Wave mechanics of a two-wire atomic beam splitter

We consider the problem of an atomic beam propagating quantum mechanically through an atom beam splitter. Casting the problem in an adiabatic representation (in the spirit of the Born-Oppenheimer approximation in molecular physics) sheds light on explicit effects due to nonadiabatic passage of the atoms through the splitter region. We are thus able to probe the fully three-dimensional structure of the beam splitter, gathering quantitative information about mode mixing, splitting ratios, and reflection and transmission probabilities.

Figure 1

Constant energy surface representing the potential in Eq. (2). The parameters chosen for this plot are $d=0.1\mu \mathrm{m}$, $B_{min}=21.3\mathrm{G}$, $B_{max}=22.5\mathrm{G}$, $B_{bz}=1.0\mathrm{G}$, $L=20\mu \mathrm{m}$. The surface contour is drawn at the energy of the lowest mode of the input arm of the beam splitter, and we define “left” and “right” arms for labeling convenience.

Figure 2

Comparison of lowest-lying Born-Oppenheimer $\left(U_0\right)$ and adiabatic $\left(U_0^{eff}\right)$ curves, for the beam splitter in Fig. 1.

Figure 3

Effective adiabatic potentials for the configuration in Fig. 1. Each curve corresponds to a different transverse mode of the beam splitter. Because of the intrinsic symmetry of the potential, even and odd modes can be treated separately. The first six even and odd modes of the structure are depicted here.

Figure 4

$P$-matrix elements coupling the first transverse even and odd modes in Fig. 3 to selected higher ones.

Figure 5

Probability for exit in the various arms of the beam splitter depicted in Fig. 1, vs incident atom velocity. This figure assumes that the atoms have entered in the lowest mode of the left arm. The velocities shown correspond to atom energies below the second threshold, thus suppressing higher mode excitations. The bottom panel is a $26\mu \mathrm{K}$ Gaussian convolution of the one above.

Figure 6

Scattering probabilities as in Fig. 5, but for the extremely nonadiabatic case $L=1\mu \mathrm{m}$. Only the convolved plot is shown, with a width of $26\mu \mathrm{K}$.

Figure 7

Total reflection probabilities for multimode beam splitter of different lengths. From top to bottom $L=1,2,5,7,15,30\mu \mathrm{m}$. The cusps around 12.5 and $13\text{cm}∕\mathrm{s}$ are the effects of the second and third thresholds becoming energetically available. The plot represents a $16\mu \mathrm{K}$ width Gaussian convolution.

Figure 8

Total transmission to higher modes for different lengths beam splitter from top to bottom $L=1,2,5,7,15,30\mu \mathrm{m}$. The plot represents a $3.25\mu \mathrm{K}$ width Gaussian convolution.