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arXiv:cond-mat/9902166 · Physical Review A

Surface of a Bose-Einstein condensed atomic cloud

Usama Al Khawaja, C. J. Pethick, H. Smith

Published Aug 1, 1999• 48 citations
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Abstract

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 ${q}^{4}\mathrm{ln}q.$ We demonstrate the usefulness of the concept of an effective surface tension for describing both static and dynamic properties of condensed atomic clouds.

DOI
10.1103/physreva.60.1507

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Here we describe a weakly interacting Bose gas on a curved smooth manifold, which is embedded in the three-dimensional Euclidean space. To this end we start by considering a harmonic trap in the normal direction of the manifold, which confines the three-dimensional Bose gas in the vicinity of its surface. Following the notion of dimensional reduction as outlined in [L Salasnich et al, Phys. Rev. A 65, 043614 (2002)], we assume a large enough trap frequency so that the normal degree of freedom of the condensate wave function can be approximately integrated out. In this way we obtain an effective condensate wave function on the quasi-two-dimensional surface of the curved manifold, where the thickness of the cloud is determined self-consistently. For the particular case when the manifold is a sphere, our equilibrium results show how the chemical potential and the thickness of the cloud increase with the interaction strength. Furthermore, we determine within a linear stability analysis the low-lying collective excitations together with their eigenfrequencies, which turn out to reveal an instability for attractive interactions.

•Jan 1, 2020•41 citations
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 \ensuremathω\ensuremath{\omega}\ensuremathω and the wave number q related by \ensuremathω2=Fq/m.{\ensuremath{\omega}}^{2}=Fq/m.\ensuremathω2=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 \ensuremathω2{\ensuremath{\omega}}^{2}\ensuremathω2 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.
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes. We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations. We then use the hydrodynamic equations to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional solutions of the time-dependent Gross-Pitaevskii equation. General formulation In the previous chapter we saw that the equilibrium structure of the condensate is described by a time-independent Schrödinger equation with a nonlinear contribution to the potential to take into account interactions between particles.
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes. We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations, which we then use to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional non-linear solutions of the time-dependent Gross–Pitaevskii equation.
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.
Abstract We study both static and dynamic properties of a weakly interacting Bose–Einstein condensate (BEC) in a quasi one-dimensional gravito-optical surface trap, where the downward pull of gravity is compensated by the exponentially decaying potential of an evanescent wave. First, we work out approximate solutions of the Gross–Pitaevskii equation for both a small number of atoms using a Gaussian ansatz and a larger number of atoms using the Thomas–Fermi limit. Then we confirm the accuracy of these analytical solutions by comparing them to numerical results. From there, we numerically analyze how the BEC cloud expands non-ballistically, when the confining evanescent laser beam is shut off, showing agreement between our theoretical and previous experimental results. Furthermore, we analyze how the BEC cloud expands non-ballistically due to gravity after switching off the evanescent laser field in the presence of a hard-wall mirror which we model by using a blue-detuned far-off-resonant sheet of light. There we find that the BEC shows significant self-interference patterns for a large number of atoms, whereas for a small number of atoms, a revival of the BEC wave packet with few matter-wave interference patterns is observed.
The stability of a Bose-Einstein condensed state of trapped ultra-cold atoms is investigated under the assumption of an attractive two-body and a repulsive three-body interaction. The Ginzburg-Pitaevskii-Gross (GPG) nonlinear Schrödinger equation is extended to include an effective potential dependent on the square of the density and solved numerically for the s−s-s−wave. The lowest frequency of the collective mode is determined through the Fourier transform of the time dependent solution and its dependences on the number of atoms and the strength of the three-body force are studied. We show that the addition of three-body dynamics can allow the number of condensed atoms to increase considerably, even when the strength of the three-body force is very small compared with the strength of the two-body force.
Here we describe a weakly interacting Bose gas on a curved smooth manifold, which is embedded in the three-dimensional Euclidean space. To this end we start by considering a harmonic trap in the normal direction of the manifold, which confines the three-dimensional Bose gas in the vicinity of its surface. Following the notion of dimensional reduction as outlined in [L Salasnich et al, Phys. Rev. A 65, 043614 (2002)], we assume a large enough trap frequency so that the normal degree of freedom of the condensate wave function can be approximately integrated out. In this way we obtain an effective condensate wave function on the quasi-two-dimensional surface of the curved manifold, where the thickness of the cloud is determined self-consistently. For the particular case when the manifold is a sphere, our equilibrium results show how the chemical potential and the thickness of the cloud increase with the interaction strength. Furthermore, we determine within a linear stability analysis the low-lying collective excitations together with their eigenfrequencies, which turn out to reveal an instability for attractive interactions.