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A coherent Ising machine (CIM) is known to deliver the low-energy states of the Ising model. Here, we investigate how well the CIM simulates the thermodynamic properties of a two-dimensional square-lattice Ising model. Assuming that the spin sets sampled by the CIM can be regarded as a canonical ensemble, we estimate the effective temperature of the spins represented by degenerate optical parametric oscillator pulses by using maximum likelihood estimation. With the obtained temperature, we confirmed that the thermodynamic quantities obtained with the CIM exhibited a phase-transition-like behavior that better matches the analytical and numerical results than the mean field one., 6 pages, 3 figures

This letter reports on an ultra-fast and accurate channel assignment approach using a coherent Ising machine (CIM) for dense wireless LAN (WLAN) systems. CIM is a laser-based high-speed hardware approach for solving Ising problems using optoelectronic features. We formulate the total-throughput maximization problem for large-scale centralized WLAN systems as a combinatorial optimization problem. Subsequently, the problem is converted into an Ising problem to be optimized via a CIM. Simulation results reveal that the CIM can improve the total throughput performance and give results in a very short time (millisecond order).

Nonlinear dynamics of spiking neural networks has recently attracted much interest as an approach to understand possible information processing in the brain and apply it to artificial intelligence. Since information can be processed by collective spiking dynamics of neurons, the fine control of spiking dynamics is desirable for neuromorphic devices. Here we show that photonic spiking neurons implemented with paired nonlinear optical oscillators can be controlled to generate two modes of bio-realistic spiking dynamics by changing the optical pump amplitude. When they are coupled in a network, we found that the interaction between the photonic neurons induces an effective change in the pump amplitude depending on the order parameter that characterizes synchronization. The experimental results show that the effective change causes spontaneous modification of the spiking modes and firing rates of clustered neurons, and such collective dynamics can be utilized to realize efficient heuristics for solving NP-hard combinatorial optimization problems., Comment: 43 pages, 9 figures

We show that two-dimensionally coupled optical parametric oscillator pulses capture the signature of the exact phase transition of the two-dimensional Ising model rather than that of a mean-field one.

Description, We constructed a large optical computer based on optical parametric oscillators coupled via measurement and feedback., Computers based on physical systems are increasingly anticipated to overcome the impending limitations on digital computer performance. One such computer is a coherent Ising machine (CIM) for solving combinatorial optimization problems. Here, we report a CIM with 100,512 degenerate optical parametric oscillator pulses working as the Ising spins. We show that the CIM delivers fine solutions to maximum cut problems of 100,000-node graphs drastically faster than standard simulated annealing. Moreover, the CIM, when operated near the phase transition point, provides some extremely good solutions and a very broad distribution. This characteristic will be useful for applications that require fast random sampling such as machine learning.

The Potts model describes Ising-model-like interacting spin systems with multivalued spin component, and ground-state search problems of the Potts model can be efficiently mapped onto various integer optimization problems thanks to the rich expression of the multivalued spins. Here, we demonstrate a solver of this model based on hybrid computation using physical and digital architectures, wherein a digital computer updates the interaction matrices in the iterative calculations of the physical Ising-model solvers. This update of interactions corresponds to learning from the Ising solutions, which allows us to save resources when embedding a problem in a physical system. We experimentally solved integer optimization problems (graph coloring and graph clustering) with this hybrid architecture in which the physical solver consisted of coupled degenerate optical parametric oscillators.

Non-orthogonal multiple access (NOMA) can multiplex different users on a single channel by applying superposition coding at the transmitter and Successive Interference Cancellation (SIC) at the receiver. It is considered to be an important technology for next generation wireless communications to achieve high throughput and large-scale connectivity. To take advantage of the effectiveness of NOMA, in addition to pairing users using criteria based on distance and power allocation, a key question is how to optimally allocate resources such as channels to users. Previous studies have discussed user pairing schemes and power allocation algorithms for NOMA systems, but the key issue is how to allocate the channels. Coherent Ising Machine (CIM) is an Ising computer that can solve NP-hard combinatorial optimization problems at high speed. CIM can run optimization algorithms based on mutually coupled neural networks and can be applied to a variety of optimization problems. In this paper, we proposed a CIM-based optimization method for user pairing and channel allocation in NOMA systems. Simulation results show that the CIM-based optimization can achieve superior performance of the NOMA system.

We demonstrated a photonic spiking neuron that can represent the bio-realistic spiking modes of Class-I and Class-II neurons. In network experiments, we observed spontaneous modification of spiking mode depending on the synchronization of clustered neurons.

