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arXiv:cond-mat/0604464 · Physical Review A

Quantum theory of an atom laser originating from a Bose-Einstein condensate or a Fermi gas in the presence of gravity

Tobias Kramer, Mirta Rodríguez

Published Jul 14, 2006• 18 citations

Abstract

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.

DOI: 10.1103/physreva.74.013611

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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.

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.

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.

A method to create paired-atom laser beams from a metastable helium atom laser via four-wave mixing is demonstrated. Radio-frequency outcoupling is used to extract atoms from a Bose-Einstein condensate near the center of the condensate and initiate scattering between trapped and untrapped atoms. The unequal strengths of the interactions for different internal states allows an energy-momentum resonance which leads to the creation of pairs of atoms scattered from the zero-velocity condensate. The resulting scattered beams are well separated from the main atom laser in the two-dimensional transverse atom laser profile. Numerical simulations of the system are in good agreement with the observed atom laser spatial profiles and indicate that the scattered beams are generated by a four-wave mixing process, suggesting that the beams are correlated.

We develop the quantum-mechanical description of output coupling of macroscopic coherent matter waves from a Bose-Einstein condensate (BEC) via a radio-frequency field in the pulsed and cw limits for both strong and weak field coupling. The theory is converted into a Fock state description to point out the stimulated (in Bose particle) nature of the output coupling. A useful analogy with the theory of molecular photodissociation is used to explain the various regimes of output coupling BECs using radiation sources. We present specific calculations of the rates of output coupling as a function of the number of Bose atoms in the condensate and the frequency, detuning, and intensity of the radio-frequency field.

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

We analyze the extraction from a trapped Bose-Einstein condensate to a nontrapping state of a coherent atomic beam and its subsequent fall, at T = 0 K. Our treatment fully takes gravity into account but neglects the mean-field potential exerted on the free falling beam by the trapped atoms. We derive analytical expressions for the output rate and the output mode of the quasicontinuous "atom laser." Comparison with experimental data of Bloch et al. [Phys. Rev. Lett. 82, 3008 (1999)] is satisfactory.