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arXiv:cond-mat/0312178 · Journal of Physics B Atomic Molecular and Optical Physics

Adiabatic association of ultracold molecules via magnetic-field tunable interactions

Krzysztof Góral, Thorsten Köhler, S. A. Gardiner, Eite Tiesinga, Paul S. Julienne

Published Aug 24, 2004• 102 citations

Abstract

We consider in detail the situation of applying a time-dependent external magnetic field to a 87 Rb atomic Bose–Einstein condensate held in a harmonic trap, in order to adiabatically sweep the interatomic interactions across a Feshbach resonance to produce diatomic molecules. To this end, we introduce a minimal two-body Hamiltonian depending on just five measurable parameters of a Feshbach resonance, which accurately determines all low-energy binary scattering observables, in particular, the molecular conversion efficiency of just two atoms. Based on this description of the microscopic collision phenomena, we use the many-body theory of Köhler and Burnett (2002 Phys. Rev. A 65 033601) to study the efficiency of the association of molecules in a 87 Rb Bose–Einstein condensate during a linear passage of the magnetic-field strength across the 100 mT Feshbach resonance. We explore different, experimentally accessible, parameter regimes, and compare the predictions of Landau–Zener, configuration interaction, and two-level mean-field calculations with those of the microscopic many-body approach. Our comparative studies reveal a remarkable insensitivity of the molecular conversion efficiency with respect to both the details of the microscopic binary collision physics and the coherent nature of the Bose–Einstein condensed gas, provided that the magnetic-field strength is varied linearly. We provide the reasons for this universality of the molecular production achieved by linear ramps of the magnetic-field strength, and identify the Landau–Zener coefficient determined by Mies et al (2000 Phys. Rev. A 61 022721) as the main parameter that controls the efficiency.

DOI: 10.1088/0953-4075/37/17/006

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We consider in detail the situation of applying a time-dependent external magnetic field to a 87 Rb atomic Bose–Einstein condensate held in a harmonic trap, in order to adiabatically sweep the interatomic interactions across a Feshbach resonance to produce diatomic molecules. To this end, we introduce a minimal two-body Hamiltonian depending on just five measurable parameters of a Feshbach resonance, which accurately determines all low-energy binary scattering observables, in particular, the molecular conversion efficiency of just two atoms. Based on this description of the microscopic collision phenomena, we use the many-body theory of Köhler and Burnett (2002 Phys. Rev. A 65 033601) to study the efficiency of the association of molecules in a 87 Rb Bose–Einstein condensate during a linear passage of the magnetic-field strength across the 100 mT Feshbach resonance. We explore different, experimentally accessible, parameter regimes, and compare the predictions of Landau–Zener, configuration interaction, and two-level mean-field calculations with those of the microscopic many-body approach. Our comparative studies reveal a remarkable insensitivity of the molecular conversion efficiency with respect to both the details of the microscopic binary collision physics and the coherent nature of the Bose–Einstein condensed gas, provided that the magnetic-field strength is varied linearly. We provide the reasons for this universality of the molecular production achieved by linear ramps of the magnetic-field strength, and identify the Landau–Zener coefficient determined by Mies et al (2000 Phys. Rev. A 61 022721) as the main parameter that controls the efficiency.

In many experiments involving conversion of quantum-degenerate atomic gases into molecular dimers via a Feshbach resonance, an external magnetic field is linearly swept from above resonance to below resonance. In the adiabatic limit, the fraction of atoms converted into molecules is independent of the functional form of the sweep and is predicted to be 100%. However, for nonadiabatic sweeps through resonance, Landau-Zener theory predicts that a linear sweep will result in a negligible production of molecules. Here we employ a genetic algorithm to determine the functional time dependence of the magnetic field that produces the maximum number of molecules for sweep times that are comparable to the period of resonant atom-molecule oscillations, $2\ensuremath{\pi}{\ensuremath{\Omega}}_{\mathit{\text{Rabi}}}^{\ensuremath{-}1}$. The optimal sweep through resonance indicates that more than 95% of the atoms can be converted into molecules for sweep times as short as $2\ensuremath{\pi}{\ensuremath{\Omega}}_{\mathit{\text{Rabi}}}^{\ensuremath{-}1}$ while the linear sweep results in a conversion of only a few percent. We also find that the qualitative form of the optimal sweep is independent of the strength of the two-body interactions between atoms and molecules and the width of the resonance.

