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Physical Review A

Quantum field theory of atoms interacting with photons. III. Scattering of weak cw light from cold samples of bosonic atoms

Li You, Maciej Lewenstein, Roy J. Glauber, J. Cooper

Published Jan 1, 1996• 47 citations
Publisher

Abstract

We study the coherent scattering of weak light from a system of cooled bosonic atoms in a trap. We derive and discuss in detail the scattering equation for such a process that describes multiple scatterings from different atoms. Our theory accounts for atom-atom interactions and determines the density profile of the atomic fields from the self-consistent Bogoliubov-Hartree method. We present two kinds of analytic solutions to the scattering equation using the on-shell approximation and the generalized diffraction theory. We discuss the validity of those approaches, and demonstrate that they both lead to similar results. Our studies indicate that the scattering cross sections exhibit non-Lorentzian shapes with narrow features close to resonance. Coherent scattering occurs mainly in the forward direction, and the angular distribution is limited by the size of the atomic sample in the trap. We present results for various ranges of relevant parameters such as trap size, interaction strength, number of atoms, and temperature, etc. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.329

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We study the coherent scattering of weak light from a system of cooled bosonic atoms in a trap. We derive and discuss in detail the scattering equation for such a process that describes multiple scatterings from different atoms. Our theory accounts for atom-atom interactions and determines the density profile of the atomic fields from the self-consistent Bogoliubov-Hartree method. We present two kinds of analytic solutions to the scattering equation using the on-shell approximation and the generalized diffraction theory. We discuss the validity of those approaches, and demonstrate that they both lead to similar results. Our studies indicate that the scattering cross sections exhibit non-Lorentzian shapes with narrow features close to resonance. Coherent scattering occurs mainly in the forward direction, and the angular distribution is limited by the size of the atomic sample in the trap. We present results for various ranges of relevant parameters such as trap size, interaction strength, number of atoms, and temperature, etc. \textcopyright{} 1996 The American Physical Society.
We discuss a method for probing a system of cold bosonic atoms in a trap using intense short laser pulses. Above the critical temperature for Bose-Einstein condensation, such a system scatters very weakly. Coherent scattering occurs primarily in the forward direction within a solid angle determined by phase matching conditions for the thermal atomic distribution. Below the critical temperature, the number of scattered photons increases dramatically and the scattered light is emitted into a solid angle determined by the size of the condensate. Quantum statistics of the atoms explicitly effect the spectrum as well as the squeezing properties of the scattered light. The theory accounts for the atom-atom interactions.
We study interactions of light with a sample of two-level atoms. Quantum statistical effects of degenerate boson and fermion gases are described by using the quantum field theory of atom-photon interactions, fully including the angular momentum degeneracies. Coupled propagation equations are given for light and matter fields, and plausible general simplifications are enumerated.^ In the limit of large atom-field detuning, multiple scattering of light and QED linewidths and line shifts may be ignored, and the propagation equations of light and matter fields may be decoupled. A general expression for the spectrum of the scattered light is derived, complete with the dependence on atomic level scheme, on various light polarizations, and on the geometry of the experiment. With these simplifications we are able to find distinct qualitative features in the spectrum of the scattered light in the case of both fermions and bosons that depend on the statistics of atoms. This is because for bosons (fermions), those scattering events in which an atom recoils to an already occupied state are enhanced (inhibited).^ As the driving light gets closer to the atomic resonance, multiple scattering and vacuum fluctuations of the electromagnetic fields should also be taken into account. We write the equations of motion for the excited state field operators by eliminating the vacuum electromagnetic field. It is always assumed that the driving external laser field is in the coherent state. We find in the limit of low light intensity, a hierarchy of equations of motion for correlation functions that contain one excited-atom field and one, two, three, etc., ground state atom fields. It is conjectured that the entire linear hierarchy may be solved by solving numerically the classical equations for the coupled system of electromagnetic fields and charged harmonic oscillators. We discuss the emergence of resonant dipole-dipole interactions and collective linewidths, and delineate the limits of validity of the column density approach in terms of non-cooperative atoms by presenting a mathematical example in which this approach is exact. ^
Based on the developed quantum microscopic theory, the interaction of weak electromagnetic radiation with dense ultracold atomic clouds is described in detail. The differential and total cooperative scattering cross sections are calculated for monochromatic radiation as particular examples of application of the general theory. The angular, spectral, and polarization properties of scattered light are determined. The dependence of these quantities on the sample size and concentration of atoms is studied and the influence of collective effects is analyzed.
This thesis presents the author's work on multiple scattering of intense laser light by cold atoms in chapter 2 and nonlinear coherent backscattering in chapter 3. The basic concepts and most important results should be understandable from reading these chapters on their own, whereas we refer to the subsequent collection of articles (P1-P11) for more detailed treatments. In chapter 2.1, we review scattering of light by a single two-level atom, with special emphasis on the fact that the atoms are nonlinear and inelastic scatterers, which will be relevant for multiple scattering. In chapter 2.2, we treat double scattering by two distant atoms, before we generalize this technique to a dilute disordered sample of many atoms in chapter 2.3. Both 2.2 and 2.3 apply in the perturbative regime of low laser intensity, where, by analyzing scattering of two photons, we clarify the different roles of nonlinearity and inelasticity concerning the coherent backscattering interference. Chapter 2.4 discusses the master equation as a complementary approach, applicable at arbitrarily high laser intensity, but only for a small number of atoms. In addition, we present an equivalent new approach, which explicitly takes advantage of the large distance between the atoms, and thereby combines the advantages of the master equation and the scattering approach. As discussed in chapter 2.5, this new approach opens the way to a non-perturbative theory for multiple scattering of intense laser light by dilute atomic clouds. Chapter 3 presents our diagrammatic theory for nonlinear coherent backscattering of classical (or single-particle quantum) waves. In chapter 3.1, we explain the most important assumptions of the theory: weak disorder and small nonlinearity, in particular existence of a stable, stationary solution of the nonlinear wave equation. In chapter 3.2, we explain why - contrary to seemingly convincing intuitive reasoning - the nonlinearity indeed changes the height of the coherent backscattering interference peak. In order to describe this effect quantitatively, a diagrammatic theory is sketched in chapters 3.3 and 3.4, where the nonlinear Born series and the nonlinear ladder and crossed diagrams are introduced. Chapter 3.5 presents the transport equations which result from the complete summation of ladder and crossed diagrams - non-perturbatively in the strength of the nonlinearity. We discuss the conditions under which the nonlinearity reduces or enhances the coherent backscattering interference, as compared to the linear case. In order to confirm the validity of the diagrammatic analysis, we compare these results with numerical simulations in chapter 3.6. We conclude with a short account of our work on speckle instabilities for nonlinear point scatterers in chapter 3.7. Finally, chapter 4 describes some future perspectives of our work, in particular the application of the diagrammatic theory to describe the impact of nonlinearity on weak localization and Anderson localization, and possible extensions to the quantum many-particle regime.