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.
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.
The field of Bose-Einstein condensation (BEC) in dilute atomic gases provides a fruitful playground to test well-developed theories of quantum fluids. Research using BECs can address open questions relating to the many-body aspects of two-component quantum liquids, namely the interaction between the hydrodynamic normal and the superfluid component at finite temperatures. After the first realization of BEC some pilot experiments have been carried out, but detailed experiments are missing. This has to be compared to the case of liquid helium II, where many experiments since the 1950s have added to our understanding of novel phenomena in quantum liquids, like collective excitations, first and second sound, and others. One of the drawbacks of liquid helium is that the interactions are so strong that a clear distinction between the two components is difficult and the interpretation of the phenomena remains ambiguous. Most experimental studies of dilute Bose-Einstein condensed gases use clouds, which are in the collisionless regime. The reason for the lack of detailed experiments in BECs to study quantum liquids and in particular the hydrodynamical aspects of it, is the limited number of atoms (typically 1--10 million) in the experiments leaving the thermal atoms virtually collisionless. Efforts to decrease the mean free path by increasing the confinement limits the lifetime of the sample, since the density is limited by three-body decay. Experiments on dilute clouds of cold atoms are generally conducted in highly asymmetric traps. In these elongated, cigar-shaped geometries the mean free path of the atoms can become much shorter than the size of the cloud in the long, axial direction, but at the same time exceeds the size in the other, radial directions. In this so-called hydrodynamic-collisionless regime the system can be considered radially collisionless and axially hydrodynamic. In our setup we have created BECs containing up to 300 million sodium atoms by evaporation of atoms in an axially strongly decompressed trap with an aspect ratio of 1:65. Hot atoms created in three-body collisions are able to leave the sample in this highly asymmetric trap, before they can heat other atoms in an avalanche. The sample is axially hydrodynamic, but due to the large aspect ratio collisionless in the radial direction. Such samples are ideal for the observation of the interaction between the superfluid and normal fluid component of the cloud. In this thesis we describe the setup which is used to reach the hydrodynamic regime and we describe a novel implementation of the well-known phase contrast imaging technique. Next, we study collective excitations in the crossover from the collisionless toward the hydrodynamic regime, we determine the heat conduction in a cold, hydrodynamic thermal cloud and observe the propagation of a thermal wave in the two-fluid regime. Furthermore, we study the friction between the superfluid and normal fluid component of the cloud. In the final chapters of this thesis we study the propagation of second sound, as well as the formation and propagation of shock waves.
At temperatures near the absolut zero, a gas, here atomic sodium vapour, with high enough density cannot be described as tiny balls moving around as in classical physics. Since the temperature is low, the atoms are so slow that the matterwave of each atom starts to extend over the size of the atom and even over the interatomic distance. Therefore, they start to interfere like waves. Quantum mechanics start to dominate the physics in this regime. Further, depending on the sort of atoms (bosons or fermions) the atoms prefer to be in the same state or avoid to be in the same state. In the case of bosons as in the thesis, if the temperature is lowered to sub micro Kelvin temperature, a new state of matter appears after a phase transition - a macroscopic, standing wave, the Bose-Einstein condensate. This leads to a new phenomena: superfluidity - frictionless flow, second sound, vorticity and coherent scattering effects to name a few. The atoms are trapped in a elongated trap as in most of the experiments in ultra cold gasses. Usually experiments are done in a regime where the atoms seldomly collide with each other while travelling from one end to the other end of the cloud. In this experiment, however, the atoms collide many times with each other when they oscillate in the trap. This means that the cloud is hydrodynamic and leads to a very different behaviour. Two different sound waves (first and second sound), heat conduction, and collisional dominated transport can be observed in this case. The fact that the gas is weakly interacting allows comparison with current theory. At very low temperatures as in the experiments described in the thesis, the Bose character strongly alters the collisions of the atoms. The outcome of the collision does not only depend on the colliding atoms, but also on the atoms near by in phase space. The experiments outlined in this thesis cover some aspects of physics involved. Vortices have been created and observed in the Bose-Einstein condensate that rested in a weak trap. Both sound modes first and second sound have been observed with slight local heating. In the regime considered, first sound, that has not been observed in a dilute Bose-Einstein condensate before, is mainly a modulation in local temperature and not density. In another experiment a particular effect is observed when the cloud is only axially hydrodynamic. When the trap is suddenly axially relaxed, the cloud creats a strip pattern that is in the radial direction and moving outward. The analysis of this experiment suggests that this is only the case when the cloud is indeed only axially hydrodynamic. The last chapter describs a spin drag experiment. In this experiment two spin species of atoms are prepared and on one of them a force is applied. Through collisions, the other species is dragged along. This effect is Bose enhanced at temperatures approaching the transition temperature to a Bose-Einstein condensate which is observed and matches recent theory.
