A two-dimensional array of Kerr-nonlinear parametric oscillators (KPOs) with local four-body interactions is a promising candidate for realizing an Ising machine with all-to-all spin couplings, based on adiabatic quantum computation in the Lechner-Hauke-Zoller (LHZ) scheme. However its performance has been evaluated for only a few KPOs. By numerically simulating more KPOs, here we show that the performance can be dramatically improved by reducing inhomogeneity in photon numbers induced by the four-body interactions. The discrepancies of the photon numbers can be corrected by tuning the detunings of KPOs depending on their positions, without monitoring their states during adiabatic time evolution. This correction can be used independent of the number of KPOs in the LHZ scheme and thus can be applied to large-scale implementation.
In this article we present a full description of the quantum Kerr Ising model---a linear optical network of parametrically pumped Kerr non-linearities. We consider the non-dissapative Kerr Ising model and, using variational techniques, show that the energy spectrum is primarily determined by the adjacency matrix in the Ising model and exhibits highly non-classical cat like eigenstates. We then introduce dissipation to give a quantum mechanical treatment of the measurement process based on homodyne detection via the conditional stochastic Schrodinger equation. Finally we identify a quantum adva…
We present a heralded state preparation scheme for driven nonlinear open quantum systems. The protocol is based on a continuous photon counting measurement of the system's decay channel. When no photons are detected for a period of time, the system has relaxed to a measurement-induced pseudo-steady state. We illustrate the protocol by the creation of states with a negative Wigner function in a Kerr oscillator, a system whose unconditional steady state is strictly positive.
We consider the model of two coupled oscillators with Kerr nonlinearities in the rotating-wave approximation. We demonstrate that for a certain set of parameters of the model, the multi-photon transitions occur between many pairs of the oscillator states simultaneously. Also, the position of the multi-photon resonances does not depend on the coupling strength between two oscillators. We prove rigorously that this is a consequence of a certain symmetry of the perturbation theory series for the model. In addition, we analyse the model in the quasi-classical limit by considering the dynamics of t…
We study a minimal model that has a driven-dissipative quantum phase transition, namely a Kerr non-linear oscillator subject to driving and dissipation. Using mean-field theory, exact diagonalization, and the Keldysh formalism, we analyze the critical phenomena in this system, showing which aspects can be captured by each approach and how the approaches complement each other. Then critical scaling and finite-size scaling are calculated analytically using the quantum Langevin equation. The physics contained in this simple model is surprisingly rich: it includes a continuous phase transition, $Z…
We study microwave response of a Josephson parametric oscillator consisting of a superconducting transmission-line resonator with an embedded dc-SQUID. The dc-SQUID allows to control the magnitude of a Kerr nonlinearity over the ranges where it is smaller or larger than the photon loss rate. Spectroscopy measurements reveal the change of the microwave response from a classical Duffing oscillator to a Kerr parametric oscillator in a single device. In the single-photon Kerr regime, we observe parametric oscillations with a well-defined phase of either $0$ or $π$, whose probability can be control…
We considered the multiphoton resonance in the periodically driven quantum oscillator with Kerr nonlinearity in the presence of weak high-order nonlinearities. Multiphoton resonance leads to the emergence of peaks and dips in the dependence of the stationary occupations of the stable states on detuning. We demonstrated that due to high-order nonlinearities, these peaks and dips acquire additional fine structure and split into several closely spaced ones. Quasiclassically, multiphoton resonance is treated as tunneling between the regions of the oscillator phase portrait, and the fine structure …
The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, dep…
We study the time evolution of two coupled quantum harmonic oscillators interacting through nonlinear optomechanical-like Hamiltonians that include cross-Kerr interactions. We employ techniques developed to decouple the time-evolution operator and obtain the analytical solution for the time evolution of the system. We apply these results to obtain explicit expressions of a few quantities of interest. Our results do not require approximations and therefore allow to study the nature and implications of the full nonlinearity of the system. Potential applications and extensions to optomechanics ar…
Quantum Kerr-nonlinear oscillator is a paradigmatic model in cavity and circuit quantum electrodynamics, and quantum optomechanics. We theoretically study the echo phenomenon in a single impulsively excited ("kicked") Kerr-nonlinear oscillator. We reveal two types of echoes, "quantum" and "classical" ones, emerging on the long and short time-scales, respectively. The mechanisms of the echoes are discussed, and their sensitivity to dissipation is considered. These echoes may be useful for studying decoherence processes in a number of systems related to quantum information processing.
