Time Circular Birefringence in Time-Dependent Magnetoelectric Media

Light traveling in time-dependent media has many extraordinary properties which can be utilized to convert frequency, achieve temporal cloaking, and simulate cosmological phenomena. In this paper, we focus on time-dependent axion-type magnetoelectric (ME) media, and prove that light in these media always has two degenerate modes with opposite circular polarizations corresponding to one wave vector , and name this effect “time circular birefringence” (TCB). By interchanging the status of space and time, the pair of TCB modes can appear simultaneously via “time refraction” and “time reflection” of a linear polarized incident wave at a time interface of ME media. The superposition of the two TCB modes causes the “time Faraday effect”, namely the globally unified polarization axes rotate with time. A circularly polarized Gaussian pulse traversing a time interface is also studied. If the wave-vector spectrum of a pulse mainly concentrates in the non-traveling-wave band, the pulse will be trapped with nearly fixed center while its intensity will grow rapidly. In addition, we propose an experimental scheme of using molecular fluid with external time-varying electric and magnetic fields both parallel to the direction of light to realize these phenomena in practice.

Axion was originally proposed as a hypothetical elementary particle 25 , while it won great interests in condensed matter physics recently because of the significant discovery that an effective quantized axion field can be induced in topological insulators when time reversal symmetry is weakly broken [7][8][9][10][11] . Actually, since Θ is a pseudoscalar, the axion-type ME coupling only exists in the systems where both the time reversal (T) and the parity (P) symmetries are broken but the combined PT symmetry is held 3 . There is no visible effect for light traveling in globally constant axion field, however, a Kerr or Faraday rotation can be detected for lights reflected or refracted by the surface of an axion medium 7,20-22 , which essentially originates from the sudden change of Θ at the spatial interface 7 . Noteworthy, a type of circular birefringence, known as Carroll-Field-Jackiw (CFJ) birefringence, can emerge in Chern-Simons modified electrodynamics 26 . And Y. Itin proved that the CFJ birefringence can be alternatively caused by a space-and-time-dependent axion field in geometric optics approximation 27,28 . The CFJ birefringence is generally anisotropic in space, whereas it reduces to isotropy when the 4-gradient ∂ Θ μ is timelike, i.e. the axion field only changes with time.
Light traveling in time-dependent media has many extraordinary properties which can be utilized to achieve frequency conversion 29,30 , temporal cloaking [31][32][33][34] , and to simulate cosmological phenomena 35,36 etc. In this paper, we focus on time-dependent axion-type ME media, and prove that light in these media always has two oppositely circularly polarized modes corresponding to one wave vector k but not limited to geometric optics approximation. The key idea of this paper is to interchange the status of space and time. We will show that the pair of TCB modes can appear simultaneously via the "time refraction" and "time reflection" of a linearly polarized incident wave at a time-discontinuous interface of the ME media. The superposition of two TCB modes causes the "time Faraday effect" which is a novel effect as a temporal counterpart of the ordinary spatial Faraday effect or optical activity. Further discussions about the propagating velocities of energy and information for TCB modes and about the time refraction and reflection of Gaussian pulse at time interfaces in ME media are also provided. Furthermore, we put forward an experimental scheme to generate the effective time-dependent axion-type ME media controlled by time-varying external electric field  and magnetic field  parallel to each other which offers a practical way to realize the novel phenomena predicted in this paper.

