Einstein-Podolsky-Rosen correlations between mechanical oscillators revealed through SU(1,1) interferometry

This paper reports the experimental observation of Einstein-Podolsky-Rosen correlations between two macroscopic mechanical oscillators coupled to a superconducting qubit, utilizing a mechanical SU(1,1) interferometer to demonstrate quantum correlations stronger than entanglement in a macroscopic regime.

The Gist

Imagine two tiny, invisible drums made of solid material, sitting a few millimeters apart on a chip. These aren't just any drums; they are so light and delicate that they can vibrate in a "quantum" way, behaving more like waves of probability than solid objects. In this experiment, researchers at ETH Zurich managed to make these two distant drums dance in perfect, spooky synchronization, even though they are separated by space and made of billions of atoms.

Here is a breakdown of what they did, using simple analogies:

1. The Setup: The Conductor and the Drums

Think of the two mechanical drums (called HBARs) as two separate musicians. Usually, getting two musicians to play in perfect sync without talking to each other is hard. To solve this, the researchers introduced a "conductor": a superconducting qubit (a type of artificial atom).

The qubit acts like a bridge. It doesn't just listen to the drums; it actively connects them. By sending specific microwave signals (like musical notes) to this conductor, the researchers could make the two drums start vibrating in a special, linked pattern.

2. The Magic Trick: "Two-Mode Squeezing"

The core of the experiment is a process called Two-Mode Squeezing (TMS).

• The Analogy: Imagine you have two balloons. Normally, if you squeeze one, it gets smaller, and the other stays the same. But in this quantum trick, when you "squeeze" the system, the balloons don't just shrink; they become perfectly correlated. If one balloon suddenly expands, the other instantly expands by the exact same amount, even if they are in different rooms.
• The Result: The researchers created pairs of vibrations (phonons) where the two drums were so linked that measuring the vibration of one told you exactly what the other was doing, with a precision that defies the normal rules of physics (the "uncertainty principle").

3. The Test: The SU(1,1) Interferometer

To prove this link was real and not just a lucky coincidence, they built a "quantum interferometer."

• The Analogy: Think of a standard interferometer (like a Mach-Zehnder) as a fork in a road where a car splits into two paths, travels, and then merges back together. If the paths are different lengths, the car arrives at a different time, creating a pattern.
• The Twist: In this experiment, instead of just splitting the path, the researchers used the "squeezing" magic to amplify the vibrations at the start and the end. It's like having a machine that creates two cars out of thin air, sends them down two paths, and then uses another machine to see how they interfere when they come back.
• The Outcome: By adjusting the timing (phase) of the microwave signals, they saw the population of vibrations in the drums go up and down in a wave pattern. This wave pattern proved that the two drums were sharing a single quantum state, not just acting like two separate drums.

4. The Big Discovery: "Steering" the Quantum State

The most exciting part is what they call EPR Steering.

• The Concept: In quantum physics, "entanglement" means two things are linked. "Steering" is a stronger, more one-sided version of this. It means that by measuring one drum, you can effectively "steer" or predict the state of the other drum with such accuracy that it seems like you are influencing it faster than light, or at least proving that the other drum didn't have a pre-determined state before you looked.
• The Achievement: The researchers showed that the link between their two drums was strong enough to pass the test for EPR steering. This is a big deal because these drums are macroscopic (visible to the naked eye if you squint, weighing about 16 micrograms, which is like a tiny grain of sand).
• Why it matters: Usually, we only see this "spooky" behavior in tiny particles like electrons or photons. Seeing it in something as "heavy" as a mechanical drum suggests that the boundary between the quantum world (weird, small stuff) and the classical world (normal, big stuff) might be blurrier than we thought.

Summary

In short, the team used a superconducting "conductor" to make two tiny, distant mechanical drums dance in a quantum waltz. They proved that these drums were so deeply connected that measuring one instantly revealed the state of the other, a phenomenon known as EPR steering. This was achieved using a clever "interferometer" setup that amplified the quantum signals, proving that even relatively large, mechanical objects can exhibit the most bizarre and powerful forms of quantum correlation.

Technical Summary

Technical Summary: Einstein–Podolsky–Rosen correlations between mechanical oscillators revealed through SU(1,1) interferometry

Problem Statement
Quantum correlations are fundamental resources for quantum advantage in computing, communication, and sensing. While continuous-variable (CV) entanglement has been demonstrated between spatially-separated nanomechanical oscillators, stronger forms of nonclassical correlations, specifically Einstein–Podolsky–Rosen (EPR) steering, have not yet been observed in the macroscopic regime. EPR steering is a strictly stronger and asymmetric form of correlation than entanglement, where a measurement on one system allows for the prediction of another with accuracy violating local uncertainty relations. Its observation is crucial for one-sided device-independent quantum protocols and for testing the boundaries between quantum and classical physics in macroscopic systems (effective masses $\sim 16$ $\mu$g). The primary challenge lies in the stringent requirements for correlation strength and coherence preservation in macroscopic mechanical systems.

