⚛️ quantum physics

Nonlocal transfer of quantized toroidal magnetic flux

The paper proposes a nonlocal experiment to coherently transfer quantized toroidal magnetic flux between spatially separated superconducting structures via a ground-state vector potential coupling, aiming to test fundamental limits of macroscopic quantum coherence and resolve signaling paradoxes.

Original authors: Adel Ali, Alexey Belyanin

Published 2026-07-08

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CC BY 4.0

Original authors: Adel Ali, Alexey Belyanin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A "Ghost" Connection Between Two Boxes

Imagine you have two special, sealed boxes (called toroids) made of a super-conducting material. Inside each box, there is a tiny, swirling magnetic field, like a miniature tornado trapped in a bottle. Crucially, these magnetic tornadoes are completely sealed inside their boxes; no magnetic field leaks out into the space between them.

Now, imagine you have a third object: a superconducting loop (a ring of wire) that passes through the holes of both boxes, linking them together like a chain.

The Experiment's Claim:
The authors propose an experiment where they can make the "tornado" in Box A disappear and instantly reappear in Box B. The magic is that no magnetic field ever travels through the empty space between the boxes. The magnetic energy jumps from one box to the other without crossing the gap.

How Does It Work? The "Invisible String"

Usually, to move something from one place to another, you need a physical bridge (like a wire carrying electricity or a wave moving through air). Here, the bridge is different.

Think of the two boxes and the linking ring as a single, giant, invisible system. The ring acts like a global rulebook or a shared constraint.

In physics, this is called a fluxoid constraint.
The Analogy: Imagine two people standing far apart, each holding a spinning top. They are connected by a single, invisible, rigid rod that passes through the center of both tops. If Person A stops their top, Person B's top must start spinning to keep the rod balanced. The "force" that makes Person B's top spin doesn't travel through the air; it is enforced by the rigid rod itself.

In this experiment, the "rigid rod" is the quantum rule that the total magnetic "twist" in the whole system must stay constant. The magnetic field stays trapped inside the boxes, but the state of the system changes so that the energy moves from one box to the other.

The "Spooky" Paradox: Faster Than Light?

This setup leads to a confusing question: If the energy moves from Box A to Box B without traveling through the space in between, does it happen instantly? If it happens instantly, isn't that faster than the speed of light?

The authors admit this looks like a paradox. If you have a huge number of particles in the linking ring, the math suggests the "swap" could happen so fast that it would beat a beam of light traveling between the boxes. This would seem to break the rules of the universe (causality), allowing you to send a message faster than light.

The Authors' Solution:
They argue that this "instant" speed is an illusion caused by a simplified math model. They propose that nature has a hidden safety valve.

The Metaphor: Imagine a crowd of people holding hands in a giant circle. If the circle is small, everyone can move in perfect unison instantly. But if the circle gets too huge, or if there are too many people, they can't all stay perfectly synchronized anymore. The "connection" breaks.
The Claim: The authors suggest that if you try to make this connection too strong (by using too many particles or making the system too big), the "quantum coherence" (the perfect synchronization) will spontaneously collapse. The system will stop behaving like a single, unified quantum object and will break apart into normal, messy pieces before it can violate the speed of light.

What Would We See in the Lab?

The paper proposes a specific test to prove this:

The Signal: Two detectors (called SQUIDs) attached to the two boxes would see the magnetic energy "ping-pong" back and forth between them in a rhythmic, correlated pattern.
The Null Test: Sensors placed in the empty space between the boxes would detect zero magnetic field. This proves the energy didn't travel through the air.
The Limit: If they try to make the system "too big" or "too dense," the rhythmic swapping should stop or get messy. This would be proof that there is a fundamental limit to how far and how long quantum connections can last.

Why Does This Matter?

The authors say this isn't just about moving magnetic fields. It's a way to test the very foundations of reality:

Is the wavefunction real? Does a quantum system stay connected across huge distances, or does it have a "maximum range" before it breaks?
Quantum Computers: If there is a fundamental limit to how large a quantum computer can be before it loses its "quantumness," this experiment could find that limit. It suggests that even if we build perfect, noise-free machines, there might be a hard wall where quantum mechanics stops working over large distances.

Summary

The paper proposes a way to move a trapped magnetic field from one superconducting ring to another without the field ever crossing the space between them. It uses a shared quantum "rule" to do this. While the math suggests this could happen instantly (faster than light), the authors hypothesize that nature prevents this by causing the connection to break if the system gets too big, effectively enforcing the speed of light limit on quantum coherence.

Technical Summary: Nonlocal Transfer of Quantized Toroidal Magnetic Flux

Problem Statement
The paper addresses a foundational question in quantum mechanics: to what extent can quantized flux excitations be coherently transferred between spatially separated regions without the magnetic field occupying the intervening space? While the Aharonov–Bohm (AB) effect establishes that electromagnetic potentials have physical significance even where local magnetic fields vanish, the authors propose an experiment to test the limits of this nonlocality in a fully quantum, macroscopic setting. Specifically, they investigate whether a quantized magnetic-field excitation confined within one toroidal superconducting structure can be coherently transferred to a distant toroid via a mediating superconducting loop, without the transfer relying on propagating magnetic fields or real excitations in the intermediate region. This setup serves as a probe for fundamental questions regarding the spatiotemporal bounds of macroscopic quantum coherence and the existence of objective wavefunction collapse.

