Rapid laser solver for the phase retrieval problem

Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields

Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ±J models, Sherrington-Kirkpatrick models, and only locally connected J1-J2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high-power efficiency Boltzmann sampling to a generalized large-scale Ising model.

Improved laser phase locking with intra-cavity adaptive optics

Phase locking of coupled lasers is severely hindered by the spread in their natural lasing frequencies. We present an intra-cavity adaptive optics method that reduces the frequency spread and thereby improves phase locking. Using an intra-cavity spatial light modulator and an iterative optimization algorithm, we demonstrate a fourfold enhancement of phase locking 450 coupled lasers, as quantified by the peak intensity and the inverse participation ratio of the far-field output distributions. We further show that the improvement is long-lasting, and suitable for phase locking of weakly coupled lasers.

Optimization with photonic wave-based annealers

Many NP-hard combinatorial optimization (CO) problems can be cast as a quadratic unconstrained binary optimization model, which maps naturally to an Ising model. The final spin configuration in the Ising model can adiabatically arrive at a solution to a Hamiltonian, given a known set of interactions between spins. We enhance two photonic Ising machines (PIMs) and compare their performance against classical (Gurobi) and quantum (D-Wave) solvers. The temporal multiplexed coherent Ising machine (TMCIM) uses the bistable response of an electro-optic modulator to mimic the spin up and down states. We compare TMCIM performance on Max-cut problems. A spatial photonic Ising machine (SPIM) convolves the wavefront of a coherent laser beam with the pixel distribution of a spatial light modulator to adiabatically achieve a minimum energy configuration, and solve a number partitioning problem (NPP). Our computational results on Max-cut indicate that classical solvers are still a better choice, while our NPP results show that SPIM is better as the problem size increases. In both cases, connectivity in Ising hardware is crucial for performance. Our results also highlight the importance of better understanding which CO problems are most likely to benefit from which type of PIM. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Massively degenerate coherent perfect absorber for arbitrary wavefronts

One of the key insights of non-Hermitian photonics is that well-established concepts such as the laser can be operated in reverse to realize a coherent perfect absorber (CPA). Although conceptually appealing, such CPAs are limited so far to a single, judiciously shaped wavefront or mode. Here, we demonstrate how this limitation can be overcome by time-reversing a degenerate cavity laser based on a unique cavity that self-images any incident light field onto itself. Placing a weak, critically coupled absorber into this cavity, any incoming wavefront, even a complex and dynamically varying speckle pattern, is absorbed with close to perfect efficiency in a massively parallel interference process. These characteristics open up interesting new possibilities for applications in light harvesting, energy delivery, light control, and imaging.

Anyonic-parity-time symmetry in complex-coupled lasers

Non-Hermitian Hamiltonians, and particularly parity-time (PT) and anti-PT symmetric Hamiltonians, play an important role in many branches of physics, from quantum mechanics to optical systems and acoustics. Both the PT and anti-PT symmetries are specific instances of a broader class known as anyonic-PT symmetry, where the Hamiltonian and the PT operator satisfy a generalized commutation relation. Here, we study theoretically these novel symmetries and demonstrate them experimentally in coupled lasers systems. We resort to complex coupling of mixed dispersive and dissipative nature, which allows unprecedented control on the location in parameter space where the symmetry and symmetry breaking occur. Moreover, tuning the coupling in the same physical system allows us to realize the special cases of PT and anti-PT symmetries. In a more general perspective, we present and experimentally validate a new relation between laser synchronization and the symmetry of the underlying non-Hermitian Hamiltonian.

Chiral States in Coupled-Lasers Lattice by On-Site Complex Potential

The ability to control the chirality of physical devices is of great scientific and technological importance, from investigations of topologically protected edge states in condensed matter systems to wavefront engineering, isolation, and unidirectional communication. When dealing with large networks of oscillators, the control over the chirality of the bulk states becomes significantly more complicated and requires complex apparatus for generating asymmetric coupling or artificial gauge fields. Here we present a new approach for a precise control over the chirality of the bulk state of a triangular array of hundreds of symmetrically coupled lasers, by introducing a weak non-Hermitian complex potential, requiring only local on-site control of loss and frequency. In the unperturbed network, lasing supermodes with opposite chirality (staggered vortex and staggered antivortex) are equally probable. We show that by tuning the complex potential to an exceptional point, a nearly pure chiral lasing supermode is achieved. While our approach is applicable to any oscillators network, we demonstrate how the inherent nonlinearity of the lasers effectively pulls the network to the exceptional point, making the chirality extremely resilient against noise and imperfections.

