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Physical Review

Atomic Response Function

Werner Brandt, Stig Lundqvist

Published Dec 1, 1963• 64 citations
Publisher

Abstract

The response function of atoms is derived approximately within the framework of the time-dependent Hartree scheme for external fields of wavelength large compared to atomic dimensions. Solutions of the resulting dispersion equation correspond to new resonances of atoms. The theory is discussed in terms of the properties of the differential oscillator-strength distribution, or photoabsorption cross section, for the Hartree model and for the statistical model of the atom. An illustrative calculation in the statistical approximation exhibits qualitatively the conditions for the occurrence of these resonances.

DOI
10.1103/physrev.132.2135

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Physical Review•Aug 10, 1969•59 citations
The response function of atoms is derived approximately within the framework of the time-dependent Hartree scheme for external fields of wavelength large compared to atomic dimensions. Solutions of the resulting dispersion equation correspond to new resonances of atoms. The theory is discussed in terms of the properties of the differential oscillator-strength distribution, or photoabsorption cross section, for the Hartree model and for the statistical model of the atom. An illustrative calculation in the statistical approximation exhibits qualitatively the conditions for the occurrence of these resonances.
The dipolar spectral distribution function, or the photoabsorption cross section, of atoms is derived in the statistical approximation starting from general dynamic equations describing the density fluctuations induced in the atom by an external field. Using a local form of the theory it is found that coherence between the motion in different parts of the atom causes a modification of the spectral function. The modification can be put in the form of a dispersion denominator, and if this denominator becomes small in certain frequency ranges, the absorption is enhanced by what can be identified as collective resonances of the atom as a whole.
Quantum-mechanical expressions for nonresonance-dipole light-scattering cross sections are related to corresponding expressions for oscillator strengths and refractive indices. In particular, Rayleigh and Raman cross sections for atoms are expressed in terms of oscillator strengths and vector-coupling coefficients. Several of our results, including the relationship between Rayleigh scattering and refractive index, differ in general from the corresponding results of classical dispersion theory. We show that these differences arise from antisymmetric components Ckja = (Ckj − Cjk)/2 of the polarizability tensor, which have been neglected in the classical analyses. Calculated Rayleigh cross sections for cesium and aluminum atoms, and Raman cross sections for aluminum atoms are presented to illustrate our results. The antisymmetric contribution is found to be substantial in all of these cross sections; its most obvious effect is to cause the depolarization (for linearly polarized incident light) to exceed 34 over extended wavelength ranges away from resonance. On the other hand, for atoms initially in states of zero angular momentum, and molecules under conditions which allow the use of Placzek’s polarizability approximation, our results agree with those of classical dispersion theory.
We consider the radiation from a system of NNN identical two-level atoms coupled to a continuum of quantized em modes, and possibly, to an external driving field near resonance. The atoms can be distributed over a region large in comparison to the resonant wavelength, but smaller than the spontaneous pulse length. Radiation rates and correlation functions are expressed in terms of expectation values of time-dependent atomic operators, which are shown to satisfy coupled first-order differential equations involving similar atomic operators and the initial radiation operators. The corresponding equations for the expectation values simplify considerably if no driving field is present. Similar results are derived for a model in which each atom is replaced by a harmonic oscillator.
By means of the retarded Green's function (response function) we derive the threshold condition, the oscillation frequency and the time-limited band width of lasers. The statistical property of the set of identical two-level atoms are taken into account on the basis of the Dicke-Haken's model. This property gives rise to atom-field correlation terms in the polarization operator. It is found that this terms influence the imaginary part of the oscillation frequency and the effect becomes large as the photon density increases or the occupancy difference tends to zero. In this respect the relation with the super-radiation is discussed. The customary approximation of considering only the interaction which is conservative for the particle number is tested. Under the resonance condition this approximation is correct up to the order of |ω k -ω 0 |/ω 0 , where ω 0 is the transition frequency of the atoms and ω k is the frequency of the free field.
General behavior of the transient response of materials excited by a stepwise optical field has been studied by means of numerical analysis based on a model of the inhomogeneously broadened two-level atomic system with relaxation obeylng the quantum mechanical equation of motion. The time dependences of the polarization and the excited state probability of atoms are investigated in detail under a wide variety of conditions including both the resonant and far off-resonant excitations. The behaviors are found to be quite different depending on the relative magnitudes of the light frequency, the center frequency and the bandwidth of transition, the Rabi frequency, the longitudinal and transverse relaxation rates. Two kinds of response times are defined to describe the rapidity of the response, and the physical parameters goveming them are detennined for a variety of cases. Another kind of response time originated from the statistical behavior of atoms is further considered and discussed. Based on these results, the range of the values of the response times are estimated in general. The results for both cases with and without the rotating wave approximation are also calculated and compared.
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.