⚛️ quantum physics

The Apparatus Strikes Back: Momentum Conservation and the Cost of Spatial Superpositions

This paper identifies a universal constraint on creating spatial superpositions of massive particles, demonstrating that momentum conservation inevitably entangles the particle with the preparation apparatus, thereby imposing strict limits on coherence based on the apparatus's mass, temperature, and anchoring that persist even for macroscopic devices.

Original authors: Lucas C. Céleri, Diogo O. Soares-Pinto, Daniel A. Turolla Vanzella

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Lucas C. Céleri, Diogo O. Soares-Pinto, Daniel A. Turolla Vanzella

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Imagine you are trying to build a giant, invisible seesaw in a quantum playground. You want to take a tiny particle and put it in two places at once—a "superposition"—so it can act like a wave. This is the holy grail for testing the limits of quantum mechanics, from sensing gravity to probing the very fabric of reality.

But here is the twist: the paper by Lucas C. Céleri and his colleagues suggests that the very machine you use to create this magic trick might be the one to ruin it.

The Heavy Hand of Momentum

Think of the particle as a lightweight acrobat and the machine (the "apparatus") as a massive, heavy trapeze artist. To get the acrobat to split into two paths, the machine has to give them a little push.

In our everyday world, if you push a feather, you barely feel a thing. But in the quantum world, everything is connected by a strict rule called momentum conservation. If the particle gets pushed one way, the machine must recoil the other way. It's like a cosmic game of "you push, I push back."

The paper argues that because the machine is made of quantum stuff too, this recoil isn't just a simple bump. It creates a "ghostly link" (entanglement) between the particle and the machine. The machine's position becomes slightly different depending on which path the particle took. If you don't keep track of the machine, the particle loses its "quantumness" and acts like a normal, boring object. The machine has essentially "snitched" on the particle's path.

The Temperature Trap

The authors calculate exactly how much this recoil hurts. They find that the machine needs to be incredibly stable and cold.

Imagine the machine is a giant, heavy block sitting in a room. If the room is warm, the block jiggles around due to heat. If the block jiggles too much, it can't tell the difference between the two recoil directions, and the particle's superposition collapses.

The paper gives us a specific "speed limit" for this jiggling. For a machine weighing 100 kg (about the weight of a large person) to successfully split a particle separated by 1 µm (one-millionth of a meter), the machine's center of mass must be incredibly quiet.

If you try to do this with a particle as heavy as the Planck mass (a theoretical limit of about $m_{Pl}$ ), the requirements become almost impossible. The paper suggests that to keep the superposition alive, you would need to cool the machine's center of mass to a temperature of $2 \times 10^{-16}$ K. That is colder than the deepest, darkest void of space. Even a heavy machine at this temperature would struggle to hold the superposition together.

The "Acoustic Horizon" Problem

You might think, "Okay, let's just make the machine heavier! A bigger machine recoils less, right?"

The paper says: Not so fast.

Imagine the machine is a giant building. When you push the particle, the "push" travels through the building like a sound wave. If the building is too big, the sound wave hasn't reached the back of the building by the time the experiment is over. The back of the building doesn't know it's supposed to recoil.

Instead of the whole building moving as one unit, only a small chunk near the push moves. The rest of the building acts like a messy, noisy environment that steals the quantum information. The paper suggests there is a "sweet spot" for the machine's size: big enough to be heavy, but small enough that the whole thing moves together before the sound wave dies out. For typical materials, this "rigid zone" is only a few meters wide.

Is the Game Rigged? (False Decoherence)

Here is the most mind-bending part. The paper asks: Is the particle actually losing its quantum magic, or are we just looking at it the wrong way?

If you treat the machine as part of the "environment" (something you ignore), the particle looks decohered. But if you treat the machine as part of the "system" (something you keep track of), the whole thing (particle + machine) is still perfectly quantum.

The authors call this "false decoherence." It's like a magician hiding a card in his sleeve. If you only look at the card, it's gone. But if you look at the whole magician, the card is still there, just in a different spot.

The paper suggests that for experiments like the Bose–Marletto–Vedral (BMV) proposal (which tries to prove gravity is quantum), we might not need to cool the machine to absolute zero. Instead, we might need to control the machine's quantum state perfectly, reversing the recoil at the end of the experiment to "un-scrunch" the entanglement. If we can do that, the momentum conservation rule doesn't kill the experiment; it just changes the rules of the game.

The Bottom Line

The main finding is that momentum conservation alone creates a universal barrier to creating superpositions of heavy particles. You don't need a weird new physics theory to explain why it's hard; you just need to remember that the machine pushing the particle has to move, too.

The paper rules out the idea that we can ignore the machine's recoil. It argues that for massive particles, the machine's movement is a fundamental constraint, not just a technical nuisance.

However, the authors are careful to say this is a conservative estimate. They assume the machine is a perfect, simple block. In reality, machines have internal vibrations (like springs inside) that will likely make the problem even worse. So, the limits they calculated are the best-case scenario. If your machine isn't perfect, the limits are even stricter.

The paper doesn't claim to have solved the problem or built a new machine. Instead, it provides a necessary condition: a checklist of how cold, how heavy, and how rigid your machine must be before you even start. If you can't meet these numbers, no amount of fancy engineering will save your quantum superposition.

