Self-energy shift of the energy levels of atomic hydrogen in photonic crystal medium

Abstract

Abstract. Corrections to the average kinetic energy of atomic electrons caused by the change in electron mass in the photonic crystal medium are investigated. Corresponding shift of energy levels of atoms placed in a photonic crystal is shown to be of order of the ordinary Lamb shift.The investigation of photonic crystals (PC) is an actual problem today. The main feature of such media is the periodic modulation of dielectric constant along one, two or three directions of space. It yields to the range of forbidden frequencies in a material called photonic band-gaps (PBG). Unusual optical properties of the PC medium stimulate a wide range of its practical applications [1]. Another interesting field of study is quantum electrodynamics (QED) in PC medium. It has been shown that modification of photon density of states (DOS) in PC's lead to the strong emitter-photon coupling [2][3][4], coherent control of spontaneous emission [5][6][7], appearance of photon-atom bound states [2,8,9] etc. Special attention was given to the modification of the Lamb shift [2,8,[10][11][12][13][14] in atoms placed in the photonic crystal medium. By definition the Lamb shift in atoms is caused by the QED corrections to the Coulumb interaction between electrons and the atomic nucleus (see figure 1a) that gives the leading order contribution to the energy of atomic states. This interaction is modified by the processes in which the virtual photons come into play (see figure 1b). The contribution to atomic energy levels from these processes is just the Lamb shift. In principle one should account for the processes of the self interaction of the atomic electrons shown in figure 1c. In the case of atoms in free space these processes contribute only to the electron mass and are subtracted by the counter-terms of the renormalization. However, as it has been shown in Ref.[15], a modification of the of the interaction of the electron with its own radiation field in a PC gives rise to the fact that contributions from the processes in a PC shown in figure 1c differ profoundly from those in free space, and, as a result, these self-energy processes can play an important role. First of all, they change the mass of an electron in the PC medium [15]. In addition they contribute to the kinetic energy of atomic electrons in the PC. The aim of this paper is to investigate this correction to the energy levels of atoms in the PC medium.The change of the value of the kinetic energy of an electron δE is the difference between its values in the PC medium and in vacuum:

Influence of the photonic crystal medium on the self-interaction of electrons

“…This in turn has a significant effect on the self-energy function of the electron. Further estimates can be made with the model of N p (ω) analogous to the one proposed in [14]. The results of our calculations of the self-energy function C p (z) displayed in figure 1 show its strong dependence on z.…”

“…Being equivalent to the Schrödinger equation in the case when the interaction in a quantum system is instantaneous, GDE allows one to extend dynamics to the case of nonlocal-in-time interactions. This equation provides a new insight into many problems in atomic physics [17][18][19][20], nuclear physics [21][22][23][24] and quantum optics [25,26]. The method allows one to take into account from the every beginning that the contribution to the Green operator G(z), which comes from the processes associated with the self-interaction of particles, has the same structure as the free Green operator G 0 (z).…”

“…The Bloch structure of the photon states in the PC medium means that these states can be expanded in a set of Bloch states. The interaction Hamiltonian describing the light-matter coupling in the PC medium is H pc I = − e me p · A pc , where p is the electron momentum, and A pc is the PC-medium quantized vector potential (for details see [14]). Let us assume that the perturbation theory is applicable in this case, and the leading order solution for M (z) is M (z) = H pc I .…”

“…being the PC medium correction to the electron mass derived in [14]. Expression (4) for δm pc can be simplified [14]:…”

“…At the same time, in Ref. [14] it has been shown that there is a novel effect in PC medium which consists in the fact that the rest mass of an electron should change its value. Actually, this effect is another consequence of the periodic changing of the photonmatter interaction in the PC medium.…”