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arXiv:cond-mat/0301182 · Physical Review A

Expansion of a quantum degenerate boson-fermion mixture

Hui Hu, Xia-Ji Liu, Michele Modugno

Published Jun 30, 2003• 22 citations
arXivHTMLPDF

Abstract

We study the expansion of an ultracold boson-fermion mixture released from an elongated magnetic trap, by using a scaling approach. We discuss in detail the role of the boson-fermion interaction on the evolution of the radial-to-axial aspect ratio of the condensate, and show that the latter depends crucially on the relative dynamics of the condensate and degenerate Fermi gas in the radial direction, which is characterized by the ratio between the trapping frequencies for fermions and bosons. The numerical solution of the scaling equations provides a reasonable agreement with the recent experiment [G. Roati et al., Phys. Rev. Lett. 89, 150403 (2002)].

DOI
10.1103/physreva.67.063614

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We study the expansion of an ultracold boson-fermion mixture released from an elongated magnetic trap, by using a scaling approach. We discuss in detail the role of the boson-fermion interaction on the evolution of the radial-to-axial aspect ratio of the condensate, and show that the latter depends crucially on the relative dynamics of the condensate and degenerate Fermi gas in the radial direction, which is characterized by the ratio between the trapping frequencies for fermions and bosons. The numerical solution of the scaling equations provides a reasonable agreement with the recent experiment [G. Roati et al., Phys. Rev. Lett. 89, 150403 (2002)].
We study the expansion of an atomic Fermi gas interacting attractively with a Bose-Einstein condensate. We observe a slower evolution of the radial-to-axial aspect ratio which reveals the importance of the mutual attraction between the two samples during the first phase of the expansion. For large atom numbers, we also observe a bimodal momentum distribution of the Fermi gas, which reflects the spatial distribution of the mixture in trap. This effect allows us to extract important information on the overlap of the two species across the collapse.
Abstract. – We discuss the effects of collisions on the expansion of an ultracold normal Fermi gas, following the sudden removal of the confining trap. Using a Boltzmann equation approach, we calculate the time dependence of the aspect ratio and the temperature increase of the expanding atomic cloud. We find that for dilute gases and standard choices of the trap parameters the aspect ratio does not deviate significantly from the predictions of ballistic expansion. Conversely, if the trap is sufficiently elongated the thermal broadening of the density distribution can be quite large, revealing that even at zero temperature collisions are effective in a Fermi gas. The expansion of an atomic gas following the sudden removal of the confining trap is known to provide valuable information on the state of the system and on the role of interactions [1]. In particular in the superfluid phase, where the macroscopic dynamics of the gas are governed by the equations of hydrodynamics, one expects anisotropic expansion if the gas is released from an anisotropic trap. This peculiar feature, which should be contrasted with the isotropic ballistic expansion exhibited by a non interacting gas, was first discussed in [2,3] in the case
We investigate the dynamics of magnetically trapped boson-fermion mixtures at very low temperature. Using the mean field approach, we derive the Gross-Pitaevskii equation for the Bose-Einstein condensate wavefunction. We apply the Thomas-Fermi approximation and the hydrodynamic approach to the degenerate Fermi gas and obtain an equation of motion for its number density perturbation. It is remarkable that coupled time-evolution equations for the mixture can be regarded as an extension of the Zakharov equations, a model for the Langmuir waves in plasma.
Interactions in an ultracold boson-fermion mixture are often manifested by elastic collisions. In a mixture of a condensed Bose gas (BEC) and spin polarized degenerate Fermi gas (DFG), fermions can mediate spin-spin interactions between bosons, leading to an effective long-range magnetic interaction analogous to Ruderman-Kittel-Kasuya-Yosida [Phys. Rev. 96, 99 (1954); Prog. Theor. Phys. 16, 45 (1956); Phys. Rev. 106, 893 (1957)] interaction in solids. We used Ramsey spectroscopy of the hyperfine clock transition in a ^{87}Rb BEC to measure the interaction mediated by a ^{40}K DFG. By controlling the boson density we isolated the effect of mediated interactions from mean-field frequency shifts due to direct collision with fermions. We measured an increase of boson spin-spin interaction by a factor of η=1.45±0.05^{stat}±0.13^{syst} in the presence of the DFG, providing clear evidence of spin-spin fermion mediated interaction. Decoherence in our system was dominated by inhomogeneous boson density shift, which increased significantly in the presence of the DFG, again indicating mediated interactions. We also measured a frequency shift due to boson-fermion interactions in accordance with a scattering length difference of a_{bf_{2}}-a_{bf_{1}}=-5.36±0.44^{stat}±1.43^{syst}a_{0} between the clock-transition states, a first measurement beyond the low-energy elastic approximation [R. Côté, A. Dalgarno, H. Wang, and W. C. Stwalley, Phys. Rev. A 57, R4118 (1998); A. Dalgarno and M. Rudge, Proc. R. Soc. A 286, 519 (1965)] in this mixture. This interaction can be tuned with a future use of a boson-fermion Feshbach resonance. Fermion-mediated interactions can potentially give rise to interesting new magnetic phases and extend the Bose-Hubbard model when the atoms are placed in an optical lattice.
After release from the trap the momentum distribution of an impenetrable gas asymptotically approaches that of a spinless noninteracting Fermi gas in the initial trap. This phenomenon is called dynamical fermionization and, very recently, has been experimentally confirmed in the case of the Lieb-Liniger model in the Tonks-Girardeau regime. We prove analytically and confirm numerically that following the removal of axial confinement the strongly interacting Bose-Fermi mixture exhibits dynamical fermionization and the asymptotical momentum distribution of each component has the same shape as its density profile at t=0t=0t=0. Under a sudden change of the trap frequency to a new nonzero value the dynamics of both fermionic and bosonic momentum distributions presents characteristics which are similar to the case of single component bosons experiencing a similar quench. Our results are derived using a product representation for the correlation functions which, in addition to analytical considerations, can be implemented numerically very easily with complexity which scales polynomially in the number of particles.
We discuss the effects of collisions on the expansion of a degenerate normal Fermi gas, following the sudden removal of the confining trap. Using a Boltzmann equation approach, we calculate the time dependence of the aspect ratio and the entropy increase of the expanding atomic cloud taking into account the collisional effects due to the deformation of the distribution function in momentum space. We find that in dilute gases the aspect ratio does not deviate significantly from the predictions of ballistic expansion. Conversely, if the trap is sufficiently elongated the thermal broadening of the density distribution due to the entropy increase can be sizeable, revealing that even at zero temperature collisions are effective in a Fermi gas.