Acceleration toward polarization singularity inspired by relativistic E×B drift

The relativistic trajectory of a charged particle driven by the Lorentz force is different from the classical one, by velocity-dependent relativistic acceleration term. Here we show that the evolution of optical polarization states near the polarization singularity can be described in analogy to the relativistic dynamics of charged particles. A phase transition in parity-time symmetric potentials is then interpreted in terms of the competition between electric and magnetic ‘pseudo’-fields applied to polarization states. Based on this Lorentz pseudo-force representation, we reveal that zero Lorentz pseudo-force is the origin of recently reported strong polarization convergence to the singular state at the exceptional point. We also demonstrate the deterministic design of achiral and directional eigenstates at the exceptional point, allowing an anomalous linear polarizer which operates orthogonal to forward and backward waves. Our results linking parity-time symmetry and relativistic electrodynamics show that previous PT-symmetric potentials for the polarization singularity with a chiral eigenstate are the subset of optical potentials for the E×B “polarization” drift.

With the universal existence of open systems 1,2 of non-equilibrium and time-dependent 3-5 potential energy, the concept of parity-time (PT) symmetry 6,7 has become a multidisciplinary topic [8][9][10][11][12][13][14][15] . PT symmetry successfully offers the special form of potentials V(x) = V * (− x), providing physical observables even for non-equilibrium systems. The existence of real observables in PT-symmetric complex potentials has opened the field of non-Hermitian quantum mechanics 6,8 , which exhibits the phase transition 6,16,17 between real and complex eigenspectra in stark contrast to a purely real eigenspectrum observed in Hermitian potentials. By utilizing optical gain-and lossmaterials in the refractive index form n(x) = n * (− x), the physics of PT-symmetric potentials has been applied to the exotic control of light flows 17,18 . The effective realization of PT symmetry has also been extended to acoustics 9,13,19 , optomechanics 10 , electronics 11 , gyrotropic systems 20,21 , and population genetics 12 . For all of these fields, critical traits of PT symmetry, e.g. unidirectionality [22][23][24] , non-Hermitian degeneracy 18 , and chirality 25,26 , impose intriguing features on wave dynamics in terms of the 'singularity' 27,28 : the coalescence of eigenstates with a chiral form 25,26,[29][30][31][32] at the exceptional point (EP, or phase transition point) 33 .
Meanwhile, it is known that relativistic electrodynamics 34 for charged particles also exhibits the inherent feature of open systems. The famous relativistic energy expression 35 , Ẽ = mc 2 /[1 − (v/c) 2 ] 1/2 , shows that observers in different frames will see different values of total energy for moving charged particles of velocity v. We note that this non-equilibrium condition results in non-Hermitian form of Hamiltonians, the necessary condition for the achievement of PT symmetry. In the context of the multidisciplinary realization of PT symmetry [8][9][10][11][12] , therefore, the link between relativistic behaviors of charged particles in electromagnetic fields and wave dynamics in PT-symmetric potentials could offer different viewpoints on the physics of EP singularity in PT-symmetric potentials.
Inspired by the polarization equation of motion from the Schrödinger-like form of Maxwell's equations 36 , here we interpret the evolution of optical states of polarizations (SOP) near the EP singularity in direct analogy to the relativistic E×B drift (the movement under orthogonal E and B fields) of charged particles 34 , which we call the relativistic E×B "polarization" drift of light. The phase transition in PT-symmetric potentials 6,16,17 is then understood in view of the competition between electric-and magnetic-'pseudo'-fields, and we prove that strong chiral conversion of optical SOP at the EP 15,16 corresponds to the accidental cancellation of the Lorentz pseudo-force on the Poincaré hemisphere. By employing this "Lorentz-force picture" in the analysis of the polarization singularity, we then extend the class of the polarization singularity in vector wave equations 25,26,[37][38][39] , revealing the existence of achiral and directional eigenstates at the EP. Our approach paves the way for the unconventional control of optical polarizations, such as anomalous directional polarizers.

