[Paper Review] Theory of matter wave beam splitters in gravito-inertial and trapping potentials

Charles Antoine, Borde, Christian J.|arXiv (Cornell University)|Dec 31, 2005

Cold Atom Physics and Bose-Einstein Condensates4 references3 citations

TL;DR

This paper develops a strong-field theoretical framework for matter wave beam splitters in combined gravitational, inertial, and trapping potentials, showing how these fields modify resonance conditions and induce dispersive effects like anomalous dispersion and velocity selection. The key contribution is a generalized 'ttt' scheme that models the triple interaction as an effective instantaneous interaction, enabling accurate atom interferometer signal calculations with full inclusion of external potential dynamics.

ABSTRACT

We present a strong field theory of matter wave splitting in the presence of various gravitational, inertial and trapping potentials. The effect of these potentials on the resonance condition (between the splitting potential and the considered effective two-level system) and on the atomic Borrmann effect is investigated in detail. The dispersive structuring of an incident atomic wave packet - due to such generalized beam splitters - is studied and modeled, and several important dynamical features of the solutions are detailed (generalized Rabi oscillations, velocity selection, anomalous dispersion, generalized Borrmann effect and anomalous gravitational bending). Finally, we show how to express this triple interaction "matter - splitting potential - gravito-inertial and trapping potentials" as an equivalent instantaneous interaction which turns out to be a very efficient tool for the modeling of atom interferometers.

Motivation & Objective

• To develop a comprehensive theoretical model for matter wave beam splitters beyond the 'infinitely thin' approximation, accounting for strong external potentials.
• To investigate how gravitational, inertial, and trapping potentials affect resonance conditions in effective two-level atomic systems.
• To model dispersive structuring of atomic wave packets, including velocity selection, anomalous dispersion, and the generalized Borrmann effect.
• To derive an equivalent instantaneous interaction (generalized 'ttt' scheme) for efficient atom interferometer signal modeling.
• To extend existing weak-field models to strong-field regimes involving time- and space-dependent potentials and non-trivial dynamics.

Proposed method

• Formulate the triple interaction between matter, splitting potential, and external fields (gravito-inertial/trapping) using a time-dependent Hamiltonian framework.
• Apply unitary transformations and passage to rotating frames to simplify the Schrödinger equation into a solvable form.
• Solve the transformed equation analytically or numerically using WKB and other approximations for different potential configurations.
• Reconstruct the physical solution in the original frame to extract group velocities and wave packet dynamics.
• Derive the generalized 'ttt' scheme as an effective instantaneous interaction that captures the full dynamics of the beam splitter.
• Model wave packet evolution using time-dependent group velocity expressions derived from the resonance condition and external potential effects.

Experimental results

Research questions

• RQ1 How do gravitational and inertial potentials modify the resonance condition between the splitting potential and the effective two-level atomic system?
• RQ2 What is the impact of non-uniform accelerations and trapping potentials on the dispersive structuring of atomic wave packets?
• RQ3 How do generalized Rabi oscillations and anomalous dispersion emerge in the presence of external fields during matter wave splitting?
• RQ4 In what way does the generalized 'ttt' scheme accurately represent the full triple interaction as an effective instantaneous process?
• RQ5 What are the quantitative effects of velocity selection and group velocity shifts due to time-dependent external potentials?

Key findings

• The resonance condition between the splitting potential and the atomic system is modified by external potentials, with corrections involving time-dependent phase and momentum terms.
• Anomalous dispersion and velocity selection arise naturally from the time- and space-dependent structure of the effective potential, leading to non-monotonic group velocity profiles.
• The generalized Borrmann effect is shown to be influenced by external fields, with enhanced transmission peaks shifted due to gravito-inertial and trapping potentials.
• Group velocities inside the beam splitter are derived from a first-order differential equation involving the effective detuning and time-dependent potential gradients.
• The WKB approximation provides a good estimate of group velocities when the time integral of the initial group velocity is neglected, simplifying the dynamics for non-uniform accelerations.
• The generalized 'ttt' scheme enables a Gaussian-based, efficient modeling of beam splitters in atom interferometers, significantly improving signal calculation accuracy over weak-field approximations.