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Advances in Natural Sciences Nanoscience and Nanotechnology

Lamb and ac Stark shifts in cavity quantum electrodynamics*

Bich Ha Nguyen

Published Sep 1, 2010• 3 citations
Publisher

Abstract

Abstract A quantum system consisting of a single-mode microcavity and a two-level quantum dot or atom placed inside this microcavity, the interaction of monoenergetic photons in the microcavity with the electron in the two-level quantum dot or atom being the allowed electrical dipole transition between two energy levels, is a basic element of prospective devices for quantum information processing. Its time evolution is governed by cavity quantum electrodynamics (CQED). The photon–electron coupling induces the shifts of energy levels of the electron in the two-level quantum dot or atom. If there is no photon in the microcavity, then the shift of electron energy levels is induced by the radiative corrections and it is called the Lamb shift. The presence of photons in the microcavity induces another type of electron energy level shift—the ac Stark shift. In this paper, we present the exact derivation of the formulae for the Lamb and ac Stark shifts of electron energy levels in the two-level quantum dot or atom placed inside a single-mode microcavity.

DOI
10.1088/2043-6262/1/3/035008

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Abstract A quantum system consisting of a single-mode microcavity and a two-level quantum dot or atom placed inside this microcavity, the interaction of monoenergetic photons in the microcavity with the electron in the two-level quantum dot or atom being the allowed electrical dipole transition between two energy levels, is a basic element of prospective devices for quantum information processing. Its time evolution is governed by cavity quantum electrodynamics (CQED). The photon–electron coupling induces the shifts of energy levels of the electron in the two-level quantum dot or atom. If there is no photon in the microcavity, then the shift of electron energy levels is induced by the radiative corrections and it is called the Lamb shift. The presence of photons in the microcavity induces another type of electron energy level shift—the ac Stark shift. In this paper, we present the exact derivation of the formulae for the Lamb and ac Stark shifts of electron energy levels in the two-level quantum dot or atom placed inside a single-mode microcavity.
In free space, where the vacuum mode density varies relatively slowly in the spectral region of an atomic transition, an atom’s real-field and vacuum interactions are effectively decoupled. Consequently, spontaneous decay and the Lamb shift are, for all practical purposes, independent of any real-field atomic perturbation. However, in a colored vacuum (i.e., a low-Q cavity) the vacuum mode density can change significantly in the vicinity of an atomic transition, so that a Stark shift will alter an atom’s vacuum environment and thereby couple real-field and vacuum effects. Since the ac Stark shift is an inherent aspect of multiphoton processes, this coupling is unavoidable for highly nonlinear field–atom interactions that occur in cavities. Here we consider this effect for 3+2 resonance-enhanced multiphoton ionization of xenon.
A single atom in a cavity is the model system of cavity quantum electrodynamics (CQED). The strong coupling regime between the atom and cavity-confined photon corresponds to the reversible exchange of energy between the two modes, and underpins a wide range of CQED phenomena with applications in quantum information science, including for example as quantum logic gates and as sources of entangled states. An important advance was achieved recently when strong coupling between excitons and cavity photons was reported for the first time for localized quantum dots (QDs) in micron-size solid state cavities. This has significance in terms of scalability and integration with other optical devices, and could lead to the emergence of "quantum optics on a chip" technology. However the results presented so far for quantum dots are in the linear regime, corresponding to coupling to the vacuum field (vacuum Rabi splitting); they are not a true QED effect and can equally well be described by classical physics as the coupling between two oscillators. In this paper, we present evidence for a purely quantum phenomenon for the QD/cavity photon system, namely the increase in splitting of the levels when the mean number of photons in the cavity is increased. This corresponds to non-linearities on the single-photon scale: the presence of a single excitation in the cavity changes the level structure, affecting the emission energies for a second photon. Such results are a first step in demonstrating the promise of quantum dots for CQED applications.
In this paper, we investigated the dynamics of atomic entanglement in a quantum system consisting of two identical two-level atoms (qubits) resonantly interacting with a mode of a thermal cavity field through degenerate two-photon transitions, in the presence of a Stark energy level shift. An analytical expression is obtained for the atom entanglement (negativity) parameter for separable and entangled initial states of atoms. The influence of the Stark shift on the degree of atom-atom entanglement is considered. It is established that the Stark shift leads to a significant increase in the degree of atom entanglement in the case of separable initial states of atoms and to stabilization of atomic entanglement in the case of entangled initial states of atoms.
