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Theory of Feshbach molecule formation in a dilute gas during a magnetic field ramp

JAMES E. JUN. WILLIAMS, Nygaard, Nicolai, Charles W. Clark

Published Jan 1, 2006• 8 citations
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Abstract

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: (1) the effective adiabatic rate constant is proportional to density; (2) 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; (3) 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, Phys. Rev. Lett. 94, 120402 (2005)] without the use of adjustable parameters.

Related papers

Theory of Feshbach molecule formation in a dilute gas during a magnetic field ramp

J E Williams, N Nygaard, C W Clark

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.

arXiv:cond-mat/0511011•New Journal of Physics•Aug 30, 2006•23 citations

Conversion efficiencies of heteronuclear Feshbach molecules

Shohei Watabe, Tetsuro Nikuni

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.

arXiv:0710.2377•Physical Review A•Jan 23, 2008•6 citations

Dynamics of Feshbach molecule production

Thomas M. Hanna

The variation of a magnetic field in the vicinity of a zero-energy resonance allows highly vibrationally excited molecules (‘Feshbach molecules’) to be produced from an ultracold atomic gas. In this thesis, we study the dynamics of this process. We begin by studying the dissociation of Feshbach molecules, showing that in the limit of a sudden jump the shape of the spectrum of dissociated atoms can act as a probe of the zero-energy resonance. For some resonances, such jumps are within reach of current experiments. We also study the intermediate region between sudden jumps and asymptotically wide, linear ramps. It is shown from a precise derivation how the latter limit leads to a universal spectrum with a shape independent of the implementation of the two-body physics, provided that the near-resonant scattering properties are correctly modelled. We then turn to the dynamics of Feshbach molecule production from thermal and condensed gases. Our microscopic quantum dynamics approach includes the exact twobody evolution as an input to the many-body calculations. We show that in the long-time limit, and the Markov limit for the interactions, the non-Markovian Boltzmann equation (NMBE) we derive for the one-body density matrix reduces to the normal Boltzmann equation. In the limit of short times and small depletion of the atomic gas, the molecule production efficiency can be calculated by thermally averaging the two-body transition probability density. This thermal averaging technique is applied to studies of the formation of Feshbach molecules using a magnetic field modulation that is near-resonant with the molecular bound state energy. The continuum is shown to have a significant effect on both the dynamics and efficiency of this process. We examine the dependence of the molecule production efficiency on the duration, amplitude and frequency of the modulation, as well as the temperature and density of the gas. This method of producing molecules is effective for a wide range of bound state energies, but requires sufficient variation of the two-body energy levels with magnetic field. Lastly, we implement the NMBE for the case of a fast linear ramp across a Feshbach resonance. The solution of this equation is made feasible by including a large part of the required computation in the kernel, which is calculated in advance. The NMBE allows predictions of the molecule production efficiency which go beyond the thermal averaging technique by accounting for the depletion and rethermalisation of the continuum. In the limit of small depletions, the two approaches give the same results. As the depletion increases, the two approaches differ due to many-body effects limiting the maximum possible molecule production efficiency. We have observed this in our simulations by considering higher-density gases. We have therefore shown the suitability and practicability of this beyond mean-field approach for application to further problems in the production of Feshbach molecules from ultracold gases.

•Jan 1, 2008•2 citations

Adiabatic association of ultracold molecules via magnetic-field tunable interactions

Krzysztof Góral, Thorsten Köhler, S. A. Gardiner +2 more

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.

arXiv:cond-mat/0312178•Journal of Physics B Atomic Molecular and Optical Physics•Aug 24, 2004•102 citations

Many-body dynamics of p -wave Feshbach-molecule production: A mean-field approach

L. Austen, L.G. Cook, M. D. Lee +1 more

We study the mean-field dynamics of $p$-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 $p$-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.

