The response of a TLS to waves, phase- and frequency-modulated by a purely Markovian process, is calculated rigorously. The spectrum of luminescence characterised by two rates of phase relaxation Gamma 1 and Gamma 2 , rather than one, as in the case of phase diffusion ( Gamma 2 =4 Gamma 1 ), is found. Saturation of TLS and its absorption of light at arbitrary correlation between successive values of its randomly and suddenly varying frequency are determined. The probability of light absorption at any strength of its interaction with TLS and at any rate of frequency modulation is found for uncorrelated frequency change.
The population and phase relaxation in TLS induced by resonant radiation with stochastic amplitude was studied within the framework of sudden modulation theory. Kinetics and stationary light absorption as well as resonance fluorescence were calculated for light with complex and real amplitude. Although the latter is a very popular model of the former, it was shown to be inappropriate at strong interaction. The real (Rayleigh) light has complex amplitude random in magnitude and phase and changing in time as a Markovian chain, correlated or non-correlated. With respect to atom response to light of medium power, the analytically studied non-correlated field was shown to be similar to strongly correlated, “stochastic” field opened to only numerical inspection.
Sudden-modulation theory is used for the calculation of stationary saturation and polarization of two-level systems. The randomly fluctuating frequency of the two-level system is assumed to be a noncorrelated, purely discontinuous Markovian process. The stationary probability of field-induced transitions is calculated exactly. It is shown that if the frequency modulation is fast (or if the field is strong enough) the transition probability coincides with that obtained in the framework of perturbation theory when the latter is developed in its integrodifferential non-Markovian version. This probability differs from the kinetic probability calculated by Markovian perturbation theory (with fluctuating frequency) in its corresponding spectral wing shape and in the field dependence of its corresponding resonance width. This field dependence is smoother than that in Markovian theory and stabilizes when the resonance width is not equal to the natural transition width. Nevertheless, in the Lorentzian approximation the free-induction decay rate in the framework of exact (or non-Markovian) theory increases with the field in nearly the same way as that in Markovian theory. Almost the same is true of the field dependence of the free-induction decay amplitude in either theory, provided that the signal correlation before and after the field is switched off is neglected. The conditions are found under which the Lorentzian approximation of the transition probability is valid. These conditions are shown to break down when the theory (in the fast-frequency-modulation limit) leads to agreement with experiments reported by De Voe and Brewer [ Phys. Rev. Lett.50, 1269– 1272 ( 1983)].
A phase-diffusion stochastic field with frequency fluctuations obeying Gaussian Markovian statistics is produced at microwave frequency and used to drive a two-level spin system. The system is tuned to the two-photon (TP) resonance and its TP-induced second-order response is measured as a function of the characteristic parameters of the frequency noise. It is found that, when the statistics of the driving field are varied from the fast- to the slow-modulation limit, the power spectrum of the nonlinear response of the system varies in a continuous way from a Lorentzian form toward a Gaussian one, and its relative width (normalized to the input spectral width) varies from 4.0\ifmmode\pm\else\textpm\fi{}0.4 to 2.0\ifmmode\pm\else\textpm\fi{}0.2. The experimental results are interpreted on the basis of a semiclassical calculation which takes into account the stochastic nature of the field. The effect of the filter action of the cavity modes is discussed. The relationship with previous experiments and theoretical calculations of TP absorption is also examined.
The effect of environmental perturbations on a system of two-level atoms is to alter the coupling between the atoms and their surroundings, which manifests itself as a change in the relaxation rates of the atoms. One of the ways in which environmental fluctuations can affect an atom is to modulate its transition frequency in a stochastic manner. We study the response of such two-level atoms to irradiation by a monochromatic laser, and report the results of numerical experiments on the spectrum of scattered light. It is observed that this spectrum is a sensitive indicator of the parameters that describe the stochastic modulation. In addition, the spectra exhibit substantially different features depending on whether the stochastic process follows a diffusion or a jump mechanism. The numerical results are based on a Monte Carlo procedure, which can be applied to stochastic processes in a wide variety of biological, chemical, and physical processes.
The time evolution of a two-level system interacting with a classical monochromatic radiation field is calculated by Fourier transforming the equations of motion. This can be done exactly if one uses the convolution of the product of two Fourier transformations. The method produces recurrence relationships for the transforms. The time-dependent amplitudes of the states and the transition probability are evaluated by using two such recurrence relationships. The time-averaged probability shows characteristic resonances at odd multiples of the ratio of the natural frequency of the system to the frequency of the radiation field. The peaks of the resonances are displaced by the Bloch-Siegert shift. At these resonances the system is completely saturated with the flipping frequency dependent on the strength of the radiation field. The time dependence and the frequency spectrum of the dipole moment is determined at the main resonance and at the three-quantum resonance. The spectrum contains the zero-frequency terms which are responsible for the coherence resonances.
The present paper treats the electromagnetically induced transparency (EIT), i.e., the absorption spectrum of a weak probe field, near-resonant to a material transition, in the presence of a strong amplitude-phase fluctuating field which couples the upper level of the transition to another excited level. The coupling field is assumed to be chaotic (i.e., a complex Gaussian process with a Lorentzian spectrum) and so strong that the material relaxation and the detuning can be neglected. In addition, we consider the Fourier transform of the spectrum
Uaa(t), which determines absorption of a probe field with a time-dependent intensity. The quantity
Uaa(t) equals a diagonal element of the average reduced evolution operator of the two-level system (TLS), coupled by the strong field. As a starting point, we use the stationary, partially averaged stochastic Schr\"odinger equation, which has the form of a set of two second-order differential equations. The theory is developed by combining analytical and numerical approaches. For the most important regime, when the rms Rabi frequency
V0 is much greater than the coupling-field bandwidth $\ensuremath{\nu},$ we obtain scaling relations for the EIT line shape and the TLS evolution. In particular, the absorption at the minimum (i.e., at the resonance frequency) scales as ${\ensuremath{\nu}}^{1/3}{V}_{0}^{\ensuremath{-}4/3},$ whereas the evolution function
Uaa(t) performs damped oscillations with the decay rate on the order of ${(V}_{0}^{2}\ensuremath{\nu}{)}^{1/3}.$ These results differ significantly from those following from the uncorrelated-jump model. The results of this paper are shown to be applicable to more general models of amplitude-phase fluctuations than the Gaussian-Markovian chaotic field.