Systems of coupled optical parametric oscillators (OPOs) forming an Ising machine are emerging as large-scale simulators of the Ising model. The advances in computer science and nonlinear optics have triggered not only the physical realization of hybrid (electrooptical) or all-optical Ising machines, but also the demonstration of quantum-inspired algorithms boosting their performances. To date, the use of the quantum nature of parametrically generated light as a further resource for computation represents a major open issue. A key quantum feature is the non-Gaussian character of the system state across the oscillation threshold. In this paper, we perform an ab initio analysis of the emergence of non-Gaussianity in the single quantum OPO with an applied external field. We model the OPO by a Lindblad master equation, which is numerically solved by a first-principles method based on exact diagonalization. Non-Gaussianity is quantified by means of three different metrics: the Hilbert-Schmidt distance, quantum relative entropy, and photon distribution. Our findings reveal a nontrivial interplay between parametric drive and applied field: (i) the increasing pump monotonically enhances non-Gaussianity and (ii) the increasing field first sharpens non-Gaussianity, and then restores the Gaussian character of the state when above a threshold value. We also report a first-principles computation in the Fock space of the distance from the mixture of coherent states, a strongly nonclassical behavior that can play a significant role in the quantum parallel search for optimization.
Systems of coupled optical parametric oscillators (OPOs) forming an Ising machine are emerging as large-scale simulators of the Ising model. The advances in computer science and nonlinear optics have triggered not only the physical realization of hybrid (electrooptical) or all-optical Ising machines, but also the demonstration of quantum-inspired algorithms boosting their performances. To date, the use of the quantum nature of parametrically generated light as a further resource for computation represents a major open issue. A key quantum feature is the non-Gaussian character of the system state across the oscillation threshold. In this paper, we perform an ab initio analysis of the emergence of non-Gaussianity in the single quantum OPO with an applied external field. We model the OPO by a Lindblad master equation, which is numerically solved by a first-principles method based on exact diagonalization. Non-Gaussianity is quantified by means of three different metrics: the Hilbert-Schmidt distance, quantum relative entropy, and photon distribution. Our findings reveal a nontrivial interplay between parametric drive and applied field: (i) the increasing pump monotonically enhances non-Gaussianity and (ii) the increasing field first sharpens non-Gaussianity, and then restores the Gaussian character of the state when above a threshold value. We also report a first-principles computation in the Fock space of the distance from the mixture of coherent states, a strongly nonclassical behavior that can play a significant role in the quantum parallel search for optimization.
Quantum states with non-Gaussian statistics generated by optical parametric oscillators (OPO) with fluctuating parameters are studied by means of the kurtosis excess of the external field quadratures. The field generated is viewed as the response of a nonlinear device to the fluctuations of pump laser amplitude and frequency, crystal temperature, and cavity detuning, in addition to quantum noise sources. The kurtosis excess has been evaluated perturbatively up to the third order in the strength of the crystal nonlinear coupling factor and the second order in the classical fluctuating parameters. Applied to the device described in Opt. Expr. 13, 948 (2005), the model has given values of the kurtosis excess in good agreement with the measured ones.
The quantum features of ultrashort-pulse optical parametric oscillators (OPOs) are investigated theoretically in the nonlinear regime near and above threshold. Viewing the pulsed OPO as a multimode open quantum system, we rigorously derive a general input-output model that features nonlinear coupling among many cavity (i.e., system) signal modes and a broadband single-pass (i.e., reservoir) pump field. Under appropriate assumptions, our model produces a Lindblad master equation with multimode nonlinear Lindblad operators describing two-photon dissipation and a multimode four-wave-mixing Hamiltonian describing a broadband, dispersive optical cascade, which we show is required to preserve causality. To simplify the multimode complexity of the model, we employ a supermode decomposition to perform numerical simulations in the regime where the pulsed supermodes experience strong single-photon nonlinearity. We find that the quantum nonlinear dynamics induces pump depletion as well as corrections to the below-threshold squeezing spectrum predicted by linearized models. We also observe the formation of non-Gaussian states with Wigner-function negativity and show that the multimode interactions with the pump, both dissipative and dispersive, can act as effective decoherence channels. Finally, we briefly discuss some experimental considerations for potentially observing such quantum nonlinear phenomena with ultrashort-pulse OPOs on nonlinear nanophotonic platforms.