The creation of quantum coherences requires a system to be anharmonic. The simplest such continuous 1D quantum system is the Kerr oscillator. It has a number of interesting symmetries we derive. Its quantum dynamics is best studied in phase space, using Wigner's distribution W and the associated Wigner phase space current J. Expressions for the continuity equation governing its time evolution are derived in terms of J and it is shown that J for Kerr oscillators follows circles in phase space. Using J we also show that the evolution's classical shear in phase space is quantum suppressed by an e…
Ising selector machine by Kerr parametric oscillators
Université Paris Cité, CNRS, Matériaux et Phénoménes Quantiques（巴黎西岱大学，法国国家科学研究中心，物质与量子现象研究所）；Sapienza University of Rome（罗马大学）；Centro Ricerche Enrico Fermi (CREF)（恩里科·费米研究中心）
We investigate the quantum synchronization phenomena of two mechanical oscillators of different frequencies in two optomechanical systems under periodically modulating cavity detunings or driving amplitudes, which can interact mutually through an optical fiber or a phonon tunneling. The cavities are filled with Kerr-type nonlinear medium. It is found that, no matter which the coupling and periodically modulation we choose, both of the quantum synchronization of nonlinear optomechanical system are more appealing than the linear optomechanical system. It is easier to observe greatly enhanced qua…
We study the temporal evolution of a coherent state under the action of a parametric oscillator and a nonlinear Kerr-like medium. We make use of the interaction picture representation and use an exact time evolution operator for the time independent part of the Hamiltonian. We approximate the interaction picture Hamiltonian in such a way as to make it a member of a Lie algebra. The corresponding time evolution operator behaves like a squeezing operator due to the temporal dependence of the oscillator's frequency. We analyze the probability amplitude and the auto correlation function for differ…
We study the effect of Kerr nonlinearity in quantum thermal machines having a Kerr-nonlinear oscillator as working substance and operating under the ideal quantum Otto cycle. We first investigate the efficiency of a Kerr-nonlinear heat engine and show that by varying the Kerr-nonlinear strength the efficiency surpasses in up to 2.5 times the efficiency of a quantum harmonic oscillator Otto engine. Moreover, the Kerr-nonlinearity makes the coefficient of performance of the Kerr-nonlinear refrigerator to be as large as 3 times the performance of quantum harmonic oscillator Otto refrigerators. Th…
We study Kerr nonlinear resonators (KNR) driven by a continuous wave field in quantum regimes where strong Kerr interactions give rise to selective resonant excitations of oscillatory modes. We use an exact quantum theory of KNR in the framework of the Fokker-Planck equation without any quantum state truncation or perturbation procedure. This approach allows non-perturbative consideration of KNR for various quantum operational regimes including cascaded processes between oscillatory states. We focus on understanding of multi-photon non-resonant and selective resonant excitations of introcavity…
Quantum theory of a three-photon Kerr parametric oscillator
PSI Center for Photon Science（瑞士保罗谢勒研究所光子科学中心）；Swiss Nanoscience Institute, University of Basel（巴塞尔大学瑞士纳米科学研究所）；Department of Physics, University of Basel（巴塞尔大学物理系）；JEIP, UAR 3573 CNRS, Collège de France, PSL Research University（法国国家科学研究中心联合研究单位3573，法兰西学院，巴黎文理研究大学）
A method of quantum annealing (QA) using Kerr-nonlinear parametric oscillators (KPOs) was proposed. This method is described by bosonic operators and has different characteristics from QA based on the transverse-field Ising model. As the first step to describe this method in the conventional framework of QA, we propose effective spin models of KPOs. The spin models are obtained via a variant of the Holstein-Primakoff transformation and are described by spin-$s$ operators. The terms for detuning, coherent driving, parametric driving, and the Kerr effect are mapped to the transverse field, longi…
Quantum computers with Kerr-nonlinear parametric oscillators (KPOs) have recently been proposed by the author and others. Quantum computation using KPOs is based on quantum adiabatic bifurcations of the KPOs, which lead to quantum superpositions of coherent states, such as Schrodinger cat states. Therefore, these quantum computers are referred to as "quantum bifurcation machines (QbMs)." QbMs can be used for qauntum adiabatic optimization and universal quantum computation. Superconducting circuits with Josephson junctions, Josephson parametric oscillators (JPOs) in particular, are promising fo…
