Vectorial Pauli algebraic approach in polarization optics. II. Interaction of light with the canonical polarization devices

A theoretical approach to the interaction between polarized light and polarization devices, based on the vectorial and pure operatorial form of the Pauli algebra, is presented. In the first part of the paper we have established the vectorial Pauli-algebraic forms of the operators corresponding to various polarization devices and states of light polarization. In this second part we give the vectorial Pauli-algebraic treatment of the interaction between the canonical polarization devices and the various forms of light polarization. Unlike the standard (Jones and Mueller) approaches, this formalism does not appeal to any matrix representation of the involved operators. This approach establishes a bridge between the Hilbert space of the density operators of the polarization states and the Poincaré space of their geometric representations and gives a rigorous justification of the handling of the interactions between the polarization states and polarization systems on the Poincaré sphere and in the Poincaré ball. In such an approach, unlike the standard ones, the three relevant quantities that characterize the interaction - the gain, the Poincaré vector of the outgoing light and its degree of polarization - result straightforwardly, in block. A generalized form of Malus’ law, for any dichroic device and partially polarized light is also obtained this way.     

Mueller Matrix Analysis of PDL Components

A detailed Mueller-Stokes analysis of an arbitrary elliptical dichroic polarizer is presented. Explicit expressions for the Jones matrix, the Stokes parameters, and the Mueller matrix for a distributed and localized dichroic polarizer acting as a polarization-dependent loss (PDL) element are derived. Application to wavelength-dependent PDL elements is discussed. The Poincare sphere representation and the application of the Mueller matrices for the study of randomly oriented concatenated PDL elements and PDL vector measurement are addressed.     

Optical Mueller matrices in terms of geometric algebra

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl_3,0 algebra and using isomorphism between Cl_3,0 and even Cl_3,1⁺ subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl_3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.     

Classification of unconventional electron-hole condensates

Heavy Fermion metals with their very anisotropic quasiparticle states may support unconventional electron-hole (Peierls) pairing in addition to unconventional two electron (Cooper) pairs in the superconducting phase. For two different nesting Fermi surface models the possible types of electron hole condensates are classified according to the symmetries of their order parameters. This is performed within a continuum representation for the electronic states near the van Hove saddle point singularities. The quasiparticle bands and the unitary transformation to Bloch states in the condensed phase are derived for the two Fermi surface models with one and two independent nesting vectors respectively. Emphasis is put on the investigation of electron-hole condensed phases with 2Q-modulated structure. It is shown that in the continuum approximation the gap equations are all equivalent and the critical field curve is calculated in the rigid band model.     

Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,R) and GL(n,R) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.     

A Real Quaternion Spherical Ensemble of Random Matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form Y=A ^?1 B, where A and B are independent N×N matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue joint probability density function and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices β=1,2,4. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure Y=A ^?1 B, where A and B are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.     

Three-particle poincaré states and SU(6) × SU(3) as a classification group for baryons

A complete set of democratic quantum numbers is introduced to classify the states of an irreducible unitary representation (IUR) of the Poincaré group obtained from the decomposition of the direct product of three IUR. Such states are identified with the baryon states constituted of three free relativistic quarks. The transformation from current to constituent quarks is then easily reobtained. Moreover, the group SU(6) × SU(3) appears naturally as a collinear classification group for baryons. Results similar to those of the symmetric harmonic oscillator quark model are obtained.     

Mueller-Stokes polarimetric characterization of transmissive liquid crystal spatial light modulator

In the twisted nematic liquid crystal spatial light modulators (TN-LCSLM), distortion of uniform twist and decrease in tilt angle of liquid crystal molecules on application of an electric field lead to amplitude and phase modulations of the transmitted or reflected wavefront, respectively. The amplitude and phase modulation characterization of TN-LCSLM using Jones calculi is simple and extensively used but does not give any information about important polarimetric parameters such as diattenuation and depolarizance. On the other hand, the characterization using Mueller calculi provides all information in terms of polarimetric properties such as diattenuation, retardance (birefringence) and depolarization. In this paper, polarimetric properties of the transmissive TN-LCSLM (HOLOEYE LC2002) are characterized measuring 17 different Mueller matrices at different addressed gray scale through Mueller Matrix Imaging Polarimeter (MMIP) at 530 nm wavelength. Lu-Chipman polar decomposition for Mueller matrix is utilized to separate out three independent Mueller matrices for diattenuation, depolarization and retardance as a function of addressed gray scale. Further, Mueller-Stokes combined formulation is used to examine the effect of depolarization present in the TN-LCSLM on six different states of polarization and evaluation of eigenpolarization states for the TN-LCSLM has been presented.     