We report on the emergence of a highly nonclassical collective behavior in quantum parametric oscillators, which we name quantum hyperspin, induced by a tailored nonlinear interaction. This is the second quantized version of classical multidimensional spherical spins, as XY spins in two dimensions, and Heisenberg spins in three dimensions. In the phase space, the quantum hyperspins are represented as spherical shells whose radius scales with the number of particles in a way such that it cannot be factorized even in the limit of large particle number. We show that the nonlinearly coupled quantum oscillators form a high-dimensional entangled state that is surprisingly robust with respect to the coupling with the environment. Such a behavior results from a properly engineered reservoir. Networks of entangled quantum hyperspins are a new approach to quantum simulations for applications in computing, Ising machines, and high-energy physics models. We analyze from first principles through ab initio numerical simulations the properties of quantum hyperspins, including the interplay of entanglement and coupling frustration.
Engineering non-classical states of the electromagnetic field is a central quest for quantum optics(1,2). Beyond their fundamental significance, such states are indeed the resources for implementing various protocols, ranging from enhanced metrology to quantum communication and computing. A variety of devices can be used to generate non-classical states, such as single emitters, light-matter interfaces or non-linear systems(3). We focus here on the use of a continuous-wave optical parametric oscillator(3,4). This system is based on a non-linear χ(2) crystal inserted inside an optical cavity and it is now well-known as a very efficient source of non-classical light, such as single-mode or two-mode squeezed vacuum depending on the crystal phase matching. Squeezed vacuum is a Gaussian state as its quadrature distributions follow a Gaussian statistics. However, it has been shown that number of protocols require non-Gaussian states(5). Generating directly such states is a difficult task and would require strong χ(3) non-linearities. Another procedure, probabilistic but heralded, consists in using a measurement-induced non-linearity via a conditional preparation technique operated on Gaussian states. Here, we detail this generation protocol for two non-Gaussian states, the single-photon state and a superposition of coherent states, using two differently phase-matched parametric oscillators as primary resources. This technique enables achievement of a high fidelity with the targeted state and generation of the state in a well-controlled spatiotemporal mode.
Parametrically driven Kerr nonlinear quantum oscillator plays a central role in quantum optics, superconducting circuits, and in engineered light-matter interactions. For the development of future quantum technologies, it is crucial to understand the behavior of this quantum system under realistic dissipative conditions. The objective of this thesis is to investigate the phase-space dynamics of a dissipative Kerr nonlinear oscillator under different driving regimes and corresponding dissipation channels, with a particular focus on the emergence of nonclassical features in phase space. The aim is also to understand how nonlinearity and different driving mechanisms influence quantum properties such as coherence, nonclassicality, and phase-space structure. The analysis was carried out using the Wigner phase-space formalism and numerical simulations of the Lindblad master equation. The results show that in the case of finite nonlinearity the coherence time is prolonged compared to the harmonic case. In the single photon regime, the interplay between the photon drive and nonlinearity can lead to coherence revivals after which the system relaxes into a squeezed steady state. In the two photon and dissipation regime, the system exhibits parity-protected dynamics, leading to very long coherence times.
Engineering non-classical states of the electromagnetic field is a central quest for quantum optics1,2. Beyond their fundamental significance, such states are indeed the resources for implementing various protocols, ranging from enhanced metrology to quantum communication and computing. A variety of devices can be used to generate non-classical states, such as single emitters, light-matter interfaces or non-linear systems3. We focus here on the use of a continuous-wave optical parametric oscillator3,4. This system is based on a non-linear χ2 crystal inserted inside an optical cavity and it is now well-known as a very efficient source of non-classical light, such as single-mode or two-mode squeezed vacuum depending on the crystal phase matching. Squeezed vacuum is a Gaussian state as its quadrature distributions follow a Gaussian statistics. However, it has been shown that number of protocols require non-Gaussian states5. Generating directly such states is a difficult task and would require strong χ3 non-linearities. Another procedure, probabilistic but heralded, consists in using a measurement-induced non-linearity via a conditional preparation technique operated on Gaussian states. Here, we detail this generation protocol for two non-Gaussian states, the single-photon state and a superposition of coherent states, using two differently phase-matched parametric oscillators as primary resources. This technique enables achievement of a high fidelity with the targeted state and generation of the state in a well-controlled spatiotemporal mode.