Results
Time circular birefringence and time Faraday effect. In time-dependent axion-type ME media, the magnetic induction obeys the wave equation . As a result, there always exists a pair of circularly birefringent modes T ± for a given wave vector k in time-dependent axion-type media: We call this effect the time circular birefringence (TCB). If Θ =  0, the two distinct equations of T ± reduce to an identical one, and the birefringent phenomenon vanishes. Thereby TCB is entirely induced by the time varying axion field. In addition, TCB happens in isotropic media, thus it is different from both the ordinary birefringence in uniaxial or biaxial crystals and the ME Jones birefringence [37][38][39][40] which are all caused by the anisotropy of materials. TCB is also different from the optical active circular birefringence (OACB), because TCB is generated from the temporal nonhomogeneity of the nonreciprocal ME media but OACB is a reciprocal magnetoelectric effect originating from the chirality of molecules. For traditional birefringent effects, two different wave numbers k correspond to one frequency. One can realize the two birefringent states just via shooting a beam onto a birefringent medium subject to the temporal-phase-matching condition = at the spatial interface. However, the temporal parts T ± (t) of the pair of TCB modes corresponding to a fixed wave number are different, and accordingly could not match the temporal phase factor of the incident wave simultaneously. This difference gives rise to a handicap for realizing this pair of circular polarized states in practice. To overcome this difficulty, we think up the idea of "time discontinuous media" by analogy with the "spatial discontinuous media" used in traditional birefringent systems, then the spatial phase factor, ⋅ e k r i , should be matched at time interfaces. Considering a time-dependent medium ε(t), μ(t), Θ (t) discontinuous at a time interface t 0 , we can get the temporal boundary Just as spatial optical wave plate devices, we analyze light propagating in a "time wave plate" with piecewise medium parameters: ε 0 , μ 0 , Θ 0 are constant when t < t 0 ; ε 1 (t), μ 1 (t), Θ 1 (t) are some continuous functions when t 0 < t < t 1 ; ε 2 , μ 2 , Θ 2 are also constant when t > t 1 , as shown in Fig. 1. For a linearly polarized incident wave = , the wave will become the sum of the two TCB modes = ∑ ± ± B B at t 0 . Moreover, there always exist two linearly independent solutions for Eq. (4) which are complex conjugates of each other: , then the general solution of Eq. (4) is their superposition: , and the two TCB states can be further separated as It can be proved that the momentums of the two branches ± B 1 and ± B 2 are always in opposite directions, i.e. one branch always propagates along the incident direction (for convenience, let it be ± B 1 ), while the other (let it be ± B 2 ) is always along the opposite. As a result, ± B 1 and ± B 2 are exactly the "time refraction" and "time reflection" of the corresponding TCB modes at the time interface t 0 (see the supplementary information for more discussions).
A simplified case is β μ Θ ( ) ≡ / >  t 0 1 1 , and ε 1 , μ 1 are both constant. Then the TCB modes are identical with the CFJ modes obtained in geometric optics approximation [26][27][28] , therefore the light splits into two plane waves . The dispersion relations of two TCB modes are ω β = ( ± )/ ± v k k k 1 , and the coefficients determined by the temporal boundary conditions are 1 . According to the dispersion relations, the two TCB modes B ± both have a forbidden band of k for traveling waves: . Outside the forbidden band, B 1 travels along the incident direction, i.e. it is the time refraction, and B 2 is the time reversal of B 1 . However, a wave should not propagate backwards through time. The practical observable is its real part which propagates opposite to the incident direction in space, therefore, B 2 is actually the time reflection. Without loss of the physical generality, a further simplification will applied in the following: ε 0 = ε 1 = ε 2 , μ 0 = μ 1 = μ 2 , Θ 1 (t 0 ) = Θ 0 , and Θ 1 (t 1 ) = Θ 2 , i.e. the medium is continuous at t 0 and t 1 but Θ  is still discontinuous.
The time dependence of media destructs the symmetry of time translation, therefore, the energy of the electromagnetic field is not conserved in general. On the other hand, the lagrangian of time dependent media shown in Eq. (2) is invariant under spatial translation, so the apparent electromagnetic momentum = ( ) × ( ) Re Re P D B must be conserved. Typically, the energy of incident wave does not equal to the total energy of the time refracted and reflected waves at the time interfaces of a time wave plate (see Fig. 2(a)), whereas the incident apparent momentum equals to the resultant momentum of the reflected and refracted waves: 2 (see the supplementary information for general proof). From a photonic point of view, the nonconservation of energy indicates ω ω ≠ ± 0 , while the conservation of momentum insures k in = k 1 = k 2 at time interfaces. This fact is different from the case of ordinary refraction and reflection at a spatial interface of two media, in which the energy is conserved, but the normal momentum to the spatial interface isn't conserved because the discontinuity of the media breaks the symmetry of spatial translation.
