Bending the waveguide

It would be very useful to be able to carry atom beams along curved paths, by bending our atom waveguide in the plane of the substrate, without significant atom loss. Here we briefly estimate three limitations on the waveguide minimum bending radius (in decreasing order of leniency): 1) the limit imposed by optical radiation leakage, 2) the limit needed for incoherent atom transport, and 3) the limit needed for coherent atom transport in the transverse groundstate. This will give us an idea of the practicality of curved atomic guides.

Firstly, whenever a dielectric optical guide has curvature, there is a loss rate (exponentially small in the curvature radius $R$), which can be viewed as tunneling out of the guide's `potential well' induced by the addition of an effective centrifugal potential. In the limit $R \gg W$, the effective potential is linear with x (the radial coordinate), and the fractional loss per radian of curvature can be estimated^8.6, for instance using the one-dimensional WKB formula, to be

$$\alpha = C \, \frac{R}{\lambda} \exp \left( -\frac{1}{6 \pi^2} \frac{\lambda^2 R}{L_x^3} \right) \, ,$$ (8.15)

where $L_x$ is the evanescent decay length in the radial direction, and $C$ is a constant of order unity. Therefore for negligible light loss at a $\pi/2$ bend we need $R > 60\pi^2 L_x^3 / \lambda^2$, typically a couple of tens of microns. This is so small chiefly because we are using an optical guide with a large refractive index step[117].

Secondly, we consider atom loss from an incoherent beam with a transverse temperature $k_B T_\bot$ (assumed small compared to the trap depth magnitude $U_{\rm max}$), and a longitudinal kinetic energy $E_\Vert$. We call the approximate spatial extent of the trap potential in the x direction $2 \xi$, and restrict ourselves to one-dimensional classical motion in this direction. When in a region of radius of curvature $R \gg \xi$, an effective centrifugal term adds to the trapping potential giving $U(x) = U_{\rm dip}(x) - 2E_\Vert (x/R)$. This causes the atoms to `slosh' towards positive x, only ever returning if there exists a point where $U(x) > -U_{\rm max}$ for $x>0$. We can estimate that this will happen if $U_{\rm max} > 2E_\Vert (\xi/R)$, giving our lower limit on $R$ as $2\xi E_\Vert / U_{\rm max}$. In our design $\xi \approx 0.5\,\mu$m, so if we choose $R=1$mm we can expect loss-free transport of a beam at a longitudinal kinetic energy up to $10^3$ times the trap depth.

Thirdly, to model coherent matter-wave propagation along a curved guide, we consider the amplitude for remaining in the transverse ground-state, having passed into a curved section and back into a straight section. If again we assume one-dimensional x motion, and assume a harmonic potential $U(x) = {\textstyle\frac{1}{2}} M \omega_x^2 (x-x_0)^2$ around the trap minimum, then the effect of curvature is to shift the minimum position from $x_0 = 0$ to $x_0 = 2 E_\Vert / M \omega_x^2 R$. If this shift is much less than the characteristic ground-state size $(\hbar/M \omega_x)^{1/2}$ then the projection at each transition will be high, resulting in high flux transmission coefficient. This gives $R \gg 2 E_\Vert / (\hbar M \omega_x^3)^{1/2}$ as our condition, which for $E_\Vert / k_B = 10\,$mK (that is, $v_\Vert = 1.1$ms$^{-1}$) and $\omega_x/2\pi = 40$kHz corresponds to $R \gg 0.45$mm. This limit is very conservative since we have not yet made use of the adiabatic condition $\Omega \ll \omega_x$ (where $\Omega \equiv v_\Vert /R$ is the rate of change of direction of the guided atom), to design a waveguide path without discontinuities in the curvature.

In conclusion, we have shown that it is possible to bend atoms both incoherently and coherently through large angles on a compact substrate structure of a few millimeters in size.