Measurement of the Casimir force between superconductors

This paper reports the observation of an intense nonlinear force on a superconducting drum resonator within a microwave optomechanical cavity, which is consistent with the Casimir force and suggests a pathway to achieving the single-phonon nonlinear regime for enhanced quantum operations.

The Gist

The Big Idea: The Invisible "Ghost" Force

Imagine you have two very smooth, flat plates floating in a vacuum, very close to each other but not touching. Even though there is nothing between them, quantum physics tells us that empty space isn't actually empty. It's filled with tiny, invisible waves of energy popping in and out of existence.

These waves push on the plates. Because the space between the plates is so narrow, fewer waves can fit inside than outside. This creates a pressure difference that pushes the plates together. This is called the Casimir force. It's like a ghostly hand gently squeezing the plates together.

Scientists have known about this force for a long time, but they have a puzzle: when they measure it between normal metals, the numbers don't quite match the math. They suspect that the "low-frequency" waves (the slow, lazy waves) might be behaving differently than expected.

The Experiment: A Superconducting Drum

To solve this puzzle, the researchers built a tiny, super-sensitive instrument. Think of it as a microscopic drum.

• The Drum: It's a thin, circular sheet of aluminum (the top plate) suspended over a fixed bottom plate.
• The Superpower: They cooled this drum down to nearly absolute zero (colder than outer space). At this temperature, the aluminum becomes a superconductor. This means electricity flows through it with zero resistance, and it changes how it interacts with those invisible quantum waves.
• The Goal: They wanted to see if the "ghostly squeeze" (Casimir force) changes when the material becomes a superconductor.

How They Measured It: The "Bouncy" Problem

Usually, to measure this force, scientists try to move the plates closer and further apart. But doing this precisely in a super-cold environment is incredibly hard.

Instead, this team used a clever trick involving nonlinear dynamics (a fancy way of saying "weird bouncing behavior").

1. The Setup: They placed the drum inside a microwave cavity (a box that traps microwave light).
2. The Push: They used microwaves to gently push the drum, making it vibrate.
3. The Observation: When the drum vibrates with a small push, it bounces at a steady, predictable rhythm. But as they pushed harder, something strange happened. The drum didn't just bounce higher; its rhythm slowed down significantly.

The Analogy: Imagine a trampoline.

• Normal behavior: If you jump lightly, you bounce up and down at a steady speed. If you jump harder, you go higher, but the speed of your bounce stays the same.
• This experiment: Imagine the trampoline gets "spongy" the harder you push. The more you jump, the slower your bounce becomes. This "softening" is a sign that a strong, invisible force is pulling the trampoline down, fighting against the springs.

What They Found

The researchers found that the drum was experiencing a massive, invisible pull that made it "soften" and slow down its rhythm.

• The Match: They compared this weird bouncing behavior to a computer model of the Casimir force. The match was perfect. The invisible force pulling the drum down was exactly what the math predicted for the Casimir force between superconductors.
• The Ruling Out: They checked all other possible reasons for this "softening" (like static electricity, tiny bumps on the metal, or the metal stretching). None of those could explain the data. The only thing that fit was the Casimir force.

Why This Matters (According to the Paper)

The paper claims two main things:

1. Proof of Concept: They successfully measured the Casimir force between superconductors by watching how it changed the drum's "bouncing rhythm," without needing to move the plates with precise mechanical arms.
2. A New Tool for Quantum Physics: Because this force is so strong in their tiny device, it creates a very powerful "nonlinearity" (that weird softening effect). The authors say this is a big deal because it might allow them to control the drum's motion at the level of a single "phonon" (a single unit of vibration). This is a long-sought goal in quantum physics, which could help build better quantum computers or sensors in the future.

Summary

In short, the scientists built a tiny, super-cold drum. They found that invisible quantum waves were pushing the drum so hard that it changed how it vibrated. By measuring this change, they proved they could detect the Casimir force between superconductors, opening a new door to studying quantum mechanics with mechanical objects.