Technical Summary: The Apparatus Strikes Back: Momentum Conservation and the Cost of Spatial Superpositions

Problem Statement
The preparation of massive particles in coherent spatial superpositions is a primary objective in modern quantum science, driven by goals ranging from testing fundamental quantum mechanics and gravity (e.g., the Bose–Marletto–Vedral proposal) to quantum-enhanced sensing. While current experimental limitations are typically attributed to environmental decoherence mechanisms (such as photon scattering, gas collisions, or gravitational radiation), this paper identifies a previously overlooked, universal constraint. The authors argue that any protocol placing a particle of mass $m$ into a spatial superposition with separation $d$ necessarily entangles the particle with the center-of-mass (CM) degree of freedom of the preparation apparatus due to the conservation of momentum and the mass dipole moment. This entanglement leads to a loss of coherence in the reduced state of the particle if the apparatus is not explicitly included as part of the quantum system.

Methodology
The authors employ a theoretical framework based on standard quantum mechanics and conservation laws:

System Definition: They model a joint system consisting of a particle (mass $m$ ) and an apparatus (mass $M$ ). The apparatus is treated as a quantum system with a CM degree of freedom, initially in a thermal state confined by an effective harmonic potential with frequency $\omega$ .
Conservation Laws: The analysis relies on the conservation of the mass dipole moment ( $\vec{\mu} = M_{tot}\vec{X}_{cm}$ ) in the center-of-mass frame. When the particle is displaced by $\pm \vec{d}/2$ , the apparatus must recoil by $\mp \vec{D}/2$ such that $m\vec{d} + M\vec{D} = 0$ .
Entanglement Derivation: The interaction is modeled as a unitary evolution where the particle's path states ( $|L\rangle, |R\rangle$ ) become correlated with opposite displacements of the apparatus CM. The reduced state of the particle is obtained by tracing out the apparatus degrees of freedom.
Coherence Quantification: The authors calculate the coherence factor $\chi_c$ , which determines the visibility of interference. This factor depends on the distinguishability of the apparatus's recoil states, derived using the thermal Gibbs state and the Mehler kernel for a harmonic oscillator.

Key Contributions and Results
The paper derives a quantitative bound on the parameters required to maintain coherence in the reduced particle state. The main result is expressed as a constraint on the product of particle mass and superposition separation ($md$):

$md \lesssim \sqrt{\frac{2\hbar M}{\omega} \tanh\left(\frac{\hbar\omega}{2k_B T}\right)}$

This inequality implies that the recoil displacement $D$ must remain smaller than the coherence length of the apparatus's CM state. The bound yields two distinct regimes:

Low Temperature ( $k_B T \ll \hbar\omega$ ): The apparatus approaches its ground state. The constraint becomes temperature-independent: $m \lesssim \sqrt{2\hbar M / (\omega d^2)}$ .
High Temperature ( $k_B T \gg \hbar\omega$ ): Thermal fluctuations increase the distinguishability of recoil states, leading to a stricter bound: $m \lesssim \sqrt{\hbar^2 M / (k_B T d^2)}$ .

Key Findings:

Universal Constraint: The bound arises solely from momentum conservation and the canonical commutation relations, independent of the specific mechanism (optical, magnetic, etc.) used to split the wavefunction.
Critical Mass: For a given apparatus, there exists a critical mass $m^*$ above which coherence is exponentially suppressed for all temperatures. For representative parameters ( $M=100$ kg, $d=1$ $\mu$ m), this critical mass is approximately $3 \times 10^{-3}$ of the Planck mass.
Planck Scale Implications: While current matter-wave interferometry (masses $\sim 10^{-14} m_{Pl}$ ) is largely unaffected, proposals involving masses near the Planck scale ( $m_{Pl}$ ) face severe constraints. Achieving coherence at the Planck scale would require apparatus temperatures ( $T \lesssim 10^{-16}$ K) and anchoring frequencies ( $\omega/2\pi \lesssim 10^{-5}$ Hz) far beyond current experimental capabilities.
Role of the Apparatus Mass: The analysis highlights a trade-off. While increasing the apparatus mass $M$ relaxes the bound, it also increases the likelihood that internal vibrational modes (which act as an environment) will be excited by the localized splitting impulse, leading to genuine, irreversible decoherence. Thus, the "optimal" apparatus is not the heaviest possible, but the heaviest that can still be treated as a single coherent quantum degree of freedom during the splitting time.

Significance and Claims
The authors position this work as identifying a "conservative estimate" or a "necessary, but not sufficient" condition for preparing massive superpositions.

False Decoherence: The paper argues that the loss of coherence described by this bound is a form of "false decoherence" (in the sense of Unruh). If the apparatus is included in the quantum system and the splitting/recombination process is reversible, the entanglement can be undone, and coherence recovered. The bound only applies if the apparatus is treated as an environment (traced out).
Interpretation of Future Experiments: The authors claim that any experimental observation of coherence suppression at large mass scales must first be disentangled from this momentum-conservation effect before it can be attributed to new physics (e.g., objective collapse models or modifications of quantum mechanics).
Limitations: The paper explicitly states that the derived bound assumes the recoil is absorbed entirely by the CM mode. In realistic scenarios, internal modes will likely decohere faster, making the actual experimental requirements even more stringent than the calculated bound.

In summary, the paper establishes that the quantum nature of the preparation apparatus and momentum conservation impose a fundamental, model-independent cost on the creation of massive spatial superpositions, which becomes a dominant constraint as one approaches the Planck mass scale.