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arXiv:1107.2360 · Physical Review A

Prototyping method for Bragg-type atom interferometers

Brandon Benton, Michael Krygier, Jeffrey Heward, Mark Edwards, Charles W. Clark

Published Oct 31, 2011• 8 citations

Abstract

We present a method for rapid modeling of new Bragg ultracold atom-interferometer (AI) designs useful for assessing the performance of such interferometers. The method simulates the overall effect on the condensate wave function in a given AI design using two separate elements. These are (1) modeling the effect of a Bragg pulse on the wave function and (2) approximating the evolution of the wave function during the intervals between the pulses. The actual sequence of these pulses and intervals is then followed to determine the approximate final wave function from which the interference pattern can be calculated. The exact evolution between pulses is assumed to be governed by the Gross-Pitaevskii (GP) equation whose solution is approximated using a Lagrangian variational method to facilitate rapid estimation of performance. The method presented here is an extension of an earlier one that was used to analyze the results of an experiment [J. E. Simsarian et al., Phys. Rev. Lett. 85, 2040 (2000)], where the phase of a Bose-Einstein condensate was measured using a Mach-Zehnder-type Bragg AI. We have developed both 1D and 3D versions of this method and we have determined their validity by comparing their predicted interference patterns with those obtained by numerical integration of the 1D GP equation and with the results of the above experiment. We find excellent agreement between the 1D interference patterns predicted by this method and those found by the GP equation. We show that we can reproduce all of the results of that experiment without recourse to an ad hoc velocity-kick correction needed by the earlier method, including some experimental results that the earlier model did not predict. We also found that this method provides estimates of 1D interference patterns at least four orders-of-magnitude faster than direct numerical solution of the 1D GP equation.

DOI: 10.1103/physreva.84.043648

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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

{N}_{c}

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.

Recent atom interferometry (AI) experiments involving Bose–Einstein condensates (BECs) have been conducted under extreme conditions of volume and interrogation time. Numerical solution of the rotating-frame Gross–Pitaevskii equation (RFGPE), which is the standard mean-field theory applied to these experiments, is impractical due to the excessive computation time and memory required. We present a variational model that provides approximate solutions of the RFGPE for a power-law potential on a practical time scale. This model is well-suited to the design and analysis of AI experiments involving BECs that are split and later recombined to form an interference pattern. We derive the equations of motion of the variational parameters for this model and illustrate how the model can be applied to the sequence of steps in a recent AI experiment where BECs were used to implement a dual-Sagnac atom interferometer rotation sensor. We use this model to investigate the impact of finite-size and interaction effects on the single-Sagnac-interferometer phase shift.

Abstract Atom interferometers using Bose–Einstein condensates are fundamentally limited by a phase diffusion process that arises from atomic interactions. Numerical solution of the Gross–Pitaevskii equation is here used to accurately calculate the diffusion rate for a Bragg interferometer. It is seen to agree with a Thomas–Fermi approximation at large atom numbers and a perturbative approximation at low atom numbers. Most experiments to date operate in the crossover region between these two regimes. Even in this case, it is found that that the diffusion time is governed in a simple way by the ratio of the healing length to the size of the condensate.

We present a general method for calculating wave-packet evolution in the context of ultracold atoms in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers based on atom clouds initially prepared in a trap in a Bose-Einstein condensate (BEC) state or in a thermal distribution. Without having to solve explicitely the Gross-Pitaevskii or the Schrodinger equation, this provides a good estimation of dynamic wave-packet properties such as size and phase across the wave-packet, and slows to predict the coherence of the interferometer. We develop a generalized Thomas-Fermi approximation for a BEC in a harmonic trap, which is valid for any atom-atom interaction strength. In particular, this approximation also provides a good estimation of the transition from a three-dimensional to a one-dimensional BEC in an elongated trap. The approximation is followed by a theory of dynamic wave-packet evolution when the trap is fully or partially switched off and the atoms are split and propagate, for example, in free space or in a matter waveguide. The method is applied for studying two examples of interferometric effects: reduction of coherence of two-state interferometers due to imperfect spatial recombination of the two interferometer arms (the Humpty-Dumpty effect) and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential). For both effects we extend the applicability of the theory to a wide range of interferometric scenarios that were not included in previous theories.

We present a generalized model for the evolution of atomic wave-packets in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers using atom clouds prepared in a trap as a Bose-Einstein condensate (BEC) or as a thermal ensemble. Predictions of the model for dynamic properties such as wave-packet size and phase are in excellent agreement with explicit numerical solutions of the non-linear Gross-Pitaevskii equations and enable fast calculations of interferometric performance in regimes where full numerical solutions become impractical. As a starting point, the static Thomas-Fermi (TF) approximation for a BEC in a harmonic trap is generalized to the whole range of atom-atom interaction strengths: from non-interacting atoms (low densities) to the standard TF limit (high atomic densities, as long as the condensate approximation still holds). In particular, this generalization allows a good estimation of atomic cloud properties along the transition from a three-dimensional to a quasi-one-dimensional BEC in an elongated trap. We then develop a theoretical model of wave-packet evolution in time-dependent conditions. The model is applicable for a wide range of dynamical problems involving evolution in time-dependent potentials and in a changing mean-field atomic repulsion due to splitting and separation of wave-packets. We use the model for studying two effects that influence interferometric coherence: imperfect spatial recombination in a two-state interferometer (the so-called "Humpty-Dumpty effect") and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential).

We numerically implement a Mach-Zehnder interferometer, where the coherence and oscillatory properties of Bose-Einstein condensates are explored and the system is modeled by the Gross-Pitaevskii equation. Several time-dependent external trapping potentials were engineered seeking the adiabatic regime which is quantified using fidelity measurements with respect to the actual ground-state of the trap. The dynamics of both conjugate variables, namely density and phase of the matter-wave function, are shown. Moreover, the density and fidelity profiles of the system are presented when the phase-shifter is switching-on and -off, being found in the presented profiles that the system exhibits three different regimes during the recombination stage among them even an orthogonal BEC to the original one is obtained. We achieve the numerical solution through an adequate implementation of the finite-difference method for the spatial discretization and a Runge-Kutta method for the time evolution.