Hermitian dynamics effectively describes many physical phenomena, including closed quantum systems and Landau-type phase transitions. However, the increasing complexity of quantum many-body systems, chiral transport, and strong system-environment coupling necessitate non-Hermitian models. A key example is non-reciprocal interaction, seemingly violating Newton's third law. While prevalent in biological systems, non-reciprocity finds applications in diverse fields such as optics, photonics, ultracold atoms, electrical circuits, and metamaterials. Arrays of mechanically interacting objects offer unique advantages for realizing unidirectional or topological transport, enhanced sensing, and topological states. Optically levitated nanoparticles, with their translational and rotational degrees of freedom, are a promising platform for quantum physics research, and recent advances enable control over arrays of these particles. This platform offers single-site readout, allowing for full reconstruction of collective degrees of freedom, and optical forces provide wide tunability of interactions. This experiment leverages previously demonstrated non-reciprocal and nonlinear light-induced dipole-dipole interactions between levitated nanoparticles to explore previously inaccessible interaction regimes.

The paper extensively reviews existing literature on non-Hermitian dynamics and non-reciprocity across various platforms. It highlights previous experimental demonstrations in classical regimes using photonic, atomic, electrical, and optomechanical systems. The authors also cite work on optically levitated nanoparticles and their use in creating particle arrays for studying quantum phenomena. Specific references are made to prior research on topological funneling of light, PT-symmetry breaking in complex optical potentials, and non-Hermitian topology in active mechanical metamaterials. The review sets the stage for this work by emphasizing the novel aspects of the tunable, non-reciprocal interactions achieved in the current experiment using optically levitated nanoparticles.

The experimental setup employs two orthogonal acousto-optical deflectors (AODs) driven by radiofrequency (RF) tones to create a 2x2 laser beam array. A slit selects beams with equal optical frequencies, and a Dove prism rotates the optical plane for easier manipulation. Two silica nanoparticles are trapped at a controlled distance in a vacuum chamber, and their motion is monitored using balanced heterodyne detection. The particle distance is tuned by varying the RF tone frequency difference. Optical phases at the traps are controlled by the RF tone phases, and particle distances are controlled to maintain equal laser frequencies. The optical interaction is tuned by adjusting the laser polarization angle using a half-wave plate (HWP). Electrostatic coupling is suppressed by discharging the particles. The theoretical model linearizes the equations of motion, averaged over one oscillation period. The linearized equations describe the system with an effectively non-Hermitian Hamiltonian that includes gas damping and diffusion. The non-Hermitian dynamics stems from the anti-reciprocal coupling, tunable via optical phase differences and particle distance. The eigenvalues of the non-Hermitian Hamiltonian are complex, defining the eigenfrequencies and damping rates. The system's exceptional points (EPs) define regions of PT symmetry breaking, where the interaction leads to correlated particle motion. For sufficiently strong coupling, a nonlinear model is developed to describe the Hopf bifurcation into the mechanical lasing phase and the resulting limit cycles. This model, which excludes thermal fluctuations, yields equations of motion for the amplitude of the collective motional state and the phase delay between the oscillators. The authors describe the experimental procedures for measuring the particle displacement amplitude and phase delay, including the use of Rice distributions to model the histograms of the oscillation envelopes.

The experiment observes two exceptional points (EPs) defining a PT symmetry-broken phase. In this phase, correlated particle motion is observed, characterized by a constant phase delay between the oscillators. The system exhibits a Hopf bifurcation into a mechanical lasing phase, where the particles move along stable limit cycles. The measured displacement amplitude and phase delay are compared with theoretical predictions from a model that includes the nonlinear dynamics. A good agreement is found, demonstrating that the observed nonlinear dynamics is well-described by the model. The mechanical lasing is interpreted as a transition analogous to a laser, where the highly populated tweezer and the Stokes sideband represent the two-level laser medium. The interference between mechanical sidebands suppresses Stokes or anti-Stokes scattering, leading to amplification of one mechanical mode and damping of the other. The lasing threshold is reached when amplification exceeds the intrinsic damping. The dependence of the limit cycle amplitude on coupling and detuning is measured and compared to theoretical predictions, showing good agreement. The transition is marked by the change in the amplitude correlation functions, switching from oscillatory behavior to bi-exponential decay as the system enters the PT-symmetry-broken phase. A non-zero limit cycle amplitude emerges above a threshold coupling, exhibiting a square-root dependence on the coupling strength. The authors also analyze the size ratios of the nanoparticles, electrostatic interactions, and optical coupling mechanisms.

The findings demonstrate a controllable transition from PT-symmetric to PT-symmetry-broken phases in a system of coupled mechanical oscillators. The observation of a mechanical lasing transition highlights the potential of non-Hermitian systems for generating coherent motion. The results are significant for understanding non-equilibrium multi-particle collective effects and could have implications for developing novel sensing applications and exploring more complex systems such as coupled arrays.

This study successfully demonstrates the experimental observation of PT symmetry breaking and a mechanical lasing transition in a system of two non-reciprocally coupled optically levitated nanoparticles. The results validate theoretical predictions of non-Hermitian dynamics and open new avenues for investigating non-equilibrium physics in larger arrays of nanoparticles. Future research could focus on extending these studies to larger arrays, probing dynamical encirclement of EPs for topological energy transfer, and exploring quantum phenomena in this platform.

The theoretical model used to describe the nonlinear regime simplifies the system by excluding thermal fluctuations. The agreement between the experimental data and the theoretical predictions, while good, is not perfect, suggesting that the effect of thermal fluctuations might be significant, especially near the transition point. The study focuses on a system of two particles; scaling up to larger arrays might introduce additional complexities. The experimental investigation is restricted to steady-state measurements; dynamic investigations of the non-Hermitian phases could reveal further insights.