Floquet engineering of nonreciprocal light-induced dipolar interactions

This paper demonstrates a Floquet-engineered toolbox for controlling nonreciprocal light-induced dipolar interactions in tweezer arrays, enabling operations like squeezing and beamsplitting to tune complex eigenfrequencies for exploring non-Hermitian many-body physics and collective quantum optomechanics.

The Gist

Imagine two tiny, invisible marbles floating in mid-air, held in place by invisible beams of light (like laser tweezers). Usually, if you push one marble, it might jiggle the other through the air, but they don't really "talk" to each other in a controlled way.

This paper describes a new way to make these two light-trapped marbles have a very specific, complex conversation with each other. The scientists didn't just let them interact naturally; they used a clever trick called Floquet engineering to program how they talk.

Here is the breakdown of what they did, using everyday analogies:

1. The Setup: Two Marbles and a Rhythm

The researchers trapped two silica nanoparticles (tiny glass balls) in separate laser beams.

• The Trick: They made the two laser beams vibrate at slightly different speeds (frequencies).
• The Result: Because the lasers are beating against each other, the "conversation" between the two marbles isn't static. It changes rhythmically, like a drumbeat that speeds up and slows down. This rhythmic change is what the paper calls Floquet engineering.

2. The "Magic" Conversation: Three New Moves

By tuning the rhythm of these lasers, the scientists could force the marbles to perform three specific quantum "dance moves" that are usually very hard to get them to do:

• The Swap (Beamsplitter): Imagine two people holding a ball. In this mode, the marbles swap their energy back and forth perfectly. If Marble A is shaking hard and Marble B is still, Marble A will slowly calm down while Marble B starts shaking, and then they switch back. It's like a perfect game of catch where they never drop the ball.
• The Squeeze (Squeezing): Imagine a balloon. Usually, if you squeeze it on the sides, it bulges out the top and bottom. In this experiment, the scientists used the light to "squeeze" the uncertainty of the marbles' movement. They made the marbles' positions more predictable (squashed flat) while making their speed less predictable (bulging out), or vice versa. This is a key tool for making ultra-precise measurements.
• The "Ghost" Partner (Negative Mass): This is the most mind-bending part. The scientists created a situation where one marble acted as if it had negative mass.
  • The Analogy: If you push a normal object, it moves forward. If you push a "negative mass" object, it moves backward.
  • In their experiment, the light forces made the two marbles behave as if one was pushing the other in the opposite direction of the force. This created a strange, unstable dance where they moved in perfect sync but in a way that defies normal physics rules (like a predator and prey chasing each other in a loop).

3. The "Dial" for Control

The most powerful tool they built is a "dial" (controlled by the distance between the marbles and the laser settings).

• They can turn the interaction from Reciprocal (Marble A pushes B, and B pushes A back equally) to Anti-Reciprocal (Marble A pushes B, but B pushes A in a way that creates the "negative mass" effect).
• They can even set the dial to a mix of both. This allows them to continuously tune the "personality" of the interaction, changing how the marbles move and how much energy they lose to the air around them.

4. Why Does This Matter?

The paper claims this creates a "toolbox" for quantum physics.

• Before this, scientists had to rely on specific, rigid setups to get these interactions.
• Now, they can program these interactions on demand using light.
• This allows them to study non-Hermitian physics (systems where energy is constantly flowing in and out, like a leaky bucket) and collective quantum mechanics (how groups of particles act as a single unit).

In Summary:
The researchers built a programmable stage where light acts as the director. By changing the rhythm and distance of the lasers, they can make two tiny particles swap energy, get squeezed into a precise state, or dance as if one has negative mass. This gives scientists a new, flexible way to build and test complex quantum machines.

Technical Summary

Technical Summary: Floquet Engineering of Nonreciprocal Light-Induced Dipolar Interactions

Problem and Motivation
Controllable interactions between quantum systems are fundamental to quantum metrology, sensing, and many-body physics. While atomic and molecular platforms utilize dipolar, Rydberg, and Coulomb interactions for spin exchange, light-mediated interactions offer interfaces over large distances, particularly in optomechanics. A specific class of light-induced interactions, known as "optical binding," arises when the radiation from one polarizable object interferes with the trapping field of another, coupling their mechanical degrees of freedom.

A notable feature of optical binding in tweezer arrays is its inherent nonreciprocity, stemming from engineered nonuniform optical phases. However, previous implementations of position coupling in these systems have been limited in their applicability to quantum protocols and non-Hermitian physics. Specifically, they lacked deterministic access to the fundamental building blocks of quantum optics: beamsplitter and two-mode squeezing operations. This work addresses the need to extend light-induced dipolar interactions into a regime capable of realizing these operations through Floquet engineering.

Methodology
The authors experimentally demonstrate a Floquet-driven approach to control light-induced dipolar interactions between two silica nanoparticles (radius $r \approx 105$ nm) trapped in optical tweezers. The system operates at a pressure of $p \approx 1.2$ mbar, resulting in a mechanical damping rate of $\gamma/2\pi \approx 570$ Hz.

