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Standing Sound Waves in a Bose-Einstein Condensate

G.A.R. van Dalum

Published Aug 26, 2013• 0 citations
OpenAlex

Abstract

We study two different subjects related to ultra-cold Bose gases. First we study the influence of the trap and the background pressure on particle loss and temperature change for trapped atom clouds. We find the number of particles to exponentially decrease over time due to the imperfect vacuum, while the rate of temperature change differs between a magnetic trap and a far off resonance optical trap. Secondly, we study the properties of a standing sound wave induced in a Bose-Einstein condensate. Phase contrast imaging is used to image the condensate in order to observe the density profile as a function of time. We find that there is coupling between the standing wave and other collective modes. There is a clear relation between the frequencies of these collective modes, and also between the frequencies of these modes and the frequency used to induce the modes. We also find that the amplitude of the standing wave decays over time.

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We study two different subjects related to ultra-cold Bose gases. First we study the influence of the trap and the background pressure on particle loss and temperature change for trapped atom clouds. We find the number of particles to exponentially decrease over time due to the imperfect vacuum, while the rate of temperature change differs between a magnetic trap and a far off resonance optical trap. Secondly, we study the properties of a standing sound wave induced in a Bose-Einstein condensate. Phase contrast imaging is used to image the condensate in order to observe the density profile as a function of time. We find that there is coupling between the standing wave and other collective modes. There is a clear relation between the frequencies of these collective modes, and also between the frequencies of these modes and the frequency used to induce the modes. We also find that the amplitude of the standing wave decays over time.
We study the propagation of a density wave in a magnetically trapped Bose-Einstein condensate at finite temperatures. The thermal cloud is in the hydrodynamic regime and the system is therefore described by the two-fluid model. A phase-contrast imaging technique is used to image the cloud of atoms and allows us to observe small density excitations. The propagation of the density wave in the condensate is used to determine the speed of sound as a function of the temperature. We find the speed of sound to be in good agreement with calculations based on the Landau two-fluid model.
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.
Sound propagation has been studied in a magnetically trapped dilute Bose-Einstein condensate. Localized excitations were induced by suddenly modifying the trapping potential using the optical dipole force of a focused laser beam. The resulting propagation of sound was observed using a novel technique, rapid sequencing of nondestructive phase-contrast images. The speed of sound was determined as a function of density and found to be consistent with Bogoliubov theory. This method may generally be used to observe high-lying modes and perhaps second sound.
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 investigate the structure and collective modes of a planar surface of a trapped Bose-Einstein condensed gas at zero temperature. In the long-wavelength limit we find a mode similar to the gravity wave on the surface of a fluid with the frequency $\ensuremath{\omega}$ and the wave number q related by ${\ensuremath{\omega}}^{2}=Fq/m.$ Here F is the force due to the confining potential at the surface and m is the particle mass. At shorter wavelengths we use a variational approach and find corrections to ${\ensuremath{\omega}}^{2}$ of order q4lnq.{q}^{4}\mathrm{ln}q.q4lnq. We demonstrate the usefulness of the concept of an effective surface tension for describing both static and dynamic properties of condensed atomic clouds.
Collective excitations of a dilute Bose gas were probed above and below the Bose-Einstein condensation temperature. The temperature dependencies of the frequency and damping rates of condensate oscillations indicate significant interactions between the condensate and the thermal cloud. Hydrodynamic oscillations of the thermal cloud were observed, constituting first sound. An antisymmetric dipolar oscillation of the thermal cloud and the condensate was studied, representing the bulk flow of a superfluid through the normal fluid. The excitations were observed in situ using non-destructive imaging techniques.