Constraints on Jones Transmission Matrices from Time-Reversal

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The fact that complex transmission below and elsewhere, it is a matter of current debate functions are measured and radiation with well-defined whether or not a chiral material as such can exhibit a polarization states used means that the simpler Jones Kerr rotation26,36–41. Another case where optics have matrix formalism can be used to analyze data. A chal- been used to find new correlated states of matter is in the lenge with TDTS is performing it in reflectance where case of La2−xSrxCuO4, where a factor of two difference phase information is difficult to recover reliably. in the a and b direction optical conductivities was found In this paper a formalism is discussed for inferring in an underdoped crystal that had only a 1% difference the existence of exotic broken symmetry states of mat- in the in-plane lattice constants42. Although this ma- ter from their electrodynamic response. This formal- terial was orthorhombic to start with (so additional 1D ism is discussed within the Jones transfer matrix ap- correlations broke no additional symmetries) this elec- proach and so is directly applicable to time-domain THz tronic anisotropy was interpreted as being due to the spectroscopy in transmission, but is more broadly ap- occurrence of unidirectional “stripe” correlations on top plicable. I discuss the consequences of discrete broken of orthorhombicity because the conductivity anisotropy symmetries on ordered states of matter including the onset was at approximately 80K despite the fact that the presence and absence of reflections, rotations, inversion, compound was orthorhombic at room temperature, the rotation-reflections and time-reversal symmetry and the spectral weight along the more conductive direction was constraints they give on Jones matrices. These con- actually suppressed relative to the other (in contrast to straints typically appear in the form of an algebra relat- expectations from the overlap integral), and the large ing matrix elements or overall constraints (transposition, scale of the effect. Even more recently, Lubashevsky unitarity, hermiticity, normality, etc.) on the form of ma- et al. 43 has found a small but significant polarization ro- trix. This formalism is applied to other more exotic sym- tation that derives from an anomalous linear dichroism metry breakings such as various forms of time-reversal in thin films whose principle axes are not aligned along symmetry breaking. As usual, the utility of symmetries those of the crystal symmetry directions. In YBa2Cu3Oy is that one can still deduce quantitative and qualitative the effect has a temperature onset that mirrors the pseu- information even when the underlying equations of mo- dogap temperature T∗ and is enhanced in magnitude in tion are unknown. underdoped samples. In x = 1/8 La2−xBaxCuO4, the effect onsets above room temperature, but shows a dramatic enhancement near a temperature scale known to II. GENERAL PROPERTIES OF JONES be associated with spin and charge ordered states. These MATRICES features are consistent with a loss of both C4 rotation and mirror symmetry in the electronic structure of the CuO2 Due to its phase sensitivity, in TDTS one typically planes in the pseudogap state. measures the complex transmission function and uses the Jones calculus to understand the interaction of light In the use of optical spectroscopies to determine broken with a material system being investigated. The com- symmetry states of matter, it is important to have a well plex amplitudes of the incident Ei field to the transmit- developed formalism for extracting the relevant informa- ted fields Et for propagation in a particular propagation tion as it is easy for one symmetry to masquerade as an- direction can be related through a frequency-dependent other. For instance, strong linear dichroism (absorption) 2 2“Jones” matrix. In the most general case, the Jones can effectively rotate the plane of polarization of light, matrix× is comprised of 4 independent complex values. In in a manner that could superficially be confused with a the basis of x y linear polarization it is magnetic state with circular birefringence (phase retarda- − tion). Both broken mirror and time-reversal symmetries i t can give circular birefringence. It is important to pre- Txx Txy Ex Ex Tˆ = i = t . (1) cisely determine what aspects of the optical anisotropy Tyx Tyy Ey Ey correspond to TRS breaking as various kinds of chiral and gyrotropic orders can give similar experimental sig- The above equation forms an eigenvalue-eigenvector natures. The frequency scale of the optical experiment is problem, where the Jones matrix has eigenvalues also an important consideration. Ordered states are generally defined in the ω 0 limit. This means that low → 1 2 frequency optical measurements may be very useful in de- κ± = (Txx + Tyy (Txx Tyy) +4TxyTyx) (2) termining the onset of a broken symmetry. In this regard 2 ± q − time-domain THz spectroscopy (TDTS) is a growing and and (unnormalized) eigenvectors important area of investigation, with important recent advances in THz polarimetry44–50. This technique generally satisfies the constraint in which these probes have 1 1 κ+−Txx , κ−−Txx . (3) frequency scales lower than the many frequency scales of Txy Txy interest. It has the ability to measure complex response functions directly without needing to resort to Kramers- 3

