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We analytically and numerically show that the Hillery-Zubairy’s entanglement criterion is satisfied both below and above the threshold of coupled non-degenerate optical parametric oscillators (NOPOs) with strong nonlinear gain saturation and dissipative linear coupling. We investigated two cases: for large pump mode dissipation, below-threshold entanglement is possible only when the parametric interaction has an enough detuning among the signal, idler, and pump photon modes. On the other hand, for a large dissipative coupling, below-threshold entanglement is possible even when there is no detuning in the parametric interaction. In both cases, a non-Gaussian state entanglement criterion is satisfied even at the threshold. Recent progress in nano-photonic devices might make it possible to experimentally demonstrate this phase transition in a coherent XY machine with quantum correlations.

The low-lying dipole strength in the 90,94Zr nuclei was investigated via (p,p′γ) at 80 MeV and (α,α′γ) at 130 MeV. The experiments, made at RCNP, used the magnetic spectrometer Grand Raiden for the scattered particles and the array CAGRA with HPGe detectors for the γ-decay. For 94Zr these are the first data for both reactions and for 90Zr these are the first data with (p,p′γ) and the first ones at high resolution for (α,α′γ). The comparison of the present results for the two nuclei with existing (γ,γ′) data shows that both nuclear probes produce an excitation pattern different than that of the electromagnetic probes. DWBA calculations were made using form factors deduced from transition densities, based on RPA calculations, characterized by a strong neutron component at the nuclear surface. A combined analysis of the two reactions was performed for the first time to investigate the isoscalar character of the 1− states in 90,94Zr. The (p,p′γ) cross section was calculated using values for the isoscalar electric dipole energy-weighted sum rule (E1 ISEWSR) obtained from the (α,α′γ) data. The isoscalar strength for 90Zr was found to exhaust 20 ± 2.5% of the EWSR in the energy range up to 12 MeV. In case of 94Zr, a strength of 9 ± 1.1% of the EWSR was found in the range up to 8.5 MeV. Although an overall general description was obtained in the studied energy intervals, not all proton cross sections were well reproduced using the isoscalar strength from (α,α′γ). This might suggest mixing of isoscalar and isovector components and that this mixing and the degree of collectivity are not the same for all the 1− states below the particle binding energy.

We propose a network of open-dissipative quantum oscillators with optical error correction circuits. In the proposed network, the squeezed/anti-squeezed vacuum states of the constituent optical parametric oscillators below the threshold establish quantum correlations through optical mutual coupling, while collective symmetry breaking is induced above the threshold as a decision-making process. This initial search process is followed by a chaotic solution search step facilitated by the optical error correction feedback. As an optical hardware technology, the proposed coherent Ising machine (CIM) has several unique features, such as programmable all-to-all Ising coupling in the optical domain, directional coupling $(J_{ij} \neq J_{ji})$ induced chaotic behavior, and low power operation at room temperature. We study the performance of the proposed CIMs and investigate how the performance scales with different problem sizes. The quantum theory of the proposed CIMs can be used as a heuristic algorithm and efficiently implemented on existing digital platforms. This particular algorithm is derived from the truncated Wigner stochastic differential equation. We find that the various CIMs discussed are effective at solving many problem types, however the optimal algorithm is different depending on the instance. We also find that the proposed optical implementations have the potential for low energy consumption when implemented optically on a thin film LiNbO3 platform.

We present analytical and numerical simulation results for squeezing, entanglement, and quantum discord in a dissipatively coupled coherent Ising machine (CIM). Both analytical solutions and numerical simulation results, which are obtained with positive-$P$, truncated-Wigner, and truncated-Husimi representations for the density operator, predict the presence of entanglement and quantum discord, below and above the threshold of CIM. The entanglement criteria and quantum discord are evaluated as a function of the dissipative coupling strength relative to the background loss. For coupled two DOPOs, while entanglement disappears as the background loss exceeds the Ising coupling strength, the quantum discord remains finite even with a large linear loss. For one-dimensional lattice of DOPOs, while entanglement disappears for DOPO pair with large distance, quantum discord remains finite.

