Generation of mechanical cat-like states via optomagnomechanics

This paper proposes a two-step optomagnomechanical protocol that generates mechanical cat-like states by first creating a squeezed mechanical state via microwave-driven magnetostriction and then subtracting phonons through conditional detection of anti-Stokes photons from a weak red-detuned optical pulse.

The Gist

Imagine you have a tiny, invisible drum made of a special magnetic crystal. In the world of quantum physics, this drum doesn't just vibrate; it can exist in a "superposition," meaning it can be vibrating in two completely opposite directions at the exact same time. This is the famous "Schrödinger's Cat" scenario, but instead of a cat being alive and dead, our drum is vibrating "left" and "right" simultaneously.

Creating this kind of "quantum drumming" in a large, visible object is incredibly hard because the environment usually forces it to pick just one direction. However, this paper proposes a clever two-step recipe to make it happen using a hybrid system that mixes magnetism, sound, and light.

Here is the story of how they do it, explained simply:

The Setup: A Magical Trio

Think of the system as a three-person band:

1. The Magician (Magnons): A magnetic field creates "magnons" (tiny magnetic waves) in a crystal.
2. The Drummer (Mechanical Oscillator): The crystal is attached to a tiny mechanical beam that can vibrate like a drum.
3. The Flashlight (Optical Cavity): A mirror setup that traps light, acting like a camera lens.

The Magician can shake the Drummer, and the Drummer can shake the Flashlight. The goal is to get the Drummer to do a quantum dance.

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Step 1: The "Squeeze" (Getting the Drum Ready)

First, the scientists need to get the drum into a special state called a squeezed state.

• The Analogy: Imagine a balloon filled with air. Normally, the air pressure is equal in all directions. If you squeeze the balloon, it gets thinner in one direction but fatter in another. You haven't changed the amount of air (energy), but you've changed how it's distributed.
• The Action: The team hits the magnetic crystal with two specific microwave pulses (like radio waves). These pulses act like a pair of hands squeezing the balloon.
• The Result: The drum's vibration becomes "squeezed." It is now very quiet in one direction and very wild in the other. It's a state of high potential, ready for the next trick, but it's still just a "fuzzy" vibration, not a quantum superposition yet.

Step 2: The "Subtraction" (The Quantum Magic Trick)

Now comes the tricky part. To turn that squeezed vibration into a "Cat State" (vibrating left and right at once), they need to perform a phonon subtraction.

• The Analogy: Imagine you have a jar of marbles (the vibrations). You want to remove exactly one marble without looking inside, but you have a special camera that only flashes if a marble leaves the jar.
• The Action:
  1. They turn off the microwaves.
  2. They shine a very weak, red-tuned laser pulse at the drum.
  3. This laser tries to "steal" a tiny bit of energy (a phonon, or a unit of vibration) from the drum and turn it into a photon (a particle of light) that flies out of the system.
• The "Herald": If a detector at the end of the laser path clicks and sees a specific type of light particle (an anti-Stokes photon), it's like the camera flashing. It tells us: "Success! We just removed exactly one vibration from the drum."
• The Magic: In the quantum world, removing a specific amount of energy from a "squeezed" state doesn't just make it quieter; it forces the drum to split into a superposition. It's as if, by taking one marble out, the jar suddenly becomes a ghost that is both full and empty at the same time.

Why is this a big deal?

Usually, making these "Cat States" requires delicate, microscopic systems like single atoms. This paper shows how to do it with a macroscopic object (a tiny, but visible, mechanical beam).

• The Advantage: They used a magnetic crystal (YIG) because it's incredibly quiet (low noise) compared to other materials. It's like trying to hear a whisper in a library (magnetic crystal) versus a rock concert (standard materials). This quietness allows them to get a much cleaner "squeezed" state to start with.
• The Outcome: By detecting the light particles, they successfully created a mechanical object that is in a superposition of two different vibrational states. They even checked the "fuzziness" of this state (using something called a Wigner function) and saw the interference patterns that prove it's a true quantum cat state.

The Big Picture

This research is like building a bridge between the microscopic quantum world and our everyday macroscopic world.

If we can keep these "quantum drums" stable, we could use them for:

1. Super-Sensitive Sensors: Detecting tiny forces (like gravitational waves) with incredible precision.
2. Testing Reality: Proving that the weird rules of quantum mechanics apply to big objects, not just atoms.
3. Quantum Computers: Using these mechanical vibrations as memory or processing units for future quantum tech.

In short, the authors figured out a way to use magnetism to squeeze a drum and light to pluck a string, resulting in a mechanical object that dances to the tune of quantum superposition.

Technical Summary

1. Problem Statement

The generation of macroscopic quantum superposition states (Schrödinger's cat states) in massive mechanical oscillators is a significant challenge in quantum physics. While cat states have been successfully generated in microscopic systems (ions, photons), creating them in macroscopic mechanical systems is difficult due to environmental decoherence and the weakness of coupling mechanisms. Existing proposals often rely on pure optomechanical systems, which struggle to satisfy the resolved-sideband limit ($\kappa \ll \omega$) required for efficient reservoir engineering and strong squeezing, particularly when the mechanical frequency is low. The authors aim to overcome these limitations by proposing a hybrid Opto-Magnomechanical (OMM) system to generate mechanical cat-like states with high fidelity.

