Quantum refraction effects in pulsar radio emission

Highly magnetized neutron stars exhibit the vacuum non-linear electrodynamics effects, which can be well described using the one-loop effective action for quantum electrodynamics. In this context, we study the propagation and polarization of pulsar radio emission, based on the post-Maxwellian Lagrangian from the Heisenberg-Euler-Schwinger action. Given the refractive index obtained from this Lagrangian, we determine the leading-order corrections to both the propagation and polarization vectors due to quantum refraction via perturbation analysis. In addition, the effects on the orthogonality between the propagation and polarization vectors and the Faraday rotation angle, all due to quantum refraction are investigated. Furthermore, from the dual refractive index and the associated polarization modes, we discuss quantum birefringence, with the optical phenomenology analogous to its classical counterpart.


INTRODUCTION
Neutron stars have strong magnetic fields on their surface from 10 8 G up to 10 15 G, and in particular magnetars have the strongest magnetic fields in the universe with 10 13 − 10 15 G, which are near or a little above the supercritical value Bc = m 2 e c 3 /e = 4.4 × 10 13 G (Olausen & Kaspi (2014); Kaspi & Beloborodov (2017)).In such strong magnetic fields, the vacuum becomes a polarized medium due to the interaction of the fields with virtual electron-positron pairs.As a consequence, a photon propagating in the strong magnetic field background can be refracted or split, which is prohibited in the classical Maxwell theory.
The critical magnetic field (or the so-called Schwinger field) is three order higher than the current highest strength achieved with ultra-intense lasers; i.e., B = 4.9 × 10 −4 Bc (Yoon et al. (2021)).Therefore, highly magnetized neutron stars will provide a celestial laboratory to test quantum electrodynamics (QED) in the strong field regime and the relevant consequences (for review and references, see Ruffini et al. (2010); Fedotov et al. (2023); Hattori et al. (2023)).Recently, the surface magnetic field for Swift J0243.6+6124 has been directly measured from the detection of cyclotron resonance scattering (Kong et al. (2022)).Also, space missions have been proposed to investigate the strong-field QED effects: the enhanced X-ray Timing and Polarimetry (eXTP) (Santangelo et al. (2019)) and the Compton Telescope project (Wadiasingh et al. (2019)).
In this paper, we study the propagation and polarization of a photon in the dipole magnetic field background of a pulsar model, based on the post-Maxwellian (PM) Lagrangian; it is, in the weak field limit, the generic form of non-Maxwellian Lagrangian for the non-linear vacuum (Sorokin (2022)) and has been used to test vacuum polarization effects in the PVLAS (Polarizzazione del Vuoto con LAser) project (Ejlli et al. (2020)).The Heisenberg-Euler-Schwinger (HES) Lagrangian of strong-field QED (Heisenberg & Euler (1936); Schwinger (1951)) is also well approximated by the PM Lagrangian below the critical field strength.Here, we develop the recent analysis of vacuum birefringence in Kim & Kim (2023) further for a practical model of pulsar emission, where the magnetic field is defined for an oblique rotator with an inclination angle and the electric field is discarded.We consider the dual refractive index and the associated polarization vectors of a probe photon (Kim & Kim (2022)) to investigate the photon propagation in the strong magnetic field background of this pulsar model.Then the leading-order corrections to both the propagation and polarization vectors due to quantum refraction are determined via perturbation analysis.Our study provides a novel, complementary approach to and an elaboration of other similar studies in Heyl & Shaviv (2000, 2002); Heyl et al. (2003); Caiazzo & Heyl (2018); Heyl & Caiazzo (2018); Caiazzo (2019); Caiazzo & Heyl (2021); Caiazzo et al. (2022).
The paper is organized as follows.In Section 2 we review the Lagrangian formalism for the non-linearity of the quantum vacuum due to strong electromagnetic fields; a brief account of the HES Lagrangian and the PM Lagrangian as its approximation is given.In Section 3 we work out the deflection of a light ray from pulsar emission due to quantum refraction; the leadingorder corrections to the propagation vector and then to the trajectory of the light ray are determined.In Section 4 we look into the dual refractive index and the associated polarization modes of the light ray under the effect of quantum refraction; the leading-order corrections to the polarization vectors for Case I (Section 4.1) and Case II (Section 4.2) are determined.In addition, the effects on the orthogonality between the propagation and polarization vectors and the Faraday rotation angle, due to quantum refraction are investigated.Furthermore, in regard to the optical phenomenology from the dual refractive index and the associated polarization modes, we discuss quantum birefringence for this pulsar emission.Then finally, we conclude the paper with discussions on other similar studies and follow-up studies.

