Optical squeezing mediated by levitated oscillators at their quantum ground state

This paper demonstrates the generation of optical squeezing below the shot-noise level by coupling a cavity field to two center-of-mass modes of a levitated nanoparticle simultaneously cooled to their quantum ground state, thereby bridging mechanical quantum control with non-classical light.

The Gist

Imagine you have a tiny, invisible marble made of glass, floating in mid-air, held up not by your hand, but by a focused beam of light (an "optical tweezer"). This marble is so small that it behaves like a quantum object, jittering around due to the laws of physics.

This paper is about a team of scientists who managed to do two very difficult things at the same time with this floating marble, and then used the result to create a special kind of "quiet" light.

Here is the story of what they did, broken down into simple concepts:

1. The Setup: A Marble in a Mirror Box

The scientists placed this floating glass marble inside a box made of mirrors (an optical cavity). They shone a laser into the box.

• The Goal: They wanted to cool the marble down until it was almost perfectly still. In the quantum world, "still" means the marble has almost zero energy left, a state called the "quantum ground state."
• The Challenge: Usually, it's hard to cool something down without making the light around it noisy, and it's hard to make the light quiet without heating the object up. It's like trying to calm a shaking dog without making the room louder.

2. The Breakthrough: Two Dancers, One Rhythm

The marble wasn't just shaking in one direction; it was wobbling in two different directions at once (side-to-side and front-to-back).

• The scientists managed to cool both of these wobbles down to the quantum ground state simultaneously. Think of it as getting two dancers to stop moving entirely at the exact same time.
• Because the marble was so cold and the mirrors were so perfect, the light bouncing off the marble and the marble's tiny movements started to "dance" together. They became linked, or hybridized.

3. The Result: Squeezing the Noise

When the light and the marble danced together, something magical happened to the light coming out of the box.

• The Problem: Normally, laser light has a natural "hiss" or static noise, called shot noise. Imagine trying to listen to a whisper in a room where the air itself is crackling with static.
• The Solution: The interaction with the cold marble allowed the scientists to "squeeze" this noise.
• The Analogy: Imagine a balloon filled with air (the noise). Usually, the air pushes out equally in all directions. "Squeezing" the light is like taking that balloon and pressing it from the sides. The air (noise) gets squished down in one direction, making it quieter than the natural vacuum of space, but it puffs out a little more in another direction.
• The Achievement: They successfully "squeezed" the light so that its noise dropped 2% below the natural limit (the vacuum level). This is called sub-shot-noise squeezing.

4. Why This Matters (According to the Paper)

The paper highlights a few key points about why this is a big deal:

• The "Impossible" Combination: In the past, scientists could either cool the marble or create this quiet light, but rarely both at once. This experiment proved you can do both.
• Two is Better Than One: They didn't just use one wobble of the marble; they used two at the same time. This shows that complex, multi-part quantum systems can work together to shape light.
• A New Tool: They created a way to map out exactly how the light behaves, showing exactly where and when the "quiet" happens.

5. What's Next? (Based Only on the Paper's Claims)

The authors note that while they achieved this, there is still room to make the light even quieter.

• They suggest that if they could improve their equipment (like catching more of the light or reducing air collisions), they could potentially make the light four times quieter than what they achieved here.
• They see this setup as a "testbed" or a playground to explore deeper quantum mysteries, such as creating entanglement (where two objects become linked so that what happens to one instantly affects the other) between different mechanical parts.

In Summary:
The scientists took a tiny, floating glass bead, froze its movements to the absolute quantum limit, and used its motion to "squeeze" a laser beam, making the light quieter than nature usually allows. They did this with two different movements at once, proving that levitated particles are a powerful new tool for quantum physics.

Technical Summary

1. Problem Statement and Motivation

The field of cavity optomechanics has historically faced a trade-off between two major milestones:

• Ground-state cooling: Requires the resolved sideband regime (mechanical frequency $\Omega \gg$ cavity linewidth $\kappa$) to efficiently remove thermal phonons.
• Ponderomotive squeezing: Typically observed in the bad-cavity regime ($\kappa \gg \Omega$), where quantum correlations in the optical field quadratures allow for sub-shot-noise fluctuations.

While both milestones have been achieved individually, they have rarely been observed simultaneously. Most demonstrations of ponderomotive squeezing rely on classical mechanical oscillators, and previous ground-state cooling experiments did not necessarily generate non-classical light. The primary challenge addressed in this work is the simultaneous observation of quantum behavior in both the mechanical and optical subsystems, specifically generating optical squeezing mediated by mechanical oscillators that are cooled to their quantum ground state (occupation number $n < 1$).

