Enhancing the performance of coherent Ising machines in the large-noise regime with a fifth-order nonlinearity.

Coherent Ising machines (CIMs), leveraging the bistable physical properties of coherent light to emulate Ising spins, exhibit great potential as hardware accelerators for tackling complex combinatorial optimization problems. Recent advances have demonstrated that the performance of CIMs can be enhanced either by incorporating large random noise or higher-order nonlinearities, yet their combined effects on CIM performance remain mainly unexplored. In this work, we develop a numerical CIM model that utilizes a tunable fifth-order polynomial nonlinear dynamic function under large noise levels, which has the potential to be implemented in all-optical platforms. We propose a normal form of a CIM model that allows for both supercritical and subcritical pitchfork bifurcation operational regimes, with fifth-order nonlinearity and tunable hyperparameters to control the Ising spin dynamics. In the benchmark studies, we simulate various sets of MaxCut problems using our fifth-order polynomial CIM model. The results show a significant performance improvement, achieving an average of 59.5% improvement in median time-to-solution (TTS) and an average of 6 times improvement in median success rate (SR) for dense Maxcut problems in the BiqMac library, compared to the commonly used third-order polynomial CIM model with low noise. The fifth-order polynomial CIM model in the large-noise regime also shows better performance trends as the problem size scales up. These findings reveal the enhancements on the computational performance of Ising machines in the large-nose regime from fifth-order nonlinearity, showing important implications for both simulation and hardware perspectives.

Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning.

Ising machines are a promising non-von-Neumann computational concept for neural network training and combinatorial optimization. However, while various neural networks can be implemented with Ising machines, their inability to perform fast statistical sampling makes them inefficient for training neural networks compared to digital computers. Here, we introduce a universal concept to achieve ultrafast statistical sampling with analog Ising machines by injecting noise. With an opto-electronic Ising machine, we experimentally demonstrate that this can be used for accurate sampling of Boltzmann distributions and for unsupervised training of neural networks, with equal accuracy as software-based training. Through simulations, we find that Ising machines can perform statistical sampling orders-of-magnitudes faster than software-based methods. This enables the use of Ising machines beyond combinatorial optimization and makes them into efficient tools for machine learning and other applications.

A poor man's coherent Ising machine based on opto-electronic feedback systems for solving optimization problems.

Coherent Ising machines (CIMs) constitute a promising approach to solve computationally hard optimization problems by mapping them to ground state searches of the Ising model and implementing them with optical artificial spin-networks. However, while CIMs promise speed-ups over conventional digital computers, they are still challenging to build and operate. Here, we propose and test a concept for a fully programmable CIM, which is based on opto-electronic oscillators subjected to self-feedback. Contrary to current CIM designs, the artificial spins are generated in a feedback induced bifurcation and encoded in the intensity of coherent states. This removes the necessity for nonlinear optical processes or large external cavities and offers significant advantages regarding stability, size and cost. We demonstrate a compact setup for solving MAXCUT optimization problems on regular and frustrated graphs with 100 spins and can report similar or better performance compared to CIMs based on degenerate optical parametric oscillators.

Understanding dynamics of coherent Ising machines through simulation of large-scale 2D Ising models.

Many problems in mathematics, statistical mechanics, and computer science are computationally hard but can often be mapped onto a ground-state-search problem of the Ising model and approximately solved by artificial spin-networks of coupled degenerate optical parametric oscillators (DOPOs) in coherent Ising machines. To better understand their working principle and optimize their performance, we analyze the dynamics during the ground state search of 2D Ising models with up to 1936 mutually coupled DOPOs. For regular as well as frustrated and disordered 2D lattices, the machine finds the correct solution within just a few milliseconds. We determine that calculation performance is limited by freeze-out effects and can be improved by controlling the DOPO dynamics, which allows to optimize performance of coherent Ising machines in various tasks. Comparisons with Monte Carlo simulations reveal that coherent Ising machines behave like low temperature spin systems, thus making them suitable for optimization tasks.

Anion dependent ion pairing in concentrated ytterbium halide solutions.