NON-LINEAR ELECTRODYNAMICS DUE TO STRONG FIELDS
Non-linear electrodynamic (NED) effects of the vacuum in the presence of strong electromagnetic fields have been studied employing the effective Lagrangian formalism.One of the most well-known NED models is the Heisenberg-Euler-Schwinger (HES) Lagrangian, which is obtained by adding the one-loop QED correction to the Maxwell Lagrangian, due to spin-1/2 fermions of mass me and charge e in electromagnetic fields of arbitrary strengths (Heisenberg & Euler (1936); Schwinger (1951)): in the convention with where L (0) (a, b) refers to the classical Maxwell Lagrangian, defined through with the Lorentz-and gauge-invariant Maxwell scalar F and pseudo-scalar and L (1) (a, b) refers to the Lagrangian of one-loop correction, However, in the weak-field limit (below the critical field strength Bc), the HES Lagrangian (1) has the leading-order contribution, the so-called post-Maxwellian (PM) Lagrangian (Euler & Kockel (1935) where η1 and η2 are parameters defined via η1/4 = η2/7 = e 4 / 360π 2 m 4 e .In some NED models, a parity violating term proportional to F G is added to the PM Lagrangian (Ni et al. (2013)).However, for the rest of the paper, our analysis is based on the PM Lagrangian; the dual refractive index as given by equation ( 9) in Section 3 is derived using this (Adler (1971); Kim & Kim (2022)).

DEFLECTION OF A LIGHT RAY DUE TO QUANTUM REFRACTION
A light ray is defined as an orthogonal trajectory to the geometrical wave-front S (x, y, z) = const., and therefore can be described by where s is an affine parameter to measure the length of the ray and n = n (r) is the refractive index given as a function of the position r on the ray (Born et al. (1999)).It can be further shown that Let n ≡ dr/ds be the unit propagation vector for the light ray emitted from a spot either at rest or in motion at a constant velocity.2Then equation (7) leads to (Born et al. (1999)) for n = const., ∇n ds n for n = const..
The expression of n for n = const.can be applied to a mechanism of how the light ray is deflected, for example, due to the quantum refraction effect in pulsar emission, as will be described below.In the presence of the effect, the refractive index n is given by a function of the position r, at which the light ray crosses a local magnetic field line in a pulsar magnetosphere; otherwise, it would simply be a constant.
According to Kim & Kim (2022), the refractive index n can be derived using the PM Lagrangian (5) as where B is the local magnetic field strength at a point in a pulsar magnetosphere, and ϑ denotes the angle between the light ray trajectory and the local magnetic field line (see Fig. 1).Here we have named Case I and Case II for two different values of the refractive index attributed to the same point in the magnetosphere; the propagation and polarization of the light ray are associated with these values.Later in Sections 4.1 and 4.2, the two polarization vectors for Case I and Case II, as given by ( 41) and (50), respectively are set to be orthogonal to each other and to the propagation vector such that the three vectors form a classical orthonormal basis.However, for the rest of this Section, we focus on Case I as there is little difference in the propagation of the light ray between the two cases.Then in Section 4 we look into the polarization of the light ray for both the cases and discuss quantum birefringence in relation to it.
One should note that the refractive index n as given by ( 9) has no dependence on the frequency of radiation.Consequently, in our entire analysis, the quantum refraction effects derived from this, on the propagation and polarization of a photon have no frequency dependence either, as can be checked with equations ( 11) and ( 40), respectively later.However, in order for equation ( 9) to be considered valid, the frequency of pulsar radiation must be significantly lower than that for excitation of the quantum vacuum (∼ 10 20 Hz; in the gamma-ray regime), which corresponds to the photon energy required to create an electron-positron pair, such that the vacuum is far from resonance.In addition, the plasma effects may be neglected if the pulsar radiation frequency is much higher than the local plasma frequency (∼ 10 9 Hz) (Petri (2016)).Therefore, our pulsar radiation can be safely assumed to cover optical to X-ray emissions (∼ 10 12 to 10 17 Hz) in this work.
One can expand the refractive index n for Case I in equation ( 9), having η1, η2 ∼ 10 −31 g −1 cm s 2 and B < Bc (critical magnetic field) ∼ 10 13 G, and thus η1B 2 , η2B 2 ≪ 1.Then it can be approximated as (Adler (1971)) which implies that the term η2B 2 sin 2 ϑ ∼ 10 −4 (B/Bc) 2 is the leading order quantum correction to n = 1 for classical optics, while O η 2 1 B 4 , η1η2B 4 , η 2 2 B 4 means the next-to-leading order terms to be ignored in our analysis.For computational purposes, the correction can be treated as the leading order perturbation with η2B 2 being a perturbation parameter.It should be noted here that n → 1, i.e., the refractive index goes back to the classical limit as B → 0 in the far field zone of the magnetosphere.