Computing scalar products via a two-terminal quantum transmission line

The scalar product of two vectors with K real components can be computed using two quantum channels, that is, information transmission lines in the form of spin-1/2 XX chains. Each channel has its own K-qubit sender and both channels share a single two-qubit receiver. The K elements of each vector are encoded in the pure single-excitation initial states of the senders. After time evolution, a bi-linear combination of these elements appears in the only matrix element of the second-order coherence matrix of the receiver state. An appropriate local unitary transformation of the extended receiver turns this combination into a renormalized version of the scalar product of the original vectors. The squared absolute value of this scaled scalar product is the intensity of the second-order coherence which consequently can be measured, for instance, employing multiple-quantum NMR. The unitary transformation generating the scalar product of two-element vectors is presented as an example.     

Vector parameterization of the Lorentz group transformations and polar decomposition of Mueller matrices

It is shown that the representation of the coherence matrix (the polarization density matrix) of beams of electromagnetic waves as a biquaternion corresponding to the four-vector of a pseudo-Euclidean space whose components are the intensity and the Stokes parameters provides a possibility of introducing the group transformations of these quantities isomorphic to SO(3.1) group. These transformations are a subset of the set of Mueller polarization matrices which, generally speaking, form a semigroup. The reduction of the semigroup of Mueller matrices to the group of transformations opens the possibility to use the vector parameterization of SO(3.1) group for interpretation of the polar decomposition of Mueller matrices. In particular, in this approach, the elements of the Mueller matrices corresponding to phase elements and polarizers turn out to be most simply and naturally related to their eigenpolarizations.     

Depolarization by aerosols: Entropy of the Amsterdam light scattering database

In this paper we apply an entropy analysis to measured scattering matrices from the Amsterdam light scattering database. We select examples of mineral aerosols from the database and use them to demonstrate differences in polarization behavior between the particle clouds using a new coherency matrix formulation. These differences are further investigated by analyzing the polarized component of the matrices via two new eigenvector parameters, which can be mapped conveniently onto the surface of a sphere, analogous to the Poincaré sphere used for wave states. We conclude by considering the potential for discriminating different aerosols on the basis of their entropy/eigenvector signatures by solving the contrast optimization problem for clouds with different scattering matrices by using a novel generalized coherency eigenvalue formulation.     

Poincaré covariance and κ-Minkowski spacetime

A fully Poincaré covariant model is constructed as an extension of the κ-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised à la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of “Poincaré covariance”.     

Fidelity estimation between two finite ensembles of unknown pure equatorial qubit states

Suppose, we are given two finite ensembles of pure qubit states, so that the qubits in each ensemble are prepared in identical (but unknown for us) states lying on the equator of the Bloch sphere. What is the best strategy to estimate fidelity between these two finite ensembles of qubit states? We discuss three possible strategies for the fidelity estimation. We show that the best strategy includes two stages: a specific unitary transformation on two ensembles and state estimation of the output states of this transformation.     

An Algebraic Characterization of Vacuum States in Minkowski Space. III. Reflection Maps

Employing the algebraic framework of local quantum physics, vacuum states in Minkowski space are distinguished by a property of geometric modular action. This property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper Poincaré group which acts covariantly on the net and leaves the state invariant. The present results and methods substantially improve upon previous work. In particular, the continuity properties of the representation are shown to be a consequence of the net structure, and surmised cohomological problems in the construction of the representation are resolved by demonstrating that, for the Poincaré group, continuous reflection maps are restrictions of continuous homomorphisms.     

