ATOMIC INTERACTIONS IN A BOSE EINSTEIN CONDENSATION A.KILIC C.YUCE

1. Ultra cold atoms offer ultimate control over external and internal degrees of freedom facilitating a very pure experimental implementation of quantum mechanical models. They are therefore an ideal testing ground for the study of quantum magnetism phenomena. One of the most interesting properties of Bose-Einstein condensates in dilute atomic gases is the existence of interactions between the constituent atoms. The mean field interaction is mediated by the elastic collisions between atoms. The well-known Gross-Pitaevskii (GP) equation can be used to define a condensate in the limit of zero temperature and neglecting all correlations between the atoms; where f, is the BEC order parameter, Vext is the trap potential energy, and the coefficient characterizes the pair-wise interatomic interactions in the BEC through the s-wave elastic scattering length,a. For controlling the BEC self-interaction, it would clearly be desirable to find some method to change the scattering length. It was proposed that the scattering length could be influenced using an external magnetic field. The magnetic field would allow one to shift the energy of a molecular bound state to near-degeneracy with the energy of a colliding pair of atoms, thereby altering the elastic scattering properties.The solution to the GP equation is the condensate order parameter, f, which for most purposes can be regarded as the macroscopic wavefunction of the BEC atoms. The GP equation has the form of a nonlinear Schrodinger equation, where the nonlinear term arises from the mean-field interaction of one atom with all of the others. The mean-field interaction term, often called the selfinteraction energy of the condensate, depends on the density of atoms and the scattering length. The GP equation predicts that the sign and magnitude of a determine the strength of the self-interaction and whether this interaction is attractive or repulsive. For attractive interaction, the number of atoms in the condensate grows until the total number of atoms in the condensate exceeds a critical value with which the BEC undergoes collapse. During the collapse, the number of atoms is decreased. If the condensate is fed by a surrounding thermal cloud, then the condensate undergoes cycles of growth and collapses. Hulet and his team at Rice University observed the growth of a condensate of trapped 7Li atoms with attractive interaction and its subsequent collapse 1. Ketterle with his team measured the rate of growth of a 23Na condensate fed by a thermal cloud 2. Growth of a Bose-Einstein condensate from thermal vapor was also experimentally realized for atoms of 87Rb 1. Some methods have been introduced theoretically to account for the growth of the BEC. The GP gain equation was used to model the growth of a BEC by Drummond and Kheruntsyan 3. They determined that, as the condensate grows, the center of mass oscillates in the trap. Furthermore, the GP gain equation is of special importance in the field of an atom laser, that is a device which produces an intense coherent beam of atoms by a stimulated process. In recent years, the GP gain equation has also been studied mathematically by some authors. The GP gain equation can also be used in the theory of optical fibers if the harmonic trap potential is assumed to be zero 3. The interaction effect in the system is determined by the s-wave scattering length. In Ref. 4, it has been shown that controlling the generation of bright and dark soliton trains from periodic waves can be achieved by the variation of the scattering length. Recent experiments have demonstrated that tuning of the s-wave scattering length can be achieved due to the Feshbach resonance 5. It offers a possibility to vary the interaction strength in ultracold atomic gases simply by applying an external magnetic field. where a is the asymptotic value of the scattering length far from resonance, B(t) is the time-dependent externally applied magnetic field, D is the width of resonance, and B0 is the resonant value of the magnetic field. The strength of the interaction can be adjusted experimentally from large negative values to large positive values. As a result, the experiments with magnetically trapped ultracold atomic gases, where the s-wave scattering length fully determines the interaction effects, have an unprecedented high level of control over the interatomic interactions. With this experimental degree of freedom, it is possible to intensively study the dynamics of BEC. We will study the changes in the number of atoms supplied by a surrounding thermal cloud due to the variations of the scattering length by the external magnetic field. By supposing the number of particles is not very large for the cigar-shaped traps, it is legitimate to use the reduced one-dimensional (1D) GP equation where w2=wz2/w2 , =/w, and aBis the Bohr radius. Here, the coordinate z and the time t are measured in units of a and 1/ w, respectively. The relationship between the macroscopic wave functions and is given by where. To extract qualitative information and to improve the understanding of the underlying physics, we will transform the GP gain equation analytically to theone with the constant scattering length. To get rid of the time-dependent scattering length fromEq.the two transformations are introduced as follows: where the dot denotes time derivation and the timedependent function f(t) is given by where Ωis a constant. REFERENCES 1. Pethick, C.J., Smith,H., “Bose Einstein Condensation in Dilute Gases”, Cambridge University Press, 2002. 2. Yuce,C.,Kilic,A., “Feshbach resonance and growth of a Bose- Einstein condensate”, PHYSICAL REVIEW A 74, 033609 2006 3. J. M. Gerton et al., Nature London 408, 692 2000. 4. F. K. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, Phys. Rev. Lett. 90, 230402 2003. 5. A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Phys. Rev. A. Anadolu University, Eskişehir, Turkey abkilic@anadolu.edu.tr cyuce@anadolu.edu.tr ANADOLU UNIVERSITY Eskisehir-TURKEY SCIENCE IS THE MOST GENUINE GUIDE IN LIFE ATOMIC INTERACTIONS IN A BOSE EINSTEIN CONDENSATIONA.KILIC C.YUCE