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 . . . . . . . . . . . . . . . . . . .
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
Dynamics play an important role in physics. From classical mechanics, astrophysics, hydrodynamics to quantum physics, the dynamics of physical systems have crucial impact on our understanding of nature. For simple configurations such as in the two-body problem the dynamics can be calculated analytically. However, things get more complicated once the number of contributors in a system increases such as in few- or many-body systems. One way to study such an ensemble is to use a model system where decisive parameters can be controlled. In this thesis I make use of an ultracold cloud of fermionic Lithium-6 atoms close to zero temperature to study dynamics of few- and many-body systems. Two experimental studies to this topic are presented. In the first study, I produced a gas of di-atomic molecules (dimers), consisting of two Lithium-6 atoms in different states, to study the reaction kinetics of the four-body process of a dimer-dimer collision. At temperatures close to absolute zero the internal and external degrees of freedom of the dimers can be well controlled. This allowed us to investigate the kinetics of a single-channel reaction. To resolve the dynamics of the process we heated a gas of molecules and tracked the state of the system as a function of time. We determined the reaction-rate constant of the process for various thermal energies which allowed us to verify the Arrhenius law for molecule-molecule reactions in the ultracold regime. Additionally, we provided first insight into the dependence of this four-body process on the particle interaction. In the second experimental work, I focused on the thermodynamical properties of the ultracold gas. Cooling the gas of Lithium-6 atoms close to zero temperature the gas undergoes a phase transition and superfluidity is established. To study the dynamics in such a gas, we used a local perturbation to produce sound waves. Depending on the excitation scheme we either excited first sound, i.e. a pressure wave, known from classical mechanics, or second sound, i.e. an entropy wave. In the study we focused on second sound and determined the second sound speed as a function of the particle interaction. We found agreement of our results with both - a numerical and an analytical description of the interaction strength dependence of the second sound speed. Besides the experimental work, a theoretical proposal for a novel imaging technique for a 2D ultracold gas in an optical lattice is presented. The scheme is based on a holographic approach and enables the detection of individual particles with only a few hundred scattered photons per atom.
Second sound is an entropy wave which propagates in the superfluid component of a quantum liquid. Because it is an entropy wave, it probes the thermodynamic properties of the quantum liquid which are determined, e.g., by the interaction strength between the particles of the quantum liquid and their temperature. Here, we study second sound propagation for a large range of interaction strengths within the crossover between a Bose-Einstein condensate (BEC) and the Bardeen-Cooper-Schrieffer (BCS) superfluid. In particular, we investigate the strongly-interacting regime where currently theoretical predictions only exist in terms of an interpolation between the BEC, BCS and unitary regimes. Working with a quantum gas of ultracold fermionic 6Li atoms with tunable interactions, we show that the second sound speed varies only slightly in the crossover regime. We gain deeper insights into sound propagation and excitation of second sound by varying the excitation procedure which ranges from a sudden force pulse to a gentle heating pulse at the cloud center. These measurements are accompanied by classical-field simulations which help with the interpretation of the experimental data. Furthermore, we determine the spatial extension of the superfluid phase and estimate the superfluid density. In the future, this may be used to construct the so far unknown equation of state throughout the crossover.