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Quantum Phase of free Fall: Part 3

Introduction

To fully grasp why measuring the quantum phase of free fall pushes the boundaries of experimental physics, one must confront the stark mathematical disparity between the forces at play. On an atom-chip, the gravitational signal is not measured directly as an absolute force; instead, it must be extracted from a system dominated by electromagnetism. This post provides the rigorous mathematical derivation of the cubic phase signature and quantifies the extreme scaling paradox that experimentalists must overcome to isolate the signature of gravity.

Mathematical Derivation of the Cubic Phase (t³)

To derive the exact phase accumulation, we evaluate a single particle of mass m moving in a uniform linear gravitational potential along the vertical axis x. The Hamiltonian operator Ĥ is given by:

$\^{H}=\frac{\^{p}^{2}}{2m}+mg\^{x}$

According to Feynman’s path integral formulation of quantum mechanics, the phase φ accumulated along a classical trajectory x(t) is proportional to the classical action S, defined as the time integral of the Lagrangian L = T – V:

$\S =\int _{0}^{T}\left(\frac{1}{2}m\.{x}(t)^{2}-mgx(t)\right)dt$

For the free-fall arm, the initial conditions at t=0 following the magnetic momentum “kick” are defined by position x(0) = 0 and initial vertical velocity ẋ(0) = v₀. The equations of motion yield the standard ballistic trajectory:

$x(t)=v_{0}t-\frac{1}{2}gt^{2}$
$\.{x}(t)=v_{0}-gt$

Substituting these kinematic equations back into the Lagrangian yields:

$L(t)=\frac{1}{2}m(v_{0}-gt)^{2}-mg\left(v_{0}t-\frac{1}{2}gt^{2}\right)$
$L(t)=\frac{1}{2}mv_{0}^{2}-2mv_{0}gt+mzg^{2}t^{2}$

Integrating this Lagrangian from t = 0 to the final interrogation time T determines the accumulated action $S_{\text{fall}}$ :

$S_{\text{fall}}=\frac{1}{2}mv_{0}^{2}T-mv_{0}gT^{2}+\frac{1}{3}mg^{2}T^{3}$

To find the relative phase shift Δφ, this action must be compared against the reference arm. The reference arm is held stationary at x=0, meaning its trajectory is x(t) = 0 and ẋ(t) = 0. The Lagrangian for the reference arm is identically zero ( $L_{\text{ref}} = 0$ ), yielding $S_{\text{ref}} = 0$ .

However, to close the interferometer loop and allow spatial recombination, the falling arm must be brought to rest relative to the reference frame. Applying a symmetric, inverted magnetic velocity pulse (“the parachute”) at T eliminates the kinetic terms. When evaluating the closed loop through a frame transformation into the freely falling coordinate system, the boundary conditions isolate the purely gravitational phase term:

$\phi (T)=\frac{S_{\text{closed}}}{\hbar }=\frac{mg^{2}T^{3}}{6\hbar }$

Path Separation Topology:
Spatial
Separation (x)
   ^
   |        /‾‾‾‾‾‾‾\           Ballistic Path (Accumulates cubic action via g)
   |       /         \
   |      /           \  <-- "Parachute" Pulse (Recombination)
   |     /             \
   |----+---------------+-----> Time (t)
   ^    Reference Path (Trapped at x=0)
"Kick" Pulse

The Scaling Paradox: Quantifying the Disparity

The mathematical elegance of the ~ T³ derivation masks an immense engineering challenge: the physical scale of the gravitational signal is microscopic compared to the electromagnetic background noise of the apparatus.

Let us evaluate this using the real physical constraints of a chip-based setup using Calcium-40 (m ≈ 6.64 × 10⁻²⁶ kg) over an interrogation time of T = 100 ms:

1. The Global Background Phase

If we calculate the absolute phase shift experienced by the falling packet relative to a hypothetical field-free vacuum, the value is massive:
$\phi (0.1)=\frac{(6.64\times 10^{-26}\text{\ kg})\cdot (9.81\text{\ m/s}^{2})^{2}\cdot (0.1\text{\ s})^{3}}{6\cdot (1.054\times 10^{-34}\text{\ J}\cdot \text{s})}\approx 10,100\text{\ radians}$

2. The Local Differential Phase (The Real Target)

Crucially, the interferometer does not measure the global field; it measures the spatial difference in gravity between the two paths. If the laser or magnetic optics split the wave function to a maximum vertical separation of h = 10 μm, the differential gravity gradient (Δ g) across that tiny window—caused by the earth’s curvature—is vanishingly small:

$\Delta g=\frac{2\cdot g\cdot h}{R_{\text{earth}}}\approx \frac{2\cdot 9.81\text{\ m/s}^{2}\cdot 10^{-5}\text{\ m}}{6.37\times 10^{6}\text{\ m}}\approx 3.08\times 10^{-10}\text{\ m/s}^{2}$

Substituting this Δ g into our phase equation yields the true spatial differential signal:
$\Delta \phi \approx \frac{m\cdot g\cdot \Delta g\cdot T^{3}}{6\hbar }\approx 3.17\times 10^{-7}\text{\ radians}$

The Overwhelm and the Modeling Trap

This value—10⁻⁷ radians—is the actual physical quantity that contains information about the structure of gravity across the separated quantum state.

Now consider the apparatus: the magnetic fields required to generate the initial acceleration (“kick”) and the final deceleration (“parachute”) induce local forces that are orders of magnitude stronger than gravity. These magnetic pulses introduce localized phase shifts in the range of tens of radians.

This creates a severe signal-to-noise paradox:

• The Target Signal: ~ 10⁻⁷ rad
• The Control Noise / Systemic Input: ~ 10¹ rad to 10² rad
• The Current Experimental Resolution: The Rubidium-87 experiments exhibit a 2.5% systematic error relative to their numerical simulations. On a total phase profile of 80 rad, this 2.5% error represents an unresolved phase variance of roughly 2 radians.

Conclusion

The numerical value of the systematic model error (2 rad) is tens of millions of times larger than the fundamental gravitational phase difference (10⁻⁷ rad) between the two spatial pathways.

Consequently, it is mathematically untenable to claim that these experiments provide a pristine, direct verification of gravitational spacetime effects on a quantum state. What is actually occurring is a highly sophisticated exercise in numerical compensation: the massive, dominant electromagnetic signatures of the chip are calculated and subtracted via post-facto software calibration (such as adjusting simulated kick currents by 0.15%). Until the systematic error threshold is driven down below the micro-radian scale, the true quantum signature of spatial gravity remains buried beneath the classical physics of the measuring device itself.

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