We report a coherent Ising machine (CIM) that can simulate the Ising model with external magnetic fields. To represent Ising spins, the CIM employs degenerate optical parametric oscillator (DOPO) pulses, which are coupled by a measurement-feedback scheme. By adding to a feedback signal a term that is proportional to a given magnetic field and an absolute value of the measured amplitude of a DOPO pulse, the networked DOPO pulses behave like Ising spins in a magnetic field. Using the CIM, we implement antiferromagnetically coupled one- and two-dimensional networks of the Ising spins in uniform magnetic fields. At the field amplitudes that agree with a theoretical prediction, we experimentally observe a transition from an antiferromagnetic state governed by spin-spin interactions to a ferromagnetic state induced by the external magnetic field. In addition, staggered magnetization measurement reveals that the CIM behaves differently depending on the network dimensionality, which is consistent with the results obtained with the analytical solution and Monte Carlo simulations, but cannot be reproduced with a simple mean-field theory. These results suggest that the CIM clearly simulates the behavior of Ising spin networks better than a mere mean-field solver and will thus become a useful tool for simulating various statistical physics problems based on the Ising model.

We present the first experimental measurement of the ensemble averages of both the kinetic and interaction energies of the three-dimensional Bose--Hubbard model at finite temperature and various optical lattice depths across weakly to strongly interacting regimes, for an almost unit filling factor. The kinetic energy is obtained through Fourier transformation of a time-of-flight signal, and the interaction energy is measured using a newly developed atom-number-projection spectroscopy technique, by exploiting an ultra-narrow optical transition of two-electron atoms. The obtained experimental results can be used as benchmarks for state-of-the-art numerical methods of quantum many-body theory. As an illustrative example, we compare the measured energies with numerical calculations involving the Gutzwiller and cluster-Gutzwiller approximations, assuming realistic trap potentials and particle numbers at nonzero entropy (finite temperature); we obtain good agreement without fitting parameters. We also discuss the possible application of this method to temperature estimations for atoms in optical lattices using the thermodynamic relation. This study offers a unique advantage of cold atom system for `quantum simulators', because, to the best of our knowledge, it is the first experimental determination of both the kinetic and interaction energies of quantum many-body system., 22 pages, 20 figures

The minimum requirements for an optical reservoir computer, a recent paradigm for computation using simple algorithms, are nonlinearity and internal interactions. A promising optical system satisfying these requirements is a platform based on coupled degenerate optical parametric oscillators (DOPOs) in a fiber ring cavity. We can expect advantages using DOPOs for reservoir computing with respect to scalability and reduction of excess noise; however, the continuous stabilization required for reservoir computing has not yet been demonstrated. Here, we report the continuous and long-term stabilization of an optical system by introducing periodical phase modulation patterns for DOPOs and a local oscillator. We observed that the Allan variance of the optical phase up to 100 ms was suppressed and that the homodyne measurement signal had a relative standard deviation of 1.4% over 62,500 round trips. The proposed methods represent important technical bases for realizing stable computation on large-scale optical hybrid computers.

Various artificial spin systems with networked physical devices are now emerging as a way to simulate the Ising model. Combinatorial optimization problems for many applications can be mapped onto ground state search problems of the Ising model and solved efficiently with the artificial spin system. Here, we report a coherent Ising machine based on 2,048 networked degenerate optical parametric oscillators. All-to-all couplings were achieved with a measurement and feedback scheme to implement arbitrary graph structures with up to 2,048 nodes. The coherent Ising machine can simulate large-scale Ising model and obtain good solutions for the optimization of a 2,000-node complete graph.

Physical annealing systems provide heuristic approaches to solving NP-hard Ising optimization problems. Here, we study the performance of two types of annealing machines--a commercially available quantum annealer built by D-Wave Systems, and measurement-feedback coherent Ising machines (CIMs) based on optical parametric oscillator networks--on two classes of problems, the Sherrington-Kirkpatrick (SK) model and MAX-CUT. The D-Wave quantum annealer outperforms the CIMs on MAX-CUT on regular graphs of degree 3. On denser problems, however, we observe an exponential penalty for the quantum annealer ($\exp(-\alpha_\textrm{DW} N^2)$) relative to CIMs ($\exp(-\alpha_\textrm{CIM} N)$) for fixed anneal times, on both the SK model and on 50%-edge-density MAX-CUT, where the coefficients $\alpha_\textrm{CIM}$ and $\alpha_\textrm{DW}$ are problem-class-dependent. On instances with over $50$ vertices, a several-orders-of-magnitude time-to-solution difference exists between CIMs and the D-Wave annealer. An optimal-annealing-time analysis is also consistent with a significant projected performance difference. The difference in performance between the sparsely connected D-Wave machine and the measurement-feedback facilitated all-to-all connectivity of the CIMs provides strong experimental support for efforts to increase the connectivity of quantum annealers., Comment: 12 pages, 5 figures, 1 table (main text); 14 pages, 12 figures, 2 tables (supplementary)

