Detection of applied and ambient forces with a matter-wave magnetic gradiometer

An atom interferometer using a Bose–Einstein condensate of ${}^{87}\mathrm{Rb}$ atoms is utilized for the measurement of magnetic-field gradients. Composite optical pulses are used to construct a spatially symmetric Mach–Zehnder geometry. By using a biased interferometer we demonstrate the ability to measure small residual forces in our system and discriminate between magnetic and inertial effects. These are a residual ambient magnetic-field gradient of $15\pm 2$ mG/cm and an inertial acceleration of $0.08\pm 0.02$ m/${\mathrm{s}}^2$. Our method has important applications in the calibration of precision measurement devices and the reduction of systematic errors.

Figure 1

(a) The interferometer beams of wavelength 780 nm cross at a half angle of $\theta ={26}^{\circ }$, creating an optical lattice of period 434 nm along the $x$ axis. We define this as our interferometer axis. The magnetic gradient coil is positioned in the $y\text{−}z$ plane and on axis with the interferometer. (b) Interferometer pulse sequence. (c). Breakdown of composite splitting and recombination pulses.

Figure 2

Gradiometer coil model of magnetic-field gradient magnitude, $|\partial B/\partial x|$, vs gradiometer coil current by application of the Biot–Savart law, with (solid black line) and without (dashed red line) applied bias field of 1.02 G along the $z$ axis. Shaded area indicates a coil position uncertainty of 1 mm.

Figure 3

(a) A gradient pulse of varying amplitude is applied before or after the reflection and an interference fringe is observed. An example fringe for $T_1=T_2=500\mu \mathrm{s}$ with the gradient pulse applied after the reflection is shown in panel (b). Error bars indicate standard deviation.

Figure 4

(a) Red solid line shows Mach–Zehnder interferometer pulse sequence. Blue dashed line shows applied magnetic gradient. (b) Simulation of atomic trajectory in the presence of 0 G/cm (dashed lines) and 2 G/cm (solid lines) static-field gradient.

Figure 5

(a) Gradiometer operating in continuous mode while applying a coil current of 1.9 A, with Eq. (8) fit to the data. (b) We convert population $P_{0\hslash k}$ to phase and fit with a linear model. The error bars indicate standard error of each data point (where not visible, these are included within the data point) while the shaded area around the straight line represents the standard deviation of the fit as a result of the uncertainty in the original fitted model. Note that the uncertainty in the phase of the data is greatly increased where the gradient of the fitted model is close to zero and for increased interferometer durations.

Figure 6

Measured acceleration for varying gradient coil currents. Error bars indicate standard deviation. The fitted curve is a model of our gradient coil with coil-atom distance, acceleration offset, and current offset as free parameters. We determine the effective coil-atom distance to be $15.0\pm 0.2$ mm.