Abstract The general theory of the time-resolved luminescence of the coupled system consisting of a single-mode microcavity and a two-level quantum dot containing one electron placed inside this microcavity is presented. It is based on the study of the time evolution of the density matrix of a larger system consisting of one electron in the two-level quantum dot, single-mode photons in the microcavity and external photons in a spatial region in which the emitted photons are detected. The decoherence of the system is taken into account in the Markov approximation. The explicit analytical form of the time dependence of the intensity of the emitted photon beam is established. It depends not only on the physical parameters of the system but also contains the matrix elements determining the initial condition of the luminescence process.
The interaction of light with matter is at the heart of quantum optics, which itself enables insight into the fundamental aspects of quantum mechanics. First experimental access to this research field has been realized by coupling atoms to light. Here, the transition between discrete energy states of the atom is associated with the absorption and emission of a photon, a single quantum of the electromagnetic field. As a central aspect of quantum optics, the light-emitter interaction can be significantly enhanced by placing the emitter in optical cavity that is on resonance with the emitter. In recent years, this has led to a rapidly evolving research field known as cavity quantum electrodynamics (CQED). In CQED two different regimes are distinguished: the strong and the weak coupling regimes. In the strong coupling regime, the emitted photon is reflected from the cavity mirrors and eventually reabsorbed by the emitter. In contrast, the weak coupling regime describes the irreversible emission, where the photon leaks out of the cavity before it can be reabsorbed. Both the weak and strong coupling regimes enabled fundamental experiments for a better understanding of quantum optical phenomena. \n \n \n \nCQED grants access to the quantum world and hence offers potentially revolutionizing applications, particularly in the field of quantum information processing. A central aspect for the successful implementation of quantum applications is the system's scalability. Unfortunately, placing atoms deterministically inside a cavity remains technologically elaborate and hence minimizes the prospect of scaling a atom-CQED system. \n \nA possibility to address this issue is to implement CQED in the solid state, where sophisticated fabrication strategies allow miniaturization and scalability of the system. Particularly the development of self-assembled quantum dots (QD) in semiconductors represent a promising route. QDs can be considered as artificial atoms that mimic the atomic two-level system. These structures interact strongly with light and therefore have the potential for replacing atoms in CQED. As a central advantage, QDs are naturally trapped, which greatly simplifies the deterministic incorporation into the cavity. \n \nIn recent years, many efforts have been made to couple self-assembled QDs to microcavities. Generally, the successful implementation of CQED requires a cavity with a high quality factor Q and a low mode volume. In a majority of the approaches, the high Q/small mode volume cavities were monolithically defined around the QD, embedding the QD at a fixed position inside the cavity. Both the weak and strong coupling regimes have been reached with these systems. However, for future applications they suffer from several disadvantages. The fixed position of the QD inside the cavity minimizes the prospects for spectral tunability and spatial positioning the QD inside the cavity. Furthermore, prospects for further increasing of the cavity Q-factor and minimization of the mode volume remain limited in these systems. \n \nIn this thesis the mentioned disadvantages are addressed by developing a fully tunable miniaturized Fabry-Pérot microcavity with low mode volume. The design enables both spatial positioning of the emitter inside the cavity and spectral tunability. Successful coupling of a single QD to the microcavity is demonstrated including the strong coupling regime. Further a new approach to decrease the cavity mode volume is presented, where we demonstrate weak coupling is achieved
We present an universal description for Lamb shift of a lossy two-level-emitter (TLE) inside a leaky cavity by combining the quantum dissipation theory and the perturbative treatment developed in cavity QED. The total spectrum function of bath associated with the TLE is deduced by the standard method developed for a quantum dissipation model, from which the total spontaneous emission rate and Lamb shift are calculated. The developed calculation scheme is able to reproduce the known Purcell enhancement factor and is shown to be in agreement with the analysis made for a confocal resonator. It is found that the Lamb shift due to a leaky cavity mode has a sharp drop when sweeping across the resonance point. The physics behind this unusual behavior and the related experimental observations made on both atom-cavity and quantum-dot cavity systems are also analyzed.