[Paper Review] An experimental test of the geodesic rule proposition for the non-cyclic geometric phase

Zhifan Zhou, Yair Margalit+3 more|arXiv (Cornell University)|Aug 8, 2019

Cold Atom Physics and Bose-Einstein Condensates22 references4 citations

TL;DR

This paper experimentally confirms the geodesic rule for non-cyclic geometric phases using a spatial SU(2) matter-wave interferometer with ultracold 87Rb atoms. By precisely controlling spin evolution on the Bloch sphere and measuring interference phases, the study unambiguously demonstrates the predicted sign change and π-phase jumps when trajectories cross the equator, validating the geometric phase as half the enclosed area bounded by the path and the shortest geodesic connecting endpoints.

ABSTRACT

The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics, and may become advantageous for quantum technology. In non-cyclic evolutions, a proposition relates the geometric phase to the area bounded by the phase-space trajectory and the shortest geodesic connecting its end points. The experimental verification of this geodesic rule proposition has remained elusive for more than three decades. Here, we report an unambiguous experimental confirmation of the geodesic rule for a non-cyclic geometric phase by means of a spatial SU(2) matter-wave interferometer, demonstrating, with high precision, the predicted phase sign change and pi jumps. We show the connection between our results and the Pancharatnam phase. Finally, we point out that the geodesic rule can be applied to obtain the red-shift in general relativity, enabling a completely new quantum tool to measure gravity.

Motivation & Objective

• To experimentally verify the geodesic rule proposition for non-cyclic geometric phases, which links the geometric phase to the area bounded by a trajectory and the shortest geodesic on the Bloch sphere.
• To resolve long-standing experimental ambiguity in observing the two key signatures of the geodesic rule: the π-phase jump and the sign change across the equator.
• To establish a robust, single-run interferometric method with a common phase reference to avoid artifacts from reference frame shifts between hemispheres.
• To demonstrate the connection between the non-cyclic geometric phase and the Pancharatnam phase in a controlled quantum system.
• To explore the application of the geodesic rule in measuring gravitational redshift via quantum interferometry.

Proposed method

• Utilizes a spatial SU(2) matter-wave interferometer based on ultracold 87Rb atoms on an atom chip, with coherent superposition preparation via radio-frequency π/2 pulses.
• Employs magnetic gradient pulses to induce differential forces on spin states |F=2, mF=1⟩ and |F=2, mF=2⟩, creating spatially separated wave packets with controlled relative phase.
• Applies a second magnetic gradient pulse to reverse relative motion and recombine wave packets, enabling interference measurement in free expansion.
• Measures the interference phase directly from the spatial pattern of atomic density after recombination, using absorption imaging with high spatial resolution.
• Controls the geometric phase by tuning the evolution angle θ and the azimuthal change Δϕ, mapping the phase to the area enclosed by the trajectory and geodesic.
• Uses a common phase reference across both hemispheres to eliminate artificial phase shifts, ensuring unambiguous detection of sign changes and π jumps.

Experimental results

Research questions

• RQ1 Does the non-cyclic geometric phase exhibit a sign change when the evolution trajectory crosses the equator on the Bloch sphere?
• RQ2 Can the predicted π-phase jump occur when the trajectory spans exactly half the circumference of a great circle?
• RQ3 Is the geometric phase quantitatively equal to half the area bounded by the non-cyclic path and the shortest geodesic connecting its endpoints?
• RQ4 How does the measured phase relate to the Pancharatnam phase in a non-cyclic evolution?
• RQ5 Can the geodesic rule be used to measure gravitational redshift via quantum interferometry?

Key findings

• The experiment unambiguously confirms the predicted π-phase jump when the Bloch vector trajectory crosses the equator, observed at T_R = 10 μs and T_R = 30 μs in the interferometric data.
• A clear sign change in the geometric phase is observed as the trajectory moves from the northern to the southern hemisphere, with phase rigidity in each hemisphere and a discontinuous jump at the equator.
• The measured geometric phase Φ_G matches the theoretical prediction given by half the enclosed area on the Bloch sphere, with high precision across varying θ and Δϕ.
• The interference patterns show rigid phase behavior within each hemisphere, with phase jumps occurring only at equatorial crossings, confirming the geodesic rule's topological nature.
• The results are consistent with the Pancharatnam phase definition, establishing a direct link between the geometric phase and the relative phase between non-orthogonal states.
• The study demonstrates the feasibility of using the geodesic rule for precision gravity measurements, as it enables a direct quantum measurement of the gravitational redshift via phase accumulation.