Transition rates of an electron confined in a quantum well due to interaction with photon (electromagnetic radiation) are approximated and compared with the exact results. The energy levels of the system were calculated using the Schrödinger equation within the effective mass approximation. Wave functions of an electron confined in a quantum well are approximated, which are in turn used to approximate optical transition rates when an electron interacts with electromagnetic radiation. The approximate wave functions exhibit close agreement with the exact functions at lower quantum states but become less accurate as the principal quantum number increases. The energy eigen values of the system increase with decreasing quantum well width and the accuracy of the approximate energy eigen values decreases for higher states. Transition rates are characterized by peaks which occur whenever the energy of the incident photon equals the energy difference between states. Since the accuracy of the approximations decreases for higher energy states, the positions of the peaks corresponding to transitions involving higher states become less accurate. Increase in temperature decreases peaks of absorption transition rates while it enhances peaks of emission transition rates, a behavior captured well by the approximations.
Transition rates of an electron confined in a quantum well due to interaction with photon (electromagnetic radiation) are approximated and compared with the exact results. The energy levels of the system were calculated using the Schrödinger equation within the effective mass approximation. Wave functions of an electron confined in a quantum well are approximated, which are in turn used to approximate optical transition rates when an electron interacts with electromagnetic radiation. The approximate wave functions exhibit close agreement with the exact functions at lower quantum states but become less accurate as the principal quantum number increases. The energy eigen values of the system increase with decreasing quantum well width and the accuracy of the approximate energy eigen values decreases for higher states. Transition rates are characterized by peaks which occur whenever the energy of the incident photon equals the energy difference between states. Since the accuracy of the approximations decreases for higher energy states, the positions of the peaks corresponding to transitions involving higher states become less accurate. Increase in temperature decreases peaks of absorption transition rates while it enhances peaks of emission transition rates, a behavior captured well by the approximations.
We calculate two-photon transition rates of exciton states in GaAs quantum wells for varying well widths. The exciton states are described in the effective-mass approximation and are obtained variationally assuming wave functions that are separable in the r coordinates. The two-photon transition rate is calculated taking explicitly into account the sum over all the intermediate states, using a variational procedure. We calculate transition rates of excitons of s symmetry as well as of p symmetry, which are allowed for polarization of the radiation beam along the growth axis and in the layer planes, respectively. From our calculations we find that transitions to s-type excitons have sufficiently large rates to be observable, whereas transitions to p-type excitons seem to have rates too small to be observable; these results are in agreement with experiment. The transition rate of s excitons increases significantly for decreasing well width, whereas the transition rate of p excitons is found to be almost independent of the well width. Moreover, our model allows us to calculate transition rates for photons of different energy. We have shown the resonant enhancement of the transition rate of s-type excitons when one of the photon energies approaches the energy of an intermediate state.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
Spontaneous light emission is central to a vast range of physical systems and is a founding pillar for the theory of light–matter interactions. In the presence of complex photonic media, the description of spontaneous light emission usually requires advanced theoretical quantum optics tools such as macroscopic quantum electrodynamics, involving quantized electromagnetic fields. Although rigorous and comprehensive, the complexity of such models can obscure the intuitive understanding of many quantum-optical phenomena. Here, we review a method for calculating spontaneous emission and other quantum-optical processes without making explicit use of quantized electromagnetic fields. Instead, we introduce the concept of transition currents, comprising charges in matter that undergo transitions between initial and final quantum states. We show how predictions that usually demand advanced methods in quantum electrodynamics or quantum optics can be reproduced by feeding these transition currents as sources to the classical Maxwell equations. One then obtains the relevant quantum observables from the resulting classical field amplitudes, without washing out quantum optical effects. We show that this procedure allows for a straightforward description of quantum phenomena, even when going beyond the dipole approximation and single emitters. As illustrative examples, we calculate emission patterns and Purcell-enhanced emission rates in both bound-electron and free-electron systems. For the latter, we derive cathodoluminescence emission and energy-loss probabilities of free electrons interacting with nanostructured samples. In addition, we calculate quantum-beat phenomena in bound-electron systems and wave function-dependent optical coherence in free-electron systems. Remarkably, the transition-current formalism captures more complex phenomena, such as many-body interference effects and super-radiance of both bound- and free-electron systems, second-order processes such as two-photon emission, and quantum recoil corrections to free-electron radiation. We review a variety of light–matter interactions in fields ranging from electron microscopy to nanophotonics and quantum optics, for which the transition-current theoretical formalism facilitates practical simulations and a deeper understanding of novel applications.
Optical properties of quantum well nanostructure for typical GaAs/AlxGa1-xAs are investigated in a new approach. By using the point canonical transformation method and numerical solution of the Schrödinger equation, the energy levels and wave functions of electrons in the quantum well confinement potential are calculated. Then, the effect of position-dependent effective mass on the intersubband optical absorption coefficient and the refractive index changes of the quantum well calculated by compact density matrix method, are studied. Results are compared with the case of constant mass. Our results show that with considering spatially varying electron effective mass linear and nonlinear optical properties increases and shifts toward the lower energies. Researches on this matter open a new field in fundamental and applied physics, and also offer a wide range of potential applications for photonic devices.
A steady-state cascade theory has been set up for radiative electron transitions. These are assumed to occur between a continuum and various excited states, as well as between any two excited states, of hydrogen-like atoms. The work contains two features which have not previously been fully taken into account: (1) Both spontaneous and induced transition probabilities have been included exactly. (2) In addition to the radiative transitions, the reverse transitions due to absorption of background radiation have also been included. The following graphical results are given: (a) The steady-state occupation probabilities of the excited states as a function of excess electron density. (b) A "sticking probability" Pn (for an electron from a level n to reach the ground state without leaving the atom) as a function of the principal quantum number n. (c) The effect of the cascade on the transition rate into the ground state. The calculation is valid for semiconductors and for the analogous astrophysical problem. Temperature dependences have also been studied. The graphs shown bear out quantitatively the expectation that Pn decreases as either the temperature or the principal quantum number increases.
The possibility of the electron oscillations in a two-well potential is investigated in the article. The Hamiltonian of the system is calculated taking into account the formulas of the James-Cummings model. The unitary dynamics is carried out using a general solution of the Schrodinger equation. The conditions for the parameters of the problem under which the oscillation takes place are shown. These conditions for the exact formulation are discrete. However, it is valuable to find out at what parameters of the problem there occur approximate oscillations, at which an incomplete transition of a particle from one well to another is observed. In the standard manuals on the quantum theory of a single particle, the Hamiltonian was considered from the very beginning without photons, so it was assumed that complete oscillations always take place. This approximate consideration corresponds to reality only with a high potential between the wells. If the potential is low, the wells actually merge into one well, so that the spectrum and eigenvalues of the Hamiltonian become similar to the spectrum and eigenvalues of a one-dimensional potential – a radically different case. Such a discrepancy occurs due to ignoring the interaction of the electron with the electromagnetic field. If such an interaction is taken into account we get the dynamics described by the Jaynes-Cummings Hamiltonian. At a low potential, the energy gap between the ground and excited energy states of the electron grows. Since the probability of photon emission is proportional to this value, the interaction with the field cannot be ignored.