Magnetically tunable Feshbach resonances were employed to associate cold diatomic molecules in a series of experiments involving both atomic Bose and two-spin-component Fermi gases. This review illustrates theoretical concepts of both the particular nature of the highly excited Feshbach molecules produced and the techniques for their association from unbound atom pairs. Coupled-channels theory provides a rigorous formulation of the microscopic physics of Feshbach resonances in cold gases. Concepts of dressed versus bare energy states, universal properties of Feshbach molecules, and the classification in terms of entrance- and closed-channel-dominated resonances are introduced on the basis of practical two-channel approaches. Their significance is illustrated for several experimental observations, such as binding energies and lifetimes with respect to collisional relaxation. Molecular association and dissociation are discussed in the context of techniques involving linear magnetic-field sweeps in cold Bose and Fermi gases and pulse sequences leading to Ramsey-type interference fringes. Their descriptions in terms of Landau-Zener, two-level mean-field, as well as beyond mean-field approaches are reviewed in detail, including the associated ranges of validity.

We investigate the production efficiency of ultracold molecules in bosonic 85 Rb and fermionic 40 K when the magnetic field is swept across a Feshbach resonance.For adiabatic sweeps of the magnetic field, the conversion efficiency of each species is solely determined by the phase space density of the atomic cloud, in contrast to a number of theoretical predictions.Our novel model for the adiabatic pairing process, developed from general physical principles, accurately predicts the conversion efficiency for both ultracold gases of bosons and of fermions.In the non-adiabatic regime our measurements of the 85 Rb molecule conversion efficiency follow a Landau Zener model, with a conversion efficiency that is characterized by the density divided by the time derivative of the magnetic field.

We investigate the dynamics of converting cold bosonic atoms to molecules when an external magnetic field is swept across a Feshbach resonance. Our analysis relies on a quantum microscopic model that accounts for many-body effects in the association process. We show that the picture of two-body molecular production depicted by the Landau-Zener model is significantly altered due to many-body effects. In the nonadiabatic regime, we derive an analytic expression for molecular conversion efficiency that explains the discrepancy between the prediction of the Landau-Zener formula and the experimental data [E. Hodby et al., Phys. Rev. Lett. 94, 120402 (2005)]. Our theory is further extended to the formation of heteronuclear diatomic molecules and gives some interesting predictions.

Pairs of trapped atoms can be associated to make a diatomic molecule using a time-dependent magnetic field to ramp the energy of a scattering resonance state from above to below the scattering threshold. A relatively simple model, parametrized in terms of the background scattering length and resonance width and magnetic moment, can be used to predict conversion probabilities from atoms to molecules. The model and its Landau-Zener interpretation are described and illustrated by specific calculations for 23Na, 87Rb and 133Cs resonances. The model can be readily adapted to Bose-Einstein condensates. Comparison with full many-body calculations for the condensate case shows that the model is very useful for making simple estimates of molecule conversion efficiencies.

We have calculated the time-dependent dynamics of two ultracold Na atoms in an atom trap where a time-dependent magnetic field

B(t)

moves a Feshbach resonance state across the energy threshold for a binary collision. Our coupled-channel scattering calculations, which reproduce the observed properties of such resonances in sodium atom collisions, can be reduced to an effective two-channel configuration-interaction model for one bound state and one continuum. The model is adapted to describe the time-dependent dynamics induced by

B(t)

for two atoms trapped either in a strongly confining single well of an optical lattice or in an optical potential in the presence of a Bose-Einstein condensate. We show that a simple Landau-Zener curve crossing model gives quantitative agreement with exact calculations of field-induced transition rates. If

B(t)

sweeps the resonance across threshold from above, two atoms in the ground state of the trap potential can be efficiently converted to translationally cold dimer molecules. If the resonance is swept from below, the atoms can be removed from the ground state and placed in hot vibrational levels of the trap. Our calculations reproduce the rapid atom loss rates observed in a Na Bose-Einstein condensate due to sweeping a Feshbach resonance state through the binary collision threshold.