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arXiv:cond-mat/0212169 · Physical Review A

Collisionless and hydrodynamic excitations of trapped boson-fermion mixtures

Xia-Ji Liu, Hui Hu

Published Feb 28, 2003• 50 citations
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Abstract

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.

DOI
10.1103/physreva.67.023613

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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 present a detailed study of the low-lying collective excitations of a spherically trapped Bose-Fermi mixture at finite temperature in the collisionless regime. The excitation frequencies of the condensate are calculated self-consistently using the static Hartree-Fock-Bogoliubov theory within the Popov approximation. The frequency shifts and damping rates due to the coupled dynamics of the condensate, noncondensate, and degenerate Fermi gas are also taken into account by means of the random-phase approximation and linear-response theory. In our treatment, the dipole excitation remains close to the bare trapping frequency for all temperatures considered, and thus is consistent with the generalized Kohn theorem. We discuss in some detail the behavior of monopole and quadrupole excitations as a function of the Bose-Fermi coupling. At nonzero temperatures we find that, as the mixture moves towards spatial separation with increasing Bose-Fermi coupling, the damping rate of the monopole (quadrupole) excitation increases (decreases). This provides us a useful signature to identify the phase transition of spatial separation.
We study the collective excitations of polarized single-component quasi-two-dimensional dipolar fermions in an isotropic harmonic trap by solving the collisional Boltzmann-Vlasov (CBV) equation via the method of moments. We study the response to monopole and quadrupole perturbations of the trap potential and investigate the dynamical character of excitations in each case. Simple analytic formulas are found using the linearized scaling ansatz approximation and accurate numerical results are obtained by satisfying the first eight moments of the CBV equation. Except for the lowest-lying monopole mode that is weakly affected by collisions, the quadrupole and the higher-order monopole modes undergo a transition from the collisionless regime to a dissipative crossover regime and finally approach the hydrodynamic regime upon increasing the dipolar interaction strength. For strong transverse confinement (2D limit), we predict the existence of a temperature window within which the characteristics of the collective modes become temperature independent. This plateau, which is a direct consequence of dipole-dipole scatterings, persists as long as the scattering energies remain in the near-threshold regime. The predictions of this work are expected to be observable in the current experiments.
This thesis theoretically deals with collective modes in the normal phase of a Bose-Fermi mixture composed of single-species bosons and single-species fermions.The mixture is experimentally established as an ultracold atomic gas.While the community studies the mixture including a superfluid phase, relevant to Bose-Einstein condensation of bosons and Cooper paring of fermions, dynamical behaviors in the normal phase have been not fully understood so that we are interested in.We consider the mixture in both spatially uniform and harmonically trapped systems with the Boltzmann equation at finite temperatures.We perform the analysis under the assumption that the physical quantities are near to equilibrium.After the linerization, we apply the moment method to the equation, obtaining a closed set of coupled equations to describe sound or oscillation.The collisional property of the system characterizes the collective modes in the sense of the hydrodynamic regime, collisionless regime, and transition between these regimes.In the uniform system, we investigate sound mode and the dynamic density structure factor.With decreasing temperature, we find that the sound velocity deviates from that of the hydrodynamic mode.Even at a temperature much lower than the Fermi temperature, the sound velocity does not reach the collisionless limit analogous to the zero sound in a Fermi system.However, we show the existence of a long-lived sound at a low temperature as distinct from the first and zero sounds.Nature of the long-lived sound seems to be originated by Pauli blocking of fermions; this statistical effect makes the interspecies collision suppress in the Bose-Fermi mixture.In the trapped system, we mainly study dipole oscillation.We consistently define the relaxation time originated by the interspecies scattering.We find that two oscillating modes in the collisionless regime have distinct fates in the hydrodynamic regime; one collisionless mode shows a crossover to a hydrodynamic in-phase mode, and the other collisionless mode shows a transition to two purely-damped modes.We show the time evolution of the three types of hydrodynamic modes with suitable initial conditions.We discuss temperature dependence of the dipole modes.We observe that the relaxation time dominates the dependence compared to the mean-field contribution.We also observe the nonmonotonic behavior of the relaxation time when Pauli blocking of fermions is not negligible.i List of Figures 1.1 Two schematic images of collisions with a period of oscillation . . . . . . . . . . . . . . . . 