We have studied ion pairing of ytterbium halide solutions. THz spectra (30-400 cm-1) of aqueous YbCl3 and YbBr3 solutions reveal fundamental differences in the hydration structures of YbCl3 and YbBr3 at high salt concentrations: While for YbBr3 no indications for a changing local hydration environment of the ions were experimentally observed within the measured concentration range, the spectra of YbCl3 pointed towards formation of weak contact ion pairs. The proposed anion specificity for ion pairing was confirmed by supplementary Raman measurements.

Mapping Hydration Water around Alcohol Chains by THz Calorimetry.

THz spectroscopy was used to probe changes that occur in the dynamics of the hydrogen bond network upon solvation of alcohol chains. The THz spectra can be decomposed into the spectrum of bulk water, tetrahedral hydration water, and more disordered (or interstitial) hydration water. The tetrahedrally ordered hydration water exhibits a band at 195 cm-1 and is localized around the hydrophobic moiety of the alcohol. The interstitial component yields a band at 164 cm-1 which is associated with hydration water in the first hydration shell. These temperature-dependent changes in the low-frequency spectrum of solvated alcohol chains can be correlated with changes of heat capacity, entropy, and free energy upon solvation. Surprisingly, not the tetrahedrally ordered component but the interstitial hydration water is found to be mainly responsible for the temperature-dependent change in ΔCp and ΔG. The solute-specific offset in free energy is attributed to void formation and scales linearly with the chain length.

Effects of multivalent hexacyanoferrates and their ion pairs on water molecule dynamics measured with terahertz spectroscopy.

The valency of aqueous solutes plays a large role in determining the extent of ion-water dynamics, which can greatly influence the chemical and physical properties of solutions. In these experiments, broadband Fourier transform terahertz spectroscopy is used to probe perturbations to the low-frequency dynamics of water molecules by three different multivalent hexacyanoferrate salts. K3Fe(CN)6, K4Fe(CN)6 and Na4Fe(CN)6 were investigated as a function of concentration up to their solubility limits using spectral subtractions and fitting with damped harmonic lineshapes. Regions with subtle nonlinearities in amplitude with respect to solute concentration provide insight into ion-pairing events. The extent of nonlinearity suggests that ion pairs are major constituents in solution for all concentrations measured and is consistent with ion-pairing observed at millimolar concentrations by potentiometric and spectroscopic measurements. A lower estimate for the number of water molecules that are influenced by each ion is obtained from the damped harmonic fits. Values of 19, 28 and 25 water molecules with perturbed dynamics are obtained for KFe(CN)62-, KFe(CN)63- and NaFe(CN)63- ion pairs, respectively. These values represent dynamical perturbations into a second solvation shell and are consistent with the long-range structural effects observed in recent aqueous nanodrop spectroscopy experiments. Furthermore, the spectral absorptions for hexacyanoferrates are in agreement with a wide range of solutes studied previously using the developing methodology for interpreting terahertz spectra.

Small chimera states without multistability in a globally delay-coupled network of four lasers.

We present results obtained for a network of four delay-coupled lasers modeled by Lang-Kobayashi-type equations. We find small chimera states consisting of a pair of synchronized lasers and two unsynchronized lasers. One class of these small chimera states can be understood as intermediate steps on the route from synchronization to desynchronization, and we present the entire chain of bifurcations giving birth to them. This class of small chimeras can exhibit limit-cycle or quasiperiodic dynamics. A second type of small chimera states exists apparently disconnected from any region of synchronization, arising from pair synchronization inside the chaotic desynchronized regime. In contrast to previously reported chimera states in globally coupled networks, we find that the small chimera state is the only stable solution of the system for certain parameter regions; i.e., we do not need to specially prepare initial conditions.

The low frequency modes of solvated ions and ion pairs in aqueous electrolyte solutions: iron(ii) and iron(iii) chloride.

