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Journal of Modern Optics

Vacuum modified resonance fluorescence in intense laser fields

Christoph H. Keitel

Published Aug 1, 1996• 24 citations
Publisher

Abstract

Resonance fluorescence is considered for a two-level atom in an intense laser field with a Rabi frequency that is an appreciable fraction of the transition frequency of the atom. The dressed sideband transitions then feel a significantly different density of modes of the vacuum, giving rise to substantial deviations from the common resonance fluorescence spectrum for reasonably strong driving fields.

DOI
10.1080/09500349608232827

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Physical Review A•Jul 17, 2000•8 citations
Resonance fluorescence is considered for a two-level atom in an intense laser field with a Rabi frequency that is an appreciable fraction of the transition frequency of the atom. The dressed sideband transitions then feel a significantly different density of modes of the vacuum, giving rise to substantial deviations from the common resonance fluorescence spectrum for reasonably strong driving fields.
We consider the resonance fluorescence of a two-level atom in a vacuum with a high intensity driving field, such that the Rabi frequency of the field is non-negligible up to frequencies comparable to the transition frequency of the atom. At these high intensities, the conventional rotating wave approximation cannot be applied to the atom-laser Hamiltonian and the modal densities of the vacuum at the dressed transition frequencies differ significantly from each other. On treating the counter-rotating part of the Hamiltonian as a perturbation and carrying out the Weisskopf-Wigner approximation in the dressed states basis, we find that the high driving intensity leads to significant deviations from the traditional Mollow resonance fluorescence spectrum. In particular we find unequally intense Mollow side-bands and that the generation of higher order triplets and harmonics is substantially enhanced by the high vacuum modal density at the corresponding high resonance frequencies.
The phenomenon of resonance fluorescence provides an interesting manifestation of the quantum theory of light and is a “real world” application of the material of the Chapters 8 and 9. In this process, a two-level atom is typically driven by a resonant continuous-wave laser field and the spectral and quantum statistical properties of the fluorescent light emitted by the atom are measured. Experimentally this can be achieved by scattering a laser off a collimated atomic beam such that the directions of the laser beam, atomic beam, and detector axis are mutually perpendicular as shown in Fig. 10.1. If the driving field is monochromatic, then at low excitation intensity the atom absorbs a photon at the excitation frequency and reemits it at the same frequency as a consequence of conservation of energy. The spectral width of the fluorescent light is therefore very narrow. The situation, however, is considerably more complicated when the excitation intensity increases and the Rabi frequency associated with the driving field becomes comparable to, or larger than, the atomic linewidth. At such intensity levels, the Rabi oscillations show up as a modulation of the quantum dipole moment and sidebands start emerging in the spectrum of the emitted radiation. This so-called dynamic Stark splitting is an interesting feature of the atom–field interaction. In addition to that, the fluorescent light exhibits certain nonclassical properties including photon antibunching and squeezing. In this chapter, we develop a theory of resonance fluorescence to explain these phenomena.
Resonance-fluorescence power spectrum of two-level atoms driven by a strong monochromatic laser field is discussed, taking into account effects due to the dipole-dipole interaction between the atoms. For two atoms confined to a region smaller than the wavelength of the driving field, we predict significant deviations of the power spectrum from the single-atom case. The two-atom spectrum is nonsymmetric, the wing peaks are shifted from their Rabi-frequency positions, and additional maxima may appear.
We examine the radiative properties of a two-level atom driven by a strong bichromatic field with frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{0}$-${\mathrm{\ensuremath{\delta}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{0}$+${\mathrm{\ensuremath{\delta}}}_{2}$, which can be asymmetrically placed about the atomic transition frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ and can have different Rabi frequencies ${\mathrm{\ensuremath{\Omega}}}_{1}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$. Applying the optical Bloch equations for the bichromatic excitation, we derive an infinite set of equations of motion for the time evolution of the atomic variables. The equations are solved numerically by matrix inversion and Laplace transforms. Using the quantum regression theorem, we then solve for the steady-state total fluorescence intensity, and the resonance-fluorescence and absorption spectra. The spectra are found to depend on the frequency difference 2\ensuremath{\delta}=${\mathrm{\ensuremath{\delta}}}_{1}$+${\mathrm{\ensuremath{\delta}}}_{2}$, the average detuning \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{0}$-1/2(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$), and the Rabi frequencies of the driving fields. For ${\mathrm{\ensuremath{\Omega}}}_{1}$=${\mathrm{\ensuremath{\Omega}}}_{2}$=\ensuremath{\Omega}, the total fluorescent intensity displays a series of maxima for \ensuremath{\Delta}=\ifmmode\pm\else\textpm\fi{}n\ensuremath{\delta}\ensuremath{\mp}${\mathrm{\ensuremath{\Omega}}}^{2}$/4\ensuremath{\delta}, where the n are the odd integers. The intensity-dependent shift from the resonances n\ensuremath{\delta} is explained as an analog of the generalized Bloch-Siegert shift. The resonance-fluorescence spectrum for \ensuremath{\Delta}\ensuremath{\ne}0 appears to contain more peaks than that for \ensuremath{\Delta}=0.This is due to the splitting of the central peak and the even sidebands into doublets. The absorption spectrum of a weak-probe beam for \ensuremath{\Delta}=0 and equal Rabi frequencies consists of a symmetric series of dispersionlike sidebands separated from ${\mathrm{\ensuremath{\omega}}}_{0}$ by integer multiples of \ensuremath{\delta}, together with a central absorption peak at ${\mathrm{\ensuremath{\omega}}}_{0}$, whose amplitude oscillates with \ensuremath{\Omega}. For \ensuremath{\Delta}\ensuremath{\ne}0 and/or ${\mathrm{\ensuremath{\Omega}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\Omega}}}_{2}$, the odd sidebands remain dispersionlike, while the central peak and the even sidebands split into absorption-emission doublets. A simple physical interpretation of the spectral features is given in terms of the dressed-atom model.
The fluorescence spectrum of a system of a large number of two-level atoms driven by a strong resonant field has been treated, so far, for atoms contained in a region that is small compared to a wavelength. For high laser intensities resonances displaced from the line center by twice the Rabi frequency were predicted1-3 in addition to the usual Stark sidebands.4 We show that the explicit spatial distribution of the two-level atoms affects the shape of the radiation spectra and leads to the appearance of additional sidebands.
Resonance fluorescence of a driven Jaynes-Cummings system is studied theoretically when the cavity is driven by a classical field of arbitrary strength. In the regime of strong atom-cavity coupling, there occur anomalous resonance peaks at $\ifmmode\pm\else\textpm\fi{}\sqrt{2}{g}^{\ensuremath{'}},\ifmmode\pm\else\textpm\fi{}\sqrt{3}{g}^{\ensuremath{'}},$ etc., with the modified vacuum Rabi frequency ${g}^{\ensuremath{'}}.$ These new peaks and the observed shift of the vacuum Rabi splitting are to be explained in the language of the quasienergy levels induced dynamically by the strong driving field. In the presence of significant damping the spectrum reflects the photon-number distribution inside the cavity, and particularly, a bimodality of the distribution results in a unique quintuplet resonance fluorescence spectrum.