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Comment on SPIN-DEPENDENT SELECTION RULES FOR DIPOLE TRANSITIONS

A.S. Shumovsky

Published Jan 1, 2006• 0 citations

Abstract

Despite the fact that properties of atomic transitions are well known since the early days of quantum mechanics, the theoretical analysis of peculiarities of the spectroscopic data still attracts a great deal of interest (e.g., see recent monograph [1]). It is well known that some nanostructures like quantum dots and quantum wells behave like twoor multi-level atoms. For example, an exciton-polariton confined in a quantum well represents an ideal two-level atom, manifesting a number of specific phenomena such as collaps and revival of Rabi oscillations [2, 3, 4]. Similar effects can also be observed in quantum dots [5, 6]. Let us stress that those “artificial atoms” are considered nowadays in the context of design of a hardware for realization of quantum computing. It has been observed that the dipole transitions in excitons and some nanostructures can manifest spin-flip effects [7, 8]. In the article by M. Saglam et al, it is shown that even the conventional Dirac model of hydrogen atom surprisingly allows spin-flip effects. The corresponding rate seems to be quite high (more than 50%).

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Despite the fact that properties of atomic transitions are well known since the early days of quantum mechanics, the theoretical analysis of peculiarities of the spectroscopic data still attracts a great deal of interest (e.g., see recent monograph [1]). It is well known that some nanostructures like quantum dots and quantum wells behave like twoor multi-level atoms. For example, an exciton-polariton confined in a quantum well represents an ideal two-level atom, manifesting a number of specific phenomena such as collaps and revival of Rabi oscillations [2, 3, 4]. Similar effects can also be observed in quantum dots [5, 6]. Let us stress that those “artificial atoms” are considered nowadays in the context of design of a hardware for realization of quantum computing. It has been observed that the dipole transitions in excitons and some nanostructures can manifest spin-flip effects [7, 8]. In the article by M. Saglam et al, it is shown that even the conventional Dirac model of hydrogen atom surprisingly allows spin-flip effects. The corresponding rate seems to be quite high (more than 50%).

We have investigated the spin dependent selection rules for dipole transitions in the hydrogen atom by using the solution of Dirac equation. We showed that some transitions allow spin flip which is not possible according to the conventional selec-tion rules. As an example we look at 1s-2p dipole transitions. The percentage of the spin flip rate in (1s-2p) is found to be 56 % which is larger that the non-flip rate. The present result is believed to serve a significant help for understanding the recent observation of the spin flip in the excitonic transitions in nanostructures.

A theoretical investigation of the three-photon transition rates $2\phantom{\rule{0.16em}{0ex}}{}^{1}{P}_{1}\ensuremath{\rightarrow}2\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}\phantom{\rule{0.28em}{0ex}},1\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}$ and $2\phantom{\rule{0.16em}{0ex}}{}^{3}{P}_{2}\ensuremath{\rightarrow}2\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}\phantom{\rule{0.28em}{0ex}},1\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}$ for the helium atom is presented. Photon energy distributions and precise values of the nonrelativistic transition rates are obtained with employment of correlated wave functions of the Hylleraas type. The possible experiments for the tests of the Bose-Einstein statistics for multiphoton systems are discussed.

Transient nonlinear optical spectroscopy, performed on excitons confined to single GaAs quantum dots, shows oscillations that are analogous to Rabi oscillations in two-level atomic systems. This demonstration corresponds to a one-qubit rotation in a single quantum dot which is important for proposals using quantum dot excitons for quantum computing. The dipole moment inferred from the data is consistent with that directly obtained from linear absorption studies. The measurement extends the artificial atom model of quantum dot excitonic transitions into the strong-field limit, and makes possible full coherent optical control of the quantum state of single excitons using optical pi pulses.

Introduction Quantum dots are often referred to as artificial atoms, since they trap carriers in discrete energy-levels due to the nanoscale three-dimensional finite potential energy well they provide. As such, dots exhibit a coherent light–matter interaction that is similar to an atom. This is evidenced by observations of atom–optics phenomena such as Rabi oscillations [43, 26], power broadening [27], Autler–Townes doublet [14, 42], Mollow triplet [42, 8], and coherent population trapping [6]. In this chapter, Rabi rotation measurements are used to examine how an exciton transition deviates from an ideal two-level atom due to its interaction with a reservoir of phonons. The neutral exciton transition may be regarded as a two-level system, or qubit, composed of the crystal ground-state ∣0〉 and a single electron-hole pair ∣ X 〉. The state-vector of a qubit can be described as a pseudo spin-half. When an oscillating electro-magnetic field resonantly excites the two-level transition it drives an oscillation in the population inversion known as a Rabi oscillation. This results from the oscillations of the driving field and the dipole of the two-level system being synchronous, such that in its rotating frame, the driving field acts as a static magnetic field that causes the pseudo-spin to rotate. Coherent control of the pseudo-spin can be achieved by applying well-defined driving fields, enabling the preparation, and manipulation of superposition states. Such coherent control concepts have found widespread use in electron spin, and nuclear magnetic resonance spectroscopy.