We present a variational model suitable for rapid preliminary design of atom interferometers in a microgravity environment. The model approximates the solution of the three-dimensional rotating-frame Gross-Pitaevskii equation as the sum of
Nc Gaussian clouds. Each Gaussian cloud is assumed to have time-dependent center positions, widths, and linear and quadratic phase parameters. We applied the Lagrangian variational method (LVM) with this trial wave function to derive equations of motion for these parameters that can be adapted to any external potential. We also present a one-dimensional (1D) version of this variational model. As an example we apply the model to a 1D atom interferometry scheme for measuring Newton's gravitational constant,
G, in a microgravity environment. We show how the LVM model can (1) constrain the experimental parameter space size, (2) show how the value of
G can be obtained from the experimental conditions and interference pattern characteristics, and (3) show how to improve the sensitivity of the measurement and construct a preliminary error budget.
The stochastic variational method (SVM) is the generalization of the variational approach to systems described by stochastic variables. In this paper, we investigate the applicability of SVM as an alternative field-quantization scheme, by considering the complex Klein–Gordon equation. There, the Euler–Lagrangian equation for the stochastic field variables leads to the functional Schrödinger equation, which can be interpreted as the Euler (ideal fluid) equation in the functional space. The present formulation is a quantization scheme based on commutable variables, so that there appears no ambiguity associated with the ordering of operators, e.g., in the definition of Noether charges.