Essential Quantum Einstein Gravity

The non-perturbative renormalisation of quantum gravity is investigated allowing for the metric to be reparameterised along the RG flow such that only the essential couplings constants are renormalised. This allows us to identify a universality class of quantum gravity which is guaranteed to be unitary, since the physical degrees of freedom are those of General Relativity with a vanishing cosmological constant. Considering all diffeomorphism invariant operators with up to four derivatives, only Newton's constant is essential at the Gaussian infrared fixed point associated to perturbative gravity. The other inessential couplings can then be fixed to the values they take at the Gaussian fixed point along the RG flow. In the ultraviolet, the corresponding beta function for Newton's constant vanishes at the interacting Reuter fixed point. The properties of the Reuter fixed point are stable between the Einstein-Hilbert approximation and the approximation including all diffeomorphism invariant four derivative terms in the flow equation. Our results suggest that Newton's constant is the only relevant essential coupling at the Reuter fixed point. Therefore, we conjecture that Quantum Einstein Gravity, the ultraviolet completion of Einstein's theory of General Relativity in the asymptotic safety scenario, has no free parameters and in particular predicts a vanishing cosmological constant

New developments in the Renormalization Group

In the first part of the thesis, we review the basics of the Exact Renormalization Group. In the central part, we design a specific choice of renormalization scheme in the context of Functional Renormalization Group to achieve the nonperturbative analogous of the MS scheme's results. Then, we study the properties of a more general family of renormalization schemes, that includes the one we previously analyze, and appears to be useful to eliminate the spurious breaking of symmetries cause by the renormalization scheme. The final part of this thesis consists of a new implementation of the Functional Renormalization Group, based on the Effective Average Action, that allows all possible field redefinitions to simplify the flow equations. Such a simplification is practically useful in reducing the complexity of the computations and has theoretical implications in disentangling the unphysical information due to intrinsic redundancies of the mathematical descriptions of Nature. We show such improvements in the context of the three-dimensional Ising model and the Quantum Einstein Gravity without matter. In particular, using the derivative expansion in both cases we impose renormalization conditions that fix the value of the inessential couplings obtaining only the flow of the essential ones. With such a renormalization scheme, which is called Minimal Essential Scheme, the propagator does not develop additional poles when the truncation of the derivative expansion is increased. This way, we can select the desired universality classes, avoiding encountering instabilities and unitarity violations

Relational observables in Asymptotically safe gravity

We introduce an approach to compute the renormalisation group flow of relational observables in quantum gravity which evolve from their microscopic expressions towards the full quantum expectation value. This is achieved by using the composite operator formalism of the functional renormalisation group. These methods can be applied to a large class of relational observables within a derivative expansion for different physical coordinate systems. As a first application we consider four scalar fields coupled to gravity to represent the physical coordinate frame from which relational observables can be constructed. At leading order of the derivative expansion the observables are the inverse relational metric and the relational scalar curvature. We evaluate their scaling dimensions at the fixed point, both in the standard renormalisation group scheme and in the essential scheme. This represents the first steps to describe running observables within asymptotic safety; this treatment can be generalised to other observables constructed from different tensors and in different physical coordinate systems.Comment: 29 pages, 2 figures; added references, corrected typo

Robustness of the derivative expansion in Asymptotic Safety

We analyse the renormalisation group flow of quantum gravity at sixth order in the derivative expansion within the background field approximation. Non-linear field redefinitions are used to ensure that only essential couplings flow. Working within the universality class of General Relativity, with a vanishing cosmological constant, redundant couplings are fixed to their values at the Gaussian fixed point. This reduces the theory space to two dynamical essential couplings given by Newton's and the Goroff-Sagnotti coupling. Furthermore, it implements the condition that no extra degrees of freedom are present beyond those of General Relativity, in contrast to higher derivative theories and derivative expansions in a conventional renormalisation scheme. We find a unique ultraviolet fixed point with a single relevant direction and analyse the phase diagram of the theory. Our results suggest resilience of the gravitational Reuter fixed point under the inclusion of higher order curvature invariants and show several signs of near-perturbativity. The regulator dependence of our results is investigated in detail and shows that qualitative and quantitative features are robust to a large extent