Technical Summary

Technical Summary: Measurement of the Casimir Force between Superconductors

Problem and Motivation
The Casimir force, arising from quantum fluctuations of the electromagnetic field, creates a nonlinear attractive force between closely spaced conductive objects. While extensively studied, a persistent discrepancy exists between theoretical predictions and experimental measurements of the Casimir force between normal metals at finite temperatures. This "Casimir puzzle" is suspected to involve the specific contribution of low-frequency modes. Superconductors offer a unique avenue to investigate this, as the transition to the superconducting state alters the material's electric permittivity and low-frequency response. However, measuring the Casimir force in superconductors is challenging due to the difficulty of precise positioning in cryogenic environments, which typically precludes the standard distance-variation experimental approach. Furthermore, existing nonlinear optomechanical systems often rely on coupling to external nonlinear systems (e.g., superconducting qubits) to access non-Gaussian quantum states, rather than possessing intrinsic nonlinearity.

Methodology
The authors present an experiment utilizing a superconducting drum resonator integrated into a microwave optomechanical cavity. The device consists of two aluminum plates separated by a vacuum gap; the top plate is a mechanically compliant drum suspended above a fixed bottom plate. The system is cooled to 10 mK in a dilution refrigerator, ensuring the aluminum is in the superconducting regime.

Instead of varying the distance, the authors probe the Casimir force by analyzing the nonlinear dynamics imparted to the resonator. The drum acts as the variable capacitance of a microwave cavity. The system is driven by two tones: a strong cavity drive at the resonance frequency ($\omega_c$) and a weaker sideband drive near the red sideband ($\omega_c - \omega_m$). The mechanical response is read out via the blue sideband ($\omega_c + \omega_m$).

To isolate the Casimir contribution, the authors:

1. Calibrated the system: They independently determined optomechanical parameters (cavity linewidths, coupling rate $g_0$, mechanical frequency $\omega_m$, and effective mass $m_{eff}$) using thermal motion measurements and cavity reflection responses.
2. Modeled the dynamics: They employed a model of a harmonic oscillator subject to a nonlinear attractive potential derived from the Casimir pressure. The equation of motion includes a term $P/(x+d)^n$, where $P$ and $n$ are calculated based on the Lifshitz formalism using the Mattis-Bardeen conductivity for superconductors.
3. Excluded alternatives: They systematically ruled out other sources of nonlinearity, including electrostatic forces from potential patches (verified via Kelvin Probe Force Microscopy), geometric nonlinearity (which typically causes hardening, whereas the observed effect is softening), and higher-order optomechanical couplings.

Key Results
The experiment observed a strong "softening" nonlinearity, where the resonant frequency decreases as the oscillation amplitude increases. This behavior manifests as a hysteresis loop in the frequency sweep, with the system jumping between stable low-amplitude and high-amplitude branches.

• Quantitative Agreement: The measured dynamics were quantitatively compatible with a model of the Casimir force between superconducting plates. Using a single fit parameter—the hypothetical unperturbed vacuum gap $d$—the model successfully reproduced the response curves across three orders of magnitude in displacement power.
• Parameter Extraction: The fit yielded an unperturbed gap $d = 18.00 \pm 0.25$ nm. This value is consistent with independent estimates derived from AFM measurements of the device at room temperature and subsequent thermal contraction simulations down to 10 mK.
• Force Magnitude: The inferred Casimir pressure at the equilibrium separation ($d' \approx 15.1$ nm) is approximately $12.0 \pm 0.6$ kPa. This corresponds to a force intensity roughly one-tenth of atmospheric pressure, significantly larger than in conventional room-temperature Casimir experiments due to the nanometer-scale separation.
• Scaling: The force followed a power-law scaling $P \propto d^{-3.193}$, consistent with the Lifshitz formalism for real materials in the specified separation range, distinguishing it from the van der Waals regime.

Significance and Claims
The paper claims to have observed a nonlinear potential in a superconducting mechanical system that is quantitatively consistent with the Casimir force, without relying on precise cryogenic positioners. The authors assert that this nonlinearity is intrinsic to the system's construction, distinguishing it from previous approaches that required coupling to external nonlinear systems.

The significance of this work lies in two main areas:

1. Casimir Physics: The device provides a sensitive platform to probe the Casimir effect in superconductors, potentially helping to resolve discrepancies in Casimir force measurements for materials with finite conductivity. The authors note that future modifications, such as adding a gate electrode, could allow for electrostatic tuning of the separation to probe changes across the superconducting transition.
2. Quantum Optomechanics: The intensity of the observed nonlinearity suggests that with a modified design, this system could operate in the single-phonon nonlinear regime. Achieving this regime is a long-standing goal that would facilitate quantum operations on mechanical resonators, such as the generation of non-Gaussian states (e.g., Fock or cat states) and the creation of mechanical qubits, purely through the system's intrinsic properties.