The core methodology involves engineering the optical frequencies of the tweezer drives to differ by a detuning $\Delta\omega$. This frequency difference causes the interference between the trapping field and the field scattered by the particles to oscillate at $\Delta\omega$, generating a time-dependent, phase-modulated optical force. Consequently, the directional coupling rates $g_{12}$ and $g_{21}$ become time-dependent:
$$g_{12} = g \cos(\Delta\omega t + \Delta\phi)$$
$$g_{21} = g \cos(\Delta\omega t + \Delta\phi + 2kd)$$
where $g$ is the maximum coupling strength, $k$ is the wavenumber, $d$ is the interparticle distance, and $\Delta\phi$ is the optical phase difference.

By tuning the optical detuning $\Delta\omega$ relative to the mechanical frequencies ($\Omega_1, \Omega_2$) and the interparticle distance $d$, the authors selectively enhance specific interaction types via the rotating-wave approximation (RWA):

1. Beamsplitter Interaction: Achieved when $\Delta\omega \approx \Omega_2 - \Omega_1$ (mechanical detuning $\Delta\Omega$).
2. Two-Mode Squeezing: Achieved when $\Delta\omega \approx \Omega_1 + \Omega_2$ (sum frequency $2\bar{\Omega}$).
3. Single-Mode Squeezing: Achieved when $\Delta\omega \approx 2\Omega_1$ or $2\Omega_2$.

The nature of the interaction (reciprocal vs. anti-reciprocal) is controlled by the distance $d$. For $kd = n\pi$, the coupling is reciprocal ($g_{12} = g_{21}$); for $kd = (2n+1)\pi/2$, the coupling is anti-reciprocal ($g_{12} = -g_{21}$). Intermediate distances yield programmable combinations of these cases.

Key Results
The authors demonstrate the following operations and phenomena:

• Beamsplitter and Two-Mode Squeezing: Spectrograms of particle motion reveal avoided crossings and mode degeneracies characteristic of beamsplitter and two-mode squeezing interactions, respectively. Coupling rates were extracted as $g/2\pi \approx 953$ Hz for reciprocal beamsplitter, $806$ Hz for reciprocal two-mode squeezing, $775$ Hz for anti-reciprocal two-mode squeezing, and $931$ Hz for anti-reciprocal beamsplitter interactions.
• Coherent Energy Exchange: By switching off feedback cooling on a charged particle and turning on the interaction, the authors observed coherent energy exchange (analogous to Rabi oscillations) between the particles. This exchange occurs in both reciprocal and anti-reciprocal configurations, confirming coherent dynamics even in the presence of dissipative anti-reciprocal coupling.
• Negative-Mass-Like Oscillator: In the anti-reciprocal regime ($kd = (2n+1)\pi/2$), the system exhibits dynamics analogous to a negative-mass oscillator. This is evidenced by the simultaneous correlation of both positions and momenta of the two particles, a signature typically associated with coupling positive- and negative-mass oscillators. This occurs without the need for atomic or cavity resonances, purely through the nonreciprocal optical forces.
• Thermal Noise Squashing: The anti-reciprocal two-mode squeezing interaction leads to the squashing of thermomechanical noise. The authors measured a parametric squeezing gain $s \approx 0.75$ and a noise reduction of $-2.4$ dB below the thermal noise level.
• Continuous Tuning of Eigenfrequencies: By scanning intermediate distances, the authors demonstrated that the real and imaginary parts of the complex eigenfrequencies can be continuously tuned. The trajectory of these eigenfrequencies in the complex plane traces a circle, allowing for the exploration of non-Hermitian effects. While exceptional points were not encircled in the specific scan shown (due to constant detuning), the authors note that varying both distance and detuning simultaneously would allow for encircling exceptional points.

Significance and Claims
The paper establishes a complete toolbox of quantum operations for time-dependent light-induced dipolar interactions. The authors claim that this platform enables:

1. Full Control: Deterministic realization of beamsplitter, single-mode, and two-mode squeezing operations in a pair of optically trapped dipoles.
2. Mechanical Frequency Stability: Unlike stationary interactions, the mechanical frequency shifts caused by optical binding disappear due to modulation, keeping frequencies constant regardless of interaction type or strength.
3. Non-Hermitian Physics: The ability to tune complex eigenfrequencies continuously provides a platform for investigating non-Hermitian many-body physics, including mode braiding and exceptional points.
4. Negative Mass Dynamics: The realization of a negative-mass-like oscillator in the rotating frame offers a new avenue for reservoir engineering, potentially enabling continuous measurement of conjugate quadratures beyond the Heisenberg limit and the creation of stationary entanglement.
5. Hybrid Systems: The demonstrated operations enable coherent control in hybrid systems involving nanoparticles and atoms or ions, as well as between atoms themselves.

The work positions Floquet engineering of nonreciprocal interactions as a significant step toward investigating out-of-equilibrium dynamics in nonreciprocal systems and advancing collective quantum optomechanics.