Lorentz-Lorenz shift in a Bose-Einstein condensate

We study the quantum field theory of light-matter interactions for quantum degenerate atomic gases at low light intensity. We argue that the contact interactions between atoms emerging in the dipole gauge may be ignored. Specifically, they are canceled by concurrent infinite level shifts of the atoms. Our development yields the classic Lorentz-Lorenz local-field shift of the atomic resonance. 03.75.Fi,42.50.Vk,05.30.Jp Typeset using REVTEX 1 The understanding of the interactions of light with matter in quantum degenerate systems has become especially topical after the first evidence for a Bose-Einstein condensate (BEC) of an atomic gas has appeared [1–3]. Recently, Andrews et al. [4] have also reported on non-destructive optical detection of a Bose condensate. In the limit of large detuning of the driving light from atomic resonance, the dynamics of the light and matter fields may be decoupled. It then turns out that the spectrum of the scattered light conveys direct signatures of atom statistics [5–7], and that under various conditions even the phase of the macroscopic wave function may be observed optically [8–11]. On the other hand, for a dense enough sample and small enough atom-field detuning, the analysis of the response of matter requires a concurrent treatment of light with its own dynamics. The result is that nearby atoms alter the optical response of each other. A large body of work in this direction [12–18] seems to have brought about the general notion that atom statistics should have only a minor effect on absorption, dispersion or diffraction of light [5,19]. Nonetheless, even for the Maxwell-Boltzmann gas and for the simplest optical properties such as refractive index, there still are no proven solutions for microscopic theories regarding a dense, near-resonance sample. In this paper we continue our rigorous quantum field theoretical analysis of light-matter interactions [5,18]. We argue that the contact interaction terms between different atoms that arise in the length gauge do not have any effect on light-matter dynamics. As a result, for a model BEC we find precisely the Lorentz-Lorenz (LL) shift [20] familiar from classical electrodynamics [21,22]. In the many-particle formalism of light-matter interactions, the Hamiltonian is often transformed into the length gauge using the Power-Zienau-Woolley transformation [23]. The electric displacement is then the basic dynamical degree of freedom, instead of the electric field, and the Hamiltonian picks up a polarization self-energy term