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Collective Modes in Bose-Fermi Mixture Gases

Asano Yoji, Yoji Asano

Published Mar 17, 2020• 0 citations
Publisher

Abstract

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 . . . . . . . . . . . . . . . . . . .

DOI
10.20604/00002968

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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 . . . . . . . . . . . . . . . . . . .
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.
Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.
We develop the moment method for Bose-Einstein condensates at finite temperatures that enable us to study collective sound modes from the hydrodynamic to the collisionless regime. In particular, we investigate collective excitations in a weakly interacting dilute Bose gas by applying the moment method to the Zaremba-Nikuni-Griffin equation, which is the coupled equation of the Boltzmann equation with the generalized Gross-Pitaevskii equation. Utilizing the moment method, collective excitations in the crossover regime between the hydrodynamic and collisionless regimes are investigated in detail. In the crossover regime, the second sound mode loses the weight of the density response function because of the significant coupling with incoherent modes, whereas the first sound shows a distinct but broad peak structure. We compare the result obtained by the moment method with that of the Landau two-fluid equations and show that the collective mode predicted by the Landau two-fluid equations well coincides with the result from the moment method even far from the hydrodynamic regime, whereas clear distinction also emerges in the relatively higher momentum regime.
The field of ultra-cold atomic physics allows probing of two-component quantum liquids with a hydrodynamic normal fluid accompanied by superfluid Bose-Einstein condensates (BEC’s), where the experimental results can be compared to theoretical predictions of the behavior of a weakly interacting two fluid system. Hydrodynamicity in quantum systems like BEC's can have influence on multiple aspects of the collective behavior of the system, such as on the oscillation frequency of quadrupole modes, multiple sound modes in the condensate, but also on high-order excitations which resemble standing sound waves. The hydrodynamicity and excitations of the ultra-cold atomic cloud are the main motivation for the experiments presented in this thesis. In the Utrecht experiment large BEC's up to a few hundred million atoms are produced, which are deeply in the hydrodynamic collisional regime. The macroscopic behavior of the cloud of atoms is dominated by inter-atomic collisions since the mean free path of the individual particles is much smaller than the size of the atomic sample and local equilibrium is established. The influence of the hydrodynamicity on the collective excitations can be investigated by measuring the quadrupole frequency of ultra-cold atomic clouds above the transition temperature Tc, where Bose-Einstein condensation occurs. This quadrupole frequency is determined by measuring the collective shape oscillations of the atomic cloud, where the oscillation frequency is influenced by inter-atomic collisions during the shape oscillations due to the hydrodynamicity of the atomic cloud. The hydrodynamic equations predict the existence of two sound modes in ultra-cold atomic gases below Tc. The speeds of sound are dependent on both the temperature and the chemical potential of the atomic cloud. For low enough temperatures or large enough chemical potential these two speeds of sound become comparable and reaching this regime experimentally allows measuring the speed of sound beyond this avoided crossing. An alternate method of measuring the speed of sound of ultra-cold atomic clouds is exploiting the analogy between a standing sound wave-pattern and high order excitations in BEC's, also referred to as Faraday waves. By exciting the wave-pattern with a frequency and measuring the wavenumber k of the wave-pattern, the speed of sound c can be extracted from the linear dispersion relation. In the process of exciting the standing wave-pattern, remarkable dynamics is discovered in the BECs. As a result of the excitation a large array of other modes are excited next to the wave-pattern. These modes are for example the dipole mode, the axial and radial quadrupole modes, and the scissor mode. This gives rise to complex dynamics in this hydrodynamic quantum mechanical system and although the ultra-cold sample is only excited with a very short pulsed excitation, these modes and the wave-pattern show a large degree of reproducibility during several seconds of time evolution.
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.
We calculate the effects of two-body interactions on the low-frequency oscillations of a normal Fermi gas confined in a harmonic trap. The mean-field contribution to the collective frequencies is evaluated in the collisionless regime using a sum-rule approach. We also discuss the transition between the collisionless and hydrodynamic regimes with special emphasis on the spin dipole mode in which two atomic clouds occupying different spin states oscillate in the opposite phase. The spin dipole mode is predicted to be overdamped in the hydrodynamic regime. The relaxation time is calculated as a function of temperature and the effects of Fermi statistics are explicitly pointed out.