We have investigated the hydration dynamics of solvated iron(ii) and iron(iii) chloride. For this, THz/FIR absorption spectra of acidified aqueous FeCl2 and FeCl3 solutions have been measured in a frequency range of 30-350 cm(-1) (≈1-10 THz). We observe a nonlinear concentration dependence of the absorption, which is attributed to the progressive formation of chloro-complexes of Fe(ii) and Fe(iii), respectively. By principal component analysis of the concentration dependent absorption spectra, we deduced the molar extinction spectra of the solvated species Fe(2+) + 2Cl(-) and FeCl(+) + Cl(-), as well as FeCl(2+) + 2Cl(-) and FeCl2(+) + Cl(-). In addition, we obtain ion association constants log KFeCl2 = -0.88(5) and log KFeCl3 = -0.32(16) for the association of Fe(2+) and Cl(-) to FeCl(+) and the association of FeCl(2+) and Cl(-) to FeCl2(+), respectively. We performed a simultaneous fit of all the effective extinction spectra and their differences, including our previous results of solvated manganese(ii) and nickel(ii) chlorides and bromides. Thereby we were able to assign absorption peaks to vibrational modes of ion-water complexes. Furthermore, we were able to estimate a minimum number of affected water molecules, ranging from ca. 7 in the case of FeCl(+) + Cl(-) to ca. 21 in the case of FeCl(2+) + Cl(-).

Amplitude-phase coupling drives chimera states in globally coupled laser networks.

For a globally coupled network of semiconductor lasers with delayed optical feedback, we demonstrate the existence of chimera states. The domains of coherence and incoherence that are typical for chimera states are found to exist for the amplitude, phase, and inversion of the coupled lasers. These chimera states defy several of the previously established existence criteria. While chimera states in phase oscillators generally demand nonlocal coupling, large system sizes, and specially prepared initial conditions, we find chimera states that are stable for global coupling in a network of only four coupled lasers for random initial conditions. The existence is linked to a regime of multistability between the synchronous steady state and asynchronous periodic solutions. We show that amplitude-phase coupling, a concept common in different fields, is necessary for the formation of the chimera states.

Publisher's Note: Amplitude-phase coupling drives chimera states in globally coupled laser networks [Phys. Rev. E 91, 040901(R) (2015)].

This corrects the article DOI: 10.1103/PhysRevE.91.040901.

The low frequency motions of solvated Mn(II) and Ni(II) ions and their halide complexes.

We have investigated the low frequency (30-350 cm(-1)) spectra of solvated MnCl2, MnBr2, NiCl2, and NiBr2. Using a chemical equilibrium model in combination with principal component analysis, we were able to dissect the spectra into molar extinction coefficients due to the solvated ions and - for MnCl2, MnBr2, and NiCl2- to extract information on the ion pair spectra. The deduced anion spectra (calculated as MnCl2-MnBr2 and NiCl2-NiBr2) are very similar and nearly identical to the anion spectra observed for LaCl3-LaBr3. The differences between the cationic contributions MnCl2-NiCl2 and MnBr2-NiBr2 indicate that the solvated cation spectra can be understood in terms of distinct resonances of the octahedrally solvated cation complex that are red-shifted for Mn(2+) compared to Ni(2+). The description of the full extinction spectra requires the introduction of an additional resonance at a center frequency of around 130 cm(-1) that we tentatively assign to hydration water. Cooperative effects are small and are reflected in a change in the band intensity. However, the center frequencies of the observed modes remain unchanged when exchanging the counter ion. Analysis of the ion pair extinction spectra supports contact ion pair formation for MnBr2 and NiCl2 and MnCl2.

From solvated ions to ion-pairing: a THz study of lanthanum(III) hydration.

Ion radius and charge density are important parameters that determine the solvation behavior in aqueous electrolyte solutions. Here, we report on high precision THz absorption measurements of solvated LaCl3 and LaBr3 using narrow-band (75-90 cm(-1)) p-Ge laser and wideband (30-350 cm(-1)) Fourier transform spectroscopy. The concentration dependent absorption up to 3.3 M shows a prominent nonlinearity indicating ion pair formation with increasing electrolyte concentration. A more detailed analysis in terms of a chemical equilibrium model allowed us to separate the ion and ion pair contributions from bulk and solvation water. Thus we were able to characterize anion and cation solvation independently. The center frequencies of the Cl(-) and Br(-) rattling modes are in agreement with those found in aqueous alkali and earth alkali halide solutions. The coupling between anion and cation hydration is found to be small. Based upon our detailed analysis we propose increasing formation of solvent shared ion pairs with increasing solute concentration. The well defined ion resonances imply that in spite of its high charge density La(3+) acts locally on the water structure. Terahertz absorption spectroscopy is found here to be an experimental tool which allows us to directly observe solute hydration shells as well as ion pair formation.