Dipole-dipole interaction in photonic-band-gap materials doped with nanoparticles Mahi R. Singh Department of Physics and Astronomy, The University of Western, Ontario, London, Canada N6A 3K7 Received 17 July 2006; published 12 April 2007 A theory of linear susceptibility has been developed in the presence of dipole-dipole interaction for photonic- band-gap PBG materials doped with an ensemble of five-level nanoparticles. An external probe laser induces dipole moments in nanoparticles. When the concentration of the particles is high, the induced dipoles interact with one another through dipole-dipole interaction. Mean-field theory is used to include the effect of dipole- dipole interaction in the calculation of susceptibility. Numerical simulations are performed for the real and imaginary susceptibilities and it is found that the system switches between inversionless and noninversionless states. In addition the system switches between absorptionless and nonabsorptionless states. These occur when the resonance energy lies near the valence-band edge of the photonic-band-gap material. The theory also predicts a polarization catastrophe in PBG materials doped with nanoparticles. DOI: 10.1103/PhysRevA.75.043809 PACS numbers: 42.50.Gy, 42.65.An, 42.50.Md, 32.80.Qk I. INTRODUCTION Recently, there is considerable interest in studying quan- tum coherence and interference phenomena in atomic gas systems. These phenomena include lasing without inversion, enhancement of refractive index, coherent photon trapping, and electromagnetically induced transparency 1–3. Quan- tum coherence and interference have also been studied in photonic and polaritonic-band-gap materials which have en- ergy gaps in their photonic energy spectra 4 –11 and many interesting effects have been predicted. For example, we have studied electromagnetically induced transparency in a photonic-band-gap material which is doped with an en- semble of four-level nanoparticles 9. Two laser beams are applied to the atom, pump and probe. It is found that the electromagnetically induced transparency effect disappears when one of the resonance frequencies lies close to the band edge. We have also studied the enhancement of the refractive index when an ensemble of three-level atoms is doped in a polaritonic-band-gap material in the presence of a single la- ser field 10. Singh and Haque 11 have studied coherent photon trapping in photonic-band-gap materials doped with three-level atoms in the presence of two laser fields. They found that the trapping effect disappears at the band edges. In these papers, the authors have considered that the density of the doped nanoparticles is small so that dipole-dipole in- teraction DDI between the particles can be neglected. The aim of this paper is to include the effect of DDI in the study of the quantum coherence and interference in photonic-band-gap PBG materials doped with multilevel nanoparticles. DDI has been studied by Dowling and Bowden 12 in a three-level atomic gas system. They found that for certain values of the atomic density there is an en- hancement of inversionless gain and absorptionless refrac- tive index. Manka et al. 13 have extended the work of Dowling and Bowden 12 to study the influence of atomic nonlinearities and predicted a density dependent switching between absorption and amplification. Calderon et al. 14 studied DDI in V-type atoms in the presence of one and two laser fields. They found that the relative phase difference between the probe and the pump fields changes the system from absorption to gain. Afansev et al. 15 have found an expression of susceptibility for V-type atoms in the presence of DDI and a bichromatic field. Some work has also been done to study the effect of DDI in photonic- and polaritonic-band-gap materials. For ex- ample, John and Quang 6 have studied self-induced trans- parency due to DDI in PBG materials doped with two-level atoms. Singh, and Singh and Haque have done some work on DDI in coherent population trapping and polarization 16. In the present paper, a PBG material is doped with an ensemble of five-level nanoparticles which are randomly dis- tributed throughout the system. A schematic diagram of the five-level atom is shown in Fig. 1. A probe laser field is applied to the system and it induces a dipole moment in each particle. When the concentration of the particles is high, the induced dipoles interact with each other through DDI. Mean- field theory is used to include the effect of DDI in the cal- culation of susceptibility. We consider that these atoms are not only interacting with each other but also interacting with the PBG material which is acting as a reservoir. The density- matrix method and linear-response theory are used to calcu- late the expression of susceptibility from which the absorp- tion coefficient and the refractive index can be calculated. Numerical simulations for the real and imaginary suscep- tibilities are performed for a doped PBG material. It is found a b c d Probe field e FIG. 1. A schematic diagram of a five-level nanoparticle. The levels are denoted by |a, |b, |c, |d, and |e. The nanoparticles are prepared as a linear combination of levels |b and |c. A probe laser couples level |a to levels |b and |c. Level |a decays to level |e and levels |b to |c decay to level |d. PHYSICAL REVIEW A 75, 043809 2007 1050-2947/2007/754/04380910 ©2007 The American Physical Society 043809-1