If light is sent through the system under test with convenience e.g. a well-defined linear polarization state in, for instance, the x direction, then the Faraday rotation (rotation of T T A B transmitted light) may be measured through the rela- xx xy = . (5) Tyx Tyy CD tion tan(θF ) = Txy/Txx. In principle, this is a complex quantity with real part of arctan(θ ) equal to the Fara- F It is useful to be able to transform the Tˆ matrix into day rotation itself and the imaginary part of arctan(θ ) F bases of different polarization states. Let E˜i and E˜t be equal to the ellipticity. See Ref. 45 for further details. the electric field vectors in a transformed basis. They are For plane waves, the spatial and temporal dependence related to the electric fields in the original basis through of the waves in free space is given in the usual way by i ˆ ˜i t ˆ ˜t multiplying these complex field amplitudes by a factor the transformation matrix E = ΛE and E = ΛE . ei(kz−ωt). More complicated cases but still within the The transmission matrix in the transformed basis is then ˆ˜ ˆ −1 ˆˆ paraxial approximation can be treated by letting the T = Λ T Λ. complex amplitudes acquire a dependence on x and y. One such matrix of practical importance is that of the We envision a plane wave incident on a slab made of a transformation to a circular base in terms of left and right particular material. This slab has various symmetries circularly polarized light that it inherits from both the particular point groups of the material as well the slab’s macroscale structure. 1 1 1 1 1 i Λ=ˆ , Λˆ −1 = − . (6) For most of our analysis I assume the slab itself has the √2 i i √2 1 i most generic shape that can be assumed without losing − generality, which is a flat cylindrical solid with rotation The transmission matrix in the circular basis in terms Rˆz(θ), Rˆy(π) and mirror Mxy, Myz, and Mzx symme- of the x y matrix elements is then tries. Where specified I will also address the case where − M symmetry is lost by additional features such as in xy 1 A + D + i(B C) A D i(B + C) the presence of a thin film on a substrate. The presence Tˆcirc = − − − 2 A D + i(B + C) A + D i(B C) of a substrate does not effect the consideration of sym- − − − (7) metries that include only a single surface, such as M or xz which connects the transmission amplitudes of R and Rˆ (π/2) , but can be important with symmetry consider- z L hand polarized light. Other transformation matrices− ations that involve “backside” transmission (for example exist− which represent other symmetry operations. These M ). xy will be discussed below. Alternative to the Jones formalism the 4 4 matrix formalism of Berreman can be used51. In this× method one explicitly treats the time and spatially varying magnetic III. CONSTRAINTS ON JONES MATRICES field as well. Although there are disadvantages of the FROM POINT SYMMETRIES expanded dimensionality of the matrix, such a method can be useful in computing explicit optical properties of materials as one does not have to resurrect the two Neumann’s principle states that the symmetry transformations of any intrinsic physical property of the crys- suppressed field variables with auxiliary equations when matching fields at boundaries. It is particularly useful in tal must include at least the symmetry transformations of the point group of that crystal52. Therefore the sym- calculating the properties of stratified media. Still when metries that states of matter possess impose symmetry only matters of symmetry are concerned the Jones matrix formalism is sufficient and I rely on it almost exclusively requirements on their Jones matrices and hence an algebraic relations among their elements. Conversely, ob- in the below discussion. In TDTS a time limited pulse is sent through this slab served symmetries of Jones matrices and the algebraic relations between their elements (or the lack thereof) can and its amplitude and phase measured. The electric field of a time-domain pulse E (t) can be written as a super- tell us about symmetries that states of matter possess or x break. In this discussion, it is important to keep in mind position of Fourier components in the usual way. For x polarized light that since Neumann’s principle is usually only applied to intrinsic properties, its application to a macroscopic property like a Jones matrix formally requires that the ∞ 1 iωt system has a generic shape that itself has all relevant Ex(t)= dωe Ex(ω). (4) √2π Z−∞ symmetries. For the most general case, one can consider a flat cylindrical solid with rotation Rˆz(θ), Rˆy(π) Here, Ex(t) is the purely real electric field propagating and mirror Mxy, Myz, and Mzx symmetries. In practice, through the system, and Ex(ω) describes the amplitude since in a TDTS measurement one measures a normal- of the complex Fourier components that comprise Ex(t). ized transmission (e.g. transmission through a sample One can apply the Jones calculus to each of these Fourier mounted to an aperture divided by transmission through components separately. Going forward I will mostly re- a bare aperture), sample shape does not matter appre- place the entries Tij in Eq. 1 with entries A,B,C,D for ciably. It may be appreciable only for samples where one 4 lateral dimension is smaller or of order the wavelength and near field effects become relevant. A B An advantage of the present approach is that it con- C (z) = Tˆ = . (11) 3 ⇒ B A siders only symmetries and the directly measured experi- − mental quantities e.g. ratios of transmitted electric fields ˆ to reference fields. The difficulty with applying symme- Rz(π) rotation symmetry (C2) imposes no constraints try constraints to the intrinsic material constants (for on the Jones matrix instance the dielectric function) and then calculating the transmission is that this may involve a number of compli- 1 0 cations and model dependent assumptions. If one were Rˆz(π)= − , 0 1 to calculate from the material constants, then for a par- − ticular symmetry one must write down the constrained ˜ −1 A B C (z) = Tˆ = Rˆz(π) TˆRˆz(π)= . (12) form of the dielectric tensorǫ ˆ, the magnetic permeability 2 ⇒ CD tensorµ ˆ, and the magnetoelectric susceptibility tensor αˆ. Then one must apply the boundary conditions and If an mirror symmetry exists with respect to the xz Fresnel equations properly in their full generality (e.g. plane accounting for magnetoelectric effects) to get the transmission matrix. This is obviously much more involved 1 0 than in the current approach. Moreover for many com- Mˆ xz = , 0 1 plex materials the precise issue of electromagnetic bound- − ary conditions can be complicated. For instance, there ˜ −1 A B Tˆ = Mˆ TˆMˆ xz = − , (13) is still ongoing discussion about the proper extension of xz CD the Fresnel equations to chiral materials26,36–41. In the − present approach, one avoids all this and concentrates and therefore the Jones matrix for a material with Mˆ xz on the experimentally measured quantities directly. The reflection symmetry (where Mˆ xz is the matrix for reflec- precise form ofǫ ˆ,µ ˆ,orα ˆ does not matter, nor does the na- tions across the xz plane) must be ture of the boundary conditions. The approach here bor- rows in part from a formalism for quasi-2D meta-material structures53. Important differences to the present case of A 0 Mˆ xz,yz = Tˆ = . (14) bulk broken symmetry states of matter will be discussed. ⇒ 0 D In general, the simplest examples come from rotation and reflection symmetries. Rotation by an angle φ is An identical form of the Jones matrix arises from Myz represented by the transformation matrix reflection symmetry as indicated above. The Tˆ matrix is diagonal if there exists any mirror plane parallel to cosφ sinφ the z axis and either thex ˆ or y axes. One can see by Rˆz(φ)= . (8) the constraints in Eqs. 10 and 14 that if a state has sinφ cosφ − both Rˆz(π/2) rotation symmetry and a mirror symmetry If a material possesses a Rˆz(π/2) rotation symmetry along x or y axes then off-diagonal elements vanish and (C4) around the z axis, its Jones matrix will be invari- diagonal elements must be equal. π By simple rotation of Eq. 14 one can show that if a ant under a 2 rotation. Applying the relevant rotation matrix to Eq. 5 gives mirror plane exists in any direction that contains the z axis then the Tˆ matrix will have the form 0 1 Rˆz(π/2) = , 1 0 A′ B′ − Mˆ nz = Tˆ = ′ ′ , (15) ˜ −1 D C ⇒ B D Tˆ = Rˆ (π/2)TˆRˆz(π/2) = − . (9) z B A − where the elements are not independent, but are related ′ 2 2 Hence the Jones matrix for a material with Rˆz(π/2) sym- by the trigonometric relations A = Acos φ + Dsin φ, metry must have the form D′ = Asin2φ+Dcos2φ, and B′ = (A D)sinφcosφ where φ is the rotation of the mirror plane− with respect to the A B laboratory axes. Eq. 14 is a special case of 15 when a C (z) = Tˆ = . (10) 4 ⇒ B A mirror plane is aligned along either x or y principle axes − and the off-diagonal elements are zero. If the material has a Rˆz(2π/3) rotation symmetry (C3) Additional constraints can be made for materials that as for instance encountered in rotations of cubic struc- possess time-reversal symmetry such that their optical tures along their [111] direction or in the corundum struc- response can be said to be reciprocal. Effects are non- ture of materials like Cr2O3 around the c axis, one has a reciprocal if they have opposite signs for the two states of Jones matrix of the form a crystal that are related to each other by time-reversal. 5

