Quantum refraction effects in pulsar emission

Neutron stars have strong magnetic fields on their surface from $10^{8}\,\mathrm{G}$ up to $10^{15}\,\mathrm{G}$, and in particular magnetars have the strongest magnetic fields in the universe with $10^{13}-10^{15}\,\mathrm{G}$, which are near or a little above the supercritical value $B_{\mathrm{c}}=m_{\mathrm{e}}^{2}c^{3}/e\hbar=4.4\times 10^{13}\,\mathrm{G}$ (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., $B=4.9\times 10^{-4}B_{\mathrm{c}}$ (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 leading-order 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 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 $m_{\mathrm{e}}$ and charge $e$ in electromagnetic fields of arbitrary strengths (Heisenberg & Euler (1936); Schwinger (1951)): in the convention with $\hbar=c=1$,

where $\mathcal{L}^{\left(0\right)}\left(a,b\right)$ refers to the classical Maxwell Lagrangian, defined through

with the Lorentz- and gauge-invariant Maxwell scalar $F$ and pseudo-scalar $G$

and $\mathcal{L}^{\left(1\right)}\left(a,b\right)$ refers to the Lagrangian of one-loop correction,

However, in the weak-field limit (below the critical field strength $B_{\mathrm{c}}$), the HES Lagrangian (1) has the leading-order contribution, the so-called post-Maxwellian (PM) Lagrangian (Euler & Kockel (1935)):¹¹1In effect, the PM Lagrangian approximates the HES Lagrangian well even up to $B=0.14B_{\mathrm{c}}$, with an error less than 1%.

where $\eta_{1}$ and $\eta_{2}$ are parameters defined via $\eta_{1}/4=\eta_{2}/7=e^{4}/\left(360\pi^{2}m_{\mathrm{e}}^{4}\right)$.

In some NED models, a parity violating term proportional to $FG$ 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)).

A light ray is defined as an orthogonal trajectory to the geometrical wave-front $\mathcal{S}\left(x,y,z\right)=\mathrm{const.}$, and therefore can be described by

where $s$ is an affine parameter to measure the length of the ray and $n=n\left(\mathbf{r}\right)$ is the refractive index given as a function of the position $\mathbf{r}$ on the ray (Born et al. (1999)). It can be further shown that

Let $\mathbf{\hat{n}}\equiv\mathrm{d}\mathbf{r/}\mathrm{d}s$ be the unit propagation vector for the light ray emitted from a spot either at rest or in motion at a constant velocity.²²2This is in contrast with our actual case, wherein the emission spot itself is under the centrifugal acceleration due to the rotation of a pulsar magnetosphere, as described by equation (13). Then equation (7) leads to (Born et al. (1999))

The expression of $\mathbf{\hat{n}}$ for $n\neq\mathrm{const.}$ 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 $\mathbf{r}$, 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 $\vartheta$ 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 ($\sim 10^{20}\,\mathrm{Hz}$; 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 ($\sim 10^{9}\,\mathrm{Hz}$) (Petri (2016)). Therefore, our pulsar radiation can be safely assumed to cover optical to X-ray emissions ($\sim 10^{12}$ to $10^{17}\,\mathrm{Hz}$) in this work.

One can expand the refractive index $n$ for Case I in equation (9), having $\eta_{1}$, $\eta_{2}\sim 10^{-31}\,\mathrm{g}^{-1}\,\mathrm{cm\,s}^{2}$ and $B<B_{\mathrm{c}}$ (critical magnetic field)$\,\sim 10^{13}\,\mathrm{G}$, and thus $\eta_{1}B^{2}$, $\eta_{2}B^{2}\ll 1$. Then it can be approximated as (Adler (1971))