Today, world-wide efforts are being undertaken to explore non-von Neumann computation methods to overcome “the end of Moore’s law”. A promising approach is the coherent Ising machine (CIM), which u...

A Bose–Einstein condensate was achieved in a stable magnetic trap on a persistent-supercurrent atom chip with a superconducting closed-loop circuit. We determined precisely the shape of the magnetic trapping potential by systematically controlling the persistent supercurrent. The condensation was verified by time-of-flight imaging and by atom number decay measurements. The measured decay rates agreed quantitatively with numerical simulations on the three-body loss process assuming all of the atoms to be a condensate. We also discuss the feasibility of creating a quasi-one-dimensional Bose gas on our atom chip.

We describe a large-scale degenerate optical parametric oscillator (DOPO) network for a coherent Ising machine that solves combinatorial optimization problems. By pumping a fiber-based phase-sensitive amplifier placed in a 1-km fiber cavity at a 2-GHz repetition frequency, we generated more than 10,000 DOPOs multiplexed in the time domain. We confirmed that the DOPO phases were discretized to {0, π} indicating that they could be used as stable artificial spins. We also implemented a one-dimensional Ising model by optically coupling adjacent DOPOs, and confirmed that the DOPOs well simulated the behavior of low-temperature spins.

Many-body effects are at the very heart of diverse phenomena found in condensed-matter physics. One striking example is the Mott insulator phase where conductivity is suppressed as a result of a strong repulsive interaction. Advances in cold atom physics have led to the realization of the Mott insulating phases of atoms in an optical lattice, mimicking the corresponding condensed matter systems. Here, we explore an exotic strongly-correlated system of Interacting Dual Mott Insulators of bosons and fermions. We reveal that an inter-species interaction between bosons and fermions drastically modifies each Mott insulator, causing effects that include melting, generation of composite particles, an anti-correlated phase, and complete phase-separation. Comparisons between the experimental results and numerical simulations indicate intrinsic adiabatic heating and cooling for the attractively and repulsively interacting dual Mott Insulators, respectively.

We investigate the finite-temperature properties of attractive three-component (colors) fermionic atoms in optical lattices using a self-energy functional approach. As the strength of the attractive interaction increases in the low temperature region, a second-order transition occurs from a Fermi liquid to a color superfluid (CSF). In the strong attractive region, a first-order transition occurs from a CSF to a trionic state. In the high temperature region, a crossover between a Fermi liquid and a trionic state is observed with increasing the strength of the attractive interaction. The crossover region for fixed temperature is almost independent of filling., Comment: 8 pages, 5 figures, Final version, Mod. Phys. Lett. B 25, 987 (2011)

Simulating a network of Ising spins with physical systems is now emerging as a promising approach for solving mathematically intractable problems. Here we report a large-scale network of artificial spins based on degenerate optical parametric oscillators (DOPO), paving the way towards a photonic Ising machine capable of solving difficult combinatorial optimization problems. We generated >10,000 time-division-multiplexed DOPOs using dual-pump four-wave mixing (FWM) in a highly nonlinear fibre (HNLF) placed in a fibre cavity. Using those DOPOs, a one-dimensional (1D) Ising model was simulated by introducing nearest-neighbour optical coupling. We observed the formation of spin domains and found that the domain size diverged near the DOPO threshold, which suggests that the DOPO network can simulate the behavior of low-temperature Ising spins., 6 pages, 3 figures. First draft in Nature Photonics (2016)

A system of ultracold atoms in an optical lattice has been regarded as an ideal quantum simulator for a Hubbard model with extremely high controllability of the system parameters. While making use of the controllability, a comprehensive measurement across the weakly to strongly interacting regimes in the Hubbard model to discuss the quantum many-body state is still limited. Here we observe a great change in the excitation energy spectra across the two regimes in an atomic Bose–Hubbard system by using a spectroscopic technique, which can resolve the site occupancy in the lattice. By quantitatively comparing the observed spectra and numerical simulations based on sum rule relations and a binary fluid treatment under a finite temperature Gutzwiller approximation, we show that the spectra reflect the coexistence of a delocalized superfluid state and a localized insulating state across the two regimes., A system of ultracold atoms in an optical lattice can be used as a quantum simulator for the Hubbard model with high controllability. Here, the authors report a laser spectroscopy study of an ytterbium ultracold bosonic gas across the weakly to strongly interacting regime.