Results
Lorentz pseudo-forces for optical polarizations. Consider the planewave propagating along the z-axis of the electrically anisotropic material, with the unity permeability (μ = 1). For the later discussion, we express the arbitrary permittivity tensor in the x-y plane, in terms of Pauli matrices 26,31 36,41 as H s = (ε 1 σ 1 + ε 2 σ 2 + ε 3 σ 3 )/(iλ) for λ = 2ε o 1/2 /k 0 and the free-space wavenumber k 0 . If we assign the symmetry axis between x and y axes, the condition of PT-symmetric potentials 15,16 requires real-valued ε o , ε 2 , and ε 3 , and imaginary-valued ε 1 . Note that imaginary-valued ε 1 corresponds to the linear dichroism 42,43 , the different dissipation for each linear polarization, while real-valued ε 2 represents the birefringence. ε 3 represents the magneto-optical change of plasma permittivity induced by an external static magnetic field 34 , and for now, we assume the nonmagnetic case of ε 3 = 0.
In this representation, the SOP of light is described by the Stokes parameters 44 S j = ψ e † ·σ j ·ψ e (j = 0, 1, 2, 3). The change of the SOP can then be expressed in view of light-matter interactions 36 by applying the governing equation dψ e /dz = H s ·ψ e and its conjugate form dψ e † /dz = ψ e † ·H s † , which leads to the Lorentz pseudo-force equation of motion for the SOP 36 (also see Supplementary Note 1) n n n n where S n = [S 1 , S 2 , S 3 ] T /S 0 is the pseudo-velocity of the 'hypothetical' charged particle corresponding to the SOP of light, and E = 2·Im[ε 1 , ε 2 , ε 3 ] T /λ and B = − 2·Re[ε 1 , ε 2 , ε 3 ] T /λ are the electric and magnetic pseudo-field in relation to imaginary-and real-parts of the permittivity, respectively. Note that Eq.
(2) provides direct analogy to the relativistic dynamics of massless charged particles with the motion equation 20,22 of ∂ t β = E + β× B − (β·E)β. In this representation of optical polarization states, the acceleration of optical SOP comes from the Lorentz pseudo-force, F ~ dS n /dz (Fig. 1). The first two terms of Eq. (2) are the counterparts of the classical electromagnetic Lorentz force E + β× B, and the 3 rd term -(S n ·E)S n corresponds to the Joule effect 34 -(β·E)β in the relativistic equation of motion. Figure 1a-c shows the effect of each component of the Lorentz pseudo-force on the SOP of propagating light, induced by nonmagnetic PT-symmetric materials (imaginary ε 1 , real ε 2 , and zero ε 3 ). While ε 2 of the birefringence derives the circulating acceleration on the Poincaré sphere (B = − 2ε 2 ·e 2 /λ, Fig. 1a, Hermitian case), ε 1 of the amplification or dissipation results in the linear drift of the SOP (E = 2·Im[ε 1 ]·e 1 /λ, Fig. 1b, non-Hermitian case). We also note that the energy variation from gain and loss materials S n ·E (Fig. 1c) provides the relativistic nonlinear acceleration of the SOP with respect to E. Consequently, with the orthogonality between pseudo-fields (E⊥ B), PT-symmetric potentials naturally satisfy the ideal E×B drift 34,45 condition to optical polarization states.  6,16,17 between real and complex eigenspectra in PT-symmetric potentials can be interpreted in terms of the E×B drift 34,45 : the competition between electric and magnetic pseudo-forces. Figure 2a,b shows the evolution of real and imaginary eigenvalues Δε eig for the Hamiltonian equation dψ e /dz = H s ·ψ e , as a function of the imaginary potential (ε i = Im[ε 1 ], ε c = ε 2 ). First, before the EP where eigenvalues are real and non-degenerate (Fig. 2a, ε i < ε c ), the magnetic pseudo-field is larger than the electric pseudo-field, resulting in the counter-directive acceleration of SOP (lower panels in Fig. 2c) to northern-/southern-hemispheres. At the EP with the coalescence (d point in Fig. 2a,b, ε i = ε c ), the equal magnitude of E = 2·ε i ·e 1 /λ and B = − 2ε c ·e 2 /λ fields derives the suppression of total Lorentz pseudo-force on the southern Poincaré sphere, especially with the zero net force at the south pole (S n = − e 3 , dS n /dz = E + S n × B − (S n ·E)S n = 2·ε i ·e 1 /λ + e 3 × 2ε c ·e 2 /λ = 0, Fig. 2d). It is emphasized that this force cancellation impedes the acceleration near the south pole of the stationary polarization, deriving the SOP convergence to perfect LCP chirality 26 . After the EP with amplifying and dissipative states (Fig. 2b, ε i > ε c ), the strong electric pseudo-field dominates the motion equation of the SOP, with the co-directive force (lower panels in Fig. 2e) to opposite hemispheres. In the context of electrodynamics analogy, the phase of eigenvalues in PT-symmetric potentials can thus be divided by the (i) B-dominant (before the EP), (ii) B = E (at the EP) and (iii) E-dominant regime (after the EP). It is worth mentioning that the stable point with the stationary polarization can also be obtained at the north pole by changing the sign of ε c (converting the fast and slow axes for the birefringence) or ε i (converting the gain and loss axes for the linear dichroism), allowing perfect RCP chirality. In terms of this Lorentz pseudo-force representation of SOP, we also note that PT-symmetric potentials 15,16 with real-valued ε 2 and imaginary-valued ε 1 are the special case of the E×B drift with specific field vectors E = 2·ε i ·e 1 /λ and B = − 2ε c ·e 2 /λ, implying the existence of unconventional polarization singularity at other SOPs (e.g. without optical spin) which will be discussed later.
E×B polarization drift in PT-symmetric potentials. We then investigate the "evolution" of the SOP, for different pseudo-forces shown in Fig. 2. Figure 3a-c shows the change of the initial SOP of (+ , RCP) and (− LCP) spins under different phases of PT symmetry, in relation to the charged particle movement at different phases 45 of the relativistic E×B drift (Fig. 3d-f). Because the magnetic pseudo-field dominates the dynamics of SOP before the EP (Fig. 3a,d, |B| > |E|), the SOP for each spin simply rotates around the B field following the S n ×B of Eq. (2). Yet, with different magnitudes of the forces in northern and southern hemispheres (Fig. 2c), the 'speed' of the SOP rotation near each pole is different, analogous to different magnetically-gyrating arcs of charged particles in the E× B drift 34,45 . The directional drift of relativistic particles along the E×B axis (Fig. 3d, toward − S 3 axis) is thus reproduced by the slow evolution of SOPs near the S n = − e 3 on the Poincaré sphere (Fig. 3a).
The extraordinary case of the relativistic E×B polarization drift is achieved at the singular state of EP (Fig. 3b,e), for the case of |B| = |E|. Because of the force cancellation near the perfectly stable south pole (S n = − e 3 for dS n /dz = 0), the (+ ) spin state converges to the (− ) spin when the state approaches the south pole through the gyration by the magnetic pseudo-field (orange line in Fig. 3b), similar to the convergence of the velocity in the motion of relativistic particles (orange line in Fig. 3e). Because the (− ) spin state is stationary, we note that the entire SOP, which can be represented in terms of the linear combination of the LCP (− spin) and RCP (+ spin), is thus converted to the LCP chiral wave. After the EP (|B| < |E|, Fig. 3c,f), the electric force is dominant, resulting in the linear acceleration mostly towards the direction of E. For all cases, it is noted that the relativistic correction from non-Hermitian Hamiltonians retains the evolution of SOP S n = [S 1 , S 2 , S 3 ] T /S 0 on the Poincaré sphere, in contrast to the classical evolutions (dotted lines in Fig. 3a-c) which do not include the third term of Eq. (2).

Realization of achiral and directional singularity.
