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arXiv:1210.7107 · Journal of the Optical Society of America B

Scalar Wigner theory for polarized light in nonlinear Kerr media

Tobias Hansson, Erik Jakob Wallin, Gert Brodin, Mattias Marklund

Published May 30, 2013• 7 citations
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Abstract

A scalar Wigner distribution function for describing polarized light is proposed in analogy with the treatment of spin variables in quantum kinetic theory. The formalism is applied to the propagation of circularly polarized light in nonlinear Kerr media, and an extended phase-space evolution equation is derived along with invariant quantities. The formalism is additionally used to analyze the modulational instability.

DOI
10.1364/josab.30.001765

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A scalar Wigner distribution function for describing polarized light is proposed in analogy with the treatment of spin variables in quantum kinetic theory. The formalism is applied to the propagation of circularly polarized light in nonlinear Kerr media, and an extended phase-space evolution equation is derived along with invariant quantities. The formalism is additionally used to analyze the modulational instability.
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We construct the Wigner distribution function for a coupled boson-fermion system. This function is regarded as a certain ``superfield'' defined on the extended phase space, the points of which are labeled by the anticommuting canonical c variables together with the ordinary ones. In this approach, the variables of a fermion and a boson are treated on a completely equal footing, and the phase-space representation is fully realized. We apply the formalism to the kinetic theory of the optical Dicke model with a two-level atom, and show how systematically a set of the generalized Fokker-Planck equations, which describes the effective dynamics of the radiation field, is derived from a single superfieldlike equation.
The analogy between polarization optics effects and the dynamics of a spin in a magnetic field is analyzed based on the fact that the Bloch equation for a spin in a magnetic field is formally identical to the evolution equation for the quasi-spin vector of a light propagating in an optically anisotropic medium. Among the effects discussed within this framework are light polarization beats in a birefringent medium; optical activity suppression in the media with linear birefringence; suppression of linear birefringence in the media with spatially modulated gyration; 'adiabatic' transformation of light polarization in media of smoothly varying anisotropy, and two- and three-pulse polarization echoes.
We present a canonical quantum theory of radiation in nonlinear media taking into account the effects of linear dispersion and absorption in a consistent way. We start from a microscopic model and apply recently developed concepts of the quantization of radiation in dispersive and absorbing linear media and extend the theory in order to include nonlinear optical processes. We derive propagation equations in space and time for the quantized radiation field, with special emphasis on the propagation of quantum-light pulses in Kerr media. In particular, the method enables us to derive systematically the noise sources that arise naturally in the nonlinear terms.
Starting from the Pauli Hamiltonian operator, we derive a scalar quantum kinetic equations for spin-1/2 systems. Here the regular Wigner two-state matrix is replaced by a scalar distribution function in extended phase space. Apart from being a formulation of principal interest, such scalar quantum kinetic equation makes the comparison to classical kinetic theory straightforward, and lends itself naturally to currently available numerical Vlasov and Boltzmann schemes. Moreover, while the quasi-distribution is a Wigner function in regular phase space, it is given by a Q-function in spin space. As such, nonlinear and dynamical quantum plasma problems are readily handled. Moreover, the issue of gauge invariance is treated. Applications (e.g. ultra-dense laser compressed targets and their diagnostics), possible extensions, and future improvements of the presented quantum statistical model are discussed.