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Quantum Phase of free Fall: Part 2 | Read also Part 1 & Part 3

Introduction

While the theoretical model of the quantum phase of free fall relies on clean, analytical solutions, translating this framework into an empirical laboratory setting introduces significant systematic perturbations. When dealing with precision measurements, the raw signals are heavily dominated by the apparatus itself. This post analyzes the specific physical noise channels—specifically Mean-Field interactions, magnetic calibration offsets, and optical read-out distortion—that bound the precision of these experiments.

Mean-Field Distortions: Quantum Self-Repulsion

A primary source of systematic error stems from the fact that a Bose-Einstein Condensate is a highly dense, interacting macro-quantum state. The constituent Rubidium-87 Atoms undergo s-wave scattering, generating a collective, repulsive potential known as the Mean-Field effect. This interaction is mathematically governed by the non-linear Gross-Pitaevskii Equation:

$\left(-\frac{\hbar ^{2}}{2m}\nabla ^{2}+V_{\text{ext}}(\mathbf{r})+g_{s}|{}\psi (\mathbf{r},t)|{}^{2}\right)\psi (\mathbf{r},t)=i\hbar \frac{\partial }{\partial t}\psi (\mathbf{r},t)$

Where $g_{s} = \frac{4\pi\hbar^2 a_{s}}{m}$ represents the coupling constant determined by the s-wave scattering length $a_{s}$.

During Phase 3 (the drift phase), the two separated components of the wave function undergo distinct geometric evolutions:

• The trapped reference packet is bounded by the chip’s magnetic geometry, constricting its physical expansion.
• The free-fall packet expands freely in three dimensions as it drops away from the chip surface.

[Initial Dense BEC] ---> Split ---> [Trapped Segment: Bounded, Constant High Density]
                               ---> [Free-Fall Segment: Rapid 3D Expansion, Dropping Density]

As the local density $\vert{}\psi\vert{}^2$ of the falling packet drops much faster than that of the trapped counterpart, a differential non-linear phase shift accumulates. If the modeling of this atomic expansion deviates even slightly from reality—due to minuscule spatial variations in the background magnetic field—the calculated Mean-Field subtraction becomes inaccurate, directly contributing to the systematically observed 2.5% deviation from pure gravitational theory.

The Magnetic Overwhelm: Signal-to-Noise Disparity

Because gravity is an exceptionally weak force at the scale of atomic masses, its pure phase signature is easily overwhelmed by the electromagnetic forces required to manipulate the atoms.

To quantify the sheer scale of this challenge, we can evaluate the physical parameters of the spatial splitting. If the two states are separated by a micro-scale height differential of $h = 10 \, \mu\text{m}$ , the difference in Earth’s gravitational acceleration ( $\Delta g$ ) between the two paths due to the Earth’s radius ( $R_{\text{Erde}}$ ) is extraordinarily small:

$\Delta g\approx \frac{2\cdot g\cdot h}{R_{\text{Erde}}}\approx 3.08\times 10^{-10}\,\text{m/s}^{2}$

This subtle differential gradient yields a theoretical phase delta on the order of $10^{-7}$ radians.

Conversely, the magnetic pulses used to execute the initial “kick” and the final “parachute” generate local acceleration equivalents that are orders of magnitude stronger than $g$ . To bridge the gap between this massive magnetic input and the minute gravitational output, the data modeling pipeline relies on post-facto parameter adjustments. In modern chip experiments, researchers must apply manual corrections to the simulated current values (e.g., a 0.15% correction to the kick current pulse) to align the numerical simulation with the empirical dataset. This demonstrates that the exact values are not derived purely from direct observation, but are highly dependent on model calibration.

Optical Artifacts in Phase Extraction

The final phase of the experiment involves converting the quantum phase information into a macroscopic data point via destructive absorption imaging. A resonant laser pulse illuminates the cloud, projecting its spatial density distribution onto a CCD sensor. This process introduces three distinct geometric errors:

Optical Error Source	Physical Mechanism	Impact on Phase Calibration
Optical Density Saturation	The 3D cloud absorbs photons non-linearly; the front layer shadows the core.	Artificial flattening of the interference fringe amplitudes.
Photon Recoil Scattering	Every absorbed photon transfers a discrete momentum vector $\hbar k$ to the atom.	Rapid, turbulent expansion during the exposure window, blurring high-frequency fringes.
Spatial Aliasing & Aberration	Micro-scale fringe spacings map onto discrete, macro-scale CCD pixels alongside lens distortion.	Geometric shifts in the recorded fringe centers, distorting the calculated phase angle.

Summary

Ultimately, the raw empirical data output of an atom-chip interferometer is the product of a deeply intertwined relationship between quantum mechanics, classical electromagnetism, and optical imaging. The true value of these experiments lies in their exceptional demonstrations of phase coherence control and systematic noise rejection. However, the presence of residual modeling errors and the necessity for manual parameter tuning highlight that isolating pure gravitational phase signatures requires highly sophisticated numerical models to separate the signal from the dominant environmental effects of the apparatus.

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