Guided-atom laser: Transverse mode quality and longitudinal momentum distribution

We analyze the outcoupling of a matter wave into a guide by a time-dependent spilling of the atoms from an initially trapped Bose-Einstein condensate. This process yields intrinsically a breakdown of the adiabatic condition that triggers the outcoupling of the wave function. Our analysis of the time-dependent engineering and manipulation of condensates in momentum space in this context enables us to work out the limits due to interactions in the mode quality of a guided-atom laser. This study is consistent with recent experimental observations of low transverse excitations of guided-atom lasers and suggests (i) an optimal strategy to realize such quasimonomode guided-atom lasers with, in addition, the lowest possible longitudinal velocity dispersion or alternatively (ii) a strategy for engineering the atomic flux of the atom laser.

Figure 1

Sketch of the optical outcoupling procedure. The intensity of the beam that ensures the longitudinal trapping is reduced progressively as a function of time which favors the spilling of the wave function outside the trap region.

Figure 2

Ratio between the longitudinal velocity dispersion of an ideal Bose-Einstein wave function as a function of time in the course of the outcoupling process and the initial velocity dispersion $\Delta v\left(0\right)=0.58$ mm s${}^{-1}$ for different outcoupling characteristic times $\tau =0,$ 1, 5, and 100 ms. The time $t_O$ at which the potential no longer has a minimum is $t_O\simeq$ 4$\tau$.

Figure 3

Wigner function of the wave function at different times $t/\tau =0$, $4.5$, and 7 for $f\left(t\right)=e^{-t/\tau }$ with $\tau =10$ ms. The wave function is evolved according to the 1D Gross-Pitaevskii wave function with an initial condition corresponding to the Thomas-Fermi regime ($N{g}_1=53{\hslash }^{3/2}{\omega }_0^{1/2}{m}^{-1/2}$).

Figure 4

Ratio between the longitudinal velocity dispersion of a 1D wave packet and the initial velocity dispersion $\Delta v\left(0\right)=0.25$ mm s${}^{-1}$ as a function of time in the course of the outcoupling process for (i) a sudden switching off of the trap (represented in units $\tau =1$ ms) and (ii) different outcoupling characteristic times: $\tau =$1, 5, and 100 ms. Gray solid lines: numerical simulation. Dashed lines: results from the analytical model in momentum space for the opening of the trap detailed in the Appendix. The breakdown of the good agreement between simulation and the model occurs when the spilling of atoms starts. The wave packet describes the wave function of a condensate initially in the Thomas-Fermi regime with an interaction parameter $N{g}_1=53{\hslash }^{3/2}{\omega }_0^{1/2}{m}^{-1/2}$.

Figure 5

Asymptotic longitudinal velocity dispersion ratio $\Delta v\left(\infty \right)/\Delta v\left(0\right)$ as a function of the characteristic time $\tau$ used for the outcoupling process with an exponential time function $f\left(t\right)=\exp (-t/\tau )$: $\gamma =1$ m s${}^{-2}$ (open diamonds), $\gamma =0.2$ m s${}^{-2}$ (open squares), and $\gamma =0.01$ m s${}^{-2}$ (open circles). Same initial conditions as in Fig. 4. The dotted lines correspond to the lower bound estimated with Eq. (5).

Figure 6

Output flux as a function of time for a time evolution function (i) $f\left(t\right)=\exp (-t/\tau )$ (dotted line) and (ii) $f\left(t\right)$ (solid line) deduced from Eq. (6) with a desired square flux $J_0$ (dashed line). Same initial parameters as in Fig. 4.

Figure 7

Transverse quadratic size as a function of time in the course of the outcoupling process for different outcoupling characteristic times $\tau =$ 1 ms (dotted line), 5 ms (dashed line), and 20 ms (solid line). The gray straight line corresponds to the quadratic size of the harmonic oscillator associated with the transverse confinement of the guide.

Figure 8

Upper bound of the contamination of the transverse degree resulting from the outcoupling as a function of the outcoupling characteristic time $\tau$. Calculations are performed for a strength parameter of $N{g}_2=267{\hslash }^2/m$ and potentials $U_{\mathrm{I}}$ (open squares and diamonds) and $U_{\mathit{II}}$ (black squares). An exponential decay $f\left(t\right)=\exp (-t/\tau )$ is chosen for open and black squares while a linear decay $f(t⩽\tau )=1-t/\tau$ has been chosen for data represented by open diamonds. The potential $U_{\mathit{II}}$, which couples strongly the longitudinal and transverse degrees of freedom, turns out to give similar results as $U_{\mathrm{I}}$ when the outcoupling time is sufficiently long ($\tau >10$ ms for our parameters).

Figure 9

Time evolution of the kinetic energy of an interacting BEC (same initial conditions as in Fig. 4) held in a harmonic trap of initial angular frequency ${\omega }_0=2\pi \times 150$ Hz when the angular frequency is weakened as a function of time according to $\omega \left(t\right)=4{\omega }_0(1-t/\tau )/5+{\omega }_0/5$: numerical simulation (solid gray line), scaling law approach [see Eq. (A6)] (dashed line). A very good agreement is observed for various values of $\tau =1,20,200$ ms.