Polarization skyrmions are usually characterized through a unit Stokes vector field, which defines a map from the transverse plane to the Poincaré (Bloch) sphere and allows one to assign a skyrmion number under appropriate boundary conditions. This description assumes complete polarization. In many optical situations, however, the local polarization state is partially polarized and is therefore described by a spectral polarization matrix rather than by a single Jones vector. We formulate a density-matrix-based effective extension of the skyrmion-number construction for statistically stationary, quasi-monochromatic paraxial fields in terms of the normalized spectral polarization matrix. Its eigenvectors define two orthogonal polarization eigenstates, corresponding to antipodal points on the sphere, while the eigenvalue imbalance is fixed by the local degree of polarization \(P({\bf r})\). Averaging the opposite skyrmion-density contributions of these two eigenpolarizations leads to an effective skyrmion number in which the usual pure-state density is weighted by \(P({\bf r})\). The resulting quantity reduces to the conventional skyrmion number for fully polarized fields and vanishes in locally unpolarized regions. In general it is not an integer topological invariant; rather, it is an integrated effective measure, constructed from local Stokes parameters, of the skyrmionic content associated with the polarized part of a mixed optical field. We illustrate the construction with an analytic Gaussian-beam example in which a degree-one polarization texture is combined with a radially varying degree of polarization.
Polarization skyrmions are usually characterized through a unit Stokes vector field, which defines a map from the transverse plane to the Poincaré (Bloch) sphere and allows one to assign a skyrmion number under appropriate boundary conditions. This description assumes complete polarization. In many optical situations, however, the local polarization state is partially polarized and is therefore described by a spectral polarization matrix rather than by a single Jones vector. We formulate a density-matrix-based effective extension of the skyrmion-number construction for statistically stationary, quasi-monochromatic paraxial fields in terms of the normalized spectral polarization matrix. Its eigenvectors define two orthogonal polarization eigenstates, corresponding to antipodal points on the sphere, while the eigenvalue imbalance is fixed by the local degree of polarization (P({\bf r})). Averaging the opposite skyrmion-density contributions of these two eigenpolarizations leads to an effective skyrmion number in which the usual pure-state density is weighted by (P({\bf r})). The resulting quantity reduces to the conventional skyrmion number for fully polarized fields and vanishes in locally unpolarized regions. In general it is not an integer topological invariant; rather, it is an integrated effective measure, constructed from local Stokes parameters, of the skyrmionic content associated with the polarized part of a mixed optical field. We illustrate the construction with an analytic Gaussian-beam example in which a degree-one polarization texture is combined with a radially varying degree of polarization.
Topological quasiparticles of light, such as classical and quantum optical skyrmions, have so far relied on fully coherent or pure quantum states whose topology is encoded in the entanglement between polarization and two-dimensional spatial modes. Here we show that skyrmionic topology can emerge directly within the density matrix of a mixed quantum state. We introduce a framework in which a coherence-Stokes vector defines a topological texture over the density matrix, enabling the realization of quantum skyrmions using only a pseudospin and a real or synthetic one-dimensional space of modes. For a single photon, the density matrix is analogous to the coherence matrix of classical light, and can be encoded using partially coherent electromagnetic fields. We further analyze the topological texture encoded in a bipartite entangled photon pair, showing how skyrmions arise not only in the full bi-photon system, but also in its reduced subspaces with any pseudospin-mode combination simultaneously. We then explore the robustness of such skyrmions to environmental noise, and discover their persistence in mixed-quantum states of multiple photons. Finally, we propose a feasible experimental route to generate and measure such skyrmions using integrated photonic networks, and suggest avenues for similar implementations in other quantum systems. Our work paves the way for robustly encoding classical information on partially coherent light or on mixed quantum states and for encoding topological charges on many-body quantum systems.
Quantum optics entails polarization properties that cannot be fully described by the classical Stokes parameters. In this work, we characterize the polarization of classical as well as quantum fields by means of a probability distribution on the Poincar\'e sphere. This serves to define the degree of polarization of a field state as the distance between the corresponding polarization distribution and the uniform distribution representing unpolarized light. We apply this definition to relevant quantum field states such as SU(2) coherent, squeezed, number, and phase states.
⟨ ⟩ The qang project describes qubits by measurement—level quantities, the qg values qgP = P [1]. This short note states two classical results in those units, implemented and checked in the qang library (pip install qang, version 0.6.0). First, a polarization state of light is a qubit: the normalized Stokes parameters are exactly the qg values, the degree of polarization P is the length of the qg vector, and a lossless Mueller matrix is the qg gate rule qg′ = qgU†PU . Malus's law becomes I = I0(1 + qgZ cos 2θ + qgX sin 2θ)/2 for any input state. Second, the — geometric phase of a closed loop on the Bloch (Poincaré) sphere is minus half the enclosed solid angle, which is Stokes' theorem for the Berry curvature. For a loop at constant qgZ the phase is (qgZ 1)/2 of a turn: it is fixed by qgZ alone. Nothing here is new physics. The contribution is the translation into qang's units and a set of 26 numerical checks, with the phase computed three independent ways.
The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.
Abstract In this paper the authors discuss an alternative way for reconstructing one-photon mixed states of a partially polarized optical field. The task is to represent the probability density distribution describing these kind of states with the Stokes parameters which also characterize the effective state of polarization. These parameters can be measured by means of the degree of polarization with an experimental setup containing a rotating linear polarizer and a circular polarizer. A thought experiment is presented which assumes that the measurement is undertaken on an analyzed beam coupled with a reference beam containing photons polarized in a well-known way. The method discussed in the paper is an alternative for the most commonly used quantum tomography approach.
We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.