Beam Quality of a Nonideal Atom Laser

We study the propagation of a noninteracting atom laser distorted by the strong lensing effect of the Bose-Einstein condensate (BEC) from which it is outcoupled. We observe a transverse structure containing caustics that vary with the density within the residing BEC. Using the WKB approximation, Fresnel-Kirchhoff integral formalism, and $ABCD$ matrices, we are able to describe analytically the atom-laser propagation. This allows us to characterize the quality of the nonideal atom-laser beam by a generalized $M^2$ factor defined in analogy to photon lasers. Finally we measure this quality factor for different lensing effects.

Figure 1

Absorption images of a nonideal atom laser, corresponding to density integration along the elongated axis of the BEC. The figures correspond to different height of rf outcoupler with respect to the bottom of the BEC: (a) $0.37\mu \mathrm{m}$, (b) $2.22\mu \mathrm{m}$, (c) $3.55\mu \mathrm{m}$. The graph above shows the rf outcoupler (dashed line) and the BEC slice (red) which is crossed by the atom laser. This results in the observation of caustics. The field of view is $350\mu \mathrm{m}\times 1200\mu \mathrm{m}$ for each image.

Figure 2

(a) Principle of the rf outcoupler: the radio frequency ${\nu }_{\mathrm{rf}}=(E_{\mathrm{BEC}}-E_{\mathrm{laser}})/h$ selects the initial position of the extracted atom-laser beam. The laser is then subjected to the condensate mean-field potential $U_{\mathrm{int}}$, to the quadratic Zeeman effect $V$, and to gravity $-mgz$. For the sake of clarity, $V$ has been exaggerated on the graph. (b) Representation of the two-dimensional theoretical treatment: (I) inside the condensate, phase integral along atomic paths determines the laser wave front at the BEC output (WKB approximation). (II) A Fresnel-Kirchhoff integral on the contour $\Gamma$ is used to calculate the stationary laser wave function at any point below the condensate. (III) As soon as the beam enters the paraxial regime, we calculate the $M^2$ quality factor and use $ABCD$ matrix formalism.

Figure 3

Beam profile after $600\mu \mathrm{m}$ of propagation for an outcoupler height of $3.55\mu \mathrm{m}$: (a) calculated in the central plane $y=0$, (b) resulting from the integration along the line of sight $y$, (c) obtained experimentally. Deviations between the curves (b) and (c) can be attributed to imperfections in the imaging process and bias fluctuations. Note that for all three profiles, the overall shape is preserved and the calculated rms width in the central plane is in agreement with the measured one within experimental errors.

Figure 4

$M^2$ quality factor vs rf outcoupler distance from the bottom of the BEC: theory (solid line), experimental points (circles). The two diamonds represent the $M^2$ for the two nonideal atom lasers shown in Figs. 1b, 1c. The rf outcoupler position is calibrated by the number of outcoupled atoms. Inset: typical fit of the laser rms size with the generalized Rayleigh formula [Eq. (5)] for rf outcoupler position $3.55\mu \mathrm{m}$.