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arXiv:cond-mat/9910442 · Physical Review A

Investigations of a two-mode atom-laser model

Jens Schneider, Axel Schenzle

Published Apr 19, 2000• 19 citations
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Abstract

Atom lasers based on rf outcoupling from a trapped Bose-Einstein condensate can be described by a set of generalized, coupled Gross-Pitaevskii equations (GPEs). If not only one but two radio frequencies are used for outcoupling, the atoms emerging from the trap have two different energies and the total wave function of the untrapped spin state is a coherent superposition which leads to a pulsed atomic beam. We present results for such a situation obtained from a one-dimensional GPE model for magnetically trapped ${}^{87}\mathrm{Rb}$ in the $F=1$ state. The wave function of the atomic beam can be approximated by a sum of two Airy functions. In the limit of weak coupling we calculate the intensity analytically.

DOI
10.1103/physreva.61.053611

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Atom lasers based on rf outcoupling from a trapped Bose-Einstein condensate can be described by a set of generalized, coupled Gross-Pitaevskii equations (GPEs). If not only one but two radio frequencies are used for outcoupling, the atoms emerging from the trap have two different energies and the total wave function of the untrapped spin state is a coherent superposition which leads to a pulsed atomic beam. We present results for such a situation obtained from a one-dimensional GPE model for magnetically trapped 87Rb{}^{87}\mathrm{Rb}87Rb in the F=1F=1F=1 state. The wave function of the atomic beam can be approximated by a sum of two Airy functions. In the limit of weak coupling we calculate the intensity analytically.
An atom laser based on RF-outcoupling can be operated in a coherently pulsed way by using two radio frequencies. Such a two-mode atom laser can be described by coupled Gross-Pitaevskii equations which are valid for very low temperatures (T≈0). We have derived analytic expressions for the output rate and the visibility of the pulsed atomic beam in the weak coupling regime. A comparison between the analytical and numerical results shows a good agreement. This proves our assumption that the outcoupling process may be seen as the coherent outcoupling of two independent atomic beams. Though our model is only one-dimensional, we expect also a good agreement with the full 3D situation of experiment; the estimate for the value of the output rate might be only in the right order of magnitude16. Summarizing, the manipulation of trapped Bose condensates with electromagnetic waves opens a wide new area for the field of coherent atom optics.
Atoms can be extracted from a trapped Bose-Einstein condensate (BEC) by driving spin-flips to untrapped states. The coherence properties of the BEC are transfered to the released atoms, creating a coherent beam of matter refered to as an atom laser. In this work, the extraction of atoms from a BEC is investigated numerically by solving a coupled set of Gross-Pitaevskii equations in up to three dimensions. The result is compared to experimental data and a semiclassical rate model. In the weak-coupling regime, quantitative agreement is reached between theory and experiment and a semiclassical rate model. In the strong-coupling regime, the atom laser enters a trapped state that manifests itself in a saturation of the rate of out-coupled atoms observed in new experimental data. The semiclassical rate model fails, but the numerical descriptions yield qualitative agreement with experimental data at the onset of saturation.
We study the dynamics of a magnetically trapped Bose-Einstein condensate in the presence of an external electromagnetic-field coupling trapped, untrapped, and antitrapped Zeeman sublevels. For large condensates an approximate analytical solution of the coupled Gross-Pitaevskii equations is given in the regime of strong outcoupling. The theory is developed for the cases of rf outcoupling within a hyperfine manifold of states, microwave outcoupling connecting states in two different hyperfine manifolds, and Raman outcoupling.
Summary form only given. We report results concerning both theoretical and experimental studies of collective excitations in connection with the atom laser operation. We use a radio-frequency field to out-couple a coherent matter-wave beam, either continuous or pulsed, from a /sup 87/Rb condensate. The latter scheme itself induces collective excitations in the atom laser. Starting from a condensate in the F=2, m/sub F/=2 state the radio-frequency field populates both trapped and untrapped Zeeman sublevels. A rich phenomenology arises including the production of multiple condensates exhibiting different dynamical behaviour, which make clear evidence of mutual interactions. The use of a static very elongated Ioffe trap allows a precise experimental investigation which can be compared with the predictions based on the solution of the time-dependent Gross-Pitaevskii equation. In the pulsed regime, the sudden change of the internal energy produces collective oscillations in the shape both for the m/sub F/=2 and the m/sub F/=1 trapped condensate. Preliminary results for the m/sub F/=2 atom laser show that the experimental behaviour is likely to be reproduced by the simulations only if mutual interactions are taken into account.
We present a three-dimensional quantum-mechanical theory of radio-frequency-outcoupled atom lasers from trapped atomic gases in the presence of the gravitational force. Predictions for the total outcoupling rate as a function of the radio frequency and for the beam wave function are given. We establish a sum rule for the energy-integrated outcoupling, which leads to a separate determination of the coupling strength between the atoms and radiation field. For a noninteracting Bose-Einstein condensate analytic solutions are derived which are subsequently extended to include the effects of atomic interactions. The interactions enhance interference effects in the beam profile and modify the outcoupling rate of the atom laser. We provide a complete quantum-mechanical solution which is in line with experimental findings and allows us to determine the validity of commonly used approximative methods. We also extend the formalism to a fermionic atom laser and analyze the effect of superfluidity on the outcoupling of atoms.
We consider the possibility of a discrete set of energies of a weakly outcoupled atom laser beam to the homogeneous Schrödinger equation with anisotropic harmonic trap in Cartesian coordinates outside the Bose–Einstein condensate region. This treatment is used because working in the cylindrical coordinates is not really possible, even though we implement the cigar-shaped trap case. The Schrödinger equation appears to replace a set of two-coupled Gross– Pitaevskii equations by enabling the weak-coupling assumption. This atom laser can be produced in a simple way that only involves extracting the atoms in a condensate from by using the radio frequency field. We initially present the relation between condensates as sources and atom laser as an output by exploring the previous work of Riou et al. in the case of theoretical work for the propagation of atom laser beams. We also show that even though the discrete energies are obtained by means of an approaching harmonic oscillator, degeneracy is only available in two states because of the anisotropic external potential