Reciprocity and time-reversal symmetry is discussed in 19 one can see that the presence of a mirror symmetry detail below. In the remainder of this section I assume along any direction gives the same symmetric form for reciprocity which should be valid for all non-magnetic the Jones matrix (for a system with a TRS symmetric materials. As will be shown below the backwards trans- reciprocal response). mission through a Jones matrix for a reciprocal material If a crystal has a 3D center of inversion (P symmetry) is equal to transpose of the Jones matrix through the and TRS then the Jones matrix also has the form of forward direction e.g. Eq. 19 as inversion is equivalent to a reflection across the xy plane and Rˆz(π) rotation around z. However, I have already shown in Eq. 12 that symmetry under b,−z A C ˆ Tˆ = , (16) Rz(π) does not constrain the Jones matrix, therefore the BD constraint that reflection symmetry across xy gives is the same as inversion. b,−z where the Tˆ refers to transmission through the “back- Rotation-reflection symmetry with an improper axis z side” of the materials with wave propagation in the z can be handled through a combination of normal rotation − direction and the underline here and below represents (Eq. 9 for Rˆz(π/2)) and the Mˆ xy operation (Eq. 18). For the presence of TRS. Most useful for the typical experi- S4 and 1 one gets the completely isotropic matrix mental situation is the case where instead of considering backwards transmission through the sample, one consid- A 0 ers forward transmission, but through a sample that has 1,S4(z) = Tˆ = . (20) ⇒ 0 A been rotated by π around the y axis. In this case the Jones matrix is Rotation-reflections with an in-plane improper axis (x or y) and 1 give

ˆ b,+z A C T = − , (17) A B BD 1,S (x) = Tˆ = . (21) − 2 ⇒ BD b,+z where now Tˆ refers to transmission through the back- Structures that cannot be mapped onto their mirror side of the materials with wave propagation in the +z images by rotations and translations are chiral54,55. Chi- direction. The minus signs in the off-diagonal part ac- rality itself generally does not put constraints on Jones counts for the fact that in the reference plane of the sam- matrices, but states that are chiral can be consistent with ple x x when performing the Rˆy(π) rotation. Note → − other symmetries that constrain the Jones matrices. For that this expression is true only for reciprocal systems. instance, chiral reciprocal materials with TRS can have We denote time-reversal invariance in this sense by the a Rˆx(π) symmetry symbol 1. For the most generic case, the forward going Jones matrix does not carry enough information to con- ˜ −1 A C strain the backwards propagation of the material com- Tˆ = Rˆ (π)TˆRˆx(π)= − , (22) x BD pletely. I will discuss non-reciprocal and time-reversal − symmetry broken systems in more detail below. But for such that materials which are reciprocal, mirror symmetry in the xy plane and Eq. 17 gives important constraints on the A B Jones matrices. A mirror operation across the xy plane 1, C (x) = Tˆ = . (23) ˆ 2 ⇒ BD is equivalent to a Ry(π) rotation followed by a mirror − reflection across the yz plane. Therefore Note that Eqs. 19 and 23 are only compatible with each other for vanishing off-diagonal elements. There- fore considering that circular optical activity is only pos- ˜ −1 −1 A C Tˆ = Mˆ Rˆy(π) TˆRˆy(π)Mˆyz = , (18) ˆ yz BD sible when off-diagonal elements of the T matrix are anti- symmetric, one can see that to generate circular optical which gives a form for the Tˆ matrix that is activity it is insufficient to break reflection symmetries only in the xz and yz planes, one must also break reflection symmetry in the xy plane. TRS chiral material can A B also exhibit a Rˆz(π/2) symmetry giving the constraint 1, Mˆ xy = Tˆ = . (19) ⇒ BD on the Jones matrix in Eq. 10. Chirality typically intro- duces anti-symmetric components to the dielectric tensor Note that this expression is strictly invalid for thin through an expansion first order in the light wavevector films mounted to substrates as the substrate will violate k. This is consistent with the fact that in these TRS ma- the Mˆ xy symmetry. In principle, the breaking of mirror terials one can switch the sign of the off-diagonal com- symmetry by a substrate can be weak, but specific cases ponents by considering transmission through the back- have to be analyzed individually. Comparing Eqs. 15 and side in which k k. Also note that no reference to → − 6 the “screw axis” direction was made to derive Eq. 23, exactly reconstruct an original wavefront if a system has showing that even when transmission is perpendicular to absorption or scattering (e.g. reflection). Absorption the screw axis, chiral materials can show optical activ- and the generation of heat violates time-reversal invari- ity (although it will be of a complicated elliptical variety ance through the 2nd law of thermodynamics. Since all without C4(z) symmetry). real materials will show some dissipation (or reflection) Note that translational invariance itself imposes no at frequencies of interest, a strict form of time-reversal constraints on Jones matrices. In the context of our cur- symmetry breaking of an optical process is generally not rent treatment, this must be the case because plane waves of interest to us. This demonstrates that one must formu- are invariant in the direction perpendicular to their prop- late a more applicable definition to be sensitive to time- agation direction. But more generally it is only the point reversal symmetry breaking in the state of matter itself. groups (crystallographic or magnetic) that determine the This can be found in the concept of reciprocity27,57–61. macroscopic symmetries properties of materials56. Al- A system may be reciprocal if one can switch source and though at first glance trivial, this point is important when detector (with the possibility of additional constraints) considering the properties of materials with space groups and get the same result. The importance of the reci- that have more complicated symmetries with glide or procity condition is in the fact that it is applicable even screw symmetries. In these cases it is only the magnetic in the presence of absorption or scattering when other point group part of the global symmetry that imposes a treatments lack “true” time-reversal symmetry. As in constraint. our discussion on point group symmetries, discrete time- Note that domain structures can average out polar- reversal invariance (formulated in terms of reciprocity) ization anisotropies. In order to see a broken symmetry puts certain important constraints on Jones matrices. optically, the symmetries must be broken globally over These were already exploited above in the discussion of the entire beam spot. Residual strain or stray field may how the Jones matrix behaves under Rˆx(π) or Mˆ xy op- provide an effective aligning field. Alternatively, exter- erations in non-magnetic materials. nal fields need to be used to create monodomain samples. A system can be said to be reciprocal if a response has These issues may be more pronounced in TDTS with its the same sign for the two states of a material that are inherently long wavelengths and larger beam size. related to each other by time-reversal (e.g. time even). A system is non-reciprocal if a response or component of the Tˆ matrix has the opposite sign for the time re- IV. TIME-REVERSAL AND RECIPROCITY versed state (time odd). For further analysis we need a relation for the Jones matrix of a system in which the sense of time is reversed. Although related formulations In any discussion on the constraints on Jones matri- 27,59,61 ces from time-reversal symmetry it is important to de- have been made previously , a particularly simple fine exactly what is meant by time-reversal symmetry as treatment exists for the case of coherent polarized waves that can be treated with this Jones formalism. Consider fortunately (or not) time marches inexorably in a single i direction in all real experiments. For plane waves trav- the situation when a wave E is incident on a sample with a Jones matrix Tˆ. The resulting wave must be ana- eling in the +z direction the electric field can be written r i(kz−ωt) lyzed by its projection on a reference wave E . Channel- as E0e where E0 is the two component vector in 62 the x and y directions. Time-reversal is trivially accom- ing Onsager , microscopic time-reversal invariance man- plished letting t t in this expression. One can alter- dates that if both the sense of time in the experiment and natively accomplish→− the equivalent of the time-reversal the sample are reversed then the result must be the same operation by the dual operation of complex conjugation e.g. of the amplitude E E∗ and letting k k. This can → →− be seen simply by the fact that the physical electric field b,−z 1 iωt ∗ −iωt r† ˆ i i∗† ˆ r∗ is E0 cos(ωt)= 2 (E0e + E0 e ). Reversing the sign E T E = E (T )E , (24) of the time (t t) in this real physical electric field is →− mathematically equivalent to taking the complex conju- where now the initial wave is the time-reversed Er∗, b,−z gation of E0. That complex conjugation is necessary for ˆ the effective time-reversal operation can alternatively be T is the Jones matrix for light propagation in the z direction through the backside of a time-reversed sys- see in the fact that it takes both it and the inversion of − the wavevector to reverse the sign of Poynting’s vector tem, and the wave is analyzed by its projection on the time-reversed wave Ei∗. Here signifies the Hermitian describing energy flow in an optical system. Heuristi- † cally, conjugation can also be seen to be necessary to adjoint and the overbar represents the operation of time- phase advance a time reversed wave sufficiently such the reversal. One can reverse the order of the inner product phase accumulated during time-reversed propagation is on the right side of this equation and take its complex exactly canceled to reconstruct the initial wave with its conjugate to get initial phase relation. Even time reversed waves from the dual operation of b,−z complex conjugation and wavevector inversion will not Er†Tˆ Ei = Er†(Tˆ )T Ei. (25) 7

