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Nonlinear Optics

Strong correlation effect in the second harmonic generation of a Bose-Einstein condensate

Alfred M. Levine, Anatoly B. Kuklov

Published Nov 27, 2002• 0 citations
Publisher

Abstract

The effect of Bose-Einstein condensation, achieved recently for alkaline gases in magnetic traps, should lead to a strong enhancement of non-linear harmonic generation. A nearly resonant light imposed on the gas of two-level atoms excites the intra-atomic transitions and causes generation of various harmonics. Normally, in the thermal state of the gas, a contribution of the inter-atomic interaction to the non-linear harmonic generation is negligible. However, if a Bose-Einstein condensate (BEC) is present, the inter-particle interaction cannot be ignored. Considering second harmonic generation, we show that, in the presence of the BEC, the intensity of the second harmonic is proportional to (aN n)/sup 2/, where N and n are the total number and the effective density of the condensed atoms, respectively, and a stands for the scattering length describing the process of scattering of two excited atoms by each other so that, after the collision, one excited atom enters the condensate. We consider the interaction between light, represented by the electric field E, and the atomic Bose field /spl Psi/ as well as the two body interaction treated within the scattering length approximation.

DOI
10.1109/nlo.1998.710245

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The effect of Bose-Einstein condensation, achieved recently for alkaline gases in magnetic traps, should lead to a strong enhancement of non-linear harmonic generation. A nearly resonant light imposed on the gas of two-level atoms excites the intra-atomic transitions and causes generation of various harmonics. Normally, in the thermal state of the gas, a contribution of the inter-atomic interaction to the non-linear harmonic generation is negligible. However, if a Bose-Einstein condensate (BEC) is present, the inter-particle interaction cannot be ignored. Considering second harmonic generation, we show that, in the presence of the BEC, the intensity of the second harmonic is proportional to (aN n)/sup 2/, where N and n are the total number and the effective density of the condensed atoms, respectively, and a stands for the scattering length describing the process of scattering of two excited atoms by each other so that, after the collision, one excited atom enters the condensate. We consider the interaction between light, represented by the electric field E, and the atomic Bose field /spl Psi/ as well as the two body interaction treated within the scattering length approximation.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
Bose-Einstein condensate of rarified atomic gases is considered as the state formed by exchange of virtual photons, resonant to the lowest levels of atoms; such representation corresponds to the Einstein opinion about an inter-influence of condensable particles. Considered interactions directly lead to the QED structure of nonlinear potential in the Gross-Pitaevskii equation. Linear momenta that correspond to the thermal energy of condensable atoms are connected to near field of particles and therefore leave atoms immovable. The estimations of these effects do not contradict the observed data; the general quantum principles predict possibility of stimulating of BEC formation by resonant irradiation. All this requires the spectroscopic investigation of BEC on different steps of formation.
With help of a Feshbach scattering resonance it is possible to change the effective two-body interaction of ultra-cold atoms from repulsive to attractive, from weak to strong. Regular field theory for the description of Bose-Einstein condensates is based on a single interaction parameter only, the scattering length. Such a description, however, is not valid anymore for the resonance regime. We formulated a field-theory that incorporated the microscopic energy-dependent two-body interactions, that also takes pair correlations into account. This theory allowed us to explain an experiment at JILA where Ramsey fringes were observed in a ^85Rb Bose-Einstein condensate by making use of a Feshbach resonance. The theory and experiment show that coherent molecules are created, a first step in the direction of molecular BEC. For non-resonant interactions, the self-energy corresponding to the atom-atom interactions is proportional to the scattering length. Close to resonance, however, this relationship no longer applies, and the energy will level off to a finite value on resonance. We investigate this resonance self-energy, and compare it with experimental results from strongly interacting Fermi systems.
Producing a substantial and stable resonant Bose-Einstein condensate (BEC) has proven to be a challenging experimental task due to heating and three-body losses that may occur even before the gas comes to thermal equilibrium. In this paper, by considering two-body correlations, we note that a sudden quench from small to large scattering lengths is not an efficient way to prepare a resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between 0 and $\ensuremath{\infty}$, which serves to maximize the transfer probability of NNN bosons of mass mmm in a harmonic trap with frequency $\ensuremath{\omega}$ to the resonant state. We find that the intermediate scattering length should be $a\ensuremath{\approx}3.16{N}^{\ensuremath{-}2/3}\sqrt{\ensuremath{\hbar}/(m\ensuremath{\omega})}$, and that it produces an optimum transition probability of $1.03{N}^{\ensuremath{-}1/6}$.
Summary form only given. The observation of atomic Bose-Einstein condensates (BECs) in magnetic and optical traps has stimulated theoretical investigation of the optical properties of these systems. Many of the current theories are based on the Heisenberg equations of motion for the radiation field and matter. These equations are nonlinear in the field operators and lead to an infinite hierarchy of equations for the atomic correlation functions. The hierarchy can be decoupled using an approximation. In an alternative development, the Heisenberg equations of motion have been simplified using an adiabatic approximation to eliminate the excited-state field operator. While some previous studies of BECs have employed the polariton approach, in these treatments neither the local field effects nor the polariton-phonon coupling are taken into account. Local field effects play an important role in a dense Bose gas (/spl rho//sup I/3//spl lambda//spl Lt/1) where, due to the resonance dipole-dipole interaction,the atoms respond to the electromagnetic field in a collective manner.In this contribution we examine the effects of both the local field and phonons using the polariton approach. The analysis is based on the quasiparticle concept.
The properties of Bose-Einstein condensed gases can be strongly altered by tuning the external magnetic field near a Feshbach resonance. Feshbach resonances affect elastic collisions and lead to the observed modification of the scattering length. However, as we report here, the observed rate of inelastic collisions was strongly enhanced in a sodium Bose-Einstein condensate when the scattering length was tuned to both larger or smaller values than the off-resonant value. These strong losses impose severe limitations for using Feshbach resonances to tune the properties of Bose-Einstein condensates.