Testing Spontaneous Collapse Models with Coulomb Mediated Squeezing

This paper proposes that detecting steady-state Coulomb-mediated squeezing in two charged nanospheres can establish robust bounds on the Continuous Spontaneous Localization (CSL) parameter that are comparable to X-ray emission limits and superior to bulk-heating constraints, while also offering short-time entanglement-based tests.

The Gist

Imagine you are trying to hear a whisper in a very noisy room. That's essentially what physicists are trying to do when they test the laws of quantum mechanics.

For over a century, we've known that tiny particles (like electrons) can exist in two places at once—a state called superposition. But as things get bigger (like a cat, or a ball, or a person), they seem to "snap" into just one place. Why?

Standard quantum theory says we just need to look at them to know where they are. But some physicists think there's a deeper, hidden rule: Spontaneous Collapse. They propose that the universe has a built-in "glitch" or a random "static" that forces big objects to pick a single location, even if no one is looking. This is called the Continuous Spontaneous Localization (CSL) model.

This paper proposes a clever new way to test if this "glitch" is real, using two tiny, charged balls floating in mid-air.

The Setup: Two Floating Dancers

Imagine two tiny spheres (nanospheres), each about the width of a virus, floating in a vacuum.

1. They are charged: Like two magnets with the same pole facing each other, they repel one another.
2. They are trapped: They are held in place by invisible electric fields (like a cage made of light).
3. They are dancing: Even when they are "still," they jitter around due to heat (thermal noise) and the strange rules of quantum mechanics.

The "Glitch" vs. The "Squeeze"

The authors are looking for a specific effect caused by the repulsion between the two balls.

• The Squeeze: Because the balls push against each other, their movement becomes "squeezed." Think of a spring. If you push two springs together, they wiggle less in the direction they are pushing. The paper predicts that this "squeezing" should make the balls move less than they normally would due to heat.
• The Glitch (CSL): Now, imagine the "Spontaneous Collapse" is real. It acts like a constant, invisible wind blowing randomly on the balls. This wind adds extra jitter.

The Big Idea: If the "glitch" (CSL) is strong, it will blow away the "squeeze." The balls will jitter too much, and the special quiet state will disappear. If the "glitch" is weak or non-existent, the balls will stay squeezed and quiet.

By measuring exactly how much the balls are jittering, the scientists can tell how strong this "glitch" is. If they see the squeeze, they know the "glitch" must be very weak.

Why This is a Game-Changer

Previous experiments tried to find this "glitch" in two main ways, but they had flaws:

1. X-ray Experiments: These looked for radiation emitted by atoms if the glitch existed. However, these experiments assume the "glitch" is a specific type of white noise (like static on a radio). If the noise is actually "colored" (like a low hum or a specific frequency), those experiments fail completely.
2. Bulk Heating: These looked at how much heat a big block of material generates from the glitch. Again, this fails if the noise isn't the right type.

The New Method's Superpower:
The two-ball setup is robust. It doesn't matter if the "glitch" is white noise or colored noise (a hum). The physics of the two balls interacting is so precise that it can detect the glitch regardless of its "color." It's like trying to find a specific thief in a crowd; previous methods only worked if the thief wore a red hat. This new method works even if the thief wears a blue hat, a green hat, or no hat at all.

The Results: Setting New Limits

The authors calculated that with realistic equipment (which we are close to building), this experiment could set the strictest limits on the "glitch" ever seen.

• It could rule out the "glitch" being as strong as some theories predict.
• It could be 10 to 100 times better than the best X-ray experiments currently available.
• Even if the "glitch" has a complex, colored nature (which makes other tests useless), this method still works.

The Short-Time Secret

The paper also mentions a "short-time" trick. If you start with the balls perfectly still (in their "ground state") and watch them for a split second before heat takes over, the repulsion between them creates a quantum link called entanglement.

• Entanglement: A spooky connection where the two balls know what the other is doing instantly.
• If the "glitch" is too strong, it breaks this link immediately.
• By seeing if this link survives for a tiny fraction of a second, they can put even tighter limits on the "glitch."

The Bottom Line

This paper proposes a new, incredibly sensitive "microphone" to listen for the universe's hidden "static."

• If they hear the static: We have to rewrite the laws of physics. Gravity might be the cause, and quantum mechanics only works for tiny things.
• If they don't hear the static (which is what they hope): We confirm that quantum mechanics works for bigger things too, and the "glitch" theories are likely wrong.

It's a bit like checking if the universe is made of solid Lego bricks (standard physics) or if there's a hidden force constantly knocking the bricks apart (collapse models). This experiment is the most precise hammer we've ever built to test the bricks.

Technical Summary

1. Problem Statement

Spontaneous collapse models, such as the Continuous Spontaneous Localization (CSL) and Diósi-Penrose (DP) models, propose modifications to the Schrödinger equation to resolve the quantum measurement problem. These models introduce a stochastic, non-linear mechanism that causes wave-function collapse, with a rate that scales with the mass of the system.

• The Challenge: While direct observation of macroscopic superpositions is technologically difficult, indirect tests (like X-ray emission and bulk heating) have placed constraints on the collapse parameters (collapse rate $\lambda_{CSL}$ and spatial resolution $r_{CSL}$).
• The Limitation: Existing non-interferometric tests (specifically X-ray emission) are highly sensitive to the assumption that the collapse noise is "white" (uncorrelated in time). If the collapse noise is "colored" (having a finite correlation time, as suggested by cosmological origins), the bounds from X-ray experiments weaken significantly or become irrelevant.
• The Goal: The authors propose a new experimental setup using two levitated charged nanospheres to constrain CSL parameters that is robust against colored-noise extensions and offers tighter bounds than current bulk-heating and X-ray experiments.