Relabeling the sense of time of the system on both TRS will be an invariance of the off-diagonal components sides of this equation one gets of Tˆ with the time-reversal operation. It is instructive to consider more restricted definitions of reciprocity. For instance, consider the constraints im- b,−z T Tˆ = (Tˆ ) . (26) plied by the hypothetical case where a time-reversed wave exactly reconstructs the initial wave. Considering incom- This gives the general result that the time-reversed Jones ing and outgoing waves related by matrix equals the transpose of the Jones matrix for light propagating through the system’s backside. If the system has macroscopic TRS, then Tˆ = Tˆ and one arrives at Tˆ Ei = Et. (29) the result used in Sec. III that for a system that has a reciprocal response and hence macroscopic TRS (1) the If time-reversed version of the final wave can exactly re- backside transmission is the transpose of the front side construct the initial wave and de Hoop reciprocity holds transmission then

b,−z 1 = Tˆ = (Tˆ )T . (27) TˆT Et∗ = Ei∗ (30) ⇒ where again the underline reminds us that this is a rela- is true where the complex conjugation of Ei and Et time tion for the Jones matrix of sample that has TRS. reverses the wave amplitudesa ` la the discussion above. A condition related to the above for interaction of an Taking the complex conjugate of Eq. 30 and substituting electromagnetic wave on a finite size scatterer based on a Eq. 29 into it for Et gives definition with incoming wave and a projection on a reference wave was first formulated by de Hoop59,61 and is the most general expression of reciprocity. The condition [TˆT ]∗Tˆ Ei = Ei (31) above is a representation in the Jones matrix formalism. This de Hoop reciprocity distinguishes a form of time- which can only be true if Tˆ is unitary. This form of reversal symmetry most applicable to real systems and reciprocity is not particularly significant from a practical experiments. It is a sufficient (but not necessary) con- perspective because unitary matrices are norm preserv- dition to say a state breaks TRS if it violates de Hoop ing and hence preclude absorption or reflections. Even reciprocity. Not all TRS breaking states will show optical purely dielectric media will induce reflections at inter- non-reciprocity. It is necessary to break TRS macroscop- faces, which destroys the unitary condition. However, ically, which generally means that the combination of the this condition does satisfy the simplest considerations for time-reversal operation and a lattice translation is not a 22 time-reversal symmetry and hence is important from a symmetry operation . conceptual point of view. One can call this form of time- Considering the above condition for de Hoop reci- reversal symmetry unitary reciprocity61. procity, a measure of time-reversal symmetry breaking b,−z b,−z Another form of reciprocity can be determined by con- is then the quantity Txy T where T refers i | − yx | yx sidering a situation where the initial wave E is time- to transmission through the backside of the materials reversed, but made to propagate through the system from with wave propagation in the z direction. As discussed − the “backside.” The system could be said to be reciprocal above, more relevant to the usual experimental situation in a restricted sense when the final waves in either case is the case where the waves are propagating in the +z di- are time reversed versions of each other. If the system is rection, but through a film that has been rotated around reciprocal in the sense of de Hoop and also reciprocal in the y axis by π. In that case the measure of symmetry b,+z b,+z this more restricted sense then a time reversed wave with breaking becomes Txy + T where the T refers i∗ | yx | yx polarization state E results in a backside transmitted to backside transmission but in a sample that has been wave Et∗ via flipped. Examples of states that violate this condition of de Hoop reciprocity are discussed in Sec. VI. In some systems it is possible to control the sense of the TˆT Ei∗ = Et∗. (32) direction of time by cooling in a magnetic field (for a state like a ferromagnet) or dual magnetic and electric fields Taking the complex conjugate of this expression gives (for a magnetoelectric-like Cr2O3). It is then a sufficient condition to say a state violates TRS if the equality [TˆT ]∗Ei = Et. (33) Er†Tˆ Ei = Er†Tˆ Ei (28) Eqs. 29 and 33 can only both be true if Tˆ = (TˆT )∗ is not satisfied. One can see that with this expression and e.g. Tˆ is Hermitian. Hence one can call this kind of trans- Eq. 26 that a necessary, but not sufficient condition for mission properties Hermitian reciprocity61. Note that 8 one should not confuse non-Hermiticity in the transmis- dominant dichroism and birifrigence have to be non- sion matrix (or scattering matrices in general) with non- aligned so that this kind of asymmetry may be most Hermiticity in the dielectric function. Non-Hermitian di- apparent in very anisotropic materials at frequencies of electric tensors are generally associated with bulk ab- strong absorption. Wave propagation of polarized light, sorption, whereas non-Hermiticity in the Tˆ matrix (or but not the explicit symmetry properties, in crystals scattering matrices more generally) are associated with with nonorthogonal eigenpolarizations has been investi- phase delay. The most obvious example of an optical gated previously66–69. Non-reciprocity of this kind has component that violates Hermitian reciprocity is a quar- been also called eigenwave non-reciprocity in the liter- ter wave plate that turns linearly polarized light into cir- ature and identified with a TRS breaking of a certain cularly polarized light. fashion70. However, it is important to emphasize that Very asymmetric transmissions for forward and back- although materials that are non-reciprocal in this sense ward transmission will be obtained in the case where may have very asymmetric interaction with light in time eigenpolarizations are non-orthogonal and the Jones ma- and space, they don’t themselves violate TRS. For in- trix is non-diagonalizable63–65. For this to be the case the stance effects of this kind are purported to exist in zinc- Jones matrix needs to be non-normal e.g. TˆTˆ† = Tˆ†Tˆ blende type semiconductors, which break inversion, but where signifies the Hermitian adjoint. A diagonalizable6 obviously not TRS71. Because they are not constrained Tˆ matrix† for a sample that obeys de Hoop reciprocity will by any useful symmetry, materials that show a violation be similar for forward and backward going transmissions of diagonal reciprocity will typically have eigenstates that with identical transmission eigenvalues. In contrast, one depend strongly on frequency. can see by inspection of Eqs. 2 and 3 that although Yet another (very) restricted form of reciprocity en- the eigenvalues of a non-diagonalizable transposed back- countered in a typical experimental geometry when one wards going Jones matrix are not different from the for- “flips” a sample to perform a transmission experiment ward ones, the eigenvector associated with a particular from the backside. If again de Hoop reciprocity is obeyed eigenvalue will change its character. It is a sufficient con- then the transmission matrix can be arrived at by us- dition for the Tˆ matrix to be normal that Txy = Tyx. ing Eq. 27 for backside transmission and performing ± Symmetric transmission of the kind found in normal a x x transformation from the π rotation around Jones matrices we can call diagonal reciprocity. around→ the− y axis a la Eq. 17. A simple example of an optical component that breaks diagonal reciprocity and results in such very asymmetric transmission is the composite optical component of a −1 A C Rˆ (π)TˆRˆx(π)= − . (35) linear polarizer combined with a quarter-wave plate that x BD is at 45◦ to the axis of the polarizer. Independent of the − initial polarization such a device produces linearly polar- If this Jones matrix is identical to the forward going ized light in transmission in one direction and circularly one then one will have the following constrained form for polarized light in the other. The Jones matrix for such a the forward going Jones matrix. composite device with the linear polarizer on the input side would be A B . (36) 1 1 i BD . (34) − √2 1 i A comparison with Eq. 23 shows that chiral states Using Eqs. 1 and 2 one can see that this matrix is non- that obey de Hoop reciprocity are also reciprocal in this normal with non-orthogonal eigenpolarizations that are sense. Materials which have only a mirror symmetry (Eq. linear (45◦) and circular (L) polarized light associated 15) will only be considered reciprocal in this sense if they with eigenvalues 1+i and 0 respectively. Note that upon are “flipped” along a mirror axis. I call this reciprocity consideration of the backwards going Tˆ matrix one has naive reciprocity. the same pair of eigenvalues 1 + i and 0, but that their Although these more restrictive formulations such as association has in a sense swapped e.g. they are associ- unitary, Hermitian, and diagonal reciprocity may have ated with circular (R) and linearly (45◦) polarized light utility in analyzing experimental results, the most in- respectively. Such a device is still reciprocal in the sense teresting cases of non-reciprocity will occur in situa- of de Hoop as the forward going transmission matrix is tions when the most generic condition of de Hoop reci- equal to the transpose of the backwards going one. This procity is violated. This will be the case for instance shows the central importance of the de Hoop condition in situations where time-reversal symmetry is broken in as such a device obviously does not break time-reversal the form of ferromagnetism, p-wave superconductivity 72 34,73–75 symmetry defined in any useful sense. (likely present in Sr2RuO4 ), magnetoelectrics , Based on our considerations of Sec. III the highest chiral spin liquids76,77 or in some proposed recent33–35or symmetry materials that break diagonal reciprocity can older19–27,78 models for the pseudogap state of the high- have is C2 around z. Generally one finds that axes of temperature superconductors. 9