Extending the special case of the E×B polarization drift derived from E = 2·ε i ·e 1 /λ and B = − 2ε c ·e 2 /λ, we now work on other types of E×B polarization drift which allow unconventional polarization singularity without optical spin, by manipulating the direction of electromagnetic pseudo-fields. The vector form of the Lorentz pseudo-force equation provides larger degrees of freedom for the intuitive control of the eigenstate at the singularity, in contrast to the fixed chiral form 25,26,[29][30][31][32] of scalar PT-symmetric equation. Although the magnetic transition of PT symmetry can also be utilized to achieve the achiral (spin-less) eigenstate at the EP (ε 3 ≠ 0, Supplementary Note 2), here we investigate the realization of achiral and directional singularity with the use of nonmagnetic chiral materials. From the constitutive relation including optical chirality 46 (or bi-isotropy) D = εE − iχH and B = μH + iχE, the condition of μ = μ 0 and diagonal ε with ε x ≠ ε y represents the nonmagnetic chiral material with general electrical anisotropy, including both birefringence (Re[ε x ] ≠ Re[ε y ] for different x-and y-wavevector) and linear dichroism 42,43    /(μ 0 ·ρ 2 ) and thus prohibits the existence of propagating waves at the EP. It is interesting to note that this restriction proves the necessity of complex potentials for obtaining the singularity; therefore we employ linear dichroism for achieving the EP for the propagating wave, by fulfilling the condition of |B| = |E|. Figure 4 shows the case of linearly-dichroic (ρ = i·ρ i ) chiral materials, which derive pseudo-fields of E(ρ, χ) = kρ i ·e 1 and B(ρ, χ) = − ωχ o ρ i ·e 2 + 2ωχ o ·e 3 with the EP condition of ρ i 2 = 4χ o 2 /(μ 0 ε o ) for |B| = |E|. The PT-symmetry-like phase transition (Fig. 4b-d) around the EP (marked with red dots in Fig. 4c,f) occurs in linearly-dichroic chiral materials from the competition between E and B (Fig. 4a), and the direction of the E×B drift is controlled by changing ρ i for the pseudo-magnetic field B, allowing the realization of the achiral singularity (S n ~ − e 2 in Fig. 4c); in sharp contrast to the case of PT-symmetric potentials. Furthermore, the obtained EP state has the directionality in its propagation due to the wavevector-dependency of Eq. (4) (Fig. 4b-d vs Fig. 4e-g, S n ~ e 2 in Fig. 4f), which originates from the broken mirror symmetry of chiral materials for forward and backward waves.
This directionality with achiral designer eigenstate at the EP allows the implementation of unconventional polarizers based on the polarization convergence at the EP. Figure 4h shows an example of the anomalous linear polarizer, operating 'orthogonal' to forward and backward waves. While the SOP of forward waves are converged to the + 45° linear polarization (Fig. 4c), the SOP of backward waves becomes − 45° linear polarization. Moreover, in contrast to the case of classical linear polarizers which perfectly reflect the orthogonally polarized waves (e.g. the y-polarized incidence to the x-polarizer), the linear-polarizing functionality in the structure of Fig. 4h operates for the 'entire' SOP due to the non-orthogonality between eigenstates.

Discussion
In summary, we found the link between the seemingly unrelated fields of PT symmetry optics and relativistic electrodynamics. This reinterpretation of PT symmetry brings insight to the singularity in polarization space, broadening the class of parity-time symmetric Hamiltonians in vector wave equations 39 . The counterintuitive achievement of the achiral and directional EP eigenstate is also demonstrated, which allows the realization of anomalous linear polarizers for randomly polarized incidences. The comprehensible understanding of the EP in terms of the dynamics of charged particles will provide a novel design methodology near the singularity: the generation of chiral waves [47][48][49] , topological photonics which has focused only on chiral states 38 , coherent wave dynamics 50 , PT-symmetry-like potentials based on causality 51,52 or supersymmetric optics 53,54 , and optical analogy of spintronics. At the same time, this classical viewpoint on relativistic electrodynamics also enables the analogy of EP dynamics in charged particle movements, the linear E× B drift toward a single direction for every initial velocity vectors.