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arXiv:cond-mat/0002145 · Physical Review A

Sum-rule approach to collective oscillations of a boson-fermion mixed condensate of alkali-metal atoms

T. Miyakawa, T. Suzuki, Hiroyuki Yabu

Published Nov 15, 2000• 53 citations
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Abstract

The behavior of collective oscillations of a trapped boson-fermion mixed condensate is studied in the sum-rule approach. Mixing angle of bosonic and fermionic multipole operators is introduced so that the mixing characters of the low-lying collective modes are studied as functions of the boson-fermion interaction strength. For an attractive boson-fermion interaction, the low-lying monopole mode becomes a coherent oscillation of bosons and fermions and shows a rapid decrease in the excitation energy towards the instability point of the ground state. In contrast, the low-lying quadrupole mode keeps a bosonic character over a wide range of the interaction strengths. For the dipole mode the boson-fermion in-phase oscillation remains to be the eigenmode under the external oscillator potential. For weak repulsive values of the boson-fermion interaction strengths we found that an average energy of the out-of-phase dipole mode stays lower than the in-phase oscillation. Physical origin of the behavior of the multipole modes against boson-fermion interaction strength is discussed in some detail.

DOI
10.1103/physreva.62.063613

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The behavior of collective oscillations of a trapped boson-fermion mixed condensate is studied in the sum-rule approach. Mixing angle of bosonic and fermionic multipole operators is introduced so that the mixing characters of the low-lying collective modes are studied as functions of the boson-fermion interaction strength. For an attractive boson-fermion interaction, the low-lying monopole mode becomes a coherent oscillation of bosons and fermions and shows a rapid decrease in the excitation energy towards the instability point of the ground state. In contrast, the low-lying quadrupole mode keeps a bosonic character over a wide range of the interaction strengths. For the dipole mode the boson-fermion in-phase oscillation remains to be the eigenmode under the external oscillator potential. For weak repulsive values of the boson-fermion interaction strengths we found that an average energy of the out-of-phase dipole mode stays lower than the in-phase oscillation. Physical origin of the behavior of the multipole modes against boson-fermion interaction strength is discussed in some detail.
Instabilities for boson-fermion mixed condensates of trapped alkali-metal atoms due to the boson-fermion attractive interaction are studied using a variational method. A stable, metastable, and unstable region as a function of the boson-fermion interaction exists. The stability condition is obtained analytically from the asymptotic expansion of the variational total energy. The lifetime of metastable states is discussed for tunneling decay and is estimated to be very long. It suggests that, except near the instability border, metastable mixed condensate should be almost stable against tunneling decay. The critical border between metastable and unstable phases is calculated numerically and is shown to be consistent with the M\o{}lmer scaling condition.
Within a scaling ansatz formalism plus Thomas-Fermi approximation, we investigate the collective excitations of a harmonically trapped boson-fermion mixture in the collisionless and hydrodynamic limit at low temperature. Both the monopole and quadrupole modes are considered in the presence of spherical as well as cylindrically symmetric traps. In the spherical traps, the frequency of the monopole mode coincides in the collisionless and hydrodynamic regime, suggesting that it might be undamped in all collisional regimes. In contrast, for the quadrupole mode, the frequency differs largely in these two limits. In particular, we find that in the hydrodynamic regime the quadrupole oscillations with equal bosonic and fermionic amplitudes generate an exact eigenstate of the system, regardless of the boson-fermion interaction. This resembles the Kohn mode for the dipole excitation. We discuss in some detail the behavior of monopole and quadrupole modes as a function of boson-fermion coupling at different boson-boson interaction strength. Analytic solutions valid at weak and medium fermion-boson coupling are also derived and discussed.
We calculate the spectrum of low-lying collective excitations in a gaseous cloud formed by a Bose-Einstein condensate and a spin-polarized Fermi gas over a range of the boson-fermion coupling strength extending from strongly repulsive to strongly attractive. Increasing boson-fermion repulsions drives the system towards spatial separation of its components (``demixing''), whereas boson-fermion attractions drives it towards implosion (``collapse''). The dynamics of the system is treated in the experimentally relevant collisionless regime by means of a random-phase approximation, and the behavior of a mesoscopic cloud under isotropic harmonic confinement is contrasted with that of a macroscopic mixture at given average particle densities. In the latter case the locations of both the demixing and the collapse phase transitions are sharply defined by the same stability condition, which is determined by the softening of an eigenmode of either fermionic or bosonic origin. In contrast, the transitions to either demixing or collapse in a mesoscopic cloud at fixed confinement and particle numbers are spread out over a range of boson-fermion coupling strength, and some initial decrease of the frequencies of a set of collective modes is followed by hardening, as evidenced by blue shifts of most eigenmodes. The spectral hardening can serve as a signal of the impending transition and is most evident when the number of bosons in the cloud is relatively large. We propose physical interpretations for these dynamical behaviors with the help of suitably defined partial compressibilities for the gaseous cloud under confinement.
We perform a random-phase approximation study of collective excitations in a Bose-Fermi mixed degenerate gas of alkali-metal atoms at T=0.T=0.T=0. The calculation is done by diagonalization in a model space composed of particle-hole type excitations from the ground state, the latter being obtained from the coupled Gross-Pitaevskii and Thomas-Fermi equations. We investigate strength distributions for different combinations of Bose and Fermi multipole (L) operators with L=0,1,2,3.L=0,1,2,3.L=0,1,2,3. Transition densities and dynamical structure factors are calculated for collective excitations. A comparison with the sum rule prediction for the collective frequency is given. The time dependent behavior of the system after an external impulse is studied.
We study the dynamical behavior of a Bose-Fermi mixture of trapped atomic gases which exhibits phase separation for a sufficiently strong repulsive interparticle interaction. For a dominantly bosonic mixture, the fermion density distribution and the associated single-particle spectra undergo a rapid change around the transition to the separated phase. Monopole strength distribution is studied in the random-phase approximation. The average energy calculated from the fermionic monopole strength shows a sharp minimum close to the transition point. The dynamical origin of the fermion single particle and collective properties is studied within a simple approximation.
We investigate the effect of unequal masses of the Bose and Fermi particles on the collective oscillations of the degenerate boson-fermion mixtures. In particular, we consider the monopole and the quadrupole modes of the oscillations and study their characters and the frequencies by using variational-sum-rule approach. We find that for both the modes there exist critical values of the ratio of boson-fermion masses below and above which the character and frequency of the respective modes are significantly different.