V. BREAKING OF SYMMETRIES will be zero for a TRS chiral system.

As established in Secs. III and IV, particular symme- VI. COMBINED SYMMETRIES AND tries establish algebraic properties of or constraints on EXAMPLES the Jones matrices. It it is then straightforward to determine what combination of matrix elements are a measure of what particular symmetry breaking. The methodology The various point group symmetries (reflection, ro- is clear. Given a symmetry and the set of constrained tation, rotation-reflection, inversion) and time reversal Jones matrices for forward and reversed propagation that symmetry can be combined to get more insight into var- results from it, by inspection one establishes what sum or ious broken symmetry states. The simplest example of difference of matrix elements will yield identically zero if a TRS breaking state is a ferromagnet with its magne- a particular symmetry is present. If one establishes that tization vector pointing along the propagation direction. ˆ the particular quantity is non-zero, this is a sufficient Both the time reversal operation and a Rx(π) rotation condition to establish that that symmetry is broken. give the same state with the magnetization reversed. Us- As a first example of this consider a tetragonal ini- ing Eq. 26, the time reversed Jones matrix is tially non-magnetic material that goes undergoes a symmetry breaking transition. In the high symmetry state, b,−z b,−z ˆ¯+z Txx Tyx the material has all mirror symmetries and symmetry T = b,−z b,−z . (38) Txy Tyy under Rˆz(π/2) rotation. The Jones matrix is then the simplest possible e.g. The Jones matrix under Rˆx(π) rotation is

A 0 b,−z b,−z Tˆ = . (37) ˆb,+z ˆ−1 ˆb,−z ˆ Txx Txy 0 A T = R (π)T Rx(π)= b,−z − b,−z . x T T − yx yy If the phase transition is to a non-magnetic orthorhom- (39) In order that these expressions are equivalent, the bic phase such that Rˆz(π/2) symmetry is broken, but the x and y mirror planes are maintained the Jones ma- Jones matrix must have the form trix will have the form of Eq. 14 with diagonal components unequal and off-diagonal components still zero. A B Tˆ = . (40) Therefore a measure of the orthorhombic order param- BD eter is the quantity Txx Tyy . This relation rests on − the assumption that| the− laboratory| x y frame coin- If the material is cubic or tetragonal with A = D, the cides with the orthorhombic axes. It is− useful to have a matrix is diagonalizable in the circular representation, more general relation. Inspection of Eq. 15 and the rela- otherwise the polarization eigenstates are elliptical. In tions that follow for the matrix elements in an arbitrary either case, this is consistent with a rotation of transmit- primed reference frame x′ y′, show that the quantity ted linearly polarized light e.g. a Faraday rotation. − Tx′x′ Ty′y′ +2iTx′y′ will be sensitive to the symmetry A very interesting application of these methods is to | − | breaking Txx Tyy in the orthorhombic reference frame. the case of magnetoelectrics. Consider the case of Cr2O3 If the phase| − transition| is to a non-magnetic chiral phase which is a material that forms in the corundum structure ˆ 6 such that all mirror symmetries are broken, but Rz(π/2) with space group D3d. Below TN = 306 K it becomes symmetry is preserved then the Jones matrix will have an commensurate antiferromagnet with the spins of the 4 the form of Eq. 10 with equal diagonal components and Cr+3 atoms per unit cell pointing along a three fold axis off-diagonal components of identical magnitude and op- in an up-down-up-down alternating manner with mag- posite sign. Therefore a measure of the chiral order pa- netic point group 3¯′m′. This structure breaks both in- rameter is the quantity Txy Tyx . This quantity will be version (P) and TRS (T), but the product of the two finite if a different transmission| − coefficient| is experienced is a symmetry. There are two degenerate ground states for R and L circularly polarized light, but zero otherwise. that are related to each other by either inversion or time- For instance it will be zero for a linearly birefringent ma- reversal. As pointed out originally by Dzyaloshinskii56 terial that gives an effective rotation of a plane of po- the magnetic symmetry of Cr2O3 allows linear magneto- larized light if the polarized light is not incident along a electric effects73–75,79–82 e.g. an electric field can cause a mirror symmetry. magnetization (with proportionality αij ) and a magnetic If the transition is to a state that breaks macroscopic field can cause a polarization. Such nonzero magneto- time-reversal symmetry (for instance to a ferromagnetic electric coefficients are a consequence of so-called “PT ” or magnetoelectric phase) the measure of macroscopic symmetry. They are indicative of inversion (P ) and TRS symmetry breaking and the order parameter becomes (T ) symmetries broken separately, but preserved under b,+z b,+z Txy Txy where again the Tyx refers to backside the combined PT symmetry. PT symmetric situations transmission| − | but in a sample that has been flipped. Al- where the P symmetry breaking is such as to support a though it will also have a Faraday rotation, this quantity chiral state, have been called “false chiral” by Barron 54. 10

