The reflected and transmitted energy flux corresponding to the linear response of a quantum well (QW) is calculated analytically for the cases of monochromatic and pulsed incident radiation normal to the QW plane. The light–matter interaction has been taken to all orders of perturbation theory, resulting in the appearance of radiative broadening in the expressions for the secondary fields. Three time scales determine the decay of the response after the pulse excitation while the quantum beats are superimposed on the exponentially decaying contributions at the frequency which (i) equals detuning from the electronic transition or (ii) corresponds to a characteristic excitation in the system.
The reflected and transmitted energy flux corresponding to the linear response of a quantum well (QW) is calculated analytically for the cases of monochromatic and pulsed incident radiation normal to the QW plane. The light–matter interaction has been taken to all orders of perturbation theory, resulting in the appearance of radiative broadening in the expressions for the secondary fields. Three time scales determine the decay of the response after the pulse excitation while the quantum beats are superimposed on the exponentially decaying contributions at the frequency which (i) equals detuning from the electronic transition or (ii) corresponds to a characteristic excitation in the system.
A theory for the response of a 2 D two-level system to irradiation by a symmetric light pulse is developed. Under certain conditions, such an electron system approximates an ideal solitary quantum well in a zero field or a strong magnetic field H perpendicular to the plane of the well. One of the energy levels is the ground state of the system, while the other is a discrete excited state with energy ℏω 0 , which may be an exciton level for H =0 or any level in a strong magnetic field. It is assumed that the effect of other energy levels and the interaction of light with the lattice can be ignored. General formulas are derived for the time dependence of the dimensionless “coefficients” of the reflection ℛ( t ), absorption A ( t ), and transmission ℐ( t ) for a symmetric light pulse. It is shown that the ℛ( t ), A ( t ), and ℐ( t ) time dependences have singular points of three types. At points t 0 of the first type, A ( t 0 )= T ( t 0 )=0 and total reflection takes place. It is shown that for γ r ≫γ, where γ r and γ are the radiative and nonradiative reciprocal lifetimes, respectively, for the upper energy level of the two-level system, the amplitude and shape of the transmitted pulse can change significantly under the resonance ω l =ω 0 . In the case of a long pulse, when γ l <γ r , the pulse is reflected almost completely. (The quantity γ l characterizes the duration of the exciting pulse.) In the case of an intermediate pulse duration γ l ≅γ r , the reflection, absorption, and transmission are comparable in value and the shape of the transmitted pulse differs considerably from the shape of the exciting pulse: the transmitted pulse has two peaks due to the existence of the point t 0 of total reflection, at which the transmission is zero. If the carrier frequency ω l of light differs from the resonance frequency ω 0 , the oscillating ℛ( t ), A ( t ), and ℐ( t ) time dependences are observed at the frequency Δω=ω l −ω 0 . Oscillations can be observed most conveniently for Δω≅γ l . The position of the singular points of total absorption, reflection, and transparency is studied for the case when ω l differs from the resonance frequency.
The reflection, transmission, and absorption of a symmetric electromagnetic pulse are calculated. The carrier frequency of the pulse is close to the frequency of direct interband transitions in a quantum well (QW). The QW energy levels are assumed to be discrete, with two closely spaced excited levels being taken into account. The QW width is assumed to be sufficiently large and comparable to the light wavelength corresponding to the pulse carrier frequency. In this case, the dependence of the momentum matrix element for the interband transition on the light wave vector should be taken into account. The refractive indices of the QW and barriers are assumed to be equal. The problem is solved for an arbitrary relation between the radiative and non-radiative lifetimes of the excited electronic states. It is shown that spatial dispersion considerably affects the shapes of the reflected and transmitted pulses. The greatest changes occur in the case where the inverse radiative lifetime is close to the difference between the frequencies of the interband transitions considered.
The optical response of radiatively coupled semiconductor multiple-quantum-well structures is investigated theoretically. It is shown that the transverse optical field leads to a coupling of exciton states within each well which causes a radiative decay and a mixing of excitonic resonances. Simultaneously, the field-induced long-ranged coupling between different wells leads to collective effects that are very sensitive to the detailed geometry of the structure. For a quantum-well spacing equal to an integer multiple of half the optical wavelength inside the medium, it is predicted that the collective effects cause a stimulated decay of electronic excitations that should be observable in either transmission or reflection geometry. On the other hand, in a quarter wavelength structure, the light-induced coupling causes an interwell energy transport and a splitting of the excitonic resonances, that should be observable as quantum beats in the time-resolved transmitted or reflected signal.
Using a perturbation theory based on the density-matrix formulation, we study the nonlinear optical responses of a noninteracting three-level model system to consecutive coherent pulsed excitations, as realized in several recent experiments involving excitonic transitions in quantum wells. The terahertz emission, which is a second-order response, and the four-wave mixing signal, a third-order response, are calculated within the rotating-wave approximation, in the presence of detuning, dephasing, and inhomogeneous broadening. We study the quantum beats in the photon echo from transient four-wave mixing experiments of a three-level system. We find that the temporal profile of the photon echo signal in the four-wave mixing experiments depends very strongly on the amount of inhomogeneous broadening of the energy levels involved. Both the position and the intensity of the peaks exhibit a smooth evolution from strong quantum beats to a conventional photon echo as the inhomogeneous broadening increases. These features from our noninteracting model system are compared with a recent experiment, and found to account for a number of experimental observations.
The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.
The time-dependent theory of the three-step absorption of three different light pulses in the electronic four-level systems is considered using the time-dependent perturbation theory. The first, the second, and the third pulse are in resonance with transitions from level 0 to level 1, from level 1 to level 2, and from level 2 to level 3, respectively. The spectral and temporal behaviour of the probability that the fourth level is excited at the moment of time t depends on the energy relaxation constants of the excited electronic levels, the frequencies of the maxima, the spectral widths (and so the durations) of the light pulses, and the time delays between the pulses. To conclude, there are 12 parameters that affect the spectra in the simple model used. We calculate and analyse the case where the frequencies of the maxima of the first and the second pulse are fixed and the frequency of the maximum of the third pulse varies. On the whole, three lines may exist, one of which corresponds to the coherent contribution and the others do not.