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Journal of Physics B Atomic Molecular and Optical Physics

Coherent atom–molecule oscillations with hyperspherical coordinates

Takaaki Sogo, D. V. Fedorov, A. S. Jensen

Published Aug 3, 2005• 1 citation
Publisher

Abstract

We investigate two-component boson systems consisting of atoms and binary bound molecules of the same atoms. We assume that a coherent superposition of atomic and molecular Bose?Einstein condensates is formed in a non-stationary state leading to oscillations between the two condensates. To facilitate the formulation, we first give analogies in neutrino oscillations applicable for two states of one particle. We then extend to many non-interacting particles in an external field treated in the mean-field approximation as for collective motion of finite systems. We then go beyond the mean field and provide a framework for interacting particles and correlated wavefunctions. The atom?molecule oscillation frequency is dominated by the molecular binding energy when the atom?atom scattering length is substantially smaller than the oscillator trap length. Comparable scattering lengths can completely change the oscillation frequency. The damping time of the oscillations decreases with atom number and scattering lengths, although a large increase can occur for special combinations of the different interactions.

DOI
10.1088/0953-4075/38/16/010

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We investigate two-component boson systems consisting of atoms and binary bound molecules of the same atoms. We assume that a coherent superposition of atomic and molecular Bose?Einstein condensates is formed in a non-stationary state leading to oscillations between the two condensates. To facilitate the formulation, we first give analogies in neutrino oscillations applicable for two states of one particle. We then extend to many non-interacting particles in an external field treated in the mean-field approximation as for collective motion of finite systems. We then go beyond the mean field and provide a framework for interacting particles and correlated wavefunctions. The atom?molecule oscillation frequency is dominated by the molecular binding energy when the atom?atom scattering length is substantially smaller than the oscillator trap length. Comparable scattering lengths can completely change the oscillation frequency. The damping time of the oscillations decreases with atom number and scattering lengths, although a large increase can occur for special combinations of the different interactions.
We investigate two-component boson systems by using their two individual average sizes and the distance between their centers as coordinates. We use zero-range interactions and include only relative sss waves. An effective potential is computed and discussed as a function of particle numbers and interaction strengths. Efimov conditions of two very large scattering lengths can arise if one component is a binary molecule. The corresponding atom-molecule interaction can then exhibit chaotic behavior. We present static and dynamic properties of an atom-molecule two-component system. We discuss the possible atom-molecule oscillations when all atoms are identical. The atomic number is then used as a collective coordinate and the cranking mass is computed. The collective oscillation frequency is estimated and compared with the trap frequency and the three-body recombination rate.
Abstract Recent advances in the precise control of ultracold atomic systems have led to the realisation of Bose–Einstein condensates (BECs) and degenerate Fermi gases. An important challenge is to extend this level of control to more complicated molecular systems. One route for producing ultracold molecules is to form them from the atoms in a BEC. For example, a two-photon stimulated Raman transition in a 87Rb BEC has been used to produce 87Rb2 molecules in a single rotational-vibrational state1, and ultracold molecules have also been formed2 through photoassociation of a sodium BEC. Although the coherence properties of such systems have not hitherto been probed, the prospect of creating a superposition of atomic and molecular condensates has initiated much theoretical work3, 4, 5, 6, 7. Here we make use of a time-varying magnetic field near a Feshbach resonance8, 9, 10, 11, 12 to produce coherent coupling between atoms and molecules in a 85Rb BEC. A mixture of atomic and molecular states is created and probed by sudden changes in the magnetic field, which lead to oscillations in the number of atoms that remain in the condensate. The oscillation frequency, measured over a large range of magnetic fields, is in excellent agreement with the theoretical molecular binding energy, indicating that we have created a quantum superposition of atoms and diatomic molecules—two chemically different species.
We create atom-molecule superpositions in a Bose-Fermi mixture of 87Rb^{87}\text{R}\text{b}87Rb and 40K^{40}\text{K}40K atoms. The superpositions are generated by ramping an applied magnetic field near an interspecies Fano-Feshbach resonance to coherently couple atom and molecule states. Rabi- and Ramsey-type experiments show oscillations in the molecule population that persist as long as $150\text{ }\ensuremath{\mu}\text{s}$ and have up to 50% contrast. The frequencies of these oscillations are magnetic-field dependent and consistent with the predicted molecule binding energy. This quantum superposition involves a molecule and a pair of free particles with different statistics (i.e., bosons and fermions) and furthers exploration of atom-molecule coherence in systems without a Bose-Einstein condensate.
We study the system of coupled atomic and molecular condensates within the two-mode model and beyond mean-field theory. Large-amplitude atom-molecule coherent oscillations are shown to be damped by the rapid growth of fluctuations near the dynamically unstable molecular mode. This result contradicts earlier predictions about the recovery of atom-molecule oscillations in the two-mode limit. The frequency of the damped oscillation is also shown to scale as N/lnN\sqrt{N}/\mathrm{ln}NN​/lnN with the total number of atoms N, rather than the expected pure N\sqrt{N}N​ scaling. Using a linearized model, we obtain analytical expressions for the initial depletion of the molecular condensate in the vicinity of the instability, and show that the important effect neglected by mean-field theory is an initially nonexponential ``spontaneous'' dissociation into the atomic vacuum. Starting with a small population in the atomic mode, the initial dissociation rate is sensitive to the exact atomic amplitudes, with the fastest (superexponential) rate observed for the entangled state formed by spontaneous dissociation.
We theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. In the case of small wave-function overlap between bilateral condensates, one can parametrize their spatial part of the wave functions in the two-mode approximation together with the time-dependent population imbalance zzz and the phase difference $\ensuremath{\phi}$ between two wave functions. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\ensuremath{\xi}$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave-function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of a fixed point. The Melnikov approach with a homoclinic solution of the derived $z, \ensuremath{\phi}$, and $\ensuremath{\xi}$ equations can successfully justify the existence of chaos. All results are consistent with the numerical solutions of the full time-dependent Gross-Pitaevskii equations.
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