We consider the excitation spectrum of a three-level atom interacting with an electromagnetic field whose two initially populated modes ω a and ω b are in resonance with two atomic transition frequencies while the remaining unpopulated modes are simultaneously coupled to the atomic states. It is found that at high photon densities, the Green's functions describing the lower and upper systems corresponding to the transition frequencies ω a and ω b respectively are-independent of each other. The spectral functions for the lower and upper systems consist of seven lorentzian lines describing the central peak and three pairs of sidebands where each pair is symmetrically located with respect to the transition frequency. For the lower system, the lines are peaked at frequencies ω = ω a , ω = ω a ± Ω a , ω = ω a ± Ω b and ω = ω a ± Ω ab with radiative widths γa 0/2, 3γa 0/4, (γa 0 + γb 0)/4 and (γa 0 + 3γb 0)/4 respectively, where Ω a and Ω b are the energy shifts of the populated modes ω a and ω b , Ω ab 2 = Ω b 2 + Ω a 2/4 and γa 0/2 is equal to the natural width of a photon spontaneously emitted from an isolated atom. For the upper system, the lines occur at ω = ω b , ω = ω b ± Ω b , ω = ω b ± Ω a and ω = ω b ± Ω ba with radiative widths (γa 0 + γb 0)/2, (γa 0 + 3γb 0)/4, (γa 0 + 2γb 0)/4, and 3γa 0/4 respectively. Relations between Ω a and Ω b are derived for which the heights of the lorentzian lines of the corresponding bands become positive or negative indicating attenuation or amplification of the signal field respectively. It is also shown that when certain relations exist between Ω a and Ω b , the signal field is attenuated in the lower system while it is simultaneously amplified in the upper system at different frequencies and vice versa. Spectral functions of the lower and upper systems are derived and discussed for equally spaced levels and equally populated modes Ω a = Ω b , as well as for the limiting case when Ω b 2 = 3Ω a 2/4 or Ω a 2 = 3Ω b 2/4.
We consider the excitation spectrum of a three-level atom interacting with an electromagnetic field whose two initially populated modes ω a and ω b are in resonance with two atomic transition frequencies while the remaining unpopulated modes are simultaneously coupled to the atomic states. It is found that at high photon densities, the Green's functions describing the lower and upper systems corresponding to the transition frequencies ω a and ω b respectively are-independent of each other. The spectral functions for the lower and upper systems consist of seven lorentzian lines describing the central peak and three pairs of sidebands where each pair is symmetrically located with respect to the transition frequency. For the lower system, the lines are peaked at frequencies ω = ω a , ω = ω a ± Ω a , ω = ω a ± Ω b and ω = ω a ± Ω ab with radiative widths γa 0/2, 3γa 0/4, (γa 0 + γb 0)/4 and (γa 0 + 3γb 0)/4 respectively, where Ω a and Ω b are the energy shifts of the populated modes ω a and ω b , Ω ab 2 = Ω b 2 + Ω a 2/4 and γa 0/2 is equal to the natural width of a photon spontaneously emitted from an isolated atom. For the upper system, the lines occur at ω = ω b , ω = ω b ± Ω b , ω = ω b ± Ω a and ω = ω b ± Ω ba with radiative widths (γa 0 + γb 0)/2, (γa 0 + 3γb 0)/4, (γa 0 + 2γb 0)/4, and 3γa 0/4 respectively. Relations between Ω a and Ω b are derived for which the heights of the lorentzian lines of the corresponding bands become positive or negative indicating attenuation or amplification of the signal field respectively. It is also shown that when certain relations exist between Ω a and Ω b , the signal field is attenuated in the lower system while it is simultaneously amplified in the upper system at different frequencies and vice versa. Spectral functions of the lower and upper systems are derived and discussed for equally spaced levels and equally populated modes Ω a = Ω b , as well as for the limiting case when Ω b 2 = 3Ω a 2/4 or Ω a 2 = 3Ω b 2/4.
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 arising from stimulated two-photon transitions between two atomic levels of the same parity. The system consists of a three-level atom in the cascade configuration interacting with two strong electromagnetic fields whose initially populated modes ωa and ωb are in resonance with the two atomic transition frequencies, respectively. The Green function formalism is used to calculate the spectral function of the system near the frequencies ω=ωa+ωb, in the limit of high photon densities. The excitation spectra consist of Lorentzian lines peaked at the frequencies ω−ωa−ωb=0, ±Ωa/√2, ±Ωb/√2, ±Ω, and ±2Ω, where Ω2=(Ω2a+Ω2b)/2 and Ωa,Ωb are the Rabi frequencies of the two laser fields, respectively. Numerical calculations for selected values of the Rabi frequencies are graphically presented and discussed. The computed two-photon spectra are compared with those arising from the third-order nonlinear mixing of frequencies near ω=ωa+2ωb and ωb+2ωa, respectively, as well as with those of the stimulated Raman spectra near ω=±(ωa−ωb).
A theory on interference effects at high photon densities has been developed for two types of a single three-level atom for which transitions occur: (i) from two different upper levels to a common lower one and (ii) from a common upper level to two different lower levels. The excitation spectrum for the interference effects for the two types of atoms results from the symmetric and antisymmetric interference between the two electronic transitions of the system, respectively. The spectral function for the symmetric modes consists of three Lorentzian lines peaked at the frequencies ω = Δ and ω = Δ ± Ω and having spectral widths of the order of γ0 and 3γ0/4, respectively, where Δ is the beat frequency, Ω is the Rabi frequency, and γ0/2 is equal to the natural linewidth for a photon spontaneously emitted from an isolated atom. The antisy mmetric spectrum consists of the peak of ω = Δ, which has a delta-function distribution indicating the stability of the mode in question, and two Lorentzian lines peaked at ω = Δ ± Ω with radiative widths of the order of γ0/2. The excitation spectrum of each type of atom contains also a Lorentzian line describing the very low frequency mode of the system, respectively.
The excitation spectrum arising from the interaction between two identical atoms (molecules), one of which is excited in the presence of a resonant strong electromagnetic field (pump field), is investigated. The excitation spectrum is found to consist of those describing the symmetric and antisymmetric modes, respectively. The form of both spectra depends on the relation between the distance R separating the atoms and the wavelength $\ensuremath{\lambda}$ of the transition from the ground state to the excited state. For $R<\ensuremath{\lambda}$, and when certain conditions prevail, the spectral function for the symmetric modes consists of three Lorentzian lines describing the central peak and the two sidebands whose radiative widths are equal to ${\ensuremath{\gamma}}_{0}$, $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, and $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, which are twice the corresponding ones arising from an isolated single atom interacting with the pump field. For the antisymmetric modes the central peak has a $\ensuremath{\delta}$-function distribution indicating the stability of the mode in question, while, when certain conditions are satisfied, the two sidebands are described by Lorentzian lines each having a radiative width of the order of $\frac{{\ensuremath{\gamma}}_{0}}{2}$, which is equal to the natural linewidth for a photon spontaneously emitted from an isolated atom. For both the symmetric and antisymmetric modes, the dipole-dipole interaction between the atoms brings about small energy shifts. For $R>\ensuremath{\lambda}$, apart from the small energy shifts caused by the dipole-dipole interaction, the spectral functions for symmetric and antisymmetric modes are similar to that for the single atom interacting with the pump field.
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.