Quantum fluctuations in a laser with two different relaxation times are considered, i.e., transverse (polarization relaxation) and longitudinal (population relaxation) in the case in which the cavity transmission band half-width is much smaller than the transverse width and much larger than the longitudinal one. The lasing frequency detuning from the transition frequency of a two-level system is assumed to be arbitrary in this case, and it is necessary to take into account the contribution of two-particle correlators into the dispersion and laser linewidth. The results are considered as applied to a semiconductor laser.
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.
A method of calculation of multiple-time correlation functions of the resonance fluorescence light from a two-level atom driven by a realistic laser field with phase and amplitude fluctuations is developed. Both kinds of fluctuations are assumed to be Gaussian. The phase fluctuations are treated in the phase-diffusion model. The averaging over the amplitude fluctuations can be performed in a closed way when the relative mean-square amplitude fluctuation is small compared with the ratio of the amplitude correlation decay rate to the characteristic atomic relaxation rate. There is no restriction to Markovian amplitude fluctuations. Arbitrary values for the mean Rabi frequency are allowed. Results are presented for the intensity, the intensity correlation function, and the spectrum of the fluorescent light. It is shown that even in the case when the relative strength of the amplitude fluctuations is small, significant effects can occur due to the finite correlation length of the fluctuations. The results for fast amplitude correlation decay correspond to the situation without amplitude fluctuations. When the correlation time of the amplitude fluctuations becomes comparable with the atomic relaxation times, the Rabi oscillations of the intensity and the intensity correlation function are nearly washed out, and non-Lorentzian line shapes for the side peaks in the spectrum are observed. It is found that the ratio of the heights of the central peak to the side peak increases with increasing correlation time.
The dependence of the multiphoton absorption probability for an atomic system on the statistical properties of the incident light is investigated. It is shown, by means of
nth-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the
(n,n)th normally ordered correlation functions of the electromagnetic field. For stationary fields, the transition probability per unit time is constant and is directly related to the reduced cross-spectral correlation function of the electromagnetic field. This expression for the transition probability is then analyzed under various conditions of coherence. Explicit calculations are carried out for thermal light and for laser light, and the transition rates are compared when each has a Lorentzian spectrum. In particular, when the width of the final atomic level is much larger than the bandwidth of the field, the transition rate for thermal light is
n! times higher than with laser light of the same mean intensity. The other extreme case, i.e., when the width of the final atomic level is much smaller than the bandwidth of the field, is also considered in detail. An equation for the density operator of the field is derived; this equation governs the change in the statistical properties of the photon system in
n-photon absorption processes. On the basis of this equation, the time-dependent properties of the photon system are then studied and some physical consequences of these equations are discussed. Some numerical solutions relating to the time dependence of the factorial moments in two-photon absorption processes are presented.
A quantum statistical theory of resonant optical phenomena is developed for the case when the fluctuations of the laser, used for exciting transitions, are important. The general equations describing the dynamics of a relaxing two-level atom (TLA) are solved exactly for different types of mean values and the two-time amplitude and intensity correlations. The phase-diffusion model is adopted for laser fluctuations. The exact results are used to analyze the effect of laser fluctuations on a number of optical effects-optical free-induction decay, adiabatic following, the spectrum of the scattered light from a relaxing TLA, the energy-absorption spectrum from a weak field, Hanle resonances, etc. In each case, the laser fluctuations are found to affect in an important way the characteristics of the above optical-resonance phenomena. For example, the spectrum of scattered radiation from a TLA has the usual three-peak structure for fields at resonance and with strengths $\ensuremath{\alpha}$ above the threshold and for $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\gamma}}_{c}$; however, now the peak heights (widths) are in the ratio
3x (
3x2), $x=\frac{[{\ensuremath{\gamma}}_{c}+\frac{({T}_{1}^{\ensuremath{-}1}+{T}_{2}^{\ensuremath{-}1})}{3}]}{[2{\ensuremath{\gamma}}_{c}+{T}_{2}^{\ensuremath{-}1}]}$, where ${\ensuremath{\gamma}}_{c}^{\ensuremath{-}1}$ is the correlation time for laser phase fluctuations. For ${\ensuremath{\gamma}}_{c}\ensuremath{\gg}\ensuremath{\alpha}$, one gets a single-peak spectrum. The laser field is treated as a second-quantized field with excitation in either a coherent state or a Fock state. The results, obtained by straightforward perturbation theory, but valid for arbitrary values of the relaxation parameters, detuning, and the laser correlation time, are also presented.