Methodology
The authors employed a circuit quantum acoustodynamics (cQAD) platform consisting of two high-overtone bulk acoustic wave resonators (HBARs) separated by $\sim 3.6$ mm, coupled to a single superconducting transmon qubit. The HBARs, fabricated from aluminum nitride on sapphire, possess distinct phonon spectra, allowing for the independent addressing of specific modes (labeled A and B) with frequencies $\omega_a/2\pi \approx 5.698$ GHz and $\omega_b/2\pi \approx 5.704$ GHz.

The experimental protocol involved the following steps:

1. Two-Mode Squeezing (TMS) Engineering: By parametrically driving the qubit with two microwave tones at frequencies $\omega_1$ and $\omega_2$ satisfying the resonance condition $\omega_1 + \omega_2 = \omega_a + \omega_b$, the system mediates a four-wave mixing process. This generates an effective TMS interaction between the phonon modes of the two HBARs, creating entangled phonon pairs.
2. Calibration: The TMS rate ($g_{\text{TMS}}$) was characterized using resonant phonon number (RPN) measurements. The qubit was used to measure the phonon population distribution, which was fitted to a thermal state distribution to extract the mean phonon number and the squeezing rate.
3. Mechanical SU(1,1) Interferometry: To witness quantum correlations without direct quadrature measurement (which is difficult in this platform), the authors implemented a mechanical SU(1,1) interferometer. This sequence consists of:
   • State Preparation: A first TMS operation (TMS1) generates the correlated state.
   • Phase Accumulation: A relative phase $\phi$ is introduced between the interferometer arms.
   • Readout: A second TMS operation (TMS2) acts as a phase-sensitive amplifier.
4. Correlation Witnessing: The output population of the interferometer is linked to a steering criterion. Specifically, the quantity $E(\phi) = (2\langle a^\dagger_{\text{out}}a_{\text{out}}\rangle + 1)/\cosh^2 r_2$ is measured. Theoretical analysis shows that $E(\phi) \geq 1$ for all non-steerable states (B$\to$A). A violation of this bound ($E < 1$) witnesses EPR steering.

Key Results

• TMS Characterization: The team successfully engineered a TMS interaction with a rate $g_{\text{TMS}}/2\pi \approx 6.8$ kHz. The measured phonon populations followed the expected thermal state distribution ($\rho_{\text{Th}}$) with mean phonon number $N_a = \sinh^2(g_{\text{TMS}}t_{\text{TMS}})$. The experimental rates agreed with theoretical predictions derived from perturbation theory and time-dependent simulations, accounting for drive-induced AC Stark shifts and higher-order effects.
• SU(1,1) Interferometer Operation: The mechanical SU(1,1) interferometer demonstrated phase-sensitive amplification. When TMS1 was active, the output population of TMS2 exhibited interference fringes dependent on the relative phase $\phi$. Crucially, for certain phases (specifically $\phi \approx \pi$), the output population was lower than the reference measurement (where TMS1 was off), a signature of the destructive interference enabled by quantum correlations.
• Observation of EPR Steering: By analyzing the interferometer fringes, the authors extracted the minimum value of the steering witness $E(\phi)$. At an optimal TMS2 duration ($t_{\text{TMS2}} \approx 11$ $\mu$s), they observed a violation of the steering criterion.
  • For the steering direction B$\to$A, 99% of Monte Carlo samples exhibited $E < 1$.
  • For the direction A$\to$B, 98% of samples exhibited $E < 1$.
  • This confirms the presence of two-way EPR steering between the two mechanical oscillators.

Significance and Claims
The paper claims the experimental observation of continuous-variable EPR correlations between two spatially-separated mechanical oscillators, each with an effective mass of $\sim 16$ $\mu$g (corresponding to $\sim 10^{17}$ atoms). This represents a significant step beyond previous demonstrations of CV entanglement in mechanical systems.

The authors state that their results:

1. Expand the cQAD Toolbox: They demonstrate the capability to engineer two-mode squeezing and realize SU(1,1) interferometry with phonon modes, adding to previously demonstrated beam splitter and single-mode squeezing operations.
2. Enable New Protocols: The observed correlations are sufficiently strong to enable quantum teleportation of continuous-variable states of motion between mechanical oscillators and serve as a resource for EPR steering-based quantum sensing.
3. Illuminate the Quantum-Classical Boundary: By witnessing EPR steering in macroscopic mechanical systems, the work sheds light on the validity of quantum mechanics at larger scales and the transition to the classical regime.

The authors remain modest regarding future applications, noting that while these correlations enable specific protocols like quantum teleportation and assisted metrology, the immediate contribution is the establishment of HBARs as a viable platform for CV quantum information processing and the demonstration of strong quantum correlations in the macroscopic regime.