As shown in Eq. (6), the refracted and reflected waves both have two circularly polarized components with different frequencies ω ± . The superposition of the two components gives rise to the time Faraday rotation (TFR), namely, the refracted and reflected waves can be rewritten as a sole polarized wave respectively  directions. Unlike ordinary magneto-optical Faraday effect or optical activity which both refer to the polarization of a wave changing circularly in its propagating direction, the TFR wave has a unique polarization in the whole space at any fixed time point, however, the polarization rotates with time. Note that the Faraday effect caused by two opposite circularly polarized CFJ waves was also discussed in Ref. 26. However, their effect is still a spatial Faraday rotation, i.e. the two superposed CFJ waves have same frequency ω but different k and the rotating angle changes with traveling distance, therefore the TFR caused by the time refraction and time reflection is entirely a novel effect distinct form their discussion. At the second time interface t 1 of the time wave plate, the secondary time refraction and reflection occur. Then the beam will split into four elliptically polarized branches, all of which can be written by 0 0 . The superscript στ distinguishes the four branches: στ = 11, 12 denote, respectively, the secondary refraction and reflection of the first refracted wave, and στ = 22, 21 denote the secondary refraction and reflection of the first reflected wave respectively. Eq. (10) shows that the Faraday rotating angle of the polarization ellipses of the four secondary branches is φ ω Δ = Δ ( − ) t t 1 0 as the waves pass through the time wave plate (see Fig. 1). And in terms of the boundary conditions at t 1 , the relative lengths of the two polarized axes satisfy For the axion field in vacuum, + v p and ± v g always exceed the speed of light c in vacuum. Though v 1 < c in real media, + v p and ± v g will be still superluminal when → k 0 for , However, neither phase velocity nor group velocity represents the true velocity of energy or information transfer, therefore the superluminal effects of these two types of velocities do not violate the causality and have been observed in various experiments [43][44][45] . By means of the average Poynting vector and energy density over a period, we also can calculate the energy transport velocities of the two TCB states On the contrary to the group velocities, ± v E are always less than v 1 . Moreover, we prove that the front velocity v f (the velocity of wave front which represents the speed of information propagation) of the two TCB modes is precisely v 1 , when only concerning the dispersion caused by the constant rate β of the ME coefficients but regardless of the dispersion of ε, μ, β with respect to wave number k (the detailed derivation is given in the supplementary information). Therefore, neither energy nor information of TCB modes propagates superluminally. The comparison of four types of velocities is shown in Fig. 2 (b).
Gaussian pulse traversing a time interface. The plane wave solutions we have discussed are widespread in the whole space. However, the time wave plate made of time dependent media should only have a finite scale in practice. We accordingly need to analyze the propagation of wave packages with finite length. Consider a Gaussian pulse still only concern the dispersion caused by β. Taking account of the temporal boundary conditions, we obtain the magnetic fields, for t > t 0 , where , ± ± B B 1 2 denote the time refraction and reflection parts respectively, and ± B 3 denotes the non-traveling wave part. The three parts of B − take the forms is the relative coordinate with respect to the center of the wave package, The time refractions and time reflections for two different circularly polarized pulses are shown in Fig. 3(a,b). Actually, this approximation is valid only when  Fig. 3(c) shows the pattern of − B as the pulse traversing the time interface. Fig. 3(d) plots the velocity of the pulse center and the width of the pulse (defined as the distance between the two edges where − B equal to 1/e times − B at the center of the pulse) changing with time. Consequently, the pulse keeps nearly fixed center after traversing the time interface, while its intensity increases with the magnitude about exp (t 2 ). The width of the pulse increases with time, and it can be characterized by α −  1 approximately as shown in Fig. 3(d).
