Optically Guided Beam Splitter for Propagating Matter Waves

We study experimentally and theoretically a beam splitter setup for guided atomic matter waves. The matter wave is a guided atom laser that can be tuned from quasimonomode to a regime where many transverse modes are populated, and propagates in a horizontal dipole beam until it crosses another horizontal beam at 45°. We show that depending on the parameters of this $X$ configuration, the atoms can all end up in one of the two beams (the system behaves as a perfect guide switch), or be split between the four available channels (the system behaves as a beam splitter). The splitting regime results from a chaotic scattering dynamics. The existence of these different regimes turns out to be robust against small variations of the parameters of the system. From numerical studies, we also propose a scheme that provides a robust and controlled beam splitter in two channels only.

Figure 1

Schematic of the matter-wave guided beam splitter setup based on two crossing dipole beams in a horizontal plane ($\mathbf{g}$ denotes the gravity vector). Atoms are outcoupled from a BEC located at point $A$ and propagate in the guide provided by beam $L_1$ towards point $B$ where beam $L_2$ crosses beam $L_1$.

Figure 2

Propagation of an atom laser in an $X$ shape guide structure. Top: Three regimes are experimentally observed depending on the power ratio, $P_2/P_1$, of the guides: (I) weak defect regime where the excitation of transverse modes downstream from the crossing increases with $P_2/P_1$, (II) splitting regime where atoms explored all the available channels, and (IIIa, IIIb) switch regime where all atoms are directed into guide $L_2$. The mean initial transverse excitation numbers of the incoming beam are $⟨n⟩\simeq 0.3$ (IIIa) and $⟨n⟩\simeq 10$ (IIIb). Middle: Experimental diagram that summarizes the observed regime depending on both the power ratio $P_2/P_1$ and the mean initial transverse populations. The boundaries of the three regimes turn out to be robust against the population of the transverse modes. The blue dotted line shows full three-dimensional classical simulations for $w_1=40\mu \mathrm{m}$, $w_2=72\mu \mathrm{m}$, wave packets started $700\mu \mathrm{m}$ before the crossing, with an acceleration of $0.2\mathrm{m}·{\mathrm{s}}^{-2}$, and a vertical misalignment of crossing beams of $2\mu \mathrm{m}$. Bottom: Numerical simulations with other parameters confirming the existence of the three regimes and the robustness of the boundaries against $n$; red solid line: quantum simulations, blue dotted line: classical simulations. Wave packets are started $150\mu \mathrm{m}$ before the crossing, with initial longitudinal velocity $10\mathrm{mm}.{\mathrm{s}}^{-1}$, for $w_1=45\mu \mathrm{m}$, $w_2=41\mu \mathrm{m}$.

Figure 3

Example of experimental splitting dynamics illustrating regime (II) (see Fig. 2). Time evolution is directly visible in the succession of absorption images (taken after a $400\mu \mathrm{s}$ time of flight). The mean velocity is $10\mathrm{mm}/\mathrm{s}$. The images show a larger density at the crossing point which is attributed to the complex dynamics that traps the atoms there for some time. Atoms that escape from this region populate all available channels.

Figure 4

Left: Colors associated with the different output channels. The arrow indicates the propagation direction of the incoming atoms. The overlap region between the guide is represented as a dark blue disk. Right: Output channels associated to different initial phase space conditions of the transverse degree of freedom in regime (II) (see Fig. 2). Each classical trajectory started at the bottom of the guide, $150\mu \mathrm{m}$ before the crossing, with initial velocity $10\mathrm{mm}·{\mathrm{s}}^{-1}$, for $w_1=45\mu \mathrm{m}$, $w_2=41\mu \mathrm{m}$, $P_2=0.143$ $W=0.8P_1$.

Figure 5

Left: Output channels for a classical trajectory started at the bottom of the guide, between $100\mu \mathrm{m}$ and $200\mu \mathrm{m}$ before the crossing, with initial longitudinal velocity $10\mathrm{mm}·{\mathrm{s}}^{-1}$, and an initial transverse velocity between $-25\mathrm{mm}·{\mathrm{s}}^{-1}$ and $25\mathrm{mm}·{\mathrm{s}}^{-1}$, with $w_1=45\mu \mathrm{m}$, $w_2=41\mu \mathrm{m}$, $P_2=0.365$ $W=2.04P_1$. Right: Center of mass motion of numerically simulated quantum wave packets. The one with the initial parameters of point $C$ ($150\mu \mathrm{m}$ before defect, no transverse velocity) is deviated in the second guide at 99.9% (green curve). In contrast, the one with the initial parameters of point $D$ ($154.75\mu \mathrm{m}$ before defect, transverse velocity of $15.5\mathrm{mm}·{\mathrm{s}}^{-1}$) ends up at 96.2% in the initial beam (blue curve).