2. Methodology

The authors propose a two-step pulse protocol utilizing a hybrid OMM system, which couples a magnon mode (in a Yttrium-Iron-Garnet, YIG, crystal), a mechanical mode (phonons), and an optical cavity mode.

System Setup

• Components: A YIG micro-bridge (magnon source) coupled to a mechanical mode via magnetostriction, and the mechanical mode coupled to an optical cavity via radiation pressure.
• Hamiltonian: The system is described by a three-mode Hamiltonian involving magnon ($m$), mechanical ($b$), and optical ($c$) operators, with coupling strengths $G_0$ (magnomechanical) and $g_0$ (optomechanical).

Step 1: Preparation of a Squeezed Mechanical State

• Mechanism: Two microwave pulses with frequencies $\omega_{\pm} = \omega_m \pm \omega_b$ are applied to the magnon mode.
• Interaction:
  • The red-detuned component ($\omega_-$) activates a beam-splitter interaction (anti-Stokes process), cooling the mechanical mode.
  • The blue-detuned component ($\omega_+$) activates a parametric down-conversion interaction (Stokes process), inducing squeezing.
• Result: By balancing these interactions ($G_- > G_+$ for stability), the mechanical mode evolves into a squeezed thermal state. The YIG crystal's low magnon dissipation rate ($\kappa_m \approx 1$ MHz) ensures the resolved-sideband limit is satisfied, enabling strong squeezing even at low mechanical frequencies.

Step 2: Phonon Subtraction via Optical Pulses

• Mechanism: After the microwave pulses cease and magnons decay, a weak, red-detuned optical pulse is sent to the optical cavity.
• Interaction: This pulse activates optomechanical anti-Stokes scattering.
• Conditional Measurement: The detection of $k$ anti-Stokes photons in the cavity output field heralds the subtraction of $k$ phonons from the mechanical squeezed state.
• Mathematical Operation: The process is modeled as a propagator $U(t)$ acting on the initial state. Detecting $k$ photons corresponds to applying the operator $b^k$ (phonon annihilation) to the mechanical state.
• Outcome: Subtracting phonons from a squeezed state transforms the Gaussian state into a non-Gaussian cat-like state (a superposition of coherent states with opposite phases).

3. Key Contributions

1. Hybrid OMM Architecture: The paper introduces a novel protocol combining magnomechanics and optomechanics. It leverages the superior coherence properties of YIG magnons (low dissipation) to generate the initial squeezing, which is then transferred to the mechanical mode.
2. Two-Step Pulse Sequence: A distinct protocol separating the squeezing generation (microwave domain) from the state engineering (optical domain) allows for independent optimization of cooling/squeezing and phonon subtraction.
3. Conditional Phonon Subtraction: The authors demonstrate how detecting specific photon numbers in the optical output can herald the creation of mechanical cat states, effectively "shaping" the mechanical wavefunction.
4. Robustness Analysis: The study includes a detailed analysis of experimental imperfections, such as thermal noise, detection efficiency, and dark counts, showing the protocol's viability under realistic conditions.

4. Key Results

• Squeezing Generation: Using experimentally feasible parameters (YIG micro-bridge, $T=10$ mK), the system achieves a mechanical squeezing degree of approximately 7 dB. The squeezing is robust against mechanical damping rates up to 1 kHz.
• Cat State Fidelity:
  • The protocol successfully generates 1-, 2-, and 3-phonon subtracted states.
  • Fidelity: The fidelity of the generated states with ideal cat states is calculated as:
    • 1-phonon subtracted: 87%
    • 2-phonon subtracted: 91%
    • 3-phonon subtracted: 94%
  • Higher phonon subtraction yields higher fidelity but lower success probability (scaling as $\tan^{2k}\theta$).
• Non-Classicality: The Wigner functions of the generated states exhibit negativity and interference fringes around the origin of phase space, confirming the non-classical, cat-like nature of the mechanical motion.
• Macroscopicity: The calculated macroscopicity measure ($I$) for the 3-phonon subtracted state is 5.7, indicating a significant macroscopic quantum superposition.
• Parameter Sensitivity:
  • Fidelity decreases with increasing residual thermal phonon number ($\bar{n}$), highlighting the need for low temperatures.
  • There is an optimal ratio of drive strengths ($G_+/G_-$) that balances cooling and squeezing, which shifts with temperature.

5. Significance

• Macroscopic Quantum Physics: This work provides a viable pathway to create and study macroscopic quantum superpositions in massive mechanical oscillators, bridging the gap between microscopic quantum phenomena and macroscopic reality.
• Testing Collapse Theories: The ability to generate high-fidelity mechanical cat states allows for rigorous tests of wavefunction collapse models (e.g., Continuous Spontaneous Localization), which predict the breakdown of superposition at macroscopic scales.
• Quantum Sensing: The generation of non-Gaussian states (cat states) is crucial for enhancing the sensitivity of quantum sensors beyond the standard quantum limit.
• Technological Advancement: By utilizing the low dissipation of YIG magnons, this protocol overcomes the limitations of pure optomechanical systems, offering a more robust platform for quantum state engineering in hybrid systems.

In conclusion, the paper presents a theoretically sound and experimentally feasible scheme to generate mechanical cat states by combining the strong nonlinearity of magnomechanics with the precision of optomechanical photon detection, opening new avenues for macroscopic quantum research.