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arXiv:quant-ph/0202166 · Journal of Optics B Quantum and Semiclassical Optics

A quantum stochastic approach to the spectrum of a two-level atom

Alberto Barchielli, Nicola Pero

Published Jul 24, 2002• 11 citations
arXivHTMLPDF

Abstract

By means of quantum stochastic calculus we construct a model for an atom with two degenerate levels which is stimulated by a circularly polarized laser and we compute its fluorescence spectrum. Let us stress that, once the model for the unitary atom–field dynamics has been given, then the spectrum is computed without further approximations. If only the absorption/emission term is included in the interaction, we reobtain the Mollow spectrum in the case of a monochromatic laser and the Kimble–Mandel spectrum in the case of a ‘phase diffusion model’ for a non-monochromatic laser. However, our model can also describe another type of light scattering, a ‘direct scattering’ due to the response of the atom as a whole, which we expect to be small but which interferes with the scattering due to the absorption/emission channel. When both the scattering channels are introduced we obtain a modification of the Mollow–Kimble–Mandel spectrum, which shares its main features with the usual case, but which presents some asymmetries even in the case of no detuning.

DOI
10.1088/1464-4266/4/5/308

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Physical Review A•Sep 18, 2008•1 citation
By means of quantum stochastic calculus we construct a model for an atom with two degenerate levels which is stimulated by a circularly polarized laser and we compute its fluorescence spectrum. Let us stress that, once the model for the unitary atom–field dynamics has been given, then the spectrum is computed without further approximations. If only the absorption/emission term is included in the interaction, we reobtain the Mollow spectrum in the case of a monochromatic laser and the Kimble–Mandel spectrum in the case of a ‘phase diffusion model’ for a non-monochromatic laser. However, our model can also describe another type of light scattering, a ‘direct scattering’ due to the response of the atom as a whole, which we expect to be small but which interferes with the scattering due to the absorption/emission channel. When both the scattering channels are introduced we obtain a modification of the Mollow–Kimble–Mandel spectrum, which shares its main features with the usual case, but which presents some asymmetries even in the case of no detuning.
Quantum stochastic differential equations have been used to describe the dynamics of an atom interacting with the electromagnetic field via absorption/emission processes. Here, by using the full quantum stochastic Schrödinger equation proposed by Hudson and Parthasarathy, we show that such models can be generalized to include other processes into the interaction. In the case of a two-level atom we construct a model on the basis of some physical requirements, the main being a balance equation on the fluxes of the ingoing and outgoing photons; in this model the atom-field interaction turns out to be due either to absorption/emission processes either to direct scattering processes, which simulate the interaction due to virtual transitions to the levels which have been eliminated from the description. To see the effects of the new terms, we consider both direct and heterodyne detection of the fluorescence light emitted by an atom stimulated by a monochromatic coherent laser and we deduce from these two detection schemes the expressions of the total, elastic and inelastic electromagnetic cross sections and the spectral distribution of the fluorescence light. The total cross section, as a function of the frequency of the stimulating laser, can present not only a Lorentzian shape, but the full variety of Fano profiles; intensity dependent widths and shifts are obtained. The fluorescence spectrum can present complicated shapes, according to the values of the various parameters; when the direct scattering is not important the usual symmetric triplet structure of the Mollow spectrum appears (for high intensity of the stimulating laser), while a strong contribution of the direct scattering process can distort such a triplet structure or can even make it to disappear.
On the basis of a stochastic model of Burshtein et al. (1967) which assumes the phases, frequencies and/or amplitudes of the radiation field to constitute a randomly time-varying jump-like Markoff process, a generalisation of Mollow's theory (1975) of the dynamic Stark effect is formulated leading to predictions for the resonant scattering spectrum for a two-level atom quite similar to those published in the work of Agarwal (1976), of Eberly (1976) and of Kimble et al. (1977) who use a different approach. Apparently, the general form of the resonant scattering spectrum in a modulated radiation field depends less strongly on the particular stochastic model of the laser light than might be expected.
The complex internal atomic structure involved in radiative transitions has an effect on the spectrum of fluctuations (noise) of the transmitted light. A degenerate transition has different properties in this respect than a pure two-level transition. We investigate these variations by studying a certain transition between two degenerate atomic levels for different choices of the polarization state of the driving laser. For circular polarization, corresponding to the textbook two-level-atom case, the optical spectrum shows the characteristic Mollow triplet for strong laser drive, while the corresponding noise spectrum exhibits squeezing in some frequency ranges. For a linearly polarized drive, corresponding to the case of a multilevel system, additional features appear in both optical and noise spectra. These differences are more pronounced in the regime of a weakly driven transition: whereas the two-level case essentially exhibits elastic scattering, the multilevel case has extra noise terms related to spontaneous Raman transitions. We also discuss the possibility to experimentally observe these predicted differences for the commonly encountered case where the laser drive has excess noise in its phase quadrature.
Summary form only given. The problem of an atom driven by a strong laser was first investigated theoretically in 1969 by B.R. Mollow. He discovered that the resonance fluorescence spectrum has a three-peaked structure with the central peak being more pronounced than the two side peaks. The common Mollow triplet fluorescence pattern can be altered in many ways, by making adjustments to the system. The system can be placed in a squeezed vacuum, the atom can be placed in a cavity, or a bichromatic field can be used instead of a monochromatic laser. These alterations all have an effect on the resulting resonance fluorescence and probe absorption spectra. Other interesting modifications occur when the atom interacts with a stochastic field as well as a laser. The previous approaches were numerical. Here we also obtain analytic solutions (in certain limits) which provide additional insight.
Just at the beginning of quantum stochastic calculus Hudson and Parthasarathy proposed a quantum stochastic Schrodinger equation linked to dilations of quantum dynamical semigroups. Such an equation has found applications in physics, mainly in quantum optics, but not in its full generality. It has been used to give, at least approximately, the dynamics of photoemissive sources such as an atom absorbing and emitting light or matter in an optical cavity, which exchanges light with the surrounding free space. But in these cases the possibility of introducing the gauge (or number) process in the dynamical equation has not been considered. In this paper we show, in the case of the simplest photoemissive source, namely a two-level atom stimulated by a laser, how the full Hudson-Parthasarathy equation allows to describe in a consistent way not only absorption and emission, but also the elastic scattering of the light by the atom. Morever, we study the differential and total cross sections for the scattering of laser light by the atom, as a function of the frequency of the stimulating laser. The resulting line-shape is very interesting. Not only a Lorentzian shape is permitted, but the full variety of Fano profiles can be obtained. The dependence of the line shape on the intensity of the stimulating laser is computed; in particular, the resonance position turns out to be intensity dependent, a phenomenon known as lamp shift.
The absorption spectrum is considered for a two-level atom coherently driven by a laser field and incoherently perturbed by stochastic noise. In the noise-dominated regime where incoherent perturbation far exceeds coherent excitation, the amplifying and dispersionlike components in the standard absorption spectrum [B. R. Mollow, Phys. Rev. A 5, 2217 (1972); F. Y. Wu et al., Phys. Rev. Lett 38, 1077 (1977)] develop into an absorption peak and a narrow dip, respectively. The modified absorption spectrum is attributed to the quantum interference induced by stochastic noise.