Magnetoelectrics can admit a number of very inter- matrix must be compatible with the constraints of the C3 esting phenomenon including non-reciprocal gyrotropic symmetry given in Eq. 11 for propagation along z. This birefringence, whereby there is a symmetric contribution gives a Tˆ matrix with zero off-diagonal components and to the dielectric tensor that rotates the axes of birefrin- identical diagonal ones. Although its been shown that 73,80 gence away from the crystal symmetry directions . magnetoelectrics like Cr2O3 show a non-reciprocal Kerr The sign of this contribution depends on the propagation rotation effect in reflection73–75, symmetry constraints vector and therefore the speeds of propagation can be dif- determine that the transmission of a single crystal slab ferent for counter-propagating beams. Some evidence for is invisible to such non-reciprocal effects e.g. no Faraday this effect has been seen in transmission in continuous effect is exhibited84. Microscopically one can consider 83 wave THz-range spectroscopy in Cr2O3 and related ef- that the light suffers a rotation of order α (where α is fects in reflection at higher optical frequencies74,75. The the mangetoelectric coefficient) upon crossing the front case of transmission is easier to analyze. Consider the surface in transmission, propagates through the sample transmission along a direction perpendicular to the 3-fold with E and H fields at a small angle deviating from 90◦ axis and with the x y reference frame of the linearly po- by α, and then undergoes a rotation with the same mag- larized light aligned− to the structural axes. The Tˆ matrix nitude at the back surface, but in the opposite direction. transformed under inversion Pˆ is Note that inversion symmetry breaking by a substrate will invalidate this cancelation and it should be possible b,−z b,−z to see a Faraday rotation φF (ns 1)α in transmission ˆb,+z ˆ−1 ˆb,−z ˆ Txx Txy ∝ − 85 T = P T P = b,−z b,−z . (41) for Cr2O3 on a substrate of index of refraction ns . An Tyx Tyy analysis of the reflection coefficient (discussed below) for The same state should be obtained under time-reversal a TRS breaking state such as this one show that it can exhibit a Kerr rotation. Even more exotic states in the form of various chi- b,−z b,−z 55 ˆ Txx Tyx ral spin liquid states can be analyzed . A “vector spin T = b,−z b,−z . (42) Txy Tyy chiral” state is said to be exhibited when the quantity 86,87 Si Sj = 0, while Si =0 . Such a state does not The Tˆ matrix must therefore have the form hbreak× TRSi 6 and despiteh i its accepted name, is not neces- sarily chiral in a rigorous sense. By itself the finite expec- tation value is not enough to determine whether such a A B Tˆ = . (43) state is chiral and hence specify its electrodynamics. The BD related quantity eij Si Sj = 0, where eij is a unit vec- h · × i 6 No other symmetries further constrain this matrix and tor connecting sites i and j, does break inversion and all so only equal off-diagonal entries are admitted, and hence mirror symmetries and so such an order parameter has it can be diagonalized by a rotation. As I have already a definite handedness. By the considerations of section assumed in my analysis that the coordinate axes them- III this state should have the transmission properties of selves are aligned along the principle structural axes of other chiral systems and can exhibit a Faraday rotation the system this shows that the optical axes of the linear (but not a Kerr rotation, see Sec. VII). dichroism/birefringence can be rotated from the coordi- Other vector spin chiral states may or may not ex- nate system. This is consistent with the phenomena of hibit polarization anisotropies depending on the details gyrotropic birefringence. We can get further insight into of their ordering. For instance, a magnetic cycloid chain, this system by realizing that the material is invariant un- consisting of coplanar spins that rotate around a perpen- der a Rˆy(π) rotation. The form of the Jones matrix is dicular axis as one moves along an axis parallel to the spin plane has Si Sj = 0, but is achiral. Two differ- then of Eq. 39, which determines that the off-diagonal h × i 6 components for forward and backwards propagation are ent senses of rotations of the spins can be generated by opposite to each other showing the sense of rotation of the inversion operation, but these states are not uniquely specified by Si Sj , as they can be interconverted by the optical axes is opposite for front and back surfaces, h × i giving evidence for the non-reciprocal nature of this ef- a Rˆz(π) rotation. Since we have already shown that a fect. Moreover, since the Tˆ matrix has the form of Eq. 43 Rˆz(π) rotations have no effect on transmission, a cycloid Rˆx(π) symmetry one can probe the non-reciprocal nature order of this form cannot be detected optically. of the state by measuring Txy on a single surface alone. A “scalar spin chiral” state is said to be exhibited when 76–78 Note that presence of TRS in the paramagnetic state the quantity Si Sj Sk = 0, while Si = 0 . A constrains (through the reciprocal condition Eq. 27) the state which exhibitsh · this× orderi 6 with theh samei sign on a off-diagonal components of the Tˆ matrix to be zero above line along the propagation direction (e.g. “ferro” order- TN . ing) breaks time-reversal symmetry, but not inversion. It For transmission along the c-axis the form of Eq. 43 is will show both Kerr and Faraday reflection. As pointed still valid because it was based on the combined symme- out in Ref. 22 states which exhibit an “antiferro” order- try PT without regards to propagation direction. How- ing (between say CuO2 planes in the cuprates as con- 26,78 ever Cr2O3 has a three fold axis along z and hence its Tˆ sidered historically ) and break inversion structurally 11 will show a Kerr rotation (but not Faraday) in a manner 26 applied to the reflection matrix shows that the time- very similar to the case of Cr2O3. reversed reflection matrix will be the transpose of the reflection matrix for the forward direction in time e.g.