Experimental design. Considering a fluid in the presence of external electric and magnetic fields, the multipolar polarizations induced by external electric or magnetic fields can cause the fluid to be anisotropic and lead to Kerr effect or Cotton-Mouton effect. More specially, a parallel pair of external electric field  and magnetic field  will induce the Jones birefringence for a light beam propagating perpendicularly to the direction of the fields [37][38][39][40] . The Jones birefringence has been shown to be a bianisotropic effect 39 . For symmetric analysis, the external electric field  is P odd, and the external magnetic field  is T odd, but the parallelism of the two fields protects the combined PT symmetry. This fact indicates the existence of the axion type ME coupling as we have mentioned. The ME coupling tensor of molecules can be expanded with respect to the external fields The coefficients of each order are determined by solving the time-dependent perturbation of the molecular hamiltonian 40 . The Boltzmann average over all orientations of diamagnetic molecules yields 37,40 where the external fields are supposed to parallel z axis, N is the number density of molecules, μ z e is the z component of the permanent molecular electric dipole moment, k B is the Boltzmann constant, and T is temperature. Since the system is symmetric with respect to z axis, the medium should retain isotropic in the x − y plane and has a uniaxial ME tensor ( ) . Thus a beam propagating perpendicularly to z axis has two Jones birefringent eigenmodes, linearly polarized along the ± π/4 directions with respect to z axis respectively, with the difference of refractive indexes However, if a transverse polarized light travels along z axis, i.e. parallel to the external fields, it will experience the isotropic axion-type ME coupling E B α Θ = ⊥ ⊥ z z . In terms of isotropic average 37 , the ME coefficient in As a result, the effective axion field can be controlled via the external electric and magnetic fields. If the product of the external fields z z E B changes with time, we could observe the TCB and correlated phenomena predicted in this paper. The schematic illustration are shown in Fig. 4.
In principle, the TCB, as well as the ME coupling, caused by the time-varying external fields can arise in all media, while its magnitude is characterized by E B β μ μα . Supposing the product of the fields varies linearly with time, the magnitude is determined by two parts, one is the intrinsic property of the medium α^, the other is the rate of field change Δ( )/Δt at room temperature and 1 atm. Adopting these experimental parameters and assuming the time interval of field change Δ t ~ 10 −9 s (the characteristic frequency of the external fields is equivalent to GHz), we can estimate the refractive-index difference of the two TCB modes Δ ∼ ± − n 10 17 . On the other hand, previous experiments for small birefringence measurements have achieved the sensitivity Δ ∼ − n 10 18 via the metrology of high finesse resonant cavity [46][47][48] , to measure the TCB effect is accordingly feasible. Since the group velocities of the two TCB modes are nearly equal for small β, we can ignore the central separation of two superposed TCB pulses during the time interval Δ t and regard them as a single pulse with the TFR φ ω Δ = Δ Δ ∼ − t 10 rad 11 which is large enough for detection as a 10 −13 rad resolution of phase shift has been achieved experimentally 49 .
If the external fields are both parallel to the propagating direction of the pulse rigorously, no other birefringent effects that can disturb the observation of TCB, e.g. Kerr or Cotton-Mouton effects, would arise. However, the time dependence of the external fields will induce fields in the x − y plane inevitably. Supposing only  z changes with time but  z is constant, the linearly varying  z induces an eddy magnetic in the area of r < 10 −2 m which is thus small enough to be ignored. For experimental setup, a big challenge is to precisely control the external fields. Theoretically, the external fields at any locations should change simultaneously in the laboratory reference system, namely the variation of E B z z at different points is spacelike, since the effective axion field Θ ^ only depends on time. In practice, the speed of light in the media is less than vacuum, thus the prerequisite could be relaxed into that the fields begin to change before the pulse arrives. If there is a slow-light system with strong ME coupling α^, then the technical requirement could be largely reduced.

Conclusion
To summarize, we demonstrate that light with a certain wave vector k always corresponds to a pair of circularly polarized modes, i.e. the TCB modes, in time-dependent axion-type ME media. We study the time refraction and time reflection of plane waves and Gaussian pulses traveling in this type of media, and predict the time Faraday effect as a consequence of the superposition of the two TCB modes. We also propose a scheme to realize TCB in practice. According to our estimations with the realistic parameters, the magnitude of TCB is observable via existing experimental techniques. As the significance but difficulty for detecting axion particles, our proposal offers an alternative way to simulate and study the interaction of light with time-dependent axion field. On the other hand, by exchanging the status of space and time, we foresee that various effects in space-dependent media would have their temporal counterparts in time-dependent media for not only electromagnetic fields but also all kinds of waves. We hope our work could inspire more research in this novel area.