arXiv:1210.2317•Physical Review A•Feb 13, 2013•3 citations

Feshbach-molecule formation in a Bose-Fermi mixture

Tyler D. Cumby, Ruth Shewmon, Ming-Guang Hu +2 more

We investigate magnetoassociation of ultracold fermionic Feshbach molecules in a mixture of ${}^{40}$K and ${}^{87}$Rb atoms, where we can create ${}^{40}$K${}^{87}$Rb molecules with a conversion efficiency as high as 45%. In the limit of fast magnetic sweeps and small molecule fraction, we find that the time scale of molecule production is accurately predicted by a parameter-free model that uses only the atomic densities, masses, and the known Feshbach resonance parameters. We find that molecule production efficiency saturates for slower magnetic sweeps, at which point the maximum number of Feshbach molecules depends primarily on the atoms' phase-space density. At higher temperatures, our measurements agree with a phenomenological model that successfully describes the formation of bosonic molecules from either Bose or Fermi gases. However, for quantum degenerate atom gas mixtures, we measure significantly fewer molecules than this model predicts.

arXiv:1212.0903•Physical Review A•Jan 7, 2013•35 citations
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: (1) the effective adiabatic rate constant is proportional to density; (2) 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; (3) 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, Phys. Rev. Lett. 94, 120402 (2005)] without the use of adjustable parameters.
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 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.
The variation of a magnetic field in the vicinity of a zero-energy resonance allows highly vibrationally excited molecules (‘Feshbach molecules’) to be produced from an ultracold atomic gas. In this thesis, we study the dynamics of this process. We begin by studying the dissociation of Feshbach molecules, showing that in the limit of a sudden jump the shape of the spectrum of dissociated atoms can act as a probe of the zero-energy resonance. For some resonances, such jumps are within reach of current experiments. We also study the intermediate region between sudden jumps and asymptotically wide, linear ramps. It is shown from a precise derivation how the latter limit leads to a universal spectrum with a shape independent of the implementation of the two-body physics, provided that the near-resonant scattering properties are correctly modelled. We then turn to the dynamics of Feshbach molecule production from thermal and condensed gases. Our microscopic quantum dynamics approach includes the exact twobody evolution as an input to the many-body calculations. We show that in the long-time limit, and the Markov limit for the interactions, the non-Markovian Boltzmann equation (NMBE) we derive for the one-body density matrix reduces to the normal Boltzmann equation. In the limit of short times and small depletion of the atomic gas, the molecule production efficiency can be calculated by thermally averaging the two-body transition probability density. This thermal averaging technique is applied to studies of the formation of Feshbach molecules using a magnetic field modulation that is near-resonant with the molecular bound state energy. The continuum is shown to have a significant effect on both the dynamics and efficiency of this process. We examine the dependence of the molecule production efficiency on the duration, amplitude and frequency of the modulation, as well as the temperature and density of the gas. This method of producing molecules is effective for a wide range of bound state energies, but requires sufficient variation of the two-body energy levels with magnetic field. Lastly, we implement the NMBE for the case of a fast linear ramp across a Feshbach resonance. The solution of this equation is made feasible by including a large part of the required computation in the kernel, which is calculated in advance. The NMBE allows predictions of the molecule production efficiency which go beyond the thermal averaging technique by accounting for the depletion and rethermalisation of the continuum. In the limit of small depletions, the two approaches give the same results. As the depletion increases, the two approaches differ due to many-body effects limiting the maximum possible molecule production efficiency. We have observed this in our simulations by considering higher-density gases. We have therefore shown the suitability and practicability of this beyond mean-field approach for application to further problems in the production of Feshbach molecules from ultracold gases.
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.
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.
We investigate magnetoassociation of ultracold fermionic Feshbach molecules in a mixture of 40{}^{40}40K and 87{}^{87}87Rb atoms, where we can create 40{}^{40}40K87{}^{87}87Rb molecules with a conversion efficiency as high as 45%. In the limit of fast magnetic sweeps and small molecule fraction, we find that the time scale of molecule production is accurately predicted by a parameter-free model that uses only the atomic densities, masses, and the known Feshbach resonance parameters. We find that molecule production efficiency saturates for slower magnetic sweeps, at which point the maximum number of Feshbach molecules depends primarily on the atoms' phase-space density. At higher temperatures, our measurements agree with a phenomenological model that successfully describes the formation of bosonic molecules from either Bose or Fermi gases. However, for quantum degenerate atom gas mixtures, we measure significantly fewer molecules than this model predicts.