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arXiv:1303.1941 · Physical Review A

Stochastic and equilibrium pictures of the ultracold Fano-Feshbach-resonance molecular conversion rate

Tomotake Yamakoshi, Shinichi Watanabe, Chen Zhang, Chris H. Greene

Published May 8, 2013• 4 citations
arXivHTMLPDF

Abstract

The ultracold molecular conversion rate occurring in an adiabatic ramp through a Fano-Feshbach resonance is studied and compared in two statistical models. One model, the so-called stochastic phase-space sampling (SPSS) [Hodby et al., Phys. Rev. Lett. 94, 120402 (2005)] evaluates the overlap of two atomic distributions in phase space by sampling atomic pairs according to a phase-space criterion. The other model, the chemical equilibrium theory (ChET) [Watabe and Nikuni, Phys. Rev. A 77, 013616 (2008)] considers atomic and molecular distributions in the limit of the chemical and thermal equilibrium. The present study applies SPSS and ChET to a prototypical system of K+K$\ensuremath{\rightarrow}$ K${}_{2}$ in all the symmetry combinations, namely Fermi-Fermi, Bose-Bose, and Bose-Fermi cases. To examine implications of the phase-space criterion for SPSS, the behavior of molecular conversion is analyzed using four distinct geometrical constraints. Our comparison of the results of SPSS with those of ChET shows that while they appear similar in most situations, the two models give rise to rather dissimilar behaviors when the presence of a Bose-Einstein condensate strongly affects the molecule formation.

DOI
10.1103/physreva.87.053604

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The ultracold molecular conversion rate occurring in an adiabatic ramp through a Fano-Feshbach resonance is studied and compared in two statistical models. One model, the so-called stochastic phase-space sampling (SPSS) [Hodby et al., Phys. Rev. Lett. 94, 120402 (2005)] evaluates the overlap of two atomic distributions in phase space by sampling atomic pairs according to a phase-space criterion. The other model, the chemical equilibrium theory (ChET) [Watabe and Nikuni, Phys. Rev. A 77, 013616 (2008)] considers atomic and molecular distributions in the limit of the chemical and thermal equilibrium. The present study applies SPSS and ChET to a prototypical system of K+K\ensuremath→\ensuremath{\rightarrow}\ensuremath→ K2{}_{2}2​ in all the symmetry combinations, namely Fermi-Fermi, Bose-Bose, and Bose-Fermi cases. To examine implications of the phase-space criterion for SPSS, the behavior of molecular conversion is analyzed using four distinct geometrical constraints. Our comparison of the results of SPSS with those of ChET shows that while they appear similar in most situations, the two models give rise to rather dissimilar behaviors when the presence of a Bose-Einstein condensate strongly affects the molecule formation.
We study the conversion efficiency of heteronuclear Feshbach molecules in population imbalanced atomic gases formed by ramping the magnetic field adiabatically. We extend the recent work [J. E. Williams et al., New J. Phys. 8, 150 (2006)] on the theory of Feshbach molecule formations to various combinations of quantum statistics of each atomic component. A simple calculation for a harmonically trapped ideal gas is in good agreement with the recent experiment [S. B. Papp and C. E. Wieman, Phys. Rev. Lett. 97, 180404 (2006)] without any fitting parameters. We also give the conversion efficiency as an explicit function of initial peak phase space density of the majority species for population imbalanced gases. In the low-density region where Bose-Einstein condensation does not appear, the conversion efficiency is a monotonic function of the initial peak phase space density, but independent of statistics of a minority component. The quantum statistics of majority atoms has a significant effect on the conversion efficiency. In addition, Bose-Einstein condensation of an atomic component is the key element determining the maximum conversion efficiency.
We solve a model that describes a stimulated conversion between ultracold bosonic atoms and molecules. The reaction is triggered by a linearly time-dependent transition throughout the Feshbach resonance. Our solution predicts a dependence, with a dynamic phase transition, of the reaction efficiency on the transition rate for both atoms-to-molecule pairing and molecular dissociation processes. We find that for the latter process with a linear energy dispersion of atomic modes, the emerging phase can have a thermalized energy distribution of noninteracting bosons with the temperature defined by the rate of the transition. This provides a simple interpretation of the phase transition in terms of the creation of equilibrium Bose-Einstein condensate.
We determine the adiabatic phase diagram of a resonantly-coupled system of Fermi atoms and Bose molecules confined in the harmonic trap by using the local density approximation. The adiabatic phase diagram shows the fermionic condensate fraction composed of condensed molecules and Cooper pair atoms. The key idea of our work is conservation of entropy through the adiabatic process, extending the study of Williams et al. [Williams et al., New J. Phys. 6, 123 (2004)] for an ideal gas mixture to include the resonant interaction in a mean-field theory. We also calculate the molecular conversion efficiency as a function of initial temperature. Our work helps to understand recent experiments on the BCS-BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field.
In the wake of successful experiments in Fermi condensates, experimental attention is broadening to include resonant interactions in degenerate Bose-Fermi mixtures. In this thesis we wish to study the equilibrium properties of the fermionic molecules that can be created in such a mixture. To this end, we first discuss the two body properties of the system, and introduce the model Hamiltonian we use to describe the resonant physics, highlighting its virtues, as well as its limitations. We then proceed by analyzing the mean field solution of this model, by studying both the equilibrium problem, and the non-equilibrium equations of motion, thus developing a powerful language to discuss the system. We then highlight the limitations of the mean-field approach, and develop a numerically tractable generalized version of this theory, which is able to correctly describe the two-body properties of the system in the low-density limit. Finally, we study the properties of the system using this generalized mean-field theory, by first analyzing the two-body scattering matrix in the many-body environment, assessing its complex poles in order to understand the stability properties of the Feshbach molecules in the gas. Secondly we solve the equilibrium equations self-consistently, to study the molecular populations and density distributions at equilibrium, as a function of external bias magnetic field.
Starting with coupled atom–molecule Boltzmann equations, we develop a simplified model to understand molecule formation observed in recent experiments. Our theory predicts several key features: (i) the effective adiabatic rate constant is proportional to density; (ii) in an adiabatic ramp, the dependence of molecular fraction on magnetic field resembles an error function whose width and centroid are related to the temperature; and (iii) the molecular production efficiency is a universal function of the initial phase space density, the specific form of which we derive for a classical gas. Our predictions show qualitative agreement with the data from Hodby et al (2005 Phys. Rev. Lett. 94 120402) without the use of adjustable parameters.
We study the mean-field dynamics of ppp-wave Feshbach-molecule production in an ultracold gas of Fermi atoms in the same internal state. We derive a separable potential to describe the low-energy scattering properties of such atoms, and use this potential to solve the mean-field dynamics during a magnetic-field sweep. Initially, on the negative scattering length side of a Feshbach resonance the gas is described by the BCS theory. We adapt the method by Szyma\ifmmode \acute{n}\else \'{n}\fi{}ska et al. [Phys. Rev. Lett. 94, 170402 (2005)] to ppp-wave interacting Fermi gases and model the conversion dynamics of the gas into a Bose-Einstein condensate of molecules on the other side of the resonance under the influence of a linearly varying magnetic field. We have analyzed the dependence of the molecule production efficiency on the density of the gas, temperature, initial value of the magnetic field, and magnetic-field ramp speed. Our results show that in this approximation molecule production by a linear magnetic-field sweep is highly dependent on the initial state.