Example: Effect of wavefront aberrations in atom interferometry

As an example, we reproduce two plots from the paper https://link.springer.com/article/10.1007/s00340-015-6138-5.

The simulation will require the following objects and parameters

• Wavefront: contains the wavefront aberrations of the interferometry lasers

• AtomicEnsemble: an ensemble of atoms which different trajectories or phase space vectors

• Detector: determines which atoms contribute to the signal

• times of the three interferometer pulses

• effective wavevector

import json

import numpy as np
import matplotlib.pyplot as plt
from functools import partial
import aisim as ais

Loading and preparing wavefront data

Wavefront aberration in multiples of \(\lambda\) = 780 nm.

Load Zernike coefficients from file:

coeff_window = {j: val for j, val in enumerate(np.loadtxt("data/wavefront.txt"))}

Creating Wavefront objects and removing piston, tip and tilt from the data:

r_wf = 10.91e-3  # radius of the available wavefront data in m

wf = ais.Wavefront(r_wf, coeff_window, zern_order="WYANT", zern_norm=None)
for n in [0, 1, 2]:
    wf.coeff[n] = 0

wf.plot(cmap="twilight_shifted", levels=50)
fig, ax = wf.plot_coeff(color="C3")

Creating an atomic ensemble

Due to the large number of parameters determining an atomic ensemble, dictionaries are used:

# Create an ensemble of atoms
atoms = ais.create_random_ensemble(
    int(1e5),
    x_dist=partial(ais.dist.position_dist_gaussian, std=3.0e-3),
    y_dist=partial(ais.dist.position_dist_gaussian, std=3.0e-3),
    z_dist=partial(ais.dist.position_dist_gaussian, std=3.0e-3),
    vx_dist=partial(ais.dist.velocity_dist_from_temp, temperature=3.5e-6),
    vy_dist=partial(ais.dist.velocity_dist_from_temp, temperature=3.5e-6),
    vz_dist=partial(ais.dist.velocity_dist_for_gaussian_velsel, pulse_duration=60e-6),
    seed=1,
)

Plotting the initial positions of the ensemble.

fig, ax = plt.subplots()
atoms.plot(ax=ax, cmap="Reds")
ax.set_aspect("equal", "box")
ax.set_xlabel("x / mm")
ax.set_ylabel("y / mm")

Text(0, 0.5, 'y / mm')

Setting up the detector

We want to calculate the dependency of the phase shift caused by wavefront aberrations on the detection area. For this reason, we set up a Detector with varying detection radius within a for-loop.

t_det = 778e-3  # time of the detection in s

Simulation the bias in gravity from wavefront aberrations

For the simulation we need the objects created above and the timing of the interferometer sequence.

T = 260e-3  # interferometer time in s
t1 = 130e-3  # time of first pulse in s
t2 = t1 + T
t3 = t2 + T

awfs = []
r_dets = np.linspace(2e-3, 10e-3, 16)
for r_det in r_dets:
    # creating detector with new detection radius
    det = ais.PolarDetector(t_det, r_det=r_det)

    det_atoms = det.detected_atoms(atoms)

    # calculate the imprinted phase for each "test atom" at each pulse. This is the computationally heavy part
    phi1 = 2 * np.pi * wf.get_value(det_atoms.calc_position(t1))
    phi2 = 2 * np.pi * wf.get_value(det_atoms.calc_position(t2))
    phi3 = 2 * np.pi * wf.get_value(det_atoms.calc_position(t3))

    # calculate a complex amplitude factor for the Mach-Zehnder sequence and calculate
    # the mean. Note that some atoms will have probed the wavefront outside of the
    # Raman beam radius and will thus have a NaN value in the phase.
    awfs.append(np.nanmean(np.exp(1j * (phi1 - 2 * phi2 + phi3))))

# factor two since the window is passed twice
g = 2 * ais.phase_error_to_grav(np.angle(awfs), T=260e-3, keff=1.610574779769e7)

We load the measured gravity data from a file and compare it to the simulation results.

data = json.load(open("data/experimental_data.json"))
r_det_data = np.array(data["r_det"])
grav = np.array(data["g"])
graverr = np.array(data["g_err"])

fig, ax = plt.subplots()
ax.plot(1e3 * r_dets, 1e9 * g, label="simulation")
ax.errorbar(1e3 * r_det_data, 1e9 * grav, yerr=1e9 * graverr, fmt="o", label="data")
ax.set_xlabel("Detection radius / mm")
ax.set_ylabel("Gravity bias / $nm/s^2$")
ax.set_ylim([-35, 5])
ax.legend()

<matplotlib.legend.Legend at 0x1238a7770>

Note that the original software used in the paper used a fixed position and velocity grid rather than a Monte-Carlo approach, so some discrepancies are to be expected.