We have numerically calculated the excitation spectra arising from the 3rd order mixing of the frequencies ωa and ωb of two laser fields interacting with a three-level atom, where each laser field resonantly couples the ground state with each excited state of the atom, respectively. In the limit of high photon densities, the excitation spectra near the reduced frequency X = (ω−ωa + 2ωb)/γ0 ≈ 0 are considered as a function of the reduced Rabi frequencies ηa and ηb of the two laser fields, respectively and γ0 is the spontaneous emission probability. For ηa < ηb the spectra consist of a doublet peaked at [Formula: see text] and its intensity is constant. When ηa = ηb, the spectra are composed of five pairs of bands peaked at [Formula: see text], and [Formula: see text]. When ηa < ηb the computed spectra consist of five pairs of bands, where the intensities of the peaks at [Formula: see text] and [Formula: see text] are positive indicating absorption, those at [Formula: see text] are negative implying amplification, and the two pairs of peaks at [Formula: see text] have positive and negative components describing the mixed process of absorption–amplification. The intensities of these bands are found to vary as (ηa/ηb)2 for (ηa/ηb) > 1 and, therefore, the intensities of the bands are immensely enhanced as the value of the ratio (ηa/ηb) increases. The computed spectra for a wide selection of Rabi frequencies are graphically presented and compared with those derived by analytical methods.
We have numerically calculated the excitation spectra arising from the 3rd order mixing of the frequencies ωa and ωb of two laser fields interacting with a three-level atom, where each laser field resonantly couples the ground state with each excited state of the atom, respectively. In the limit of high photon densities, the excitation spectra near the reduced frequency X = (ω−ωa + 2ωb)/γ0 ≈ 0 are considered as a function of the reduced Rabi frequencies ηa and ηb of the two laser fields, respectively and γ0 is the spontaneous emission probability. For ηa < ηb the spectra consist of a doublet peaked at [Formula: see text] and its intensity is constant. When ηa = ηb, the spectra are composed of five pairs of bands peaked at [Formula: see text], and [Formula: see text]. When ηa < ηb the computed spectra consist of five pairs of bands, where the intensities of the peaks at [Formula: see text] and [Formula: see text] are positive indicating absorption, those at [Formula: see text] are negative implying amplification, and the two pairs of peaks at [Formula: see text] have positive and negative components describing the mixed process of absorption–amplification. The intensities of these bands are found to vary as (ηa/ηb)2 for (ηa/ηb) > 1 and, therefore, the intensities of the bands are immensely enhanced as the value of the ratio (ηa/ηb) increases. The computed spectra for a wide selection of Rabi frequencies are graphically presented and compared with those derived by analytical methods.
We have studied the excitation spectrum arising from the optical mixing of the frequencies of a strong bichromatic field interacting with a three-level atom, where the two initially populated modes ωa and ωb are equal to the two atomic transition frequencies, respectively. In the limit of high photon densities, the excitation spectrum near the frequency ω = ωa – 2ωb has been calculated as a function of the parameter η = Ωa2/Ωb2, where Ωa and Ωb are the Rabi frequencies of the two laser fields, respectively. For [Formula: see text] and for weak fields for which [Formula: see text], the spectral function describes a Lorentzian line peaked at the frequency ω = ωa – 2ωb and has a width of the order of γ0, where γ0/2 is the natural width for a two-level atom. When Ωb2 > γ02 and [Formula: see text], the band at ω = ωa – 2ωb splits into two bands described by two Lorentzian lines peaked at ω = ωa – 2ωb ± Ωb/√2 and have spectral widths of the order of 3γ0/4. The ratio of the height of the band ω = ωa – 2ωb to the height ω = ωa – 2ωb ± Ωb/2 is 3:2. The probability amplitudes for both bands take large negative values indicating that optical amplification of the signal field may be expected to occur at these frequencies. When Ωa = Ωb = Ω, η = 1, and for Ω2 < γ02, the spectral function describes a single band at ω = ωa – 2ωb while for Ω2 > γ02, the single band splits into five pairs of bands which are separated from the frequency ω = ωa – 2ωb by frequency shifts which are equal to: ± Ω/√2, [Formula: see text], ± Ω, ± Ω√2, and ± Ω√3, respectively, and have spectral widths of the order of 3γ0/4. For [Formula: see text] and for laser fields for which Ωa2 > γ02 and Ωb2 > γ02, the spectral function consists of three pairs of bands. The probability amplitudes for these bands vary linearly with η and may take large values for [Formula: see text]. A complete discussion of the excitation spectrum as well as a graphical representation of the derived results has been given.
