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Quanta splitting is an essential generator of Gaussian entanglement, apparently the most commonly occurring form of entanglement in linearised systems. In general, such Gaussian entanglement results from the strong pumping of a nonlinear process with the highly coherent and low-noise external drive necessary for the linearisation. In contrast, recent experiments involving efficient trilinear processes in trapped ions and superconducting circuits have opened the complementary possibility to test the splitting of a few thermal quanta without a linearisation process. Stimulated by such small thermal energy, the strong degenerate trilinear coupling generates large amounts of nonclassicality, detectable by more than 3 dB of distillable quadrature squeezing, in contrast with the nondegenerate trilinear coupling. Substantial entanglement can be generated in tandem with the nonclassicality generation via linear coupling to a third mode, achieving the entanglement potential of the nonclassical states. Such entanglement, although not captured by the covariance matrix, grows with the mean number of split thermal quanta; something which does not happen with Gaussian entanglement. We extend our analysis to higher-order degenerate quanta splitting processes to demonstrate that this non-Gaussian entanglement is already common for strongly nonlinear systems at the low-energy thermal level. We thus shed light on the generic entanglement mechanism for nonlinear bosonic systems operating in such low-energy regimes., Comment: 5 pages, 4 figures + Supplementary Material

This evidence review synthesises the best available international evidence regarding approaches to behaviour in schools to: (1) Produce an overarching model or framework explaining why school pupils may misbehave (Review 1); (2) Review the effectiveness of classroom-based approaches to behaviour trialled in robust research studies (Review 2); and (3) Review the effectiveness of school-wide approaches to behaviour and identify the gaps for robust research studies to explore frequently used strategies (Review 3). [For the related guidance report, see ED612208.]

Laser cooled ions trapped in a linear Paul trap are long-standing ideal candidates for realizing quantum simulation, especially of many-body systems. The properties that contribute to this also provide the opportunity to demonstrate unexpected quantum phenomena in few-body systems. A pair of ions interacting in such traps exchange vibrational quanta through the Coulomb interaction. This linear interaction can be anharmonically modulated by an elementary coupling to the internal two-level structure of one of the ions. Driven by thermal energy in the passively coupled oscillators, which are themselves coupled to the internal ground states of the ions, the nonlinear interaction autonomously and unconditionally generates entanglement between the mechanical modes of the ions. We examine this counter-intuitive thermally induced entanglement for several experimentally feasible model systems, and propose parameter regimes where state of the art trapped ion systems can produce such phenomena. In addition, we demonstrate a multiqubit enhancement of such thermally induced entanglement., Comment: 11+9 pages, 4+11 figures

The Education Endowment Foundation (EEF) commissioned a team from the University of Plymouth and the University of Exeter to conduct: (1) a review of the best available evidence about what schools can do to improve children's learning and attainment via parent engagement, and (2) supplementary research to understand schools' current practices and perceptions of parental engagement. Given the focus of the EEF on closing the attainment gap there was particular interest in how best to support parents from more disadvantaged backgrounds. The review summarises the best available international research evidence on parent engagement and presents actionable recommendations for practice that could be incorporated in EEF guidance. The study has two main parts. The first of these, an evidence review, aims to synthesise the best current international evidence on parental engagement in children's learning. The second part of the study, which involved the survey of schools and interviews with school leaders, sought to identify what schools in England are doing currently to support parental engagement. This element is designed to help understand how far schools are currently prioritising parental engagement, what types of approaches they adopt, how they target their efforts, and how far practice matches the best available evidence as shown in the evidence review. The intention was to identify areas where there is potential for shifting practice towards more evidence-based approaches, and to inform possible resources that could be developed to support schools. [For the related guidance report, "Working with Parents to Support Children's Learning. Guidance Report," see ED612191.]