A coherent Ising machine (CIM) is a network of optical parametric oscillators (OPOs), in which the "strongest" collective mode of oscillation at well above threshold corresponds to an optimum solution of a given Ising problem. When a pump rate or network coupling rate is increased from below to above threshold, however, the eigenvectors with a smallest eigenvalue of Ising coupling matrix [J_ij] appear near threshold and impede the machine to relax to true ground states. Two complementary approaches to attack this problem are described here. One approach is to utilize squeezed/anti-squeezed vacuum noise of OPOs below threshold to produce coherent spreading over numerous local minima via quantum noise correlation, which could enable the machine to access either true ground states or excited states with eigen-energies close enough to that of ground states above threshold. The other approach is to implement real-time error correction feedback loop so that the machine migrates from one local minimum to another during an explorative search for ground states. Finally, a set of qualitative analogies connecting the CIM and traditional computer science techniques are pointed out. In particular, belief propagation and survey propagation used in combinatorial optimization are touched upon., Comment: 32 pages and 7 figures

The relaxation of binary spins to analog values has been the subject of much debate in the field of statistical physics, neural networks, and more recently quantum computing, notably because the benefits of using an analog state for finding lower energy spin configurations are usually offset by the negative impact of the improper mapping of the energy function that results from the relaxation. We show that it is possible to destabilize trapping sets of analog states that correspond to local minima of the binary spin Hamiltonian by extending the phase space to include error signals that correct amplitude inhomogeneity of the analog spin states and controlling the divergence of their velocity. Performance of the proposed analog spin system in finding lower energy states is competitive against state-of-the-art heuristics., Revised version. Supplementary material included

A correction to this article has been published and is linked from the HTML version of this article.

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)

We benchmark the OPO-based Coherent Ising Machine (CIM) against the D-Wave quantum annealer for Ising problems up to size N = 200. The CIM exhibits exponential speedup relative to D-Wave for Ising problems with dense connectivity.

We will present the basic concept, operational principle and experimental performance of a novel computing machine with the network of degenerate optical parametric oscillators. The developed machine has 2048 qubits with all-to-all connections.

Recently, the coherent Ising machine (CIM) as a degenerate optical parametric oscillator (DOPO) network has been researched to solve Ising combinatorial optimization problems. We formulate a theoretical model for the CIM with discrete-time measurement feedback processes, and perform numerical simulations for the simplest network, composed of two degenerate optical parametric oscillator pulses with the anti-ferromagnetic mutual coupling. We evaluate the extent to which quantum coherence exists during the optimization process.

The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive $P({\alpha},{\beta})$ representation of the density operator and the master equation. The theory is composed of the c-number stochastic differential equations for describing open dissipative quantum dynamics and the replicator dynamics equations for handling measurement-induced collapse of the density operator. We apply the present theory to simulate two simple Ising spin models and elucidate the unique features of this computing machine., Comment: 9 pages, 5 figures

We present an experimental scheme of implementing multiple spins in a classical XY model using a non-degenerate optical parametric oscillator (NOPO) network. We built an NOPO network to simulate a one-dimensional XY Hamiltonian with 5000 spins and externally controllable effective temperatures. The XY spin variables in our scheme are mapped onto the phases of multiple NOPO pulses in a single ring cavity and interactions between XY spins are implemented by mutual injections between NOPOs. We show the steady-state distribution of optical phases of such NOPO pulses is equivalent to the Boltzmann distribution of the corresponding XY model. Estimated effective temperatures converged to the setting values, and the estimated temperatures and the mean energy exhibited good agreement with the numerical simulations of the Langevin dynamics of NOPO phases., 15 pages, 6+2 figures

In this paper we introduce and analyze a system design for quantum-dot-based qubits that simultaneously supports scalable one-qubit and two-qubit gates, and single-shot qubit measurement. All three key processes (one-qubit gates, two-qubit gates, and qubit measurement) rely on the interaction between the electron in each quantum dot and exciton polaritons formed in a quantum well situated near the quantum dots. A key feature of our proposed system is the use of polariton traps, which we show enhances the quantum-dot--quantum-well interaction by a factor of 10 and consequently results in $100\ifmmode\times\else\texttimes\fi{}$ faster two-qubit gates. We also introduce a one-qubit gate that is based on a combination of optical and microwave control, which is supported in the same device and system configuration as the other operations, in contrast to the conventional one-qubit gate that is based on all-optical control.