We provide a numerical method to calculate comprehensively the microwave and the laser spectra of ultracold bosonic atoms in optical lattices at finite temperatures. Our formulation is built up with the sum rules, up to the second order, derived from the general principle of spectroscopy. The sum rule approach allows us to discuss the physical origins of a spectral peak shift and also a peak broadening. We find that a spectral broadening of superfluid atoms can be determined from number fluctuations of atoms, while that of normal-state atoms is mainly attributed to quantum fluctuations resulting from hopping of atoms. To calculate spectra at finite temperatures, based on the sum rule approach, we provide a two-mode approximation assuming that spectra of the superfluid and normal state atoms can be calculated separately. Our method can properly deal with multi-peak structures of spectra resulting from thermal fluctuations and also coexisting of the superfluid and the normal states. By combining the two-mode approximation with a finite temperature Gutzwiller approximation, we calculate spectra at finite temperatures by considering realistic systems, and the calculated spectra show nice agreements with those in experiments., 12 pages, 4 figures

We describe the enhanced magnetic transition temperatures $T_c$ of two-component fermions in three-dimensional layered Lieb lattices, which are created in cold atom experiments. We determine the phase diagram at half-filling using the dynamical mean-field theory. The dominant mechanism of enhanced $T_c$ gradually changes from the (delta-functional) flat-band to the (logarithmic) Van Hove singularity as the interlayer hopping increases. We elucidate that the interaction induces an effective flat-band singularity from a dispersive flat (or narrow) band. We offer a general analytical framework for investigating the singularity effects, where a singularity is treated as one parameter in the density of states. This framework provides a unified description of the singularity-induced phase transitions, such as magnetism and superconductivity, where the weight of the singularity characterizes physical quantities. This treatment of the flat-band provides the transition temperature and magnetization as a universal form (i.e., including the Lambert function). We also elucidate a specific feature of the magnetic crossover in magnetization at finite temperatures., 7 pages, 4 figures, to appear in PRA

We investigate the Mott transitions in two-orbital Hubbard systems. Applying the dynamical mean field theory and the self-energy functional approach, we discuss the stability of itinerant quasi-particle states in each band. It is shown that separate Mott transitions occur at different Coulomb interaction strengths in general. On the other hand, if some special conditions are satisfied for the interactions, spin and orbital fluctuations are equally enhanced at low temperatures, resulting in a single Mott transition. The phase diagrams are obtained at zero and finite temperatures. We also address the effect of the hybridization between two orbitals, which induces the Kondo-like heavy fermion states in the intermediate orbital-selective Mott phase., 21 Pages, 17 Figures, to appear in Progress of Theoretical Physics (YKIS2004 Proceedings)

On the basis of a resonating valence-bond description, we investigate the excitation spectrum caused by the Raman process in cuprate two-leg spin ladders with a cyclic four-spin interaction. Owing to a cyclic four-spin interaction, the S = 0 three-triplet excitation energy is reduced and becomes smaller than the S = 0 two-triplet excitation energy around the end of the Brillouin zone. Possible interpretations for the change in the lowest-lying mode are discussed. We show that such a change in the lowest-lying mode yields the shoulder structure at the lower edge in the phonon-assisted optical absorption spectrum.

We theoretically study magnetic properties of two-component cold fermions in half-filled multilayer Lieb optical lattices, i.e., two, three, and several layers, using the dynamical mean-field theory. We clarify that the magnetic properties of this system become quite different depending on whether the number of layers is odd or even. In odd-number-th layers in an odd-number-layer system, finite magnetization emerges even with an infinitesimal interaction. This is a striking feature of the flatband ferromagnetic state in multilayer systems as a consequence of the Lieb theorem. In contrast, in even-number layers, magnetization develops from zero on a finite interaction. These different magnetic behaviours are triggered by the flat bands in the local density of states and become identical in the limit of the infinite-layer (i.e., three-dimensional) system. We also address how interlayer hopping affects the magnetization process. Further, we point out that layer magnetization, which is a population imbalance between up and down atoms on a layer, can be employed to detect the emergence of the flat-band ferromagnetic state without addressing sublattice magnetization., 11 pages, 9 figures

We investigate an infinite-dimensional two-orbital Hubbard model with different bandwidths by means of the self-energy functional approach. By calculating the quasi-particle weight and the entropy at finite temperatures, we discuss how thermal fluctuations affect the stability of an intermediate phase, called the orbital-selective Mott phase, which has localized as well as itinerant bands.

