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Journal of the Physical Society of Japan

Dynamics of Two-State Systems: Time-Varying Polarization Devices

Tiberiu S. Tudor

Published Jan 27, 2012• 5 citations
Publisher

Abstract

The problem of the time-varying polarization devices is analyzed in the general frame of the dynamics of the two-state systems. A vectorial and pure operatorial Pauli algebraic approach, conjugated with a Poincaré geometric one is adopted. The general equation of evolution of the density operator of the light polarization state interacting with a time-varying birefringent device is established this way and applied in analyzing the modification of the polarization and spectral structure of light by modulation.

DOI
10.1143/jpsj.81.024006

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The problem of the time-varying polarization devices is analyzed in the general frame of the dynamics of the two-state systems. A vectorial and pure operatorial Pauli algebraic approach, conjugated with a Poincaré geometric one is adopted. The general equation of evolution of the density operator of the light polarization state interacting with a time-varying birefringent device is established this way and applied in analyzing the modification of the polarization and spectral structure of light by modulation.
We consider the dynamical behavior of light in a cavity that contains birefringent elements with time-dependent settings. The evolution of the two-dimensional polarization vector can be described by an equation of the Schr\"odinger type if the intracavity elements change little on the time scale of the round-trip time. A coupled-mode analysis is given, which models the multimode dynamics of the optical system. We show that a description in terms of isolated two-level systems is justified if the optical band gap is small.
The quantum theory of polarizing light and the Stokes operators are applied to the study of the evolutions of polarization states of light in the processes of i nteraction with atoms in the case with or without Kerr medium. The non_classic b ehavior in the fluctuations of Stokes parameters for the polarizing light and th eir squeezing are investigated. The polarization ellipse, the degree of polariza tion of the field in quantum optics and the signal_to_noise ratio in the Stokes parameters are discussed. It is shown that the modulated oscillations appear in the evolutions of Stokes parameters and their fluctuations. These oscillations c ollapse and revive intermittently. The polarization ellipse does not vary in the interaction with atoms but the degree of polarization will oscillate. Kerr medi um changes these oscillations distinctly.
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.
The operatorial theory of optical polarization devices can provide some of the clearest examples in which von Neumann's model of orthogonal measurement is violated. This paper analyses the global and spectral properties of some two- and three-layer optical polarizers of non-orthogonal kind by calculating the dyadic expressions of their operators in a Dirac-algebraic form. These constitute some of the most expressive examples of non-Hermitian operators corresponding to generalized observables and it is expected that the theory of generalized quantum measurement will take advantage of the simplicity of their physical realization.
A unified treatment of the Pauli algebraic forms of the linear operators defined on a unitary linear space of two dimensions over the field of complex numbers C(1) is given. The Pauli expansions of the normal and nonnormal operators, unitary and Hermitian operators, orthogonal projectors, and symmetries are deduced in this frame. A geometrical interpretation of these Pauli algebraical results is given. With each operator, one can associate a generally complex vector, its Pauli axis. This is a natural generalization of the well-known Poincaré axis of some normal operators. A geometric criterion of distinction between the normal and nonnormal operators by means of this vector is established. The results are exemplified by the Pauli representations of the normal and nonnormal operators corresponding to some widespread composite polarization devices.