Note that in general trajectory reconstructions break down during the final cycle of an atom transit. This breakdown occurs because (i) the angular momentum cannot be estimated well once the atom has escaped from bound orbit and (ii) the dominant escape mode is a rapid (less than one orbital period) burst of heating in the x direction which invalidates our assumption that x-motion can be neglected in the reconstructions.
Atoms are trapped via their interaction with single-photon strength fields inside the cavity. This interaction creates an effective potential U(rho,x) which can be calculated from the steady-state solution to the quantum master equation. The steepness of the potential in the x direction (along the cavity axis) causes a trapped atom to be tightly confined to within about 50 nm of a field antinode, which allows us to neglect motion in the x direction. Confinement in the (y,z)=(rho,theta) direction is less tight, and the atom orbits in this plane with characteristic distance from the center set by the cavity field's Gaussian waist w0=14 microns.
Since cavity transmission is altered as an atom's radial position rho changes, we can use measured transmission signals to determine rho(t) for an orbiting atom. Knowledge of the effective potential U(rho) then allows us to estimate the angular momentum of the orbits and thus the angular position theta(t) for a full two-dimensional reconstruction of the atomic motion.
Non-conservative (diffusive) forces change the angular momentum over time; our reconstruction algorithm estimates an angular momentum for each quarter-orbital period and then performs a smooth interpolation between these piecewise estimates. [Note that for an elliptical orbit, the atom's orbital period is twice the period of the resulting oscillations in rho and thus in cavity transmission. Hence the angular momentum estimates are every half-period of the transmission oscillations but every quarter-period of the atomic orbits.] The validity of the reconstruction algorithm is tested by applying it to simulated transits and comparing the reconstructed and "actual" trajectories.
Below we display a sampling of trajectories reconstructed from simulated transmission data, with shot noise and technical noise added. Each trajectory is displayed on a 30 micron square, with an accompanying transmission trace shown below it. Reconstructed trajectories (green) are compared with actual trajectories (gray). From an ensemble of such trajectories we obtain an estimate for the error in reconstruction, which is indicated by the green circle at the start of each trajectory.
Trajectory reconstructions are subject to three basic ambiguities: the initial angle, the overall sign of the angular momentum, and the specific standing wave antinode in which the orbit is confined. Below we adjust the initial angle and the sign of the angular momentum for best agreement with the "actual" trajectory.
The reconstruction algorithm must fail for atom transits with nearly linear orbits (those passing closest to the origin), since for these cases the diffusive change in angular momentum per orbit is a large fraction of the total angular momentum. Such transits -- characterized by distinct transmission oscillations that consistently reach the maximum allowed transmission -- should not be reconstructed. Attempted reconstructions fail, giving trajectories with sharp corners and unphysical kinks near the origin.
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