Keeping equation ( 10) in mind, from equation ( 8) one can determine the deflection of the light ray to leading order via where means the leading order quantum correction to the quantity (• • • ), led by η1B 2 or η2B 2 , and On the other hand, by n[0] we mean the unperturbed (classical) propagation direction of the light ray.A classical trajectory of the light ray traced by n[0] is as represented by the red dashed line as in Fig. 1, wherein it appears to be a straight line, being projected onto the xz-plane.
In our pulsar emission model, a light ray of curvature radiation is emitted from the dipole magnetic field: where µ is the magnetic dipole moment and α denotes the inclination angle between the rotation axis and the magnetic axis, and the light ray is tangent to the field line at the emission point (xo, yo, zo) = (ro sin θo, 0, ro cos θo) (see Fig. 1).At the same time, however, our pulsar magnetosphere rotates, and the magnetic field lines get twisted due to the magneto-centrifugal acceleration on the plasma particles moving along the field lines (Blandford & Payne (1982)).Taking into consideration this magnetohydrodynamic (MHD) effect due to rotation, one can describe classically the propagation direction of the light ray, Neutron star Observer Figure 1.A cross-sectional view of a pulsar magnetosphere with the dipole magnetic field lines (green) around a neutron star.The vertical dashed line (black) and the inclined solid line (red) represent the rotation axis and the magnetic axis, respectively.α between these axes denotes the inclination angle.The scale of the unity in this graph is equivalent to the neutron star radius ∼ 10 6 cm.The red dotted line represents the trajectory curve of the light ray traced by n[0] as projected onto the xz-plane.(Credit: Kim & Trippe (2021), reproduced with modifications.) which must line up with the particle velocity in order for an observer to receive the radiation, as (Gangadhara ( 2005)) where on the right-hand side B ≡ B/ |B| and the second term accounts for the centrifugal acceleration, with Ω ≡ Ωez and Ω being a pulsar rotation frequency, and with c being the speed of light and cos θ ′ ≡ cos α cos θ + sin α sin θ cos φ.However, during the rotation the azimuthal phase changes by φ ∼ Ωt, while our light ray has propagated a distance by s ∼ ct.We describe the propagation of the light ray with the consideration of the MHD effect above, assuming φ to be very small; e.g., φ 10 −1 is considered for a millisecond pulsar with Ω ∼ 10 2 Hz, during the time of rotation t 10 −3 s, such that s 10 7 cm, which corresponds to the propagation distance within about 10 times the neutron star radius.Then, for equation ( 13) we take only the leading order expansions of B (ro, θo, φ) and β (ro, θo, φ) in φ from equations ( 12) and ( 14), respectively, and obtain n and where we have considered Ωro/c φ, e.g., for a millisecond pulsar with Ω ∼ 10 2 Hz and ro ∼ 10 6 cm, such that O (Ωro/c) 2 O (φ (Ωro/c)) O φ 2 , all to be ignored in our analysis, and have substituted φ = Ωs/c in equation ( 17), the leading order rotational effect to be considered in our analysis.Now, integrating n[0] = dr/ds with respect to s, the unperturbed (classical) trajectory of the light ray can be derived: where nx[0] , nz[0] and ny[0] are given by equations ( 15)-( 17), respectively, and the emission point is (xo, yo, zo) = (ro sin θo, 0, ro cos θo).Note that the classical trajectory of the light ray approximates to a three-dimensional parabolic curve in the limit φ ≪ 1; this results from ny[0] growing linearly with s while nx[0] and nz[0] being constants. 3n equation ( 10) ϑ must be defined as the angle between the classical trajectory of the light ray traced by n[0] and the local magnetic field line B since sin ϑ is considered to be unperturbed in view of equation ( 10) (see Fig. 1).Then from equations ( 12) and ( 13) one can express taking the leading order expansion in φ.In the case of the PM Lagrangian model, one can determine the leading order correction to n by means of equations ( 10), ( 12) and ( 21): where ρ ≡ √ x 2 + z 2 with x = ρ sin θ and z = ρ cos θ.For computational convenience, equation ( 22) can be rewritten in Cartesian coordinates by substituting sin θ = x/ √ x 2 + z 2 and cos θ = z/ √ x 2 + z 2 : Using equation( 11), one can easily compute the x and z components of δn [1] : where δn [1] is given by equation ( 23), and in order to simplify our calculations we have exploited the relation, which is due to equations ( 18)-(20).