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.

We use a network of 160 optical parametric oscillators, formed using fiber-coupled PPLN in a ~320-m fiber loop, to build an annealing machine that can find solutions to Ising problems with arbitrary spin connectivity.

A quantum simulator is a purposeful quantum machine that can address complex quantum problems in a controllable setting and an efficient manner. This chapter introduces a solid-state quantum simulator platform based on exciton-polaritons, which are hybrid light-matter quantum quasi-particles. We describe the physical realization of an exciton-polariton quantum simulator in semiconductor materials (hardware) and discuss a class of problems, which the exciton-polariton quantum simulators can address well (software). A current status of the experimental progress in building the quantum machine is reviewed, and potential applications are considered.

Recently a new type of system exhibiting spontaneous coherence has emerged -- the exciton-polariton condensate. Exciton-polaritons (or polaritons for short) are bosonic quasiparticles that exist inside semiconductor microcavities, consisting of a superposition of an exciton and a cavity photon. Above a threshold density the polaritons macroscopically occupy the same quantum state, forming a condensate. The lifetime of the polaritons are typically comparable to or shorter than thermalization times, making them possess an inherently non-equilibrium nature. Nevertheless, they display many of the features that would be expected of equilibrium Bose-Einstein condensates (BECs). The non-equilibrium nature of the system raises fundamental questions of what it means for a system to be a BEC, and introduces new physics beyond that seen in other macroscopically coherent systems. In this review we focus upon several physical phenomena exhibited by exciton-polariton condensates. In particular we examine topics such as the difference between a polariton BEC, a polariton laser, and a photon laser, as well as physical phenomena such as superfluidity, vortex formation, BKT (Berezinskii-Kosterlitz-Thouless) and BCS (Bardeen-Cooper-Schrieffer) physics. We also discuss the physics and applications of engineered polariton structures., 15 pages, 5 figures. Unrefereed version according to Nature Physics guidelines

Here we report photon antibunching and magneto-spectroscopy of a single electron spin bound to a fluorine donor in a ZnMgSe/ZnSe QW nanostructure. The results confirm the presence of an optically controllable lambda-system which allows the optical manipulation of the electron bound to the neutral fluorine donor as a spin qubit. Moreover, we achieved optical spin pumping of the qubit by resonant excitation of each of the four allowed transitions of the lambda system. We verified the spin transfer by detecting single photons when the bound electron decays into the opposite spin state. The results presented here constitutes an elegant initialization and the read-out procedure of the electron spin qubit bound to a fluorine donor which are prerequisite for coherent optical control of an impurity based solid-state spin qubit.

Conventional semiconductor laser emission relies on stimulated emission of photons1, 2, which sets stringent requirements on the minimum amount of energy necessary for its operation3, 4. In comparison, exciton–polaritons in strongly coupled quantum well microcavities5 can undergo stimulated scattering that promises more energy-efficient generation of coherent light by ‘polariton lasers’3, 6. Polariton laser operation has been demonstrated in optically pumped semiconductor microcavities at temperatures up to room temperature7, 8, 9, 10, 11, 12, and such lasers can outperform their weak-coupling counterparts in that they have a lower threshold density12, 13. Even though polariton diodes have been realized14, 15, 16, electrically pumped polariton laser operation, which is essential for practical applications, has not been achieved until now. Here we present an electrically pumped polariton laser based on a microcavity containing multiple quantum wells. To prove polariton laser emission unambiguously, we apply a magnetic field and probe the hybrid light–matter nature of the polaritons. Our results represent an important step towards the practical implementation of polaritonic light sources and electrically injected condensates, and can be extended to room-temperature operation using wide-bandgap materials.

