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arXiv:cond-mat/0310498 · Physical Review A

Scissors modes of two-component degenerate gases: Bose-Bose and Bose-Fermi mixtures

Mirta Rodríguez, P. Pedri, Päivi Törmä, Luis Santos

Published Feb 26, 2004• 12 citations
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Abstract

We investigate the scissors modes in binary mixtures of degenerate dilute quantum gases for both Bose-Bose and Bose-Fermi mixtures. For the latter we consider both the superfluid and normal hydrodynamic and collisionless regimes. We analyze the dependence of the frequencies of the scissors modes and their character as a function of the Bose-Fermi coupling and the trap geometry. We show that the scissors mode can reveal a clear trace of the hydrodynamic behavior of the Fermi gas.

DOI
10.1103/physreva.69.023617

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We investigate the scissors modes in binary mixtures of degenerate dilute quantum gases for both Bose-Bose and Bose-Fermi mixtures. For the latter we consider both the superfluid and normal hydrodynamic and collisionless regimes. We analyze the dependence of the frequencies of the scissors modes and their character as a function of the Bose-Fermi coupling and the trap geometry. We show that the scissors mode can reveal a clear trace of the hydrodynamic behavior of the Fermi gas.
We evaluate the frequencies of scissors modes for density and concentration fluctuations in a vapor of fermionic atoms placed in two hyperfine levels inside a spherical harmonic trap. Both the superfluid and normal states are considered, with inclusion of the interactions at the random-phase level. Two main results are obtained: (i) the transition to the superfluid state is signaled by the disappearance of soft transverse modes of the normal fluid in the collisionless regime; and (ii) the eigenfrequency of the density fluctuations in the superfluid coincides with that of the normal fluid in the collisional regime. The latter property is related to the opening of the gap in the single-pair spectrum.
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 investigate the oscillation of a dilute atomic gas generated by a sudden rotation of the confining trap (scissors mode). This oscillation reveals the effects of superfluidity exhibited by a Bose-Einstein condensate. The scissors mode is also investigated in a classical gas above Tc{T}_{c}Tc​ in various collisional regimes. The crucial difference with respect to the superfluid case arises from the occurrence of low frequency components, which are responsible for the rigid value of the moment of inertia. Different experimental procedures to excite the scissors mode are discussed.
The scissors mode plays a crucial role in the study of the unitary Fermi gas. In this paper, we simulate the scissors mode by solving the hydrodynamic equations with appropriate initial conditions, and then extract the flow field of the gas. The flow fields in different regimes are found to be essentially different. The characteristic differences in the flow patterns between the superfluid and the normal viscous fluid are presented. Irrotational flow signals superfluidity, while rotational one indicates normal hydrodynamical behavior. These different characteristics can be visualized by the velocity portraits of the flow, which provide an intuitive way to discriminate the states of the Fermi gas.
The study of the scissors mode gives important information about trapped gases. Based on viscous hydrodynamics, a set of differential equations which describes the dynamics of the trapped unitary Fermi gas is formulated. With relevant initial conditions, the solution of these equations reproduces the angular oscillation, i.e. the scissors mode. Besides the frequency and damping, detailed information on the time evolution of the density and velocity distribution can be obtained. This approach provides an effective way to study the scissors mode.
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.