Inserting equations ( 27) and ( 28) into equations ( 24) and ( 25) through equation ( 23), and substituting s = roλ in equation ( 17), one can express nx, nz and ny in terms of λ: where nx[0] and nz[0] refer to equations ( 15) and ( 16), respectively, and Bo ≡ µ 3 cos 2 (θo − α) + 1 1/2 /r 3 o denotes the magnitude of the magnetic field at the initial point (xo, yo, zo) = (ro sin θo, 0, ro cos θo).From equations ( 29)-( 31) it is evident that n is no longer a unit vector; n Further, by integrating n = dr/ds = (nx, ny, nz) with respect to s, with nx, nz and ny given by equations ( 29)-( 31), one can construct a trajectory curve of the light ray: where For example, with ro = 2 × 10 6 cm, θo = 60 • , α = 45 • , Ω = 2π × 10 2 Hz and η2B 2 o ≈ 4.29 × 10 −5 , we plot a trajectory of our light ray (X/ro, Y /ro, Z/ro) for 0 ≤ λ ≤ 10 on a logarithmic scale, as shown in Fig. 2a.Also, for intuitive visualization, in Fig. 2b is plotted the trajectory on a linear scale, with the quantum refraction effect fairly exaggerated by η2B 2 o ∼ 10 4 , which is 10 9 times as large as an actual order ∼ 10 −5 .Note, in particular, that the trajectory is deflected from a straight line as viewed in the xz-plane (due to the quantum refraction effect), and at the same time that it follows a parabolic path in another plane perpendicular to the xz-plane (due to the rotational effect of the pulsar magnetosphere); therefore, the light ray follows a three-dimensional twisted curve.In Appendix A we provide a detailed discussion of the properties of this curve in reference to the Frenet-Serret formulas (Spivak (1999)).

CHANGE OF POLARIZATION OF A LIGHT RAY DUE TO QUANTUM REFRACTION
In Section 3 we have separated Case I and Case II for the two different values of the refractive index n attributed to the same magnetosphere, depending on the propagation and polarization of the light ray associated with them, as given by equation ( 9), according to the PM Lagrangian model (Kim & Kim (2022)).In accordance with our perturbation analysis, the refractive index can be approximated via expansion as (Adler (1971)) for Case II, ( 36) . From this, one can see that all the results obtained through the perturbation analysis in Section 3 for Case I can be recycled for Case II simply by replacing η2 by η1.That is, the propagation vector and the trajectory curve of our light ray for Case II shall be given by the same expressions as equations ( 29)-( 31) and ( 32)-( 34), respectively, but with η2 replaced by η1.
In this Section we work out the two polarization vectors of the light ray for Case I and Case II, associated with the dual refractive index given by (36); in contrast with the propagation of the ray, there is a distinct difference between them.In relation to this, we discuss quantum birefringence for our pulsar emission at the end.

For Case I
According to Born et al. (1999), the propagation of the unit polarization vector ε can be described by the equation: where S (x, y, z) = const.represents the geometrical wave-front.Substituting equation ( 6) into equation ( 37), we get Now, in view of equation ( 10), we find where δn [1] refers to equation ( 22).Then plugging this into equation ( 38) and inspecting the orders of both sides, one can derive where ε [0] denotes the classical polarization vector and δε [1] is the leading (first) order quantum correction to it, and n[0] refers to the classical propagation vector.This equation describes how quantum refraction affects the propagation of our polarization vector along the path of the light ray by means of perturbation.
One possible way of prescribing the polarization vector classically, with the consideration of the rotational effect is where nx[0] , nz[0] and ny[0] are given by equations ( 15)-( 17), respectively.It can be easily checked out that ε [0] is orthogonal to the propagation vector, Following Kim & Kim (2022), the initial polarization vector associated with n for Case I in accordance with equation ( 9) can be expressed as which has been adapted from its original expression in Kim & Kim (2022) to the geometry of our rotating magnetosphere, with the consideration of equation ( 41). 4 Then we may separate the classical part, and the quantum correction, The polarization vector with the first order correction due to the quantum refraction effect can be obtained in a similar manner as in Sect 3. Integrating equation ( 40) with respect to s = roλ, and combining this with equation (41), and using equations ( 23), ( 26), ( 27), ( 28), ( 41) and ( 44), we finally have ε = εxex + εyey + εzez with where nx[0] , nz[0] and ny[0] are given by equations ( 15), ( 16) and ( 31), respectively, and Bo = µ 3 cos 2 (θo − α) + 1 1/2 /r 3 o , and I (λ) refers to equation ( 35).In Fig. 3 is plotted the change in the polarization vector (∆εx, ∆εy, ∆εz) ≡ (εx (λ) , εy (λ) , εz (λ))| λ 0 against 0 ≤ λ ≤ 10 on a logarithmic scale, wherein ro = 2 × 10 6 cm, θo = 60 4 Originally, in Kim & Kim (2022) the initial polarization vector associated with n for Case I is given by εo  From this plot, one can see that the total polarization vector changes drastically along the x-axis and z-axis near the beginning of the propagation of our light ray due to the quantum refraction effect, while the classical polarization vector changes only along the y-axis due to the rotational effect, as can be seen from equation (41).