Exciton-polaritons in semiconductor microcavities have become a model system for the studies of dynamical Bose-Einstein condensation, macroscopic coherence, many-body effects, nonclassical states of light and matter, and possibly quantum phase transitions in a solid state. These low-mass bosonic quasiparticles can condense at comparatively high temperatures up to 300 K, and preserve the fundamental properties of the condensate, such as coherence in space and time domain, even when they are out of equilibrium with the environment. Although the presence of a confining potential is not strictly necessary in order to observe Bose-Einstein condensation, engineering of the polariton confinement is a key to controlling, shaping, and directing the flow of polaritons. Prototype polariton-based optoelectronic devices rely on ultrafast photon-like velocities and strong nonlinearities exhibited by polaritons, as well as on their tailored confinement. Nanotechnology provides several pathways to achieving polariton confinement, and the specific features and advantages of different methods are discussed in this review. Being hybrid exciton-photon quasiparticles, polaritons can be trapped via their excitonic as well as photonic component, which leads to a wide choice of highly complementary trapping techniques. Here, we highlight the almost free choice of the confinement strengths and trapping geometries that provide powerful means for control and manipulation of the polariton systems both in the semi-classical and quantum regimes. Furthermore, the possibilities to observe effects of the polariton blockade, Mott insulator physics, and population of higher-order energy bands in sophisticated lattice potentials are discussed. Observation of such effects could lead to realization of novel polaritonic non-classical light sources and quantum simulators.

Although exciton–polariton lasers have been experimentally demonstrated in a variety of material systems, robust practical implementations are still challenging. Similarities with atomic Bose–Einstein condensates make the system suitable for chip-based quantum simulators for non-trivial many-body physics.

In a standard semiconductor laser, electrons and holes recombine via stimulated emission to emit coherent light, in a process that is far from thermal equilibrium. Exciton-polariton condensates -- sharing the same basic device structure as a semiconductor laser, consisting of quantum wells coupled to a microcavity -- have been investigated primarily at densities far below the Mott density for signatures of Bose-Einstein condensation. At high densities approaching the Mott density, exciton-polariton condensates are generally thought to revert to a standard semiconductor laser, with the loss of strong coupling. Here, we report the observation of a photoluminescence sideband at high densities that cannot be accounted for by conventional semiconductor lasing. This also differs from an upper-polariton peak by the observation of the excitation power dependence in the peak-energy separation. Our interpretation as a persistent coherent electron-hole-photon coupling captures several features of this sideband whereas many remain elusive. Understanding the observation will lead to a development in non-equilibrium many-body physics., 18 pages, 5 figures + 23 pages, 9 figures

Unlike conventional photon lasing, in which the threshold is limited by the population inversion of the electron-hole plasma, the exciton lasing generated by exciton-exciton scattering and the polariton lasing generated by dynamical condensates have received considerable attention in recent years because of the sub-Mott density and low-threshold operation. This paper presents a novel approach to generate both exciton and polariton lasing in a strongly coupled microcavity (MC) and determine the critical driving requirements for simultaneously triggering these two lasing operation in temperature

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)

Bose-Einstein condensation (BEC) of dilute atomic gases and dense exciton-polaritons provides unique experimental platforms for the simulation of quantum many-body systems in various trap and lattice structures. Atomic BEC is suitable for exploration of the thermal equilibrium and steady state properties of isolated many-body systems, while exciton-polariton BEC is suitable for study of the nonequilibrium and transient properties of open dissipative many-body systems. In this chapter, we will review the fundamental properties of these distinct Bose-Einstein condensates to provide a basis for later discussions of various quantum simulation experiments using cold atoms and exciton-polaritons.

An optical parametric oscillator network driven by a quantum measurement-feedback circuit, composed of optical homodyne detectors, analog-to-digital conversion devices and field programmable gate arrays (FPGA), is proposed and analysed as a scalable coherent Ising machine. The new scheme has an advantage that a large number of optical coupling paths, which is proportional to the square of a problem size in the worst case, can be replaced by a single quantum measurement-feedback circuit. There is additional advantage in the new scheme that a three body or higher order Ising interaction can be implemented in the FPGA digital circuit. Noise associated with the measurement-feedback process is governed by the standard quantum limit. Numerical simulation based on c-number coupled Langevin equations demonstrate a satisfying performance of the proposed Ising machine against the NP-hard MAX-CUT problems.

We review high-orbital exciton-polariton condensation experiments in various two-dimensional lattices. The dynamical nature of exciton-polaritons spontaneously forms condensates at non-zero momentum, resulting from the competition between the finite lifetime and the cooling time. We describe the basics of exciton-polariton condensation, methods used to create lattices, and identification of their orbital order via photoluminescence in real and momentum spaces. We discuss the current status of high-orbital exciton-polariton condensates and the implications towards the bosonic quantum simulators.