1 1.2 Measurement of the zero sound in pure liquid 3 He . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Transition between the first sound and the zero sound in a two-component Fermi gas . . . 3 1.4 Dipole oscillations of the two-component gas of 40 K . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Frequencies of dipole mode in the two-component gas of 40 K . . . . . . . . . . . . . . . . . 4 1.6 Damping times of dipole mode in the two-component gas of 40 K . . . . . . . . . . . . . . . 5 1.7 Dipole oscillations of the mixture of 87 Rb and 40 K in the hydrodynamic regime . . . . . . .6 1.8 Dipole oscillations of the mixture of 87 Rb and 40 K in the deeper hydrodynamic regime . . .6 1.9 Dipole oscillations of the mixture of 87 Rb and 40 K in the collisionless regime . . . . . . . .7 2.1 Results obtained by solving the coupled moment equations for sound mode . . . . . . . . .26 3.1 Time evolution of the center-of-mass positions and velocity fields . . . . . . . . . . . . . .39 3.2 Dipole modes as a function of the relaxation time with N B = N F . . . . . . . . . . . . . . .40 3.3 Dipole modes as a function of the relaxation time with N B /N F = 0.2 . . . . . . . . . . . . .41 3.4 Ratio of the displacement of the center of mass to the velocity field as a function of the relaxation time . . . . . . . . . . . . . . . . . . .
We study the collective modes of a confined gaseous cloud of bosons and fermions with mutual attractive interactions at zero temperature. The cloud consists of a Bose-Einstein condensate and a spin-polarized Fermi gas inside a spherical harmonic trap and the coupling between the two species is varied by increasing either the magnitude of the interspecies sss-wave scattering length or the number of bosons. The mode frequencies are obtained in the collisional regime by solving the equations of generalized hydrodynamics and are compared with the spectra calculated in the collisionless regime within a random-phase approximation. We find that, as the mixture is driven towards the collapse instability, the frequencies of the modes of fermionic origin show a blue shift which can become very significant for large numbers of bosons. Instead the modes of bosonic origin show a softening, which becomes most pronounced in the very proximity of collapse. Explicit illustrations of these trends are given for the monopolar spectra, but similar trends are found for the dipolar and quadrupolar spectra except for the surface (n=0)(n=0)(n=0) modes which are essentially unaffected by the interactions.
Dipole oscillation is studied in a normal phase of a trapped Bose-Fermi-mixture gas composed of single-species bosons and single-species fermions. Applying the moment method to the linearized Boltzmann equation, we derive a closed set of equations of motion for the center-of-mass position and momentum of both components. By solving the coupled equations, we reveal the behavior of dipole modes in the transition between the collisionless regime and the hydrodynamic regime. We find that two oscillating modes in the collisionless regime have distinct fates in the hydrodynamic regime: one collisionless mode shows a crossover to a hydrodynamic in-phase mode, and the other collisionless mode shows a transition to two purely damped modes. The temperature dependence of these dipole modes are also discussed.
By using a sum-rule approach we investigate the transition between the hydrodynamic and the collisionless regime of the collective modes in a one-dimensional (1D) harmonically trapped Bose gas. Both the weakly interacting gas and the Tonks-Girardeau limits are considered. We predict that the excitation of the dipole compression mode is characterized in the high-temperature collisionless regime by a beating signal of two different frequencies (${\ensuremath{\omega}}_{z}$ and $3{\ensuremath{\omega}}_{z}$), while in the high-temperature collisional regime, the excitation consists of a single frequency ($\sqrt{7}{\ensuremath{\omega}}_{z}$). This behavior differs from the case of the lowest breathing mode whose excitation consists of a single frequency ($2{\ensuremath{\omega}}_{z}$) in both regimes. Our predictions for the dipole compression mode open promising perspectives for the experimental investigation of collisional effects in 1D configurations.