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Open Systems & Information Dynamics

Density Matrix Description of Fast and Slow Light Propagation in Sodium Vapour

Abu Mohamed Alhasan

Published Aug 13, 2009• 4 citations
Publisher

Abstract

The dual-colour excitation for D1 transition is analysed taking into account the hyperfine structure of sodium atomic vapour. The advancement of the probe and restoring fields are calculated through different time average schemes. The first scheme depends on the arrival time of the pulse. The other one depends explicitly on the arrival time of photons. The evolution of atomic states is described by the Liouville-von Neumann equation for the density matrix of the dressed atom. The spatiotemporal behaviour of radiation fields is described by the reduced Maxwell's field equations. We monitor the populations of atomic levels and calculate the scattered light intensity. The mean arrival time of photons is suitably defined in terms of the scattered light intensity and the total scattered light at a given distance.

DOI
10.1142/s1230161209000098

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The dual-colour excitation for D1 transition is analysed taking into account the hyperfine structure of sodium atomic vapour. The advancement of the probe and restoring fields are calculated through different time average schemes. The first scheme depends on the arrival time of the pulse. The other one depends explicitly on the arrival time of photons. The evolution of atomic states is described by the Liouville-von Neumann equation for the density matrix of the dressed atom. The spatiotemporal behaviour of radiation fields is described by the reduced Maxwell's field equations. We monitor the populations of atomic levels and calculate the scattered light intensity. The mean arrival time of photons is suitably defined in terms of the scattered light intensity and the total scattered light at a given distance.
A quantum-electrodynamic (QED) description of the resonant interaction of monochromatic light with the 32{3}^{2}32S1/2{\mathit{S}}_{1/2}S1/2​(F'=2)\ensuremath{\rightarrow}32{3}^{2}32P3/2{\mathit{P}}_{3/2}P3/2​(F=3,2,1) hyperfine transitions of the sodium D2{D}_{2}D2​ line is formulated in terms of the Heisenberg atomic operator. Off-diagonal state coherences and all relaxation terms are included. It is found that the equations for the populations, optical coherences, and certain state coherences form a closed subset of the total system of equations. This subset is small enough to be computed numerically. The results of calculations with this model are compared with three previously developed semiclassical descriptions using the density operator. One of the semiclassical models was developed to describe the interaction of weak light with the atomic transition while another is suitable in the case of high-intensity light. For both \ensuremath{\pi} and \ensuremath{\sigma} excitation it is shown that the QED calculated values for the time-averaged, excited-level population probability converge to those of each of the semiclassical models in the appropriate limit of light intensity. For homogeneous broadening, results show that the optimum light intensity to obtain the largest fraction of atoms in the 32{3}^{2}32P3/2{P}_{3/2}P3/2​ level after transversing a 1-mm-diam laser beam is 1.15 mW/mm2{\mathrm{mm}}^{2}mm2 for \ensuremath{\pi} excitation and 1.43 mW/mm2{\mathrm{mm}}^{2}mm2 for \ensuremath{\sigma} excitation.
The atomic collision parameters ?, ?, ?, ?, P, |P| and |�| were determined from measurements of the pseudo Stokes parameters for superelastic scattering of 20 eV incident electrons at angles of 10� and 15� from the 32P312 excited state of sodium. Two depolarising factors K and K' for the pseudo Stokes parameters are defined for it and a laser excitation respectively to account for the depolarising influence due to optical pumping in the hyperfine structure of the 32P3t2 excited state. A theoretical model of the laser excitation of the sodium transition is developed to determine K and K'. The theory developed is a quantum -electrodynamic description of the resonant interaction of monochromatic light with the 32S in(F'=2) <-> 32P3a(F=3,2,1) hyperfine transitions of the sodium D2 line formualated in terms of the Heisenberg atomic operator. Off diagonal state coherences and all relaxation terms are included. It is found that the equations for the populations, optical coherences and certain state coherences form a closed subset of the total system of equations. This subset is sufficiently small to enable numerical computation. As well as K and K', the model is used to calculate the probability of occupation of the various atomic states and the polarisation of the sideways scattered fluorescence from the laser excited atoms. The results of this model are compared with the predictions of four other models, a rate equation model which includes only the F=3 excited state, a semiclassical model based on the density operator, a low light intensity approximation to the semiclassical model and a high intensity model which uses a transformed basis for the states. For both it and a laser excitation it is shown that the QED calculated values converge to the values predicted by the high and low intensity approximations in the appropriate limits.
The semi-classical method for calculating the observable properties of light scattered from an optically thin atomic vapour is placed in the context of the well known input-output relations for random signals transmitted through a linear system. The autocorrelation of the scattered field amplitude, the statistic from which the mean intensity spectral density and intensity fluctuations may be deduced, is expressed in terms of the autocorrelation of the incident optical field and the autocorrelation of the scattering system; these latter are quantities that can be readily measured or calculated. An appendix contains a classical and a semi-classical calculation of the scattering system autocorrelation function for a vapour of atoms optically excited from a J=0 level under the conditions of a level crossing experiment. In the course of this calculation the relationship between the matrix element of the electric dipole moment operator and the corresponding classical quantity is carefully examined.
This paper describes a comparable study on the influence of frequency-sweep on the discrete and continuous phase distributions associated with pulse excitations for a double-lambda atomic system in alkali-metal vapours with a hyperfine structure. The excitation dynamics provided a different scheme of sigmoidal types, and the optical pulses were assumed to be Gaussians. We focused on the set-up of electromagnetically induced transparency (EIT). The phases of optical fields show similarities to discrete square wave distributions influenced by ramping. The results showed significant control of discrete phase distributions and temporal ramping by the implementation of sigmoidal membership functions. The relevant equations are the reduced Maxwell equations for the radiation fields, and the density matrix equation in the Liouville space governs the time evolution.