Nonclassical correlations provide a resource for many applications in quantum technology as well as providing strong evidence that a system is indeed operating in the quantum regime. Optomechanical systems can be arranged to generate nonclassical correlations (such as quantum entanglement) between the mechanical mode and a mode of travelling light. Here we propose automated optimization of the production of quantum correlations in such a system, beyond what can be achieved through analytical methods, by applying Bayesian optimization to the control parameters. A two-mode optomechanical squeezing experiment is simulated using a detailed theoretical model of the system and the measurable outputs fed to the Bayesian optimization process. This then modifies the controllable parameters in order to maximize the non-classical two-mode squeezing and its detection, independently of the inner workings of the model. We focus on a levitated nano-sphere system, but the techniques described are broadly applicable in optomechanical experiments, and also more widely, especially where no detailed theoretical treatment is available. We find that in the experimentally relevant thermal regimes, the ability to vary and optimize a broad array of control parameters provides access to large values of two-mode squeezing that would otherwise be difficult or intractable to discover via analytical or trial and error methods. In particular we observe that modulation of the driving frequency around the resonant sideband allows for stronger nonclassical correlations. We also observe that our optimization approach finds parameters that allow significant squeezing in the high temperature regime. This extends the range of experimental setups in which non-classical correlations could be generated beyond the region of high quantum cooperativity., Comment: 24 pages, 11 figures, including the 3 appendices and bibliography

Nonclassical optomechanical correlations enable optical control of mechanical motion beyond the limitations of classical driving. Here we investigate the feasibility of using pulsed cavity-optomechanics to create and verify nonclassical phase-sensitive correlations between light and the motion of a levitated nanoparticle in a realistic scenario. We show that optomechanical two-mode squeezing can persist even at the elevated temperatures of state-of-the-art experimental setups. We introduce a detection scheme based on optical homodyning that allows revealing nonclassical correlations without full optomechanical state tomography. We provide an analytical treatment using the rotating wave approximation (RWA) in the resolved-sideband regime and prove its validity with a full numerical solution of the Lyapunov equation beyond the RWA. We build on parameters of current experiments for our analysis and conclude that the observation of nonclassical correlations is possible today., Comment: 5 + 5 pages, 3 figures, 1 table. Comments welcome!

The measurement based, or one-way, model of quantum computation for continuous variables uses a highly entangled state called a cluster state to accomplish the task of computing. Cluster states that are universal for computation are a subset of a class of states called graph states. These states are Gaussian states and therefore require that the homodyne detection (Gaussian measurement) scheme is supplemented with a non-Gaussian measurement for universal computation, a significant experimental challenge. Here we define a new non-Gaussian class of states based on hypergraphs which satisfy the requirements of the Lloyd-Braunstein criteria while restricted to a Gaussian measurement strategy. Our main result is to show that, taking advantage of the intrinsic multimode nonlinearity, a hypergraph consisting of 3-edges can be used to apply a three-mode operation to an input three-mode state. As a special case, this technique can be used to apply the cubic phase gate to a single mode., Comment: 7 pages, 2 figures

Processing quantum information on continuous variables requires a highly nonlinear element in order to attain universality. Noise reduction in processing such quantum information involves the use of a nonlinear phase state as a non-Gaussian ancilla. A necessary condition for a nonlinear phase state to implement a nonlinear phase gate is that noise in a selected nonlinear quadrature should decrease below the level of classical states. A reduction of the variance in this nonlinear quadrature below the ground state of the ancilla, a type of nonlinear squeezing, is the resource embedded in these non-Gaussian states and a figure of merit for nonlinear quantum processes. Quantum optomechanics with levitating nanoparticles trapped in nonlinear optical potentials is a promising candidate to achieve such resources in a flexible way. We provide a scheme for reconstructing this figure of merit, which we call nonlinear squeezing, in standard linear quantum optomechanics, analysing the effects of mechanical decoherence processes on the reconstruction and show that all mechanical states which exhibit reduced noise in this nonlinear quadrature are nonclassical., Comment: 9 pages + Appendix, 2 figures