VII. EXTENSION OF JONES MATRIX T FORMALISM TO OTHER GEOMETRIES AND Rˆ = Rˆ . (44) EXPERIMENTS It follows that if the material has TRS and hence the reflection matrix doesn’t change under the time-reversal With some modifications the above arguments can be ˆ ˆ extended to other geometries and experiments. In reflec- operation (e.g. R = R), then the off-diagonal compo- tion the loss of symmetry across the free surface means nents of the reflection matrix must be equal. There- particular consideration must be given. Although the fore a material with TRS cannot exhibit a Kerr rotation point group symmetry constraints applied to the trans- as that necessitates anti-symmetric off-diagonal compo- mission matrix when only a single surface is considered nents in the reflection matrix. A somewhat related, but apply equally to the normal incidence reflection matrix, much more involved proof, for generic geometry can be there can be additional issues that arise due to the lack of found in Ref. 26. The absence of a Kerr effect except symmetry at the interface88. Moreover the de Hoop reci- in the presence of TRS breaking is relevant for the in- procity relation for TRS I found above was derived specif- terpretation of recent experiments on cuprate superconductors that have seen a Kerr rotation onset in under- ically for the case of transmission. Similar constraints on 29–31 reflection require different consideration. doped compounds . Such experiments have been interpreted as being consistent with the onset of a chiral A practical matter that arises when considering reflec- phase, but this appears to not be a possible explanation tion, is the long standing controversy regarding whether (at least at the level of linear response and in thermal or not a Kerr-like rotation can occur in reflection from equilibrium). See also Hosur et al. 32,40 , Cho et al. 41 for a chiral TRS media26,37–41. A straightforward applica- further discussion. tion of Fresnel’s equation using the bulk (antisymmetric) Most optical experiments do not measure complex dielectric constant predicts that a Kerr rotation should transmission coefficients directly. For experiments like be observed if there is finite dissipation36. However it Fourier transform infrared reflectivity (FTIR)1,2, Sagnac is clear that one cannot simply export the bulk dielec- interferometry18, or spectroscopic ellipsometry89 the tric constants to the surface as their antisymmetry arises light intensity is detected. In this case one has to make from an expansion of the dielectric constant in odd pow- use of the Mueller-Stokes matrix formalism, where the ers in the photon wavevector k. Such an expansion can- optical resonse is given by a real 4 4 “Mueller” ma- not be strictly valid at a surface and one must retain the trix operating on a real 4 component× “Stokes” vector. full spatial dependence of the dielectric constant. More- The Stokes vector is given in terms of the electric field over, in chiral systems, the issue of boundary conditions intensities in different directions and polarizations e.g. is complicated by a certain freedom in choice of ǫ and µ. Gyrotropy (e.g. dependence of dielectric constant on 2 2 k) can be included only in ǫ or in µ or both. There Ex + Ey | |2 | |2 is a disagreement within different calculational schemes, Ex Ey  | 2| − | | 2  . (45) as some observable quantities, such as reflectivity, turn E45◦ E−45◦ | | 2− | 2 | out to be dependent on these choices. For instance an  El Er  alternative treatment using essentially magnetoelectric-  | | − | |  like “Casimir” material relations37, finds that no rotation The Mueller matrix represents the intra- and intercon- should be occur. Other work says that both approaches version of these polarization states. The Mueller ma- are essentially valid and correctly applying terms involv- trix formalism is more general than the Jones formal- ing derivatives of the non-local susceptibilities gives ro- ism as its reliance on measured intensities means that tation in reflection38. Even more recent work concludes it can characterize non-coherent unpolarized light. Most that the differential reflection upon normal incidence relevant experiments however can utilize polarized light should vanish39. and in this case it is possible to convert from the Jones The present formalism however provides a simple proof to Mueller matrices for analysis. This then suggests a that materials that preserve TRS cannot exhibit a Kerr method for analyzing the symmetry properties of ma- rotation. First, one realizes that in the simplest normal terials being probed by such a technique; analyze their incidence reflection geometry with source and detector symmetry properties and apply the relevant constraints spatially coincident, the “forward” going reflection ma- to the simpler 2 2 Jones matrices first and then convert trix must be the same as the “backward” going reflection the Jones matrices× to Mueller matrices. This method matrix. Switching the source and the detector gives no should be completely valid as long as the sample under change to any physical quantity, because “forward” going test is not depolarizing (which is typically the case). and “backward” going are the same thing. Therefore a The Jones-to-Mueller matrix conversion is performed similar treatment of de Hoop reciprocity leading to Eq. by calculating the light intensities from the electric fields. 12

