[Paper Review] Coupling gravitomagnetism-spin and Berry's phase

Abel Camacho|arXiv (Cornell University)|Jun 3, 2002

Relativity and Gravitational Theory4 citations

TL;DR

This paper proposes a quantum mechanical experiment to detect gravitomagnetism via Berry's geometric phase, using a spin-1/2 particle in the gravitomagnetic field of a rotating sphere. The key result is that the geometric phase depends only on the rotation angle and time, not on the field strength, enabling detection without measuring tiny perturbations.

ABSTRACT

Resorting to Berry's phase, a new idea to detect, at quantum level, the gravitomagnetic field of any metric theory of gravity, is put forward. It is found in this proposal that the magnitude of the gravitomagnetic field appears only in the definition of the adiabatic regime, but not in the magnitude of the emerging geometric phase. In other words, the physical parameter to be observed does not involve, in a direct way, (as in the usual proposals) the tiny magnitude of the gravitomagnetic field.

Motivation & Objective

• To address the lack of experimental evidence for spin–gravitomagnetic coupling in metric theories of gravity.
• To overcome the challenge of detecting extremely weak gravitomagnetic fields via conventional methods.
• To explore whether geometric phases can provide a more sensitive probe of gravitomagnetism at the quantum level.
• To test the validity of extending orbital angular momentum–gravitomagnetic coupling to spin systems.
• To propose an experimental framework that isolates the geometric phase from dynamic contributions, avoiding reliance on measuring small field strengths.

Proposed method

• Model the gravitomagnetic field of a rotating sphere using the PPN formalism with parameters Δ₁ and Δ₂, valid for any metric theory of gravity.
• Define the Hamiltonian for a spin-1/2 particle as H = -S·B, where B is the gravitomagnetic field derived from the PPN formalism.
• Introduce a time-dependent gravitomagnetic field by rotating the angular momentum vector J around a fixed axis with angular velocity ω.
• Apply the adiabatic theorem to ensure the spin state remains aligned with the instantaneous direction of B, assuming ω₁ >> ω.
• Calculate the Berry phase using the time-ordered integral of the geometric phase factor from the evolving spin state.
• Derive the final phase expression, showing it depends only on the rotation angle and θ, not on the magnitude of the gravitomagnetic field.

Experimental results

Research questions

• RQ1 Can Berry’s geometric phase be used to detect gravitomagnetism without measuring the small magnitude of the gravitomagnetic field?
• RQ2 Does the geometric phase in a spin-1/2 system coupled to a rotating gravitomagnetic field depend on the field strength?
• RQ3 How does the adiabatic condition relate to the physical parameters of the system, such as mass, radius, and rotation rates?
• RQ4 Can this setup distinguish between different metric theories of gravity (e.g., GR vs. Brans–Dicke) via the phase shift?
• RQ5 Is the assumption of spin–gravitomagnetic coupling physically valid, and can it be tested using this geometric phase approach?

Key findings

• The Berry phase acquired by a spin-1/2 particle in a rotating gravitomagnetic field is independent of the field strength ω₁, depending only on the rotation angle and the angle θ between the rotation axis and the angular momentum vector.
• The geometric phase after one full cycle (t = 2π/ω) is given by γ₊(t) = -2π[1 - sin²(θ)/(2 + 6cos²(θ) - 4cos(θ)√(1 + 3cos²(θ)))], which is a purely geometric quantity.
• The adiabatic regime is defined by the condition ω₁ >> ω, which translates to (7Δ₁ + Δ₂)/2 × (GJ/(c²r³)) >> ω, showing that field strength directly affects the validity of the adiabatic approximation.
• The experimental setup involves splitting a beam of spin-1/2 particles, allowing one beam to remain non-interacting while the other adiabatically follows the rotating field, leading to a measurable interference phase.
• The final interference pattern contains a phase factor ζ that includes both dynamic and geometric contributions, with the geometric part being observable even if the dynamic phase is large.
• The method provides a way to test the spin–gravitomagnetic coupling assumption without requiring measurement of tiny orbital or spin shifts, offering a robust quantum alternative to classical detection methods.