Essential renormalisation group

We propose a novel scheme for the exact renormalisation group motivated by the desire of reducing the complexity of practical computations. The key idea is to specify renormalisation conditions for all inessential couplings, leaving us with the task of computing only the flow of the essential ones. To achieve this aim, we utilise a renormalisation group equation for the effective average action which incorporates general non-linear field reparameterisations. A prominent feature of the scheme is that, apart from the renormalisation of the mass, the propagator evaluated at any constant value of the field maintains its unrenormalised form. Conceptually, the scheme provides a clearer picture of renormalisation itself since the redundant, non-physical content is automatically disregarded in favour of a description based only on quantities that enter expressions for physical observables. To exemplify the scheme's utility, we investigate the Wilson-Fisher fixed point in three dimensions at order two in the derivative expansion. In this case, the scheme removes all order $\partial^2$ operators apart from the canonical term. Further simplifications occur at higher orders in the derivative expansion. Although we concentrate on a minimal scheme that reduces the complexity of computations, we propose more general schemes where inessential couplings can be tuned to optimise a given approximation. We further discuss the applicability of the scheme to a broad range of physical theories

A linear photonic swap test circuit for quantum kernel estimation

Among supervised learning models, Support Vector Machine stands out as one of the most robust and efficient models for classifying data clusters. At the core of this method, a kernel function is employed to calculate the distance between different elements of the dataset, allowing for their classification. Since every kernel function can be expressed as a scalar product, we can estimate it using Quantum Mechanics, where probability amplitudes and scalar products are fundamental objects. The swap test, indeed, is a quantum algorithm capable of computing the scalar product of two arbitrary wavefunctions, potentially enabling a quantum speed-up. Here, we present an integrated photonic circuit designed to implement the swap test algorithm. Our approach relies solely on linear optical integrated components and qudits, represented by single photons from an attenuated laser beam propagating through a set of waveguides. By utilizing 2$^3$ spatial degrees of freedom for the qudits, we can configure all the necessary arrangements to set any two-qubits state and perform the swap test. This simplifies the requirements on the circuitry elements and eliminates the need for non-linearity, heralding, or post-selection to achieve multi-qubits gates. Our photonic swap test circuit successfully encodes two qubits and estimates their scalar product with a measured root mean square error smaller than 0.05. This result paves the way for the development of integrated photonic architectures capable of performing Quantum Machine Learning tasks with robust devices operating at room temperature

Quantum fields without wick rotation

We discuss the calculation of one-loop effective actions in Lorentzian spacetimes, based on a very simple application of the method of steepest descent to the integral over the field. We show that for static spacetimes this procedure agrees with the analytic continuation of Euclidean calculations. We also discuss how to calculate the effective action by integrating a renormalization group equation. We show that the result is independent of arbitrary choices in the definition of the coarse-graining and we see again that the Lorentzian and Euclidean calculations agree. When applied to quantum gravity on static backgrounds, our procedure is equivalent to analytically continuing time and the integral over the conformal factor

SiN integrated photonic components in the Visible to Near-Infrared spectral region

Integrated photonics has emerged as one of the most promising platforms for quantum applications. The performances of quantum photonic integrated circuits (QPIC) necessitate a demanding optimization to achieve enhanced properties and tailored characteristics with more stringent requirements with respect to their classical counterparts. In this study, we report on the simulation, fabrication, and characterization of a series of fundamental components for photons manipulation in QPIC based on silicon nitride. These include crossing waveguides, multimode-interferometer-based integrated beam splitters (MMIs), asymmetric integrated Mach-Zehnder interferometers (MZIs) based on MMIs, and micro-ring resonators. Our investigation revolves primarily around the Visible to Near-Infrared spectral region, as these devices are meticulously designed and tailored for optimal operation within this wavelength range. By advancing the development of these elementary building blocks, we aim to pave the way for significant improvements in QPIC in a spectral region only little explored so far.Comment: 13 pages, 10 figure

Brillouin nonlinearity characterizations of a high refractive index silicon oxynitride platform

Silicon oxynitride (SiON) is a low-loss and versatile material for linear and nonlinear photonics applications. Controlling the oxygen-to-nitrogen (O/N) ratio in SiON provides an effective way to engineer its optical and mechanical properties, making it a great platform for the investigation of on-chip optomechanical interactions, especially the stimulated Brillouin scattering (SBS). Here we report the Brillouin nonlinearity characterization of a SiON platform with a specific O/N ratio (characterized by a refractive index of $n=1.65$). First, we introduce this particular SiON platform with fabrication details. Subsequently, we discuss various techniques for the on-chip Brillouin nonlinearity characterizations. In particular, we focus on the intensity-modulated pump-probe lock-in amplifier technique, which enables ultra-sensitive characterization. Finally, we analyze the Brillouin nonlinearities of this SiON platform and compare them with other SiON platforms. This work underscores the potential of SiON for on-chip Brillouin-based applications. Moreover, it paves the way for Brillouin nonlinearity characterization across various material platforms