Following Azzam and Bashara89 and starting from the Jones formalism, which gives Tˆ Ei = Et I begin by calculating S = ACˆ (49)

where ∗ ∗ ∗ ∗ ∗ Et E = Tˆ Ei Tˆ E = (Tˆ Tˆ )(Ei E ). (46) ⊗ t ⊗ i ⊗ ⊗ i ∗ ˆ ˆ∗ where the Et Et and T T are the direct products 100 1 yielding a 4 row⊗ vector and⊗ a 4 4 matrix respectively. 10 0 1 Aˆ =   . (50) The E ﬁeld products yields the “coherence”× vector. 011− 0  0 i i 0   −  ∗ ExEx ∗ Substituting into Eq. 48 the inverse of Eq. 49, we get  ExEy  C = E E = ∗ . (47) ⊗ EyEx  ∗  EyE ∗ −1  y  St = Aˆ(Tˆ Tˆ )Aˆ Si. (51)   ⊗ Therefore So therefore in the Mueller matrix expression St = MSˆ i the Mueller matrix Mˆ is given by Mˆ = Aˆ(Tˆ ⊗ ˆ ˆ∗ Tˆ∗)Aˆ−1. Using the direct product expression for Tˆ Tˆ∗ Ct = (T T )Ci. (48) ⊗ ⊗ and the transformation Aˆ and its inverse one can express By inspection of Eq. 45 and Eq. 47 one can express the Mueller matrix Mˆ in terms of the components of the the Stokes vector in terms the coherency vector as Jones matrix (as deﬁned in Eq. 5) as

1 (AA∗ + BB∗ + CC∗ + DD∗) 1 (AA∗ BB∗ + CC∗ DD∗) Re(AB∗)+ Re(CD∗) Im(AB∗) Im(CD∗) 2 2 − − − −  1 (AA∗ + BB∗ CC∗ DD∗) 1 (AA∗ BB∗ CC∗ + DD∗) Re(AB∗) Re(CD∗) Im(AB∗)+ Im(CD∗)  2 2 . Re(AC∗)+−Re(BD−∗) Re−(AC∗) −Re(BD∗) Re(AD∗)+− Re(BC∗) −Im(AD∗) Im(BC∗)  ∗ ∗ ∗ − ∗ ∗ ∗ − ∗ − ∗   Im(AC )+ Im(CD ) Im(AC ) Im(CD ) Im(AD )+ Im(BC ) Re(AD ) Re(BC )  − −  (52)

The symmetry constraints on the Jones matrices dis- can still deduce quantitative and qualitative information cussed in Secs. III and IV can be directly applied to Eq. even when the underlying equations of motion are un- 52 and with a great deal of algebra give constraints on known. A number of explicit examples were given that the Mueller matrices. are of current relevance to the study of correlated broken symmetry states of matter. One important consequence of this formalism is the demonstration that Kerr rotation VIII. CONCLUSIONS must be absent in time-reversal symmetric chiral materials. In this work I discuss a Jones transfer matrix formalism for inferring the existence of exotic broken symmetry states of matter from their electrodynamic response. IX. ACKNOWLEDGMENTS This formalism is directly applicable to time-domain THz spectroscopy in transmission, but is more broadly ap- I would like to thank R. Valdes Aguilar, N. Drichko, plicable. I discussed the consequences of discrete bro- A. Fried, A. Kapitulnik, S. Kivelson, Y. Lubashevsky, C. ken symmetries on ordered states of matter including Morris, L. Pan, S. Raghu, O. Tchernyshyov, A. Turner, the presence and absence of reﬂections, rotations, inver- C. Varma, and V. Yakovenko for useful discussions on re- sion, rotation-reﬂections and time-reversal symmetries lated topics. This research was supported by the Gordon and the constraints they give on Jones matrices. These and Betty Moore Foundation through Grant GBMF2628 constraints typically appear in the form of an algebra and the DOE-BES through DE-FG02-08ER46544. relating matrix elements or overall constraints (transposition, unitarity, hermiticity, normality, etc.) on the form of matrix. As usual, the utility of symmetries is that one 13

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