Tools for designing atom interferometers in a microgravity environment

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.

Figure 1

1D atom interferometry scheme for measuring $G$. (a) A BEC is formed in the presence of a harmonic trap plus source mass (SM). (b, c) Initial split phase 1: the condensate is split by laser light and the two condensate pieces are allowed to fly apart with initial speeds $\pm v$ until they stop, at which point the harmonic trap is turned off. (d) Initial split phase 2: the two condensate pieces now experience differential gravitational accelerations due to the SM during a wait time, $T_W$. (e) Initial split phase 3: the harmonic trap is turned on, pushing the condensate pieces back together. (f, g) Final split: once they overlap they are split again, creating four clouds—two that fly apart with approximate speeds $\pm 2v$ and two that are motionless except for the small relative velocity developed during the wait time.

Figure 2

Comparison of the evolution of the condensate density during the “initial split” phase of the AI scheme [see Figs. 1–1]. The left panel shows the LVM (red, solid lines) and GPE (blue, dashed lines) results at four different times as the condensate pieces are flying apart and the trap is on. Note that each curve consists of two symmetrically placed peaks and only the left peak is given a time label. The middle panel shows their evolution at the beginning and end of the period when the trap is off. The right panel shows the evolution after the trap is turned back on and the condensate pieces re-overlap. The conditions for this case are $N={10}^4{}^{87}\mathrm{Rb}$ atoms, the trap potential frequency is ${\omega }_T/2\pi =1\mathrm{Hz}$, the wait time is $T_W=500\mathrm{ms}$, the initial velocity is $v=3.14\times {10}^{-3}$ m/s, and there is no source mass present.

Figure 3

Cloud width vs time, $w_1\left(t\right)$, during the IS from the numerical solution of the full system of equations, (33a) and (33b). The conditions are $M_{\text{SM}}=100$ kg, $x_{\text{SM}}=0.1$ m, $v=9.14\times {10}^{-2}$ m/s, ${\omega }_T/2\pi =1$ Hz, and $T_W=10$ s. The vertical dotted lines indicate the end times of the three phases of the IS.

Figure 4

Interference patterns for three different wait times showing how the width of the central fringe decreases as $T_W$ increases. (a) $T_W=5$ s, ${\lambda }_f=163.0\mu \mathrm{m}$. (b) $T_W=10$ s, ${\lambda }_f=38.6\mu \mathrm{m}$. (c) $T_W=20$ s, ${\lambda }_f=9.48\mu \mathrm{m}$. Except for the wait times the conditions are the same as in Fig. 3: $M_{\text{SM}}=100$ kg, $x_{\text{SM}}=0.1$ m, $v=9.14\times {10}^{-2}$ m/s, ${\omega }_T/2\pi =1$ Hz. The resulting values for $G$ are (a) $G=6.13\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$, (b) $G=6.47\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$, and (c) $G=6.59\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$.