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Physical Review A

Quantum theory of propagation of elliptically polarized light through a Kerr medium

G. S. Agarwal, Ravinder Rupchand Puri

Published Nov 1, 1989• 46 citations
Publisher

Abstract

We consider quantum-mechanically partially polarized light propagating through a Kerr-like medium. Using the usual form of the induced polarization P=A(E\ensuremath{\cdot}${\mathrm{E}}^{\mathrm{*}}$)E+B(E\ensuremath{\cdot}${\mathrm{E})\mathrm{E}}^{\mathrm{*}}$, the theory is formulated in terms of an effective Hamiltonian which is quartic in terms of the operators for two orthogonally polarized modes. Exact solutions in closed form for the Heisenberg equations of motion are obtained. These solutions are used to evaluate the physical behavior of various observables as the field propagates through a nonlinear medium. We also present explicit results for the time evolution of the input coherent and Fock states of the field. We show the generation of states that are macroscopic superposition of coherent states. We also find that if the input field is completely polarized, then due to quantum effects the output field becomes partially polarized. This is in contrast to the classical prediction and can have an important bearing on questions like topological phases of light propagating through a nonlinear medium. Numerical results for the energy in each mode, the correlation between two modes, and the higher-order correlations are presented. The input photon statistics is found to make a considerable difference in the dynamics.

DOI
10.1103/physreva.40.5179

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We consider quantum-mechanically partially polarized light propagating through a Kerr-like medium. Using the usual form of the induced polarization P=A(E\ensuremath{\cdot}E∗{\mathrm{E}}^{\mathrm{*}}E∗)E+B(E\ensuremath{\cdot}E)E∗{\mathrm{E})\mathrm{E}}^{\mathrm{*}}E)E∗, the theory is formulated in terms of an effective Hamiltonian which is quartic in terms of the operators for two orthogonally polarized modes. Exact solutions in closed form for the Heisenberg equations of motion are obtained. These solutions are used to evaluate the physical behavior of various observables as the field propagates through a nonlinear medium. We also present explicit results for the time evolution of the input coherent and Fock states of the field. We show the generation of states that are macroscopic superposition of coherent states. We also find that if the input field is completely polarized, then due to quantum effects the output field becomes partially polarized. This is in contrast to the classical prediction and can have an important bearing on questions like topological phases of light propagating through a nonlinear medium. Numerical results for the energy in each mode, the correlation between two modes, and the higher-order correlations are presented. The input photon statistics is found to make a considerable difference in the dynamics.
Phase properties of elliptically polarized light propagating through a nonlinear Kerr medium are considered within the framework of the Pegg-Barnett Hermitian phase formalism. The joint phase probability distribution function for the phases of two orthogonal modes describing elliptical polarization of the field is calculated and its evolution discussed and illustrated graphically. The marginal phase probability distribution for the individual phases are also calculated and discussed. Analytical formulae for phase expectation values and variances are derived for the individual phases as well as for the phase difference. It is shown that in the course of propagation the correlation between the phases of the two modes builds up. This correlation is responsible for lowering phase difference variance. The expressions for the sine and cosine functions and their variances of the individual phases as well as the phase difference are obtained and discussed. The effect of randomization of individual phases and the phase difference, which is a purely quantum effect, is shown to appear during propagation. The relation between phase randomization and degradation of the degree of polarization of the light is established.
The phase properties of two modes of elliptically polarized light propagating in a nonlinear Kerr medium with dissipation are studied. Exact analytical formulas for the joint phase probability distribution, the marginal phase distributions, and the individual mode phase operators, as well as the phase-difference operator expectation values and variances, the phase correlation function, and the expectation value and the variance of the phase-difference cosine, are obtained. Some of the results are illustrated graphically to show explicitly the effect of dissipation on the evolution of the phase properties of the field.
In this chapter we review a number of quantum features of optical fields associated with propagation of intense light through nonlinear, isotropic media. We begin with the classical description of field propagation, introducing the nonlinear polarization of the medium which enters as the source term into the approximate field equations obtained from the Maxwell equations in the slowly varying amplitude approximation. This establishes the classical background for further quantum considerations. Next, we quantize the field and construct the effective Hamiltonian from which we get equations of motion for the quantum fields. We discuss a number of quantum effects such as photon antibunching, squeezing, the formation of Schrodinger cats and kittens, changes in field polarization due to the quantum nature of the field, as well as the quantum description of the field phase. The characteristic feature of quantum evolution — the periodicity — is strongly affected by dissipation. We give exact analytical formulae describing the quantum evolution of the field, including dissipation. We have collected results illustrating various aspects of quantum evolution and we believe that this review, although far from complete, will be a useful source of information on the subject.
The problem of the formation of discrete superpositions of coherent states when elliptically polarized light is propagating through a nonlinear Kerr medium is considered, It is shown that superpositions with any number of components can be obtained if the evolution time is taken as a fraction MIN of the period, where M and N are mutually prime integers. Exact analytical formulae for finding the superposition coeffi- cients are given. It is shown that the coupling between the two circularly polarized components of the ellpitically polarized light caused by the asymmetry of the nonlinear properties of the medium can suppress the number of components in the superposition from N' to N, if the asymmetry parameter takes appropriate values. The phase distri- bution function P(.9,, e.) for the two-mode field is obtained according to the new Pegg-Barnett phase formalism. This function exhibits a well resolved, multi-peak struc- ture, clearly indicating the formation of the discrete superpositions of coherent states. Examples of the phase distribution function for several superposition states are illustrated graphiczlly. showing in a very spectacular way the formation of such superpositions.
We predict theoretically and demonstrate experimentally an ellipticity-dependent nonlinear magneto-optic rotation of elliptically polarized light propagating in a medium with atomic coherence. We show that this effect results from hexadecapole and higher-order moments of the atomic coherence, and is associated with an enhancement of Kerr and higher-order nonlinearities accompanied by suppression of the other linear and nonlinear susceptibility terms of the medium. These nonlinearities might be useful for quantum signal processing. In particular, we report an observation of enhancement of the polarization rotation of elliptically polarized light resonant with the ${5S}_{1/2}F=\stackrel{\ensuremath{\rightarrow}}{2}{5P}_{1/2}F=1$ transition of 87Rb.{}^{87}\mathrm{Rb}.87Rb.
The quantum theory of light propagation in a nonlinear Kerr medium is applied to calculate the Stokes parameters and their variances in the process of light propagation. Exact quantum formulae are derived for the expectation values of the Stokes operators and thus for the azimuth θ and ellipticity η of the beam. The role of quantum fluctuations in light polarization characteristics is discussed. The periodic behaviour of quantum evolution of the light polarization is revealed explicitly. It is shown that the degree of polarization is diminished at early stages of each period of the evolution but then reverts to its initial state of complete polarization at the end of the period. The variances of the Stokes parameters are also periodic and intensity-dependent; however, they never fall below their coherent state values.