2. Methodology and Experimental Setup

The researchers utilized a levitated optomechanics platform, which offers a versatile environment for sensing and quantum control.

• System: A silica nanoparticle is confined in an optical tweezer and placed at the center of a high-finesse optical cavity.
• Coupling: The system operates in the coherent scattering regime. The particle scatters light from the tweezer into the cavity mode, creating a strong coupling between the particle's center-of-mass motion and the cavity field.
• Mechanical Modes: The experiment focuses on the two transverse center-of-mass modes ($x$ and $y$) of the nanoparticle.
  • Frequencies: $\Omega_x/2\pi \approx 121$ kHz, $\Omega_y/2\pi \approx 109$ kHz.
  • Cooling: The system operates in the resolved sideband regime ($\kappa/2\pi = 57$ kHz, where $\kappa \ll \Omega_i$).
  • State: Both modes were cooled to mean occupation numbers of $n_x \approx 0.55$ and $n_y \approx 0.74$, well below the unity threshold (ground state).
• Detection:
  • The cavity output field was analyzed using Balanced Heterodyne Detection (BHD).
  • Unlike standard homodyne detection, BHD measures both quadratures simultaneously. The authors implemented a numerical lock-in amplifier in post-processing to reconstruct the phase-sensitive detection and the full spectral covariance matrix.
  • Regime: The experiment operated in the strong coupling regime, where the mechanical modes hybridize with the optical mode, leading to avoided crossings in the spectrum.

3. Key Contributions and Analysis

• Full Spectral Covariance Matrix Reconstruction: The authors reconstructed the complete spectral covariance matrix ($V_{QP}$) of the optical field, mapping fluctuations of the amplitude ($Q$) and phase ($P$) quadratures as a function of frequency and detection phase.
• Modeling Phase Noise: A discrepancy was observed between the experimental data and theoretical models, particularly in off-diagonal elements. The authors attributed this to residual phase noise. They introduced a model averaging the covariance matrix over a Gaussian-distributed phase fluctuation ($\theta$), which successfully reconciled the theory with the experiment.
• Hybridization Effects: The analysis highlighted how the strong coupling regime causes the mechanical modes to hybridize with the optical mode, resulting in a double avoided crossing structure in the power spectral densities (PSD), distinct from simple Lorentzian peaks.

4. Key Results

• Sub-Shot-Noise Squeezing: The experiment successfully demonstrated optical squeezing below the shot-noise level.
  • Frequency Band: Consistent squeezing was observed in the 70–95 kHz band.
  • Magnitude: The lowest measured variance was 0.98 (normalized to vacuum fluctuations), representing a 2% reduction below the vacuum noise floor.
  • Optimal Spectrum: By optimizing the detection phase at every frequency, the optimal squeezing spectrum confirmed squeezing regions both below and above the mechanical resonances (around 80 kHz and 150 kHz).
• Ground-State Mediation: Crucially, this squeezing was mediated by two mechanical oscillators simultaneously in their quantum ground state. This confirms that quantum correlations can be generated and transferred even when the mechanical system is not classical.
• Efficiency Limitations: The observed squeezing was limited by the total detection efficiency ($\eta \approx 0.32$). The authors note that the ideal squeezing available at the cavity output would be approximately double the observed value (4%) if detection losses were removed.

5. Significance and Future Outlook

• Bridging Quantum Domains: This work bridges the gap between purely mechanical quantum control (ground-state cooling) and non-classical optical fields (squeezing), demonstrating a platform capable of multimode quantum interactions.
• Platform Readiness: The results validate levitated optomechanics as a robust platform for generating entanglement between multiple mechanical modes and for using mechanical oscillators as local quantum memories.
• Pathways for Improvement: The authors outline several routes to enhance squeezing:
  • Detection Efficiency: Switching to single-ended cavity configurations and improving optical collection could increase observed squeezing by a factor of 4 or more.
  • Heating Reduction: Improving vacuum conditions to reduce gas collisions would lower occupation numbers further.
  • Advanced Readout: Implementing Quantum Non-Demolition (QND) measurements and variational readout schemes could mitigate noise penalties inherent in heterodyne detection.

In conclusion, this paper establishes a new regime in cavity optomechanics where the collective quantum behavior of multiple ground-state mechanical oscillators shapes the properties of an optical field, paving the way for advanced quantum metrology and quantum state transfer protocols.