Controlling Nonlinear Interaction in a Many-Mode Laser by Tuning Disorder

A many-mode laser with nonlinear modal interaction could serve as a model system to study many-body physics. However, precise and continuous tuning of the interaction strength over a wide range is challenging. Here, we present a unique method for controlling lasing mode structures by introducing random phase fluctuation to a nearly degenerate cavity. We show numerically and experimentally that as the characteristic scale of phase fluctuation decreases by two orders of magnitude, the transverse modes become fragmented and the reduction of their spatial overlap suppresses modal competition for gain, allowing more modes to lase. The tunability, flexibility, and robustness of our system provides a powerful platform for investigating many-body phenomena.

Experimental Observation of Phase Transitions in Spatial Photonic Ising Machine

Statistical spin dynamics plays a key role in understanding the working principle for novel optical Ising machines. Here, we propose the gauge transformation for a spatial photonic Ising machine, where a single spatial phase modulator simultaneously encodes spin configurations and programs interaction strengths. Using gauge transformation, we experimentally evaluate the phase diagram of a high-dimensional spin-glass equilibrium system with 100 fully connected spins. We observe the presence of paramagnetic, ferromagnetic, and spin-glass phases and determine the critical temperature T_{c} and the critical probability p_{c} of the phase transitions, which agree well with the mean-field theory predictions. Thus, the approximation of the mean-field model is experimentally verified in the spatial photonic Ising machine. Furthermore, we discuss the phase transition in parallel with solving combinatorial optimization problems during the cooling process and identify that the spatial photonic Ising machine is robust with sufficient many-spin interactions even when the system is associated with optical aberrations and measurement uncertainty.

Vortices in Nonequilibrium Photon Condensates

We present a theoretical study of vortices in arrays of photon condensates. Even when interactions are negligible, as is the case in current experiments, pumping and losses can lead to a finite vortex core size. While some properties of photon condensate vortices, such as their self-acceleration and the generation of vortex pairs by a moving vortex resemble those in lasers and interacting polariton condensates far from equilibrium, in several aspects they differ from previously studied systems: the vortex core size is determined by the balance between pumping and tunneling, the core appears oblate in the direction of its motion, and new vortex pairs can spontaneously nucleate in the core region.

Demonstration of chip-based coupled degenerate optical parametric oscillators for realizing a nanophotonic spin-glass

The need for solving optimization problems is prevalent in various physical applications, including neuroscience, network design, biological systems, socio-economics, and chemical reactions. Many of these are classified as non-deterministic polynomial-time hard and thus become intractable to solve as the system scales to a large number of elements. Recent research advances in photonics have sparked interest in using a network of coupled degenerate optical parametric oscillators (DOPOs) to effectively find the ground state of the Ising Hamiltonian, which can be used to solve other combinatorial optimization problems through polynomial-time mapping. Here, using the nanophotonic silicon-nitride platform, we demonstrate a spatial-multiplexed DOPO system using continuous-wave pumping. We experimentally demonstrate the generation and coupling of two microresonator-based DOPOs on a single chip. Through a reconfigurable phase link, we achieve both in-phase and out-of-phase operation, which can be deterministically achieved at a fast regeneration speed of 400 kHz with a large phase tolerance.

Improved Phase Locking of Laser Arrays with Nonlinear Coupling

An arrangement based on a degenerate cavity laser for forming an array of nonlinearly coupled lasers with an intracavity saturable absorber is presented. More than 30 lasers were spatially phase locked and temporally Q switched. The arrangement with nonlinear coupling was found to be 25 times more sensitive to loss differences and converged five times faster to the lowest loss phase locked state than with linear coupling, thus providing a unique solution to problems that have several near-degenerate solutions.