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Journal of Experimental and Theoretical Physics

Transient scattering of resonance radiation in a two-level system

A. A. Panteleev, V. Roerich, A. N. Starostin

Published Feb 1, 2003• 5 citations
Publisher

Abstract

A new method is developed for the description of transfer and scattering of a time-dependent laser pulse, which is based on a hierarchy of Bogoliubov-Born-Green-Kirkwood-Yvone (BBGKY) equations for reduced density matrices. The systems of equations are obtained for the atomic and photon density matrices and the correlation operator in the second-order Born approximation. The system of equations describes stimulated and spontaneous scattering of laser radiation in an optically thin medium, where reabsorption of spontaneous radiation can be neglected. The method proposed in the paper yields for the first time the correct expressions for nonstationary resonance fluorescence spectra, which do not give the negative values for fluorescence intensity. The expressions obtained for the resonance emission spectra coincide in limiting cases with those obtained within the framework of the quantum regression theorem.

DOI
10.1134/1.1560395

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Scientific Reports•Nov 12, 2021•1 citation
A new method is developed for the description of transfer and scattering of a time-dependent laser pulse, which is based on a hierarchy of Bogoliubov-Born-Green-Kirkwood-Yvone (BBGKY) equations for reduced density matrices. The systems of equations are obtained for the atomic and photon density matrices and the correlation operator in the second-order Born approximation. The system of equations describes stimulated and spontaneous scattering of laser radiation in an optically thin medium, where reabsorption of spontaneous radiation can be neglected. The method proposed in the paper yields for the first time the correct expressions for nonstationary resonance fluorescence spectra, which do not give the negative values for fluorescence intensity. The expressions obtained for the resonance emission spectra coincide in limiting cases with those obtained within the framework of the quantum regression theorem.
Scattering of resonance radiation in a dense two-level medium is considered with allowance for the local field effects. The system is studied in the framework of the mean-field approximation for a modified hierarchy of the Bogoliubov-Born-Green-Kirkwood-Yvon equations for reduced density matrices of an ensemble of atoms and modes of a quantized electromagnetic field. The local field correction is consistently derived from the initial Hamiltonian of the system. The problem of scattering of a short rectangular pulse from a quasi-point sample is considered numerically and theoretically. The possibility of a multimode spectrum of scattered radiation with frequencies multiple of the Rabi frequency and with other intermediate frequencies is demonstrated. The relative intensities of spectral lines are determined.
A simple theoretical model is formulated which describes dynamical features of the interaction of two-level atoms with a continuum of radiation oscillators. In particular, this paper is concerned with resonance scattering by single atoms in the presence of an external classical field, and spontaneous emission by a system of identical atoms. In the case of resonance scattering, the response of an atom to rapid changes of the amplitude of the external field is studied. The model for spontaneous emission by a system of atoms is based on the assumptions that all atoms have the same resonance frequency, and that the atoms are contained in a volume whose extension is small compared with the wavelength of the emitted radiation. The complete evolution of a system of mmm atoms is determined by a set of coupled first-order linear differential equations, and explicit solutions are presented for the values m=1,2,3,4,and8m=1,2,3,4, \mathrm{and} 8m=1,2,3,4,and8. Pair correlations of the atoms and photon-number fluctuations are discussed. An approximation for resonance fluorescence by a system of atoms with a broad distribution of resonance frequencies is included in an Appendix.
Abstract A practical approach to understanding light scattering processes in the condensed phase is presented. First a theoretical discussion of the various parameters in resonance light scattering is given and there is a special emphasis on our simplified approach to this question. As with other approaches the two level density matrix in association with a quantum regression theorem leads to two types of emission from an ensemble of molecules at finite temperature (T2 ≠ 2T1). We have called one of these fluorescence and the other sharper emission Raman scattering because it results directly from the presence of a coherent polarization in the medium. A brief discussion of time dependent scattering is also given. An experiment that reproduces the main features of the theory is described for resonance scattering from azulene in naphthalene at 30 K. For all values of the laser frequency (near resonance) both a sharp (Raman) and diffuse (fluorescent) component occurs in the emission spectrum. These spectra when interpreted by the simplified theory yield a value of T2 = 0.7 ps from fluorescence linewidth, and T2 = 0.8 ps from the incoherent to coherent scattering ratio, for the 0–0 two level system of azulene in naphthalene. A discussion of practical aspects of the influence of inhomogeneous line broadening is also presented.
A quasi-two-level system is a generalization of a two-level system where each of the single levels is replaced by a nearly degenerate multiplet. We develop the theory of the photon echo, the stimulated echo, and fluorescence emission for such systems when they are subjected to pulses of electromagnetic radiation that are short compared to the reciprocal (in frequency units) of the widths of the multiplets. The observable in each of these experiments is related to equilibrium time-correlation functions of the system in the absence of the radiation field. The correlation functions associated with these experiments involve more than two dynamical operators and more than two times, unlike the correlation functions for such processes as low-power optical absorption and neutron scattering. We introduce an exactly solvable factorization model for the transition moments, which includes as special cases the situation where either of the multiplets is a single state. We also consider the unitary model, which was introduced by Mims, and which leads to considerable simplification of the correlation functions. Several examples of these models are discussed to investigate the features of time-dependent fluorescence experiments and coherent optical transient experiments. In general, there is no simple connection between the time dependence of the decay of a photon echo and the Fourier transform of the low-power homogeneous absorption spectrum.
The perturbative density matrix formalism is used to study the time-dependent resonance Raman scattering (RRS) and resonance hot luminescence (RHL) components of the light-induced spontaneous emission process that has been called resonance secondary radiation (RSR). A three-level system coupled in a stochastic way to a bath is used to study the RSR emission driven by a coherent laser pulse with a variable Gaussian width in time. The spectral filter in a light detection system is explicitly included in the detection step of the problem. Observable time and frequency resolved RSR spectra are then calculated numerically. The spectral and temporal properties of the time-dependent RSR spectrum are very sensitive to the width of the driving pulse when the pulse width is of the same order of magnitude as the important system dephasing constants. In this domain the RRS contribution to the RSR spectrum weakens relative to the RHL as the light pulse is shortened. In the limit that the laser pulse becomes ‘‘delta function-like’’ the RSR spectrum consists essentially of RHL. When appropriate, the numerical results are compared to the stationary RSR problem as well as to the case where the RSR emission is driven by an exponential pulse. The analytical solutions to both of these cases are presented.
Despite many years of research into Raman phenomena, the problem of how to include both spontaneous and stimulated Raman scattering into a unified set of partial differential equations persists. The issue is solved by formulating the quantum dynamics in the Heisenberg picture with a rigorous accounting for both time- and normal-ordering of the operators. It is shown how this can be done in a simple, straightforward way. Firstly, the technique is applied to a two-level Raman system, and comparison of analytical and numerical results verifies the approach. A connection to a fully time-dependent Langevin operator method is made for the spontaneous initiation of stimulated Raman scattering. Secondly, the technique is demonstrated for the much-studied two-level atom both in vacuum and in a lossy dielectric medium. It is shown to be fully consistent with accepted theories: using the rotating wave approximation, the Einstein A coefficient for the rate of spontaneous emission from a two-level atom can be derived in a manner parallel to the Weisskopf-Wigner approximation. The Lamb frequency shift is also calculated. It is shown throughout that field operators corresponding to spontaneous radiative terms do not commute with atomic/molecular operators. The approach may prove useful in many areas, including modeling the propagation of next-generation high-energy, high-intensity ultrafast laser pulses as well as spontaneous radiative processes in lossy media.