We have studied the excitation spectrum arising from the interaction of a two-level atom with two strong electromagnetic fields whose initially populated modes ω a and ω b are near resonance with the atomic transition frequency In the limit of high photon densities the Green's function of the system describing the electronic transition in question has been derived in a closed form and includes processes arising from the interaction of seven photon modes of both laser fields. The spectral function of the system is calculated when the frequency ω a of the one laser field is in resonance with the atomic transition frequency while the frequency ω b of the other field is varied. The excitation spectrum is found to consist, apart from the central peak at the excitation frequency ω a , of several lorentzian lines which are peaked in the neighbourhood of the frequencies ω a ± Δ, ω a ± 2Δ, ω a ± 3Δ and ω a ± 4Δ, where Δ = ω a - ω b and describe the type of optical mixing of light modes which is of the third, fifth, seventh and ninth order, respectively. Additional peaks appear in the spectra, which result from the interaction between the frequency modes in question. The probability amplitudes describing the relative intensities of the peaks in question are expressed as functions of the parameters η a = Ω a /Δ and η b = Ω b /Δ, where Ω a and Ω b are the Rabi Frequencies of the two laser fields, respectively. It is found that when η a ⩾ 1 and η b ⩾ 1 the expression for the relative intensities of the bands in question are remarkably enhanced and take extremely large positive or negative values. Results of numerical calculations for a wide range of values of η a and η b when η a = η b ⩾ 1 are presented graphically and the complete excitation spectrum is discussed in detail. The method may be useful to investigate the behaviour of excited states as well as those close to the limit of ionization.
We have considered the excitation spectra of three-photon processes arising from the interaction of a three-level atom, in the cascade configuration, with two strong electromagnetic fields whose initially populated modes ω a and ω b are near resonance with the atomic transition frequencies ω21 and ω32, respectively. In the limit of high photon densities, expressions for the Green's functions are derived in a closed form describing third order nonlinear processes in the region of frequencies ω = ω a + 2ω b and ω = ω b + 2ω a , respectively. Numerical calculations are then performed and the computed spectra are presented graphically for different values of the Rabi frequencies Ω a and Ω b , and detunings ζ a and ζ b , respectively. The excitation spectrum near ω = ω a + 2ω b is calculated as a function of the parameters η a = Ω a /γ0, η b = Ω b /γ0, ζ a = (ω21 - ω a )/γ0 and ζ b = (ω32 - ω b )/γ0 where γ0 is the spontaneous transition probability. With no detuning the spectral intensities vary slowly with respect to variations of the Rabi frequencies and are shifted by amounts governed by the field strengths η a and η b . In the presence of detunings, asymmetries, characteristics of such off-resonance situations, arise enhancing certain spectral peaks while diminishing the intensity of others.
We have calculated numerically the excitation spectra arising from the interaction of a two-level atom with two strong electromagnetic fields. In the limit of high photon densities, the excitation spectra are considered when one of two laser fields is at resonance while the frequency of the other laser field is varied as well as when both laser fields are off resonance. At resonance, the spectra consist of a central peak and several sidebands peaked at the frequencies ω = ω a ± m(ω a - ω b ), corresponding to the multi-frequency mixing of both laser fields where m is a positive integer. Sidebands also appear at fractional multiples of frequencies and are due to contributions arising from energy shifts induced by the laser fields. The computed spectra at resonance are discussed and compared with the observed ones. The off-resonance spectra are computed for a wide range of values of Rabi frequencies and detunings, and are presented graphically. It is found that in the off-resonance case, mixing of a larger number of light frequencies occurs from that in the resonance case. The intensity of the peaks in the off-resonance case are immensely enhanced in comparison with those at resonance. The enhancement of the intensities is in agreement with the observed ones and its occurrence is theoretically explained. It is shown that if the detunings are properly chosen then the desired peak describing the required mixing of frequencies of the laser fields can be immensely enhanced. The method may be useful to investigate multi-frequency resonances of the rotation or rotation-vibration spectra of excited states or excited states near their ionization continua.