In this study, we propose a method for generating cluster states of atoms in an optical lattice. By utilizing the quantum properties of Wannier orbitals, we create an tunable Ising interaction between atoms without inducing the spin-exchange interactions. We investigate the cause of errors that occur during entanglement generations, and then we propose an error-management scheme, which allows us to create high-fidelity cluster states in a short time.

We review our theoretical analysis of repulsively interacting three-component fermionic atoms in optical lattices. We discuss quantum phase transitions at around half filling with a balanced population by focusing on Mott transitions, staggered ordering, and superfluidity. At half filling (with 3/2 atoms per site), characteristic Mott transitions are induced by the anisotropic interactions, where two-particle repulsions between any two of the three colors have different strengths. At half filling, two types of staggered ordered states appear at low temperatures depending on the anisotropy of the interactions. As the temperature increases, phase transitions occur from the staggered ordered states to the unordered Mott states. Deviating from half filling, an exotic superfluid state appears close to a regime in which the Mott transition occurs. We explain the origin of these phase transitions and present the finite-temperature phase diagrams., Comment: 24 pages, 16 figures

A network of optical parametric oscillators is used to simulate classical Ising and XY spin chains. The collective nonlinear dynamics of this network, driven by quantum noise rather than thermal fluctuations, seeks out the Ising / XY ground state as the system transitions from below to above the lasing threshold. We study the behavior of this "Ising machine" for three canonical problems: a 1D ferromagnetic spin chain, a 2D square lattice, and problems where next-nearest-neighbor couplings give rise to frustration. If the pump turn-on time is finite, topological defects form (domain walls for the Ising model, winding number and vortices for XY) and their density can be predicted from a numerical model involving a linear "growth stage" and a nonlinear "saturation stage". These predictions are compared against recent data for a 10,000-spin 1D Ising machine., 34 pages, 15 figures. Preprint of an article submitted for consideration in International Journal of Modern Physics B. [Copyright 2016 World Scientific Publishing Company] [www.worldscientific.com/worldscinet/ijmpb]

We propose a method for generating high-fidelity multipartite spin-entanglement of ultracold atoms in an optical lattice in a short operation time with a scalable manner, which is suitable for measurement-based quantum computation. To perform the desired operations based on the perturbative spin-spin interactions, we propose to actively utilize the extra degrees of freedom (DOFs) usually neglected in the perturbative treatment but included in the Hubbard Hamiltonian of atoms, such as, (pseudo-)charge and orbital DOFs. Our method simultaneously achieves high fidelity, short operation time, and scalability by overcoming the following fundamental problem: enhancing the interaction strength for shortening operation time breaks the perturbative condition of the interaction and inevitably induces unwanted correlations among the spin and extra DOFs., 9 pages, 11 figures

We study the ground state properties of spin-1 bosons in a two-dimensional optical lattice, by applying a variational Monte Carlo method to the S=1 Bose-Hubbard model on a square lattice at unit filling. A doublon-holon binding factor introduced in the trial state provides a noticeable improvement in the variational energy over the conventional Gutzwiller wave function and allows us to deal effectively with the inter-site correlations of particle densities and spins. We systematically show how spin-dependent interactions modify the superfluid-Mott insulator transitions in the S=1 Bose-Hubbard model due to the interplay between the density and spin fluctuations of bosons. Furthermore, regarding the magnetic phases in the Mott region, the calculated spin structure factor elucidates the emergence of nematic and ferromagnetic spin orders for antiferromagnetic ($U_2>0$) and ferromagnetic ($U_2, 5 pages, 5 figures, to appear in Journal of the Physical Society of Japan

We investigate the superfluid state of repulsively interacting three-component (color) fermionic atoms in optical lattices. When the anisotropy of the three repulsive interactions is strong, atoms of two of the three colors form Cooper pairs and atoms of the third color remain a Fermi liquid. An effective attractive interaction is induced by density fluctuations of the third-color atoms. This superfluid state is stable against changes in filling close to half filling. We determine the phase diagrams in terms of temperature, filling, and the anisotropy of the repulsive interactions., 5 pages, 6 figures