DFK Consult

Quantum Phase of free Fall: Part 1

Introduction

Measuring the gravitational effect on a quantum system requires tracing the evolution of a coherent wave function across an asymmetric spacetime path. In modern atom-interferometry experiments, an ultra-cold atomic cloud—typically a Bose-Einstein Condensate (BEC) of Rubidium-87—is used to monitor the phase shift induced by acceleration. This post deconstructs the exact mathematical formalism governing the so-called quantum phase of free fall, isolating the pure physics from external environmental variables.

The Analytical Formulation of the Phase

In a standard field-theoretic framework, the evolution of a non-relativistic quantum state $\vert{}\psi(t)\rangle$ under a linear gravitational potential is dictated by the Schrödinger equation:

$i\hbar \frac{\partial }{\partial t}|{}\psi (t)\rangle =\left(\frac{\^{p}^{2}}{2m}+m\cdot g\cdot \^{x}\right)|\psi (t)\rangle$

When a wave packet is split into two spatially distinct trajectories—one actively held static via localized magnetic trapping potentials (the reference arm) and one propelled into a ballistic trajectory (the free-fall arm)—a relative phase difference $\Delta\phi$ accumulates.

By integrating the classical action along the respective paths or applying a gauge transformation between the laboratory frame and the freely falling frame, we yield the definitive cubic time dependency ( $t^{3}$ ) of the accumulated phase:

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

Where:

• $m$ represents the mass of the test particle ( $Rubidium-87 \approx 1.44 \times 10^{-25} \text{ kg}$ )
• $g$ is the local gravitational acceleration ( $\approx 9.81 \text{ m/s}^2$ )
• $T$ is the total interrogation or drift duration
• $\hbar$ is the reduced Planck constant ( $\approx 1.054 \times 10^{-34} \text{ J}\cdot\text{s}$ )

  Relative Phase φ [rad]
      ^
      |                                      *  Max Accumulation
      |                                  *
      |                             *
      |                         *
      |                     *
      |                *
      |           *
      |      *
      +------------------------------------------> Time T [ms]
     0.0                                        2.0

The Cubic Signature as a Frequency Filter

The cubic scaling ( $\sim T^{3}$ ) is the absolute core of the experiment’s methodology. In an operational physics laboratory, absolute isolation of gravity is impossible. Stray ambient electromagnetic fields and mechanical micro-vibrations introduce distinct phase shifts:

• Linear shifts ( $\sim T$ ): Typically induced by static energy offsets or uniform Zeeman shifts.
• Quadratic shifts ( $\sim T^{2}$ ): Associated with constant velocity deltas or uniform electric field gradients.

Because the gravitational free fall forces a parabolic displacement ( $x \sim t^2$ ), its integration into the action yields a strictly cubic phase signature. The data analysis pipeline treats this mathematical property as a static frequency filter, mapping the raw data to a cubic polynomial to extract the gravitational component while subtracting lower-order noise.

The Geometric Asymmetry of the Setup

The system is fundamentally non-symmetrical, introducing specific boundary conditions that complicate the pure mathematical idealization:

1. The Reference Path (Localized Stasis): The sub-ensemble assigned to the reference path is subject to continuous magnetic confinement. The state is locked to the spatial coordinates of the atom-chip, meaning its phase evolution is strictly tied to the laboratory’s local electromagnetic potential.
2. The Ballistic Path (Kinetic Separation): The falling sub-ensemble receives a discrete momentum kick via a localized magnetic field gradient. It translates through space, sampling the spatial spatial variations of the earth’s gravitational field before a secondary, inverted pulse (“the parachute”) matches its momentum back to the reference frame for recombination.

The phase measured at the output is therefore not a measurement of an isolated, free particle, but rather a differential reading of the dynamic correlation between a trapped state and a transient state across an asymmetric spatial path.

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cheers