We have considered the fluorescent spectra arising from the interaction of a four-level atom with two strong electromagnetic fields whose initially populated modes ω a and ω b are in resonance with three of the atomic transition frequencies. Green's function formalism has been used in the limit of high photon densities to calculate the excitation spectra near the frequencies ω = ±(2ω a - ω b ). Expressions for the total relative intensity of the system have been derived as a function of the parameter ζ = Ω2 b /Ω2 a , where Ω a and Ω b are the Rabi frequencies of the two laser fields, respectively. For ζ = 0, the spectra consist of a central peak at ω = 2ω a - ω b and a pair of sidebands at ω = 2ω a - ω b ± Ω a √2, both having spectral widths of the order of γ0, where γ0 is the spontaneous emission probability. The relative intensity of the central peak for Ω a ≠ 0 is reduced by 25 per cent of its original value obtained for Ω a = 0. For ζ ≠ 0, the relative intensity of the central peak increases slowly as ζ increases while that of the pair of sidebands at ω = 2ω b - ω b ± Ω a √2 is reduced by a factor of two of its given value for ζ = 0. This destructive interference is due entirely to the presence of the second laser field with Ω b ≠ 0. A new pair of sidebands, which arise from the cooperative effect of both laser fields, is peaked at ω = 2ω a - ω b ± Ω a √(2 + 0·5ζ) and has a spectral width equal to γ0. A set of four pairs of sidebands also appear at ω = 2ω a - ω b ± Cj (ζ)Ω a with j = 1, 2, 3, 4, respectively, which have spectral widths of the order of 3γ0/2. The existence of these pairs of sidebands depends strongly on the value of the parameter ζ. Numerical estimates for the coefficients Cj (ζ) and the relative intensities of these sidebands are tabulated for a wide range of values of ζ and discussed in detail.
We have studied the fluorescent spectra arising from the interaction of a four-level atom with a strong bichromatic field and a weak signal field simultaneously. The atomic system consists of two upper excited states \ensuremath{\Vert}3〉 and \ensuremath{\Vert}4〉 and two lower states \ensuremath{\Vert}1〉 and \ensuremath{\Vert}2〉 and the two laser fields, whose frequency modes ${\ensuremath{\omega}}_{a}$ and ${\ensuremath{\omega}}_{b}$ are initially populated, operate between the states \ensuremath{\Vert}2〉\ensuremath{\leftrightarrows}\ensuremath{\Vert}4〉 and \ensuremath{\Vert}2〉\ensuremath{\leftrightarrows}\ensuremath{\Vert}3〉, respectively. Using the Green-function formalism and a model Hamiltonian, where all the free and interacting fields are quantized, we have studied the transitions \ensuremath{\Vert}4〉\ensuremath{\leftrightarrows}\ensuremath{\Vert}1〉 and \ensuremath{\Vert}3〉\ensuremath{\leftrightarrows}\ensuremath{\Vert}1〉 of the signal field in the limit of high photon densities of both laser fields. The expression of the spectral function for the \ensuremath{\Vert}1〉\ensuremath{\rightarrow}\ensuremath{\Vert}4〉 transition describes stimulated one-photon and three-photon processes, respectively. The one-photon spectra near the frequency \ensuremath{\omega}\ensuremath{\approxeq}${\ensuremath{\omega}}_{41}$ of the signal field consist of a central peak and two pairs of sidebands whose intensities take always positive and negative values indicating that attenuation and amplification (stimulated emission) of the signal field takes place at the frequencies in question, respectively. The spectral function for the three-photon process near the frequency \ensuremath{\omega}\ensuremath{\approxeq}${\ensuremath{\omega}}_{41}$-2${\ensuremath{\omega}}_{a}$ describes two pairs of sidebands whose intensities are always negative implying that strong amplification of the signal field occurs. The computed one-photon and three-photon spectra are presented graphically for different values of Rabi frequencies and detunings of both laser fields, respectively. Then using a classical description of both laser fields, the spectral functions describing one-photon and three-photon processes are calculated and the derived results are compared with the corresponding ones when the fields are quantized. It is found that the results obtained when the fields are quantized and in the limit of high photon densities describe the classical as well as the quantum nature of the photons which is lost in the classical treatment.