Initial wavefunction dependence on atom interferometry phases

In this paper we present a mathematical procedure to analytically calculate the output signal of a pulsed atom interferometer in an inertial field. Using the well-known ABCD xi method we take into account the full wave dynamics of the atoms with a first order treatment of the wavefront distortion by the laser pulses. Using a numerical example we study the effect of both the length of the beam splitting laser pulses and of the width of the initial spatial distribution of the atoms. First, we find that in a general inertial field the interferometer only has a limited window in terms of the initial width (centered around 100 mu m in the example calculation) in which interference fringes are visible at all. This effect is caused by the inevitable statistical spread in atomic parameters, such as initial position and momentum, and the dependence of the interferometer phase on these. In the optimum case, the useful range of the initial width is formed by the range in which both the spatial distribution and the diffraction limited momentum spread are small enough to avoid large phase differences over the atomic wavefunction. As a second result we find that the interferometer phase depends strongly on the length of the laser pulses and, to a smaller extent, on the initial width of the atomic cloud. This spatial dependency is relatively small (similar to 10(-5) rad) and justifies semiclassical approximations, as used in other calculations, for most experiments. New high-accuracy experiments, however, will come in the range where this effect is no longer negligible.