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A Theoretical Development of the Optical Zeeman Effect

M. W. Evans

Published Nov 1, 1990• 0 citations
OpenAlex

Abstract

A theory of the optical Zeeman effect is developed in terms of the angular polarisability, **i, a fundamental property of both atoms and molecules. It is shown that **i is negative to motion reversal T and parity inversion P, and forms an interaction hamiltonian with the T negative, P positive conjugate product pi(i) of a circularly polarised pump laser. This activates the T negative **i, which is the axial vector equivalent of the imaginary part of the atomic or molecular electronic polarisability, an antisymmetric polar tensor. The quantity pi(i) is proportional to the square of the electric field strength of the laser, and disappears if the latter is not circularly polarised. This distinguishes the optical Zeeman effect from the experimentally measured optical Autler Townes effect, which is mediated by the symmetric, T positive, real part of the dynamic electronic polarisability. The optical Zeeman effect is accompanied by circular birefringence and the bulk magnetisation of the inverse Faraday effect.

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A theory of the optical Zeeman effect is developed in terms of the angular polarisability, **i, a fundamental property of both atoms and molecules. It is shown that **i is negative to motion reversal T and parity inversion P, and forms an interaction hamiltonian with the T negative, P positive conjugate product pi(i) of a circularly polarised pump laser. This activates the T negative **i, which is the axial vector equivalent of the imaginary part of the atomic or molecular electronic polarisability, an antisymmetric polar tensor. The quantity pi(i) is proportional to the square of the electric field strength of the laser, and disappears if the latter is not circularly polarised. This distinguishes the optical Zeeman effect from the experimentally measured optical Autler Townes effect, which is mediated by the symmetric, T positive, real part of the dynamic electronic polarisability. The optical Zeeman effect is accompanied by circular birefringence and the bulk magnetisation of the inverse Faraday effect.
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 general theory of a Zeeman laser given in part I is used to analyze laser operation in axial and transverse magnetic fields. The analysis is illustrated by graphs of intensities and beat frequencies versus cavity detuning and magnetic field strength. Among the topics discussed are frequency-locking effects due to cavity anisotropy, the dependence of competition between polarization modes on magnetic field strength for various atomic angular momenta, and the measurement of atomic decay rates and ggg values using a laser in a magnetic field.
The theory of quantum electrodynamics is employed in the description of linear and \nnonlinear optical effects. We study the effects of using a two energy level \napproximation in simplifying expressions obtained from perturbation theory, \nequivalent to truncating the completeness relation. However, applying a two-level \nmodel with a lack of regard for its domain of validity may deliver misleading results. \nA new theorem on the expectation values of analytical operator functions imposes \nadditional constraints on any atom or molecule modelled as a two-level system. We \nintroduce measures designed to indicate occasions when the two-level approximation \nmay be valid. \nAnalysis of the optical angular momentum operator delivers a division into spin \nand orbital parts satisfying electric-magnetic democracy, and determine a new \ncompartmentalisation of the optical angular momentum. An analysis is performed on \nthe recently rediscovered optical chirality, and its corresponding flux, delivering \nresults proportional to the helicity and spin angular momentum in monochromatic \nbeams. A new polarisation basis is introduced to determine the maximum values that \nan infinite family of optical helicity- and spin- type measures may take, and disproves \nrecent claims of ‘superchiral light’. A theoretical description of recent experiments \nrelate helicity- and spin- type measures to the circular differential response of \nmolecules, and show that nodal enhancements to circular dichroism relate only to \nphoton number-phase uncertainty relation and do not signify ‘superchiral’ regions. \nThe six-wave mixing of optical vortex input, in nonlinear media, demonstrates the \nquantum entanglement of pairs of optical vortex modes. The probability for each \npossible output pair displays a combinatorial weighting, associated with Pascal’s \ntriangle. \nA quantum electrodynamic analysis of the effect of a second body on absorption \ncan be extended by integrating over all possible positions of the mediator molecules, \nmodelling a continuous medium. This provides links with both the molecular and \nbulk properties of materials.
Birefringence and dichroism in atomic systems are active areas of research. Circular dichroism is typically observed in chiral atoms, molecules (such as DNA), and optically asymmetric materials. These phenomena enable polarization-based selectivity and high-precision measurements with applications in optical systems, biomedical diagnostics, and advanced photonics. Meanwhile, negative-refractive-index materials offer exotic applications in optics and photonics, including superresolution imaging, invisibility cloaking, and subwavelength imaging. Here we identify a route to achieving birefringence and dichroism in a left-handed system. We investigate the optical activity of an atomic left-handed medium, analyzing the interaction of left- and right-circularly polarized light with different atomic states while incorporating local field corrections. Notably, our results demonstrate simultaneous left-handedness for both polarization states. Furthermore, we find that electric birefringence and dichroism are significantly enhanced with increasing control field strength, while magnetic birefringence and dichroism can be fully controlled via strong laser coupling. These findings have promising implications in fields at the interface of left-handedness and dichroism.
We demonstrate that when laser light is circularly polarized and properly detuned, the ac Stark shift of an alkali-metal atom in its ground state takes the form of a pure Zeeman shift. The condition is satisfied when the laser frequency is between the D1D1D1 and D2D2D2 transition frequencies, with the size of the detuning from the D2D2D2 resonance twice that from the D1D1D1 resonance. The direction of the effective magnetic field is along the laser propagation axis, and its magnitude is proportional to the vector polarizability of the atom and to the laser intensity. We use stimulated Raman spectroscopy on an optically pumped slow atomic beam from a magneto-optical trap to measure the vector polarizability and saturated absorption spectroscopy to show that the scalar polarizability vanishes at the particular detuning.