2. Methodology

The authors propose a theoretical framework involving two identical, harmonically trapped, charged nanospheres interacting via a static Coulomb potential.

• System Setup:
  • Two nanospheres of mass $m$ and charges $q$ and $\eta q$ (where $\eta = \pm 1$) are separated by a mean distance $d$.
  • They are trapped at frequency $\omega$.
  • The system is analyzed in terms of Common (COM) and Differential (relative) motional modes.
• Dynamics:
  • The evolution is governed by a Master Equation including the system Hamiltonian, dissipative thermal noise, and collapse-induced diffusion.
  • Key Insight: The Coulomb interaction induces a frequency shift in the differential mode ($\omega_d$).
    • For repulsive interaction ($\eta=1$), $\omega_d > \omega$.
    • This shift leads to squeezing of the position variance in the steady state below the thermal level ($\sigma_{Z_d Z_d} < N/2$), even at high temperatures.
  • Collapse Effect: Spontaneous collapse introduces a diffusive noise term ($D_{CSL}$) that counteracts this squeezing. If the collapse rate is too high, the predicted squeezing is washed out.
• Two-Mode Advantage:
  • Unlike single-particle experiments where one must distinguish collapse diffusion from thermal noise, this two-particle setup allows for the independent estimation of thermal dissipation and collapse diffusion by comparing the common and differential modes.
  • The differential mode is sensitive to the difference in collapse noise ($D_{CSL}^{diff} = D_{11} - D_{12}$), while the common mode is sensitive to the sum.

3. Key Contributions

A. Steady-State Squeezing as a Probe

The authors derive a condition where the observation of steady-state position variance reduction (squeezing) in the differential mode places an upper bound on the collapse diffusion coefficient:
$$D_{CSL}^d < \frac{N m \gamma \hbar \omega \delta^2}{k_B T}$$
Where $\delta^2$ represents the frequency shift due to Coulomb coupling. This allows for constraints on $\lambda_{CSL}$ that are tighter than bulk-heating limits by several orders of magnitude.

B. Robustness Against Colored Noise

A critical contribution is demonstrating that these bounds remain valid even if the collapse noise is colored (non-Markovian) with a cutoff frequency $\Omega$.

• X-ray emission bounds degrade severely if $\Omega \sim 10^{12}$ Hz because the high-frequency components of the noise (which drive X-ray emission) are suppressed.
• The mechanical experiment proposed here operates at low frequencies ($\omega \ll \Omega$), meaning the colored noise effectively behaves as white noise in the relevant regime. Thus, the bounds derived are robust against plausible colored-noise extensions of collapse models.

C. Short-Time Entanglement Test

The authors also analyze the short-time regime ($t \ll 1/\omega$) for nanospheres initialized in the ground state.

• They show that a weak Coulomb interaction can generate entanglement between the two spheres.
• The presence of collapse-induced diffusion destroys this entanglement.
• Observing this entanglement provides an alternative, stringent constraint on $\lambda_{CSL}$ that is also robust against colored noise.

D. Diósi-Penrose (DP) Model Analysis

The paper analyzes the applicability of this setup to the DP model. They conclude that while theoretically possible, constraining the DP parameter ($R_0$) is currently infeasible due to the requirement for extremely low damping, extremely low temperature, and extremely small frequency shifts ($\delta^2 \sim 10^{-7}$), which are beyond current experimental capabilities.

4. Results

• CSL Bounds:
  • For realistic experimental parameters (e.g., $T = 10 \mu K$, $\gamma = 10^{-4} s^{-1}$), the proposed setup can constrain $\lambda_{CSL}$ to $2.09 \times 10^{-15} s^{-1}$ (at $r_{CSL} = 10^{-7} m$).
  • This is an order of magnitude better than the first X-ray emission experiments and surpasses the most recent XENONnT collaboration results for specific parameter choices.
  • Even with more ambitious parameters ($T = 1 \mu K$), bounds can reach $2.09 \times 10^{-17} s^{-1}$.
• Colored Noise Impact:
  • Under a colored noise model with $\Omega \sim 10^{12}$ Hz, the XENONnT bound degrades to $\lambda_{CSL} < 7.09 \times 10^{-3} s^{-1}$ (a loss of ~10 orders of magnitude).
  • In contrast, the mechanical squeezing/entanglement bounds remain largely unaffected, making them the most reliable current limits.
• Entanglement Constraints:
  • For a short-time entanglement test with $Q \sim 10^{11}$ and $T=1K$, the bound is $\lambda_{CSL} < 1.7 \times 10^{-12} s^{-1}$.
  • With lower temperatures ($1 mK$) and weaker coupling, this could reach $4 \times 10^{-14} s^{-1}$.

5. Significance

• Superior Sensitivity: The proposed two-nanosphere Coulomb-mediated setup offers the most stringent bounds on the CSL model to date, surpassing both interferometric and non-interferometric (X-ray/bulk heating) methods for specific parameter regimes.
• Theoretical Robustness: It addresses a major loophole in collapse model testing: the assumption of white noise. By utilizing low-frequency mechanical modes, the experiment remains valid even if the collapse mechanism is driven by a cosmological field with colored noise.
• Experimental Feasibility: The required parameters (mass, charge, trap frequency, cooling to micro-Kelvin, and high mechanical quality factors) are within the reach of current or near-future levitated optomechanics technology.
• New Physics Probe: The setup not only tests collapse models but also offers a pathway to test semi-classical gravity models (like Schrödinger-Newton) by witnessing squeezed thermal states and entanglement mediated by weak interactions.

In summary, this paper presents a theoretically rigorous and experimentally viable proposal to test the foundations of quantum mechanics, offering a definitive method to constrain spontaneous collapse models that is immune to the spectral assumptions that limit current X-ray based tests.