We present the quantum theory of coherent Ising machines based on networks of degenerate optical parametric oscillators (DOPOs). In a simple model consisting of two coupled DOPOs, both positive-$P$ representation and truncated Wigner representation predict quantum correlation and inseparability between the two DOPOs in spite of the open-dissipative nature of the system. Here, we apply the truncated Wigner representation method to coherent Ising machines with thermal, vacuum, and squeezed reservoir fields. We find that the probability of finding the ground state of a one-dimensional Ising model increases substantially as a result of reducing excess thermal noise and squeezing the incident vacuum fluctuation on the out-coupling port., Comment: 30 pages, 14 figures

‘In this chapter, we introduced and analyze techniques that allow truly secure secret key sharing over long distances, using public, open channels, where the laws of quantum mechanics ensure the security of the long distance key sharing – an idea generally referred to as the essence of a quantum repeater. We describe several proof-of-principle experiments where technology based on self-assembled quantum dots is used as the backbone of a future quantum repeater.’

We develop reduced models that describe half-harmonic generation in a synchronously-pumped optical parametric oscillator above threshold, where nonlinearity, dispersion, and group-velocity mismatch are all relevant. These models are based on (1) an eigenmode expansion for low pump powers, (2) a simulton-like sech-pulse ansatz for intermediate powers, and (3) dispersionless box-shaped pulses for high powers. Analytic formulas for pulse compression, degenerate vs. nondegenerate operation, and stability are derived and compared to numerical and experimental results., Comment: 22 pages, 24 figures. Submitted to Phys. Rev. A

Long-distance quantum teleportation and quantum repeater technologies require entanglement between a single matter quantum bit (qubit) and a telecommunications (telecom)-wavelength photonic qubit. Electron spins in III-V semiconductor quantum dots are among the matter qubits that allow for the fastest spin manipulation and photon emission, but entanglement between a single quantum-dot spin qubit and a flying (propagating) photonic qubit has yet to be demonstrated. Moreover, many quantum dots emit single photons at visible to near-infrared wavelengths, where silica fibre losses are so high that long-distance quantum communication protocols become difficult to implement. Here we demonstrate entanglement between an InAs quantum-dot electron spin qubit and a photonic qubit, by frequency downconversion of a spontaneously emitted photon from a singly charged quantum dot to a wavelength of 1,560 nanometres. The use of sub-10-picosecond pulses at a wavelength of 2.2 micrometres in the frequency downconversion process provides the necessary quantum erasure to eliminate which-path information in the photon energy. Together with previously demonstrated indistinguishable single-photon emission at high repetition rates, the present technique advances the III-V semiconductor quantum-dot spin system as a promising platform for long-distance quantum communication.

Electron spin in quantum dots are extensively studied as a qubit for quantum information processing. However, the coherence of electron spin is deleteriously influenced by nuclear spin. Quantum-dot holes are a potential alternative. Full control over hole-spin qubits is now achieved using picosecond lasers.

We experimentally studied the transient properties of polariton relaxation dynamics through the oscillatory behavior of an exciton-polariton condensate. The oscillations show characteristic pump po...

Many proposed photonic quantum networks rely on matter qubits to serve as memory elements1,2. The spin of a single electron confined in a semiconductor quantum dot forms a promising matter qubit that may be interfaced with a photonic network3. Ultrafast optical spin control allows gate operations to be performed on the spin within a picosecond timescale4,5,6,7,8,9,10,11,12,13,14, orders of magnitude faster than microwave or electrical control15,16. One obstacle to storing quantum information in a single quantum dot spin is the apparent nanosecond-timescale dephasing due to slow variations in the background nuclear magnetic field15,16,17. Here we use an ultrafast, all-optical spin echo technique to increase the decoherence time of a single quantum dot electron spin from nanoseconds to several microseconds. The ratio of decoherence time to gate time exceeds 105, suggesting strong promise for future photonic quantum information processors18 and repeater networks1,2. An ultrafast, all-optical spin echo technique is used to increase the decoherence time of a single quantum dot electron spin from nanoseconds to several microseconds. The ratio of decoherence time to gate time exceeds 105, suggesting strong promise for future photonic quantum information processors and repeater networks.