Aharonov-Bohm Effect for Cooper Pairs in Kerr Spacetime: Gravitomagnetic Phase Shifts from Frame Dragging

Abstract

The unification of quantum mechanics and general relativity remains among the most profound challenges in fundamental physics. Here we investigate a novel quantum probe of strong-field gravity: the gravitomagnetic Aharonov-Bohm (AB) effect for Cooper pairs propagating in Kerr spacetime. The frame-dragging induced by a rotating black hole (BH) generates an effective vector potential through the off-diagonal metric component $g_{tφ}$, which couples to the macroscopic phase of the superconducting condensate. We derive the gauge-invariant AB phase shift $Δθ= (4πm^* Ma/\hbar)(1/r_2 - 1/r_1)$ for an interferometer with arms at radii $r_1$ and $r_2$, where $m^* = 2m_e$ is the Cooper pair mass and $a$ is the BH spin parameter. Remarkably, the predicted phases reach $|Δθ| \sim 10^{24}$ radians for Sgr~A* and $\sim 10^{27}$ radians for M87*, reflecting the enormous gravitomagnetic flux near supermassive BHs. We analyze the dependence on interferometer geometry, demonstrate that tidal disruption of Cooper pairs is negligible at distances $r \gtrsim 10\,r_s$, and establish connections to the geometric Berry phase. Although direct experimental realization remains beyond current technology due to the vast distances involved, our framework provides quantitative predictions linking quantum coherence to spacetime curvature, complementing recent observations of gravitational AB phases in atom interferometry.

Figures (10)

• Figure 1: Gravitomagnetic potential $g_{t\phi}$ as a function of $r/M$ at the equatorial plane for spin parameters $a/M = 0.1, 0.3, 0.5, 0.7, 0.9$, and $0.99$. The potential becomes more negative (stronger frame dragging) for higher spin and smaller radii. At large $r$, all curves approach zero as $g_{t\phi} \sim -2Ma/r$.
• Figure 2: Angular dependence of the gravitomagnetic potential $g_{t\phi}$ for $a/M = 0.9$ at radii $r = 2M, 3M, 5M, 10M$, and $20M$. The potential vanishes at the poles and reaches its extremum at the equatorial plane, following the $\sin^2\vartheta$ dependence.
• Figure 3: Three-dimensional surface plot of the gravitomagnetic potential $g_{t\phi}(r, \vartheta)$ for $a/M = 0.9$. The deep well near $r \approx 2M$ and $\vartheta = \pi/2$ indicates where frame-dragging effects are strongest.
• Figure 4: Gravitomagnetic flux $\Phi_g$ as a function of $r/M$ for spin parameters $a/M = 0.1, 0.3, 0.5, 0.7, 0.9$, and $0.99$. The flux determines the AB phase shift through the relation $\Delta\theta = (m^*/\hbar)\Phi_g$.
• Figure 5: Frame-dragging angular velocity $\Omega_{\text{ZAMO}}$ as a function of $r/M$ for spin parameters $a/M = 0.1, 0.3, 0.5, 0.7, 0.9$, and $0.99$. Each curve starts just outside the corresponding horizon radius $r_+ = M + \sqrt{M^2 - a^2}$. For the near-extremal case $a/M = 0.99$, the angular velocity reaches $\Omega \approx 0.42$ near the horizon.