By equations ( 29)-( 31) and ( 45)-( 47) one can inspect the orthogonality between the propagation and polarization vectors, n and ε: where nx[0] and nz[0] are given by equations ( 15) and ( 16), respectively.This implies that the quantum refraction effect results in breaking the orthogonality at the leading order in η2B 2 o .However, the departure from the orthogonality remains constant under this effect, being determined at the leading order in η2B 2 o solely by the initial conditions for quantum refraction.For example, with ro = 2 × 10 6 cm, θo = 60 • , α = 45 • and a usual value of η2B 2 o ≈ 4.29 × 10 −5 , we find the departure value to be sin −1 (n • ε) ≈ 3.02 × 10 −5 rad.

Quantum birefringence
From equations ( 48) and (55) above, one can note the following: the polarization vector partly has a longitudinal component (i.e., a component parallel to the propagation vector) for Case I, whereas it is substantially perpendicular to the propagation vector for Case II.This is because given the conditions for pulsar emission as above, we have sin −1 n • ε (Case I) ≈ 3.02 × 10 −5 rad, which is small but comparable to the perturbation parameter η2B 2 o ≈ 4.29 × 10 −5 , and therefore not negligible, while sin −1 n • ε (Case II) ≈ 1.05 × 10 −10 rad is practically negligible compared to the perturbation parameter η1B 2 o ≈ 2.45 × 10 −5 .The two different polarization modes, together with the dual refractive index n as given by ( 36), are entirely due to the quantum refraction effect.These optical properties can be considered to define 'quantum birefringence' as the phenomenology involved is analogous to classical birefringence.
Classically, birefringence is a well-known phenomenon in crystal optics, but the quantum birefringence considered here has a notable difference from the crystal birefringence.The modes in crystal birefringence are determined by solving the characteristic equation Λij εj = 0, where εi represents the mode polarization vector and the matrix Λij is given by with n being the refractive index of the medium for the propagation of the probe light and ni being the principal refractive indices of the crystal (Fowles (1975)).It is assumed that the coordinate axes are aligned with the principal axes of the crystal, and the probe light's propagation direction is (sin θ, 0, cos θ).Note that the principal indices are determined solely by the material properties, irrespective of the probe light's propagation direction.
In fact, one can reproduce the characteristic matrix for the PM Lagrangian with a uniform magnetic field (Kim & Kim If you want to present additional material which would interrupt the flow of the main paper, it can be placed in an Appendix which appears after the list of references.quantum refraction effect, being led by the parameter η2B 2 o .In Fig. A1a is plotted the curvature κ (dimensionless, multiplied by ro) against 0 ≤ λ ≤ 10 (with the substitution s = roλ in equation (A7)), and in Fig. A1b is plotted the corresponding curve (X/ro, 0, Z/ro), which is projected onto to the xz-plane.Here we assume ro = 2 × 10 6 cm, θo = 60 • , α = 45 • , Ω = 2π × 10 2 Hz and η2B 2 o ∼ 10 4 (10 9 times as large as an actual order ∼ 10 −5 ; fairly exaggerated for intuitive visualization).Note the two points marked for κmin and κmax in each plot; in particular, the trajectory is curved the most downward and the most upward at the former and the latter points, respectively.
This paper has been typeset from a T E X/L A T E X file prepared by the author.

Figure 2 .
Figure 2. (a) A trajectory of the light ray (X/ro, Y /r o, Z/ro) plotted against 0 ≤ λ ≤ 10 on a logarithmic scale; the red solid curve and the blue dashed curve represent the total trajectory (classical trajectory + quantum correction) and the classical trajectory, respectively, (b) The trajectory plotted on a linear scale for intuitive visualization, with the quantum refraction effect fairly exaggerated by η 2 B 2 o ∼ 10 4 , which is 10 9 times as large as an actual order ∼ 10 −5 .