Fields on the Poincaré Group: Arbitrary Spin Description and Relativistic Wave Equations

In this paper, starting from a pure group-theoretic point of view, we develop an approach to describing particles with different spins in the framework of a theory of scalar fields on the Poincaré group. Such fields can be considered as generating functions for conventional spin-tensor fields. The case of two, three, and four dimensions are elaborated in detail. Discrete transformations C, P, T are defined for the scalar fields as automorphisms of the Poincaré group. We classify the scalar functions, and obtain relativistic wave equations for particles with definite spin and mass. There exist two different types of scalar functions (which describe the same mass and spin), one related to a finite-dimensional nonunitary representation and the other to an infinite-dimensional unitary representation of the Lorentz subgroup. This allows us to derive both usual finite-component wave equations for spin-tensor fields and positive-energy, infinite-component wave equations.     

单光分束器的量子变换算符研究

针对理想情况下单光分束器,得到了其对应的变换矩阵.用相干态表象和有序算符内积分技术,给出其对应的幺正变换算符.纠缠态的实验制备方案,说明单光分束器在量子信息源研究中具有十分重要的作用.     

Applications of the Poincaré sphere in polarization metrology

The Poincaré sphere representation of polarization states is used to derive two auxiliary equations for phase retardance measurements. These equations, in addition to another two previously derived equations, allow for extending the range of validity of a model for calibrating phase plates in pairs. In another application, the sphere is used to explain a new method for identifying the principal axes of two birefringent phase plates during their calibration.     

Vectorial Pauli algebraic approach in polarization optics. I. Device and state operators

This paper inscribes on the line of the efforts (sketched in the Introduction) in elaborating theoretical approaches alternative to the traditional Jones and Mueller matrix calculi in polarization optics. The more abstract, compact and elevated forms of linear algebra are not fully exploited yet in the polarization optics. A vectorial and pure operatorial Pauli algebraic approach to the interaction between the polarized light and the polarization optical systems is given. This is the most compact, adequate and elegant calculus corresponding to the well-known geometric handling of the polarization states and their interaction with the polarization devices on the Poincaré sphere. In this first paper, we deduce the Pauli algebraic vectorial forms of the operators corresponding to the orthogonal and nonorthogonal polarization devices and to all the states of light polarization. In the next paper we shall give the vectorial Pauli algebraic analysis of the interaction between the whole hierarchy of these devices and the various forms of polarized light.     

基于穆勒矩阵的模拟溢油样品荧光偏振特性研究

为了对模拟溢油样品在不同偏振态激发下诱导荧光的偏振特性进行研究,借鉴了穆勒矩阵椭偏仪的原理和结构,搭建了基于旋转波片原理的模拟溢油样品激光诱导荧光椭偏实验装置。通过特征值校准方法对该装置进行校准,获得了宽波段下偏振状态调制矩阵W(λ)和偏振状态分析矩阵A(λ)的确切调制状态,并基于荧光光谱强度矩阵Flu(λ)分别建立了轻、中、重质原油样品和柴油样品的荧光穆勒矩阵。通过极化分解方法对荧光穆勒矩阵进行分解后发现,不同样品荧光光谱的退偏振性质差异十分显著。柴油样品荧光穆勒矩阵的退偏振系数Δ(λ)没有明显的波长响应性,在荧光光谱范围内始终保持较高的退偏值,而三种原油样品的退偏系数Δ(λ)则随波长增大逐渐上升,其中,中质原油样品退偏系数随波长的变化幅度小于重质样品,超过轻质样品;就不同样品的退偏值来看,轻质原油样品最高,重质样品最低,中质原油样品介于二者之间,柴油样品的荧光退偏值略低于轻质原油样品,介于轻质和中质原油样品之间。将基于荧光穆勒矩阵极化分解后的结果与线偏振激发下样品荧光光谱的正交偏振实验结果进行对比,发现两种实验方法获得的退偏系数具有较高的吻合程度。实验还发现,四种模拟溢油样品荧光穆勒矩阵所包含的双向衰减和相位延迟性质都很微弱,不具有明显的差异。