Pancharatnam phase as an entanglement witness for quantum gravity in dual Stern-Gerlach interferometers

TL;DR

The paper addresses whether gravity can generate entanglement between spatially separated quantum superpositions, which would indicate gravity as a quantum field source. It analyzes a setup of dual spin-1/2 Stern-Gerlach interferometers and uses the Pancharatnam phase as a witness to distinguish semiclassical from quantum gravity. The key result is that semiclassical gravity produces a $\pi$ phase jump at a geodesic singularity, whereas quantum gravity yields a continuous phase with the jump suppressed by a factor $\\xi$, illustrating how entanglement affects geometric phase in gravity-mediated interferometry. This work offers a concrete, albeit challenging, tabletop route to probe quantum features of gravity and clarifies how entanglement and geometric phases differentiate gravity models.

Abstract

Entanglement plays a central role in the fundamental tests and practical applications of quantum mechanics. Because entanglement is a feature unique to quantum systems, its observations provide evidence of quantumness. Hence, if gravity can generate entanglement between quantum superpositions, this indicates that quantum amplitudes are field sources and that gravity is quantum. I study the dual spin-one-half Stern-Gerlach interferometers and show that the Pancharatnam phase is a tool that qualitatively distinguishes semiclassical from quantum gravity. The semiclassical evolution is equivalent to that of two independent interferometers in an external field. In this case, a phase jump was observed, as expected from the geodesic rule, which dictates the noncyclic evolution in the Bloch sphere. By contrast, in the quantum case, the quantum amplitudes are the sources of the gravitational field, inducing entanglement between the two interferometers, and the phase is continuous.

Figures (5)

• Figure 1: Illustration of the geodesic rule for the Pancharatnam phase (which includes purely geometric and dynamical components): During the evolution from points K to L on the Bloch sphere, the geometric component of the Pancharatnam phase is given by minus half the spherical area enclosed by the closed curve formed by curve KL (blue line) and the shortest geodesic joining L to K. An orientation is given by the normal to the plane KOL. The geodesic rule is illustrated for a trajectory along the equatorial plane. The curve KL is shown by the blue line, and the shortest geodesic joining L to K by the dashed red line. (A) When $\widehat{\text{KOL}} < \pi$, the shortest geodesic joining L to K coincides with curve KL and the enclosed area on the Bloch sphere, and the Pancharatnam phase reduces to zero. (B) When $\widehat{\text{KOL}} > \pi$, curve KL and the shortest geodesic joining L to K form an equatorial circle, the enclosed spherical area is half of the Bloch sphere whose magnitude is $2\pi$ and the geometric phase is $-\pi$. There is a sudden change in the geodesic when $\widehat{\text{KOL}}$ crosses $\pi$ resulting in the jump of the Pancharatnam phase.
• Figure 2: Sketch of the experiment proposed in Ref. bose: two nanoparticles with equal masses $m$ and spins one-half are released from a trap at $t=0$. After a short time, an equal-weight superposition of each mass was created by an RF pulse of duration $\Delta t_{RF}$. The superposition of the nanoparticles can evolve under mutual gravity. They were then recombined by another RF pulse. The trap release time and $\Delta t_{RF}$ (grey area) are negligible compared to the free-fall time of the two superpositions. Here, I illustrate two possibilities. (A) Quantized gravity: quantum amplitudes are sources of field, and each superposition is sensitive to the gravitational field created by each component of the other superposition (note that the self-gravity term, which in principle should be included, does not induce a phase difference); (B) semiclassical gravity: the quantum amplitudes are not direct sources of gravitational field, and each superposition evolves under a single field created by the effective mass of the other superposition (green oval). In both cases, the interactions are indicated by dotted orange lines.
• Figure 3: Interferometric phase and visibility: (A) The Pancharatnam phase in the dual SGI experiment proposed in Ref. bose. The black dashed line corresponds to the semiclassical configuration discussed in Fig. \ref{['fig:sketch']}A. The continuous blue line corresponds to the quantum configuration of Fig. \ref{['fig:sketch']}B. (B) The visibility in the dual SGI experiment proposed in Ref. bose. The black dashed line corresponds to the semiclassical configuration discussed in Fig. \ref{['fig:sketch']}A. The continuous blue line corresponds to the quantum configuration of Fig. \ref{['fig:sketch']}B.
• Figure 4: The Pancharatnam phase in the dual SGI experiment proposed in Ref. bose near the semiclassical singularity shown in Fig. \ref{['fig:phase-vis']}. The system is brought at $\delta \phi_0$ from the singularity by a magnetic gradient pulse and then let evolve under the gravitational interaction. The black dashed line corresponds to the semiclassical interferometer configuration discussed in Fig. \ref{['fig:sketch']}A. The continuous blue line corresponds to the quantum configuration of Fig. \ref{['fig:sketch']}B. (A) $\delta \phi_0=\pi/10$; (B) $\delta \phi_0=\pi/100$.
• Figure 5: The Pancharatnam phase in the dual SGI experiment proposed in Ref. bose near the semiclassical singularity shown in Fig. \ref{['fig:phase-vis']}. The system is brought at $\delta \phi_0$ from the singularity by a magnetic gradient pulse and then let evolve under the gravitational interaction. Here we use Eq. \ref{['eq:cq']} with values of $\xi$ near to the classical limit $\xi_c=1$ to illustrate the occurence of a large phase difference quantum gravity while $\delta \phi$ is varied in a narrow range near the jump point. The semiclassical phase remains nearly constant for the same values of $\delta \phi$.