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arXiv:cond-mat/0303230 · New Journal of Physics

The Josephson frequency of resonantly coupled atomic and molecular condensates

R. A. Duine, Henk T. C. Stoof

Published Jun 13, 2003• 7 citations
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Abstract

Motivated by recent experiments (Claussen et al 2003 Preprint cond-mat/0302195) we investigate the magnetic-field dependence of the Josephson frequency of coherent atom–molecule oscillations near a Feshbach resonance. Far off resonance this frequency is determined by two-atom physics. However, the Josephson frequency is relatively large in this case and a description purely in terms of scattering lengths turns out to be inadequate. In particular, we have to include the effective range parameter of the interatomic interaction in our calculations. Close to resonance, the frequency deviates from the two-body results due to various many-body shifts. Considering also the many-body effects, we find perfect agreement with the experimental results over the entire range of magnetic field.

DOI
10.1088/1367-2630/5/1/369

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Motivated by recent experiments (Claussen et al 2003 Preprint cond-mat/0302195) we investigate the magnetic-field dependence of the Josephson frequency of coherent atom–molecule oscillations near a Feshbach resonance. Far off resonance this frequency is determined by two-atom physics. However, the Josephson frequency is relatively large in this case and a description purely in terms of scattering lengths turns out to be inadequate. In particular, we have to include the effective range parameter of the interatomic interaction in our calculations. Close to resonance, the frequency deviates from the two-body results due to various many-body shifts. Considering also the many-body effects, we find perfect agreement with the experimental results over the entire range of magnetic field.
Using the theoretical formalism of R A Duine and H T C Stoof (arXiv: cond-mat/0303230v) 22 March 2003), we have theoretically evaluated magnetic field and density dependence of the Josephson frequency in coherence atom-molecule oscillation in Bose-Einstein condensed gas.Our theoretically obtained results show that Josephson frequency increases with both magnetic field for constant condensate density and also with condensate density for fixed magnetic field.Our theoretically evaluated results are in good agreement with the experimental data and also with other theoretical workers.Using the theoretical formalism of Markes Greiner etal (Nature, 26 Nov (2005)) , we have studied the emergence of molecular Bose-Einstein condensate from a Fermi gas of 6 Li atoms.The study has been performed with the help of Feshbach resonance.We have evaluated molecular condensate fraction N0/N as a function of (T/TC) and as a function of (T/TF).We have also evaluated scattering length 'a' in unit of a0, of mixture of two lowest hyperfine states of 6 Li atom, temperature (µK) all as a function of magnetic field (mT) near the crossing of the An evaluation …
We study the many-body effects on coherent atom-molecule oscillations by means of an effective quantum field theory that describes Feshbach-resonant interactions in Bose gases in terms of an atom-molecule Hamiltonian. We determine numerically the many-body corrections to the oscillation frequency for various densities of the atomic condensate. We also derive an analytic expression that approximately describes both the density and magnetic-field dependence of this frequency near the resonance. We find excellent agreement with experiment.
We discuss the frequency and visibility of atom-molecule Ramsey fringes observed in recent experiments by Claussen et al. [Phys. Rev. A 67, 060701 (2003)]. In these experiments a 85Rb^{85}\mathrm{Rb}85Rb Bose-Einstein condensate was exposed to a sequence of magnetic field pulses on the high-field side of the 155G155\phantom{\rule{0.3em}{0ex}}\mathrm{G}155G Feshbach resonance. The observed oscillation frequencies largely agree with the theoretically predicted magnetic field dependence of the binding energy of the highest excited diatomic vibrational state, except for a small region very close to the singularity of the scattering length. Our analytic treatment of the experiment, as well as our dynamical simulations, follow the magnitude of the measured oscillation frequencies as well as the visibilities of the Ramsey fringes. We show that significant deviations from a purely binary dynamics, with an associated binding frequency, occur when the spatial extent of the molecular wave function becomes comparable with the mean distance between the atoms in the dilute gas. The experiments thus clearly identify the conditions under which diatomic molecules may be identified as a separate entity of the gas or, conversely, when the concept of binary physics in a many-body environment is bound to break down.
Atom–molecule coherence in a Bose-Einstein condensate (BEC) has been demonstrated. Sudden changes were made to the magnetic field near a Feshbach resonance such that oscillations between atomic and molecular states were excited. The frequency of these oscillations was measured over a large range of magnetic fields and was found to be in excellent quantitative agreement with the predicted energy difference between two colliding atoms and the bound molecular state. This agreement indicates that we have created a quantum superposition of atoms and diatomic molecules, which are chemically different species. 1
We present semianalytical many-body results for energies and excitation frequencies for an inhomogeneous Bose-Einstein condensate over a wide range of atom numbers NNN for both small sss-wave scattering lengths, typical of most laboratory experiments, and large scattering lengths, achieved by tuning through a Feshbach resonance. Our dimensional perturbation treatment includes two-body correlations at all orders and yields analytical results through first order by taking advantage of the high degree of symmetry of the condensate at the zeroth-order limit. Because NNN remains a parameter in our analytical results, the challenge of calculating energies and excitation frequencies does not rise with the number of condensate atoms. In this proof-of-concept paper the atoms are confined in a spherical trap and are treated as hard spheres. Our many-body calculations compare well to Gross-Pitaevskii results in the weakly interacting regime and depart from the mean-field approximation as the density approaches the strongly interacting regime. The excitation frequencies provide a particularly sensitive test of beyond-mean-field corrections. For example, for N=2000N=2000N=2000 atoms and an experimentally realized large scattering length of $a=0.433{a}_{ho}\phantom{\rule{0.3em}{0ex}}({a}_{ho}=\sqrt{\ensuremath{\hbar}∕m{\ensuremath{\omega}}_{ho}})$ we predict a 7575%75 shift from the mean-field breathing mode frequency.
We derive the atom-molecule theory for an atomic gas near a broad Feshbach resonance, where the energy dependence of the atom-molecule coupling becomes crucial for understanding experimental results. We show how our many-body theory incorporates the two-atom physics exactly. In particular, we calculate the magnetic moment of a two-component gas of 6Li^{6}\mathrm{Li}6Li atoms for a wide range of magnetic fields near the broad Feshbach resonance at about 834G834\phantom{\rule{0.3em}{0ex}}\mathrm{G}834G. We find excellent agreement with the experiment of Jochim et al. [Phys. Rev. Lett. 91, 240402 (2003)].