ages, and distances, we could generate statistically signicant predictions for the all-sk y ux of axion-induced photons. This w ould allo w for direct comparison with unresolv ed background radiation in radio and X-ray surv e ys, setting more rob ust, population-a v eraged constraints on axion paramet ers. Furthermore, our results can be incorporated into spectral modeling codes for indi vidual NS. By calculating the e xpected axion-con v ersion photon ux as a function of ener gy and adding it to standard magnetospheric emission models, we can search for spectral anomalies or e xcesses that could be attrib uted to axions. This approach is particularly promising for interpreting data from ne xt-generation X-ray observ atories (e.g., Athena) and radio telescopes lik e the SKA and MeerKA T [9], [45]. 3.6. V alidity of the adiabatic appr oximation and non-adiabatic transitions Our deri v ation of the con v ersion probability (31) assumes a homogeneous en vironment with a constant mixing matrix. Ho we v er , in a realistic NS magnetosphere, the magnetic eld strength ⃗ B n and plasma frequenc y ω p are functions of position. The system’ s e v olution is then go v erned by a position-dependent Hamiltonian M ( r ) . A k e y question is whet her the adiabatic approximation is v alid. This approximation holds when the en vironment changes slo wly compared to the system’ s internal oscillation frequenc y . The condition for adiabatically is that the mixing angle θ ( r ) changes little o v er an oscillation length L osc as in (33) [33]. γ = | dθ /dr | L − 1 osc ≪ 1 (33) Where the oscillation length is L osc = 2 π / | λ 2 − λ 1 | = 2 π / ∆ e , and ∆ ef f = q 4 ⃗ B 2 n g 2 a γ γ ω 2 + m 2 a − ω 2 p 2 / 2 ω is the eigen v alue dif ference from our model. – When adiabaticall y holds γ ≪ 1 : the system smoothly follo ws an instantaneous mass eigenstate. In this re gime, the con v ersion probability can be calculated using the Landau-Zener formula for le v el crossing. If the axion passes through a resonance (where ω p ( r ) ≃ m a , making ∆ a γ γ ≃ ∆ p ), the adiabatic con v ersion probability can be v ery high. Int J Adv Appl Sci, V ol. 15, No. 1, March 2026: 355–371 Evaluation Warning : The document was created with Spire.PDF for Python.