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Physical Review A

Analytic approximate expression for the spectral distribution of the emission from a slab of resonant two-level atoms prepared by an ultrashort δ pulse

Jamal T. Manassah

Published May 14, 2012• 6 citations
Publisher

Abstract

Using the slowly varying envelope approximation in space and time and the expression for the cooperative Lamb shift, I obtain an approximate analytic expression for the spectral distribution of the radiation from a slab of two atoms that was initially weakly excited by an ultrashort resonant pulse. The closed-form expression obtained reproduces very accurately the numerical exact results computed from the eigenmode analysis for the emission from the front face of the slab.

DOI
10.1103/physreva.85.055801

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Using the slowly varying envelope approximation in space and time and the expression for the cooperative Lamb shift, I obtain an approximate analytic expression for the spectral distribution of the radiation from a slab of two atoms that was initially weakly excited by an ultrashort resonant pulse. The closed-form expression obtained reproduces very accurately the numerical exact results computed from the eigenmode analysis for the emission from the front face of the slab.
Using eigenmode analysis I calculate in the linear regime the emission characteristics from a slab of two-level resonant atoms excited by a delta-pulse propagating normal to the surface of the slab. I show that the qualitative features of the emitted spectra in the forward and backward directions to be different from each other; I also find the effects of the total linewidth of the resonance line and of the deviation in the spatial periodicity of the initial atomic polarization from k 0, where k 0 = ω0/c and ω0 is the two level resonant frequency, on the spectra. The dominant spatial mode of the system is identified as the key parameter for determining the forward emission spectral width.
This Brief Report gives the spectral distribution of the supercontinuum generated from a system of initially partially excited two-level atoms. The expansion in the eigenfunctions of the one-dimensional Li\'enard-Wiechert kernel which was successful in obtaining previously the results in the linear regime is generalized to the nonlinear regime. The detailed spectral distribution of the emitted radiation is computed without invoking either the phenomenological simplified physical model for the active medium or the mathematical approximations commonly used in treating propagation in absorbing media. The numerical solutions presented are those of the complete set of the coupled Maxwell-Bloch equations.
Interaction with a multimode quantized radiation field of an assembly of NNN two-level atoms contained in a small volume is considered for the special cases of (i) highly excited, (ii) weakly excited, and (iii) superradiant atomic assemblies, and solutions of equations of motion are obtained in closed form. For (i) and (ii) we show that, if the atomic assembly is in the Dicke state ∣r,m>|r,m>∣r,m>, the spontaneously emitted radiation is Lorentzian, and its spectral width and the numerical value of the Lamb shift are r+∣m∣r+|m|r+∣m∣ times that for a single excited atom. For (i) the sign of Lamb shift is opposite to that for (ii) and for a single excited atom. Change in the frequency shift gives chirping in the pulse radiation emitted spontaneously by an assembly of atoms. We obtain the reduced density operator for a single mode and for the complete spontaneous emitted radiation, and discuss its characteristic function, weight function in diagonal representation, coherence functions, and normalized variance. We find that the emitted radiation is chaotic if all atoms are excited or if the assembly is weakly excited and is in thermal equilibrium. For a superradiant assembly, the radiation is coherent; it is only amplitude coherent if the initial atomic density operator is nondiagonal. We also study stimulated emission and scattering for the cases (i) and (ii). For case (iii) these phenomena do not take place.
Previous numerical work on the spatiotemporal distributions of the atomic polarization and of the difference in levels population of a one-dimensional system of two-level atoms initially inverted strongly suggested that the dynamics of the slab atomic ensemble in the linear regime of superradiance may best be treated by an eigenmode analysis. This method proved effective in obtaining both the time dependence of the atomic polarization and the spectral distribution of the emitted radiation from the slab in a number of physically interesting problems in the linear regime. In the present paper, I use the eigenfunction expansion technique to reduce the Maxwell-Bloch system of partial differential equations into a system of coupled first-order ordinary differential equations whose solutions clearly detail the contributions of the different modes to the system dynamics in the nonlinear regime of superradiant emission. I compute also the delay in time of the peak of the superradiant burst and its magnitude as functions of the ratio of the longitudinal decay rate to the transverse decay rate of the system.
Propagation and amplification of an ultrashort laser pulse in a two-level medium is numerically investigated with the use of the finite-difference time-domain (FDTD) procedure simultaneously solving Maxwell and Schrodinger equations with no assumptions characteristic of slowly varying envelope approximation. Comparison of the results of FDTD numerical simulations with the predictions of the pulse area theorem has demonstrated that the developed numerical procedure provides an adequate description of pulse evolution in a two-level medium. Amplification of ultrashort light pulses in a two-level medium is analyzed. Two methods of improving the gain of ultrashort pulses in a two-level medium by modulating the spatial distribution of dipole moments of resonant transitions and using initially chirped frequency-detuned pulses are explored.