Methodology
The authors propose a theoretical framework and experimental protocol utilizing superconducting circuits. The core setup consists of two spatially separated toroidal superconducting flux elements (toroids) linked by a common superconducting loop.

Hamiltonian Formulation: The system is modeled by defining a total linked flux $\Phi_{link}$ that includes the fluxes from both toroids ( $\hat{\phi}_2, \hat{\phi}_3$ ) and a bias flux, constrained by the superconducting phase quantization condition of the linking loop. The inductive energy of the loop generates an effective interaction term in the Hamiltonian, coupling the collective topological flux of the toroids rather than their local magnetic field profiles.
Model Regimes: The analysis proceeds through two regimes:
1. Harmonic Oscillators: Treating the toroidal flux modes as harmonic oscillators to derive normal mode frequencies and susceptibility, demonstrating that the coupling arises from the global fluxoid constraint.
2. Two-Level Flux Qubits: Projecting the system onto the two lowest energy states of each toroid (using Josephson junctions for nonlinearity) to derive an effective Ising-type interaction ( $J_{zz}\sigma_z^2\sigma_z^3$ ).
Causality and Collapse Analysis: To address the apparent paradox of instantaneous Hamiltonian changes in the reduced description, the authors introduce a "charged-ring" model where the mediating loop is replaced by $N$ charged bosons. They analyze the scaling of the exchange rate with particle number $N$ . They then propose a phenomenological "causal collapse criterion," suggesting that if the exchange time predicted by the unitary Hamiltonian falls below the light-crossing time of the system, the global spatial coherence of the mediator must break down. This is modeled via a master equation with a density-dependent stochastic collapse term that suppresses off-diagonal long-range order in the mediator.

Key Contributions

Nonlocal Flux Transfer Protocol: The paper proposes a specific experimental architecture where a localized flux excitation is transferred between remote toroids mediated solely by a vector potential coupling (fluxoid constraint), with the magnetic field strictly confined to the toroidal cores.
Experimental Signatures: The authors identify distinct observables for this interaction:
- Correlated, time-resolved oscillatory flux signals measured by SQUIDs attached to the remote toroids.
- A "null test" confirming the absence of magnetic field leakage in the space between the toroids, distinguishing the effect from conventional stray-field crosstalk.
- Mode splitting and coupling strength dependence on the relative linking orientation ( $s_2s_3$ ).
Causal Collapse Hypothesis: The paper formulates a specific hypothesis regarding the limits of macroscopic quantum coherence. It posits that the unitary evolution of a many-body wavefunction mediating nonlocal interactions is bounded by a critical spacetime volume. If the density of the mediating condensate exceeds a critical threshold ( $n_c \propto M^*c/\hbar$ ), the global phase coherence required for the interaction is lost, preventing the exchange from occurring faster than the speed of light.
Resolution of Signaling Paradox: The authors clarify that while the reduced Hamiltonian suggests nonlocal coupling, the full electrodynamic description (including Faraday fields generated by flux changes) preserves relativistic causality. The apparent paradox serves as a diagnostic tool to test whether fundamental collapse mechanisms prevent the realization of the "instantaneous" effective Hamiltonian in the large-N limit.

Results

Coupling Mechanism: The analysis confirms that the linking loop creates an effective interaction $H_{int} \propto \hat{\phi}_2 \hat{\phi}_3$ mediated by the global fluxoid constraint, independent of local mutual inductance.
Exchange Dynamics: In the two-level qubit regime, the system exhibits Rabi-like oscillations between flux states with a rate $J_{zz}$ . In the bosonic ring model, the exchange rate scales linearly with the number of particles $N$ ( $J_N \propto N$ ).
Critical Density: The authors derive a critical linear density $n_c \approx M^*c/\hbar$ (approx. $8.2 \times 10^{11} \text{ m}^{-1}$ for a superconductor). If the participating superfluid density exceeds this value, the projected unitary dynamics would imply superluminal exchange.
Collapse Model: The proposed master equation (Eq. 21) demonstrates that above the critical density, the off-diagonal elements of the one-body density matrix (spatial coherence) decay exponentially, effectively suppressing the collective exchange before the system violates causality. The model distinguishes itself from mass-density collapse models by using gauge-invariant condensate current and fluxoid variables as the collapse operators.

Significance
The paper claims significance in two primary areas:

Experimental Testbed: It provides a concrete, technologically feasible platform using superconducting circuits to test foundational aspects of quantum mechanics, specifically the interplay between gauge potentials, field confinement, and macroscopic coherence. The proposed experiment allows for the direct observation of nonlocal flux exchange without magnetic field occupation of the intervening space.
Foundational Probe: The setup offers a method to empirically investigate whether spatiotemporal quantum coherence is fundamentally bounded. By testing the density-dependent loss of coherence, the experiment could constrain phenomenological collapse models and address whether there are intrinsic limits to the scalability of large-scale quantum computers, independent of engineering noise sources. The authors emphasize that a null result (no suppression) would constrain collapse theories, while an observed suppression would motivate a more complete microscopic theory of macroscopic quantum limits.