Autonomous quantum error correction has gained considerable attention to avoid complicated measurements and feedback. Despite its simplicity compared with the conventional measurement-based quantum error correction, it is still a far from practical technique because of significant hardware overhead. We propose an autonomous quantum error correction scheme for a rotational symmetric bosonic code in a four-photon Kerr parametric oscillator. Our scheme is the simplest possible error correction scheme that can surpass the break-even point -- it requires only a single continuous microwave tone. We …
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…
Kerr parametric oscillators (KPOs), which can be implemented with superconducting parametrons possessing large Kerr nonlinearity, have been attracting much attention in terms of their applications to quantum annealing, universal quantum computation and studies of quantum many-body systems. It is of practical importance for these studies to realize fast and accurate tunable coupling between KPOs in a simple manner. We develop a simple scheme of fast tunable coupling of KPOs with high tunability in speed and amplitude using the fast transitionless rotation of a KPO in the phase space based on th…
Motivated by quantum experiments with nanomechanical systems, the evolution of a Kerr oscillator with focus on creation of states with a negative Wigner function is investigated. Using the phase space formalism, results are presented that demonstrate an asymptotic behavior in the large squeezing regime for the negativity of a squeezed vacuum state under unitary evolution. The analysis and model are extended to squeezed vacuum states of open systems, adding the decoherence effects of damping and dephasing. To increase experimental relevance, the regime of strong damping is considered. These eff…
We study the effect of Kerr anharmonicity on the symmetry breaking phenomena of coupled quantum oscillators. We study two types of symmetry-breaking processes, namely the inhomogeneous steady state (or quantum oscillation death state) and quantum chimera state. Remarkably, it is found that Kerr nonlinearity hinders the process of symmetry-breaking in both the cases. We establish our results using direct simulation of quantum master equation and analysis of the stochastic semiclassical model. Interestingly, in the case of quantum oscillation death, an increase in the strength of Kerr nonlineari…
Quantum annealing aims to solve combinatorial optimization problems mapped on to Ising interactions between quantum spins. A critical factor that limits the success of a quantum annealer is its sensitivity to noise, and intensive research is consequently focussed towards developing noise-resilient annealers. Here we propose a new paradigm for quantum annealing with a scalable network of all-to-all connected, two-photon driven Kerr-nonlinear resonators. Each of these resonators encode an Ising spin in a robust degenerate subspace formed by two coherent states of opposite phases. The fully-conne…
Continuous-variable systems realized in high-coherence microwave cavities are a promising platform for quantum information processing. While strong dynamic nonlinear interactions are desired to implement fast and high-fidelity quantum operations, static cavity nonlinearities typically limit the performance of bosonic quantum error-correcting codes. Here we study theoretical models of nonlinear oscillators describing superconducting quantum circuits with asymmetric Josephson-junctions loops. Treating the nonlinearity as a perturbation, we derive effective Hamiltonians using the Schrieffer-Wolff…
A Kerr nonlinear parametric oscillator (KPO) can stabilize a quantum superposition of two coherent states with opposite phases, which can be used as a qubit. In a universal gate set for quantum computation with KPOs, an $R_x$ gate, which interchanges the two coherent states, is relatively hard to perform owing to the stability of the two states. We propose a method for a high-fidelity $R_x$ gate by exciting the KPO outside the qubit space with parity-selective transitions, which can be implemented by only adding a driving field. In this method, the utilization of higher effective excited state…
Quantum squeezing amplification with a weak Kerr nonlinear oscillator
Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学深圳量子科学与工程研究院）；International Quantum Academy（国际量子科学院）；Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学广东省量子科学与工程重点实验室）；Department of Physics, Southern University of Science and Technology（南方科技大学物理系）
Spin Model for Quantum Annealing with Kerr Parametric Oscillators
Parity Quantum Computing Germany GmbH（Parity量子计算德国有限公司）；Parity Quantum Computing GmbH（Parity量子计算有限公司）；Secure System Platform Research Laboratories, NEC Corporation（NEC公司安全系统平台研究实验室）；NEC-AIST Quantum Technology Cooperative Research Laboratory（NEC-AIST量子技术联合研究实验室）；Institute for Theoretical Physics, University of Innsbruck（因斯布鲁克大学理论物理研究所）
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…