which implies that the term $\eta_{2}B^{2}\sin^{2}\vartheta\sim 10^{-4}\left(B/B_{\mathrm{c}}\right)^{2}$ is the leading order quantum correction to $n=1$ for classical optics, while $\mathcal{O}\left(\eta_{1}^{2}B^{4},\eta_{1}\eta_{2}B^{4},\eta_{2}^{2}B^{4}\right)$ 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 $\eta_{2}B^{2}$ being a perturbation parameter. It should be noted here that $n\rightarrow 1$, i.e., the refractive index goes back to the classical limit as $B\rightarrow 0$ 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 $\delta\left(\cdots\right)_{\left[1\right]}$ means the leading order quantum correction to the quantity $\left(\cdots\right)$, led by $\eta_{1}B^{2}$ or $\eta_{2}B^{2}$, and $\mathcal{O}\left({}_{\left[2\right]}\right)$ is a shorthand expression for $\mathcal{O}\left(\eta_{1}^{2}B^{4},\eta_{1}\eta_{2}B^{4},\eta_{2}^{2}B^{4}\right)$. On the other hand, by $\mathbf{\hat{n}}_{\left[0\right]}$ we mean the  unperturbed (classical) propagation direction of the light ray. A classical trajectory of the light ray traced by $\mathbf{\hat{n}}_{\left[0\right]}$ 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 $\mu$ is the magnetic dipole moment and $\alpha$ 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 $\left(x_{\mathrm{o}},y_{\mathrm{o}},z_{\mathrm{o}}\right)=\left(r_{\mathrm{o}}\sin\theta_{\mathrm{o}},0,r_{\mathrm{o}}\cos\theta_{\mathrm{o}}\right)$ (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, 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 $\mathbf{\hat{B}}\equiv\mathbf{B/}\left|\mathbf{B}\right|$ and the second term accounts for the centrifugal acceleration, with $\mathbf{\Omega}\equiv\Omega\mathbf{e}_{z}$ and $\Omega$ being a pulsar rotation frequency, and

with $c$ being the speed of light and $\cos\theta^{\prime}\equiv\cos\alpha\cos\theta+\sin\alpha\sin\theta\cos\phi$.

However, during the rotation the azimuthal phase changes by $\phi\sim\Omega t$, while our light ray has propagated a distance by $s\sim ct$. We describe the propagation of the light ray with the consideration of the MHD effect above, assuming $\phi$ to be very small; e.g., $\phi\lesssim 10^{-1}$ is considered for a millisecond pulsar with $\Omega\sim 10^{2}\,\mathrm{Hz}$, during the time of rotation $t\lesssim 10^{-3}\,\mathrm{s}$, such that $s\lesssim 10^{7}\,\mathrm{cm}$, 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 $\mathbf{\hat{B}}\left(r_{\mathrm{o}},\theta_{\mathrm{o}},\phi\right)$ and $\beta\left(r_{\mathrm{o}},\theta_{\mathrm{o}},\phi\right)$ in $\phi$ from equations (12) and (14), respectively, and obtain $\mathbf{\hat{n}}_{\left[0\right]}=\hat{n}_{x\left[0\right]}\mathbf{e}_{x}+\hat{n}_{y\left[0\right]}\mathbf{e}_{y}+\hat{n}_{z\left[0\right]}\mathbf{e}_{z}$, expressed in Cartesian coordinates:

and

where we have considered $\Omega r_{\mathrm{o}}/c\lesssim\phi$, e.g., for a millisecond pulsar with $\Omega\sim 10^{2}\,\mathrm{Hz}$ and $r_{\mathrm{o}}\sim 10^{6}\,\mathrm{cm}$, such that $\mathcal{O}\left(\left(\Omega r_{\mathrm{o}}/c\right)^{2}\right)\lesssim\mathcal{O}\left(\phi\left(\Omega r_{\mathrm{o}}/c\right)\right)\lesssim\mathcal{O}\left(\phi^{2}\right)$, all to be ignored in our analysis, and have substituted $\phi=\Omega s/c$ in equation (17), the leading order rotational effect to be considered in our analysis.

Now, integrating $\mathbf{\hat{n}}_{\left[0\right]}=\mathrm{d}\mathbf{r/}\mathrm{d}s$ with respect to $s$, the unperturbed (classical) trajectory of the light ray can be derived:

where $\hat{n}_{x\left[0\right]}$, $\hat{n}_{z\left[0\right]}$ and $\hat{n}_{y\left[0\right]}$ are given by equations (15)-(17), respectively, and the emission point is $\left(x_{\mathrm{o}},y_{\mathrm{o}},z_{\mathrm{o}}\right)=(r_{\mathrm{o}}\sin\theta_{\mathrm{o}},0,\allowbreak r_{\mathrm{o}}\cos\theta_{\mathrm{o}})$. Note that the classical trajectory of the light ray approximates to a three-dimensional parabolic curve in the limit $\phi\ll 1$; this results from $\hat{n}_{y\left[0\right]}$ growing linearly with $s$ while $\hat{n}_{x\left[0\right]}$ and $\hat{n}_{z\left[0\right]}$ being constants.³³3Being projected onto the $xz$-plane, the curve appears to be a straight line, as represented by the red dashed line in Fig. 1.

