By means of the retarded Green's function (response function) we derive the threshold condition, the oscillation frequency and the time-limited band width of lasers. The statistical property of the set of identical two-level atoms are taken into account on the basis of the Dicke-Haken's model. This property gives rise to atom-field correlation terms in the polarization operator. It is found that this terms influence the imaginary part of the oscillation frequency and the effect becomes large as the photon density increases or the occupancy difference tends to zero. In this respect the relation with the super-radiation is discussed. The customary approximation of considering only the interaction which is conservative for the particle number is tested. Under the resonance condition this approximation is correct up to the order of |ω k -ω 0 |/ω 0 , where ω 0 is the transition frequency of the atoms and ω k is the frequency of the free field.
By means of the retarded Green's function (response function) we derive the threshold condition, the oscillation frequency and the time-limited band width of lasers. The statistical property of the set of identical two-level atoms are taken into account on the basis of the Dicke-Haken's model. This property gives rise to atom-field correlation terms in the polarization operator. It is found that this terms influence the imaginary part of the oscillation frequency and the effect becomes large as the photon density increases or the occupancy difference tends to zero. In this respect the relation with the super-radiation is discussed. The customary approximation of considering only the interaction which is conservative for the particle number is tested. Under the resonance condition this approximation is correct up to the order of |ω k -ω 0 |/ω 0 , where ω 0 is the transition frequency of the atoms and ω k is the frequency of the free field.
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 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.
The simultaneous action of a rf perturbation between the Zeeman sublevels of an atomic transition, which is also producing laser oscillation, is considered. The complete Hamiltonian of the system is made time-independent by appropriate unitary transformations, and the corresponding steady-state solutions of the density matrix are deduced using appropriate rate constants. Results for stationary atoms indicate a dependence of the threshold for oscillation on the rf perturbation, together with rf resonance effects which increase the laser intensity some two or three times for suitable transitions. The increase in intensity is greatest for rf perturbations between the upper Zeeman levels, which also have the smaller resonance width, because of the longer lifetimes involved. Such effects will be small unless there is a reasonable difference in the populations of the Zeeman levels in the absence of the rf perturbation. Appropriate laser transitions, cavity tuning, and polarization characteristics must thus be used to obtain maximum effect, and examples are given. The atomic motion is included in the theory in an approximate way, and similar conclusions are derived as regards the variations of laser intensity. However, no dependence of the threshold on the rf perturbation is then indicated, the former being determined by the relatively large Doppler width. Finally, some consideration is given to cases where more than one laser transition occurs between the Zeeman levels.
Starting from the microscopic Hamiltonian of the three-level model and making use of the slowly varying amplitude approximation and the two-photon rotating wave approximation, we derive an effective Hamiltonian for the two-level two photon model under conditions where the two-photon process dominates the other radiartive processes, so that an exact relation between the three-level model and the two-level two-photon model is established. The effective Hamiltonian includes a term come from the optical Stark effect which was neglected in the conventional treatment. The coefficient of this term may have the same order of magnitude as the effective coupling constant. Based on this effective Hamiltonian, the standard method of Haken's laser theory is applied to study the threshold, the mean photon number, the self-induced frequency shift, the photon statistics and the fluctuation in steady states. Results are compared with previous ones which neglect the Strk effect.
The adequate description of the dynamic properties of macroscopic two-level systems interacting with the electromagnetic field is of particular interest in connection with the problem of choosing the working regime and active medium for the realization of single-pass resonator-free superradiative lasers. Since the derivation of an exact kinetic equation for such systems is a very complicated mathematical problem, different simplifying assumptions of physical nature are generally employed. There is, for example, the approach based on a Markov master equation developed in a series of papers by Bonifacio et al. [1-6]. The dynamics of the photon system with allowance for processes of incoherent pumping was studied on the basis of Zwanzig's approach [7] by Haake and Mandel [8, 9]. The Bogolyubov-Krylo v method of solution of nonlinear equations was used by Willis [10J. Some important estimates for the conversion times and other physical characteristics relating to the problem of creating a gamma laser were obtained by Khokhlov et al. [11-14] (see also the review [15]). Recently, exploitation of the deep analogies in the mathematical formulation of the model problems in the theory of superradiance and in polaron theory made it possible to obtain an exact generalization of the kinetic equation for superradiative systems [161 on the basis of the method of eliminating the boson variables developed in [17, 18]. It proved possible to obtain a number of physical results, in particular, to determine the magnitudes of the constant and collective frequency shift in the presence of inhomogeneous Lorentz broadening under conditions of cooperative spontaneous radiation. In following papers [19, 20], which were based on the generalized kinetic equation of [16], an equation for the single-particle distribution function was constructed. Comparison of this equation with Pauli's master equation made it possible to determine the characteristic relaxation times. There is however a simpler and more direct way of determining the probability F_ of transition of a radiator to an excited state with absorption of a photon and the probability F+ of transition to the de-excited state with emission of a photon (x~-F~ -', z are the characteristic relaxation times). It is associated with the use of the method of two-time temperature-depe ndent Green's functions. This method, which was introduced into statistical mechanics by Bogolyubov and Tyablikov [21], plays an important part in the theory of nonequilibrium processes (see, for example, [22, 23]). In the present paper, we investigate the dynamical properties of superradiative systems by means of two-time temperature-depe ndent Green's functions and, in particular, we obtain relations for the characteristic relaxation times. For clarity~ we consider the simplest model, characterized by the Hamiltonian