The preparation of nonclassical states of mechanical motion conclusively proves that control over such motion has reached the quantum level. We investigate ways to achieve nonclassical states of macroscopic mechanical oscillators, particularly levitated nanoparticles. We analyze the possibility of the conditional squeezing of the levitated particle induced by the homodyne detection of light in a pulsed optomechanical setup within the resolved sideband regime. We focus on the regimes that are experimentally relevant for the levitated systems where the ground-state cooling is not achievable and the optomechanical coupling is comparable with the cavity linewidth. The analysis is thereby performed beyond the adiabatic regime routinely used for the bulk optomechanical pulsed systems. The results show that the quantum state of a levitated particle could be squeezed below the ground state variance within a wide range of temperatures. This opens a path to test for the first time nonclassical control of levitating nanoparticles beyond the ground state., Comment: 10 pages, 2 figures, close to published version

Universal quantum computation encoded over continuous variables can be achieved via Gaussian measurements acting on entangled non-Gaussian states. However, due to the weakness of available nonlinearities, generally these states can only be prepared conditionally, potentially with low probability. Here we show how universal quantum computation could be implemented unconditionally using an integrated platform able to sustain both linear and quadratic optomechanical-like interactions. Specifically, considering cavity opto- and electro-mechanical systems, we propose a realisation of a driven-dissipative dynamics that deterministically prepares the required non-Gaussian cluster states -- entangled squeezed states of multiple mechanical oscillators suitably interspersed with cubic-phase states. We next demonstrate how arbitrary Gaussian measurements on the cluster nodes can be performed by continuously monitoring the output cavity field. Finally, the feasibility requirements of this approach are analysed in detail, suggesting that its building blocks are within reach of current technology., Comment: 19 pages, 10 figures

Reservoir engineering enables the robust preparation of pure quantum states in noisy environments. We show how a new family of quantum states of a mechanical oscillator can be stabilized in a cavity that is parametrically coupled to both the mechanical displacement and the displacement squared. The cavity is driven with three tones, on the red sideband, on the cavity resonance and on the second blue sideband. The states so stabilized are (squeezed and displaced) superpositions of a finite number of phonons. They show the unique feature of encompassing two prototypes of nonclassicality for bosonic systems: by adjusting the strength of the drives, one can in fact move from a single-phonon- to a Schrodinger-cat-like state. The scheme is deterministic, supersedes the need for measurement-and-feedback loops and does not require initialization of the oscillator to the ground state. As such, it enables the unconditional preparation of nonclassical states of a macroscopic object., Comment: 13 pages, 9 figures. Accepted version

We consider opto- and electro-mechanical quantum systems composed of a driven cavity mode interacting with a set of mechanical resonators. It has been proposed that the latter can be initialized in arbitrary cluster states, including universal resource states for Measurement Based Quantum Computation (MBQC). We show that, despite the unavailability in this set-up of direct measurements over the mechanical resonators, computation can still be performed to a high degree of accuracy. In particular, it is possible to indirectly implement the measurements necessary for arbitrary Gaussian MBQC by properly coupling the mechanical resonators to the cavity field and continuously monitoring the leakage of the latter. We provide a thorough theoretical analysis of the performances obtained via indirect measurements, comparing them with what is achievable when direct measurements are instead available. We show that high levels of fidelity are attainable in parameter regimes within reach of present experimental capabilities., Comment: 12 pages, 8 figures

We introduce a scheme to reconstruct an arbitrary quantum state of a mechanical oscillator network. We assume that a single element of the network is coupled to a cavity field via a linearized optomechanical interaction, whose time dependence is controlled by a classical driving field. By designing a suitable interaction profile, we show how the statistics of an arbitrary mechanical quadrature can be encoded in the cavity field, which can then be measured. We discuss the important special case of Gaussian state reconstruction, and study numerically the effectiveness of our scheme for a finite number of measurements. Finally, we speculate on possible routes to extend our ideas to the regime of single-photon optomechanics., Comment: 11 pages, 6 figures; v2: appendix added, minor modifications, close to published version