In equation (10) $\vartheta$ must be defined as the angle between the classical trajectory of the light ray traced by $\mathbf{\hat{n}}_{\left[0\right]}$ and the local magnetic field line $\mathbf{B}$ since $\sin\vartheta$ 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 $\phi$. 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 $\rho\equiv\sqrt{x^{2}+z^{2}}$ with $x=\rho\sin\theta$ and $z=\rho\cos\theta$. For computational convenience, equation (22) can be rewritten in Cartesian coordinates by substituting $\sin\theta=x/\sqrt{x^{2}+z^{2}}$ and $\cos\theta=z/\sqrt{x^{2}+z^{2}}$:

Using equation(11), one can easily compute the $x$ and $z$ components of $\delta\mathbf{\hat{n}}_{\left[1\right]}$:

where $\delta n_{\left[1\right]}$ is given by equation (23), and in order to simplify our calculations we have exploited the relation,

which is due to equations (18)-(20).

To provide further convenience for equation (23), we may re-parametrise the variables $x$ and $z$ given by equations (18) and (19) using a dimensionless parameter $\lambda\geq 0$, defined via $s=r_{\mathrm{o}}\lambda$:

where $\hat{n}_{x\left[0\right]}$ and $\hat{n}_{z\left[0\right]}$ refer to equations (15) and (16), respectively. It should be noted here that the value of $\lambda$ is restricted by the condition $\phi\sim\Omega t=\Omega r_{\mathrm{o}}\lambda/c\ll 1$; from this it follows that $\lambda\ll c/\left(\Omega r_{\mathrm{o}}\right)$. For example, for a millisecond pulsar with $\Omega\sim 10^{2}\,\mathrm{Hz}$, we shall have $\lambda\ll 10^{2}$, given $r_{\mathrm{o}}\sim 10^{6}\,\mathrm{cm}$.

Inserting equations (27) and (28) into equations (24) and (25) through equation (23), and substituting $s=r_{\mathrm{o}}\lambda$ in equation (17), one can express $\hat{n}_{x}$, $\hat{n}_{z}$ and $\hat{n}_{y}$ in terms of $\lambda$:

where $\hat{n}_{x\left[0\right]}$ and $\hat{n}_{z\left[0\right]}$ refer to equations (15) and (16), respectively, and $B_{\mathrm{o}}\equiv\mu\left(3\cos^{2}\left(\theta_{\mathrm{o}}-\alpha\right)+1\right)^{1/2}/r_{\mathrm{o}}^{3}$ denotes the magnitude of the magnetic field at the initial point $\left(x_{\mathrm{o}},y_{\mathrm{o}},z_{\mathrm{o}}\right)=\left(r_{\mathrm{o}}\sin\theta_{\mathrm{o}},0,r_{\mathrm{o}}\cos\theta_{\mathrm{o}}\right)$. From equations (29)-(31) it is evident that $\mathbf{\hat{n}}$ is no longer a unit vector; $\mathbf{\hat{n}\cdot\hat{n}}\approx 1+\mathcal{O}\left({}_{\left[1\right]}\right)+\mathcal{O}\left(\left(\Omega r_{\mathrm{o}}/c\right)^{2},_{\left[2\right]}\right)$.

Further, by integrating $\mathbf{\hat{n}}=\mathrm{d}\mathbf{r/}\mathrm{d}s=\left(\hat{n}_{x},\hat{n}_{y},\hat{n}_{z}\right)$ with respect to $s$, with $\hat{n}_{x}$, $\hat{n}_{z}$ and $\hat{n}_{y}$ given by equations (29)-(31), one can construct a trajectory curve of the light ray:

where

For example, with $r_{\mathrm{o}}=2\times 10^{6}\,\mathrm{cm}$, $\theta_{\mathrm{o}}=60^{\circ}$, $\alpha=45^{\circ}$, $\Omega=2\pi\times 10^{2}\,\mathrm{Hz}$ and $\eta_{2}B_{\mathrm{o}}^{2}\approx 4.29\times 10^{-5}$, we plot a trajectory of our light ray $\left(X/r_{\mathrm{o}},Y/r_{\mathrm{o}},Z/r_{\mathrm{o}}\right)$ for $0\leq\lambda\leq 10$ 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 $\eta_{2}B_{\mathrm{o}}^{2}\sim 10^{4}$, which is $10^{9}$ times as large as an actual order $\sim 10^{-5}$. 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)).

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))

where $\eta_{1}B^{2}$, $\eta_{2}B^{2}\lesssim 10^{-4}\ll 1$ and $\mathcal{O}\left({}_{\left[2\right]}\right)$ is a shorthand expression for $\mathcal{O}\left(\eta_{1}^{2}B^{4},\eta_{1}\eta_{2}B^{4},\eta_{2}^{2}B^{4}\right)$. 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 $\eta_{2}$ by $\eta_{1}$. 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 $\eta_{2}$ replaced by $\eta_{1}$.

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.