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Journal of Experimental and Theoretical Physics Letters

Generalized expression for the natural width of the radiation spectrum of quantum oscillators

A. S. Chirkin, Arkadi V. Chipouline

Published Apr 1, 2011• 15 citations
Publisher

Abstract

A simple method for calculating the linewidth of sources of stimulated coherent radiation caused by spontaneous noise and thermal fluctuations has been proposed in the framework of the two-level model of the active medium. An expression for the spectral width has been derived beyond the adiabatic approximation under an arbitrary relation between relaxation times in a self-sustained oscillation system, taking into account a finite spectral band of spontaneous radiation. The result is applicable for a wide class of coherent radiation sources.

DOI
10.1134/s0021364011030039

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A simple method for calculating the linewidth of sources of stimulated coherent radiation caused by spontaneous noise and thermal fluctuations has been proposed in the framework of the two-level model of the active medium. An expression for the spectral width has been derived beyond the adiabatic approximation under an arbitrary relation between relaxation times in a self-sustained oscillation system, taking into account a finite spectral band of spontaneous radiation. The result is applicable for a wide class of coherent radiation sources.
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Quantum fluctuations in a laser with two different relaxation times are considered, i.e., transverse (polarization relaxation) and longitudinal (population relaxation) in the case in which the cavity transmission band half-width is much smaller than the transverse width and much larger than the longitudinal one. The lasing frequency detuning from the transition frequency of a two-level system is assumed to be arbitrary in this case, and it is necessary to take into account the contribution of two-particle correlators into the dispersion and laser linewidth. The results are considered as applied to a semiconductor laser.
The problem of induced and spontaneous emission is investigated for an atomic two-level system with incident beams of radiation which are either in a coherent state or in a stationary state (contain a definite number of photons). The treatment is fully quantum-mechanical, and is confined to the case where the frequency spectrum of the incident beam is narrow compared to the natural linewidth of the system. It is shown that, under such conditions, the spontaneous emission for frequencies within the narrow band of the incident radiation is sharply reduced compared to the prediction of the natural lineshape. It is shown that a hole is burned in the natural lineshape within the narrow frequency band, thus effectively quenching the spontaneous emission at some frequency within the band. This effect is shown to occur both for the coherent and stationary beams. Quantities proportional to the induced and spontaneous probability amplitudes and the lifetimes are computed for times comparable to and long compared to the free lifetime of the state. An expression is found for the spectrum of the emergent radiation in terms of these quantities. Its physical meaning is briefly discussed. The density operator of the field for all times is given.
A theory is developed of stimulated scattering of nonmonochromatic radiation whose spectral width is much greater than the spontaneous scattering line width or the frequency shift of the Stokes wave. It is shown that scattering of nonmonochromatic incident radiation under coherent conditions may take place as efficiently as for monochromatic pump radiation. The results apply to various types of stimulated scattering, in particular, to stimulated thermal scattering and stimulated scattering in media which are in thermodynamic nonequilibrium.
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.
The spectral densities of Einstein coefficients for absorption and stimulated emission in a two-level quantum system are not equal to each other beyond the absorption (emission) line if the homogeneous broadening caused by interaction with a thermostat is much larger than the natural width. In this study, a relationship between these coefficients is derived on the basis of general thermodynamic requirements. This relationship is associated with new effects such as population inversion upon the nonresonant absorption of continuous radiation and radiation amplification without population inversion.