BV and Sobolev homeomorphisms between metric measure spaces and the plane

We show that given a homeomorphism $f:GrightarrowOmega$ where $G$ is a open subset of $mathbb{R}^2$ and $Omega$ is a open subset of a $2$-Ahlfors regular metric measure space supporting a weak $(1,1)$-Poincare inequality, it holds $fin BV_{operatorname{loc}}(G,Omega)$ if and only $f^{-1}in BV_{operatorname{loc}}(Omega,G)$. Further if $f$ satisfies the Luzin N and N$^{-1}$ conditions then $fin W^{1,1}_{operatorname{loc}}(G,Omega)$ if and only if $f^{-1}in W^{1,1}_{operatorname{loc}}(Omega,G)$.

Smooth homeomorphic approximation of piecewise affine homeomorphisms

Given any $f$ a locally finitely piecewise affine homeomorphism of $Omega subset rn$ onto $Delta subset rn$ in $W^{1,p}$, $1leq p < infty$ and any $epsilon >0$ we construct a smooth injective map $tilde{f}$ such that $|f-tilde{f}|_{W^{1,p}(Omega,rn)} < epsilon$.

Classification of strict limits of planar BV homeomorphisms

We present a classification of strict limits of planar BV homeomorphisms. The authors and S. Hencl showed in a previous work cite{CHKR} that such mappings allow for cavitations and fractures singularities but fulfill a suitable generalization of the INV condition. As pointed out by J. Ball cite{B}, these features are physically expected by limit configurations of elastic deformations. In the present work we develop a suitable generalization of the emph{no-crossing} condition introduced by De Philippis and Pratelli in cite{PP} to describe weak limits of planar Sobolev homeomorphisms that we call emph{BV no-crossing} condition, and we show that a planar mapping satisfies this property if and only if it can be approximated strictly by homeomorphisms of bounded variations.

Approximation of planar Sobolev $W^{2,1}$ homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms

Given a Sobolev homeomorphism $fin W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.

Diffeomorphic approximation of Planar Sobolev Homeomorphisms in rearrangement invariant spaces

Let $Omegasubseteqmathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $fin W^{1}X(Omega,mathcal{R}^2)$ be a homeomorphism between $Omega$ and $f(Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(Omega,mathcal{R}^2)$.

Comparison between the non-crossing and the non-crossing on lines properties

In the recent paper [2], it was proved that the closure of the planar diffeomorphisms in the Sobolev norm consists of the functions which are non-crossing (NC), i.e., the functions which can be uniformly approximated by continuous one-to-one functions on the grids. A deep simplification of this property is to consider curves instead of grids, so considering functions which are non-crossing on lines (NCL). Since the NCL property is way easier to check, it would be extremely positive if they actually coincide, while it is only obvious that NC implies NCL. We show that in general NCL does not imply NC, but the implication becomes true with the additional assumption that $det(Du)>0$ a.e., which is a very common assumption in nonlinear elasticity.

Generating Spatially Entangled Itinerant Photons with Waveguide Quantum Electrodynamics

Realizing a fully connected network of quantum processors requires the ability to distribute quantum entanglement. For distant processing nodes, this can be achieved by generating, routing, and capturing spatially entangled itinerant photons. In this work, we demonstrate the deterministic generation of such photons using superconducting transmon qubits that are directly coupled to a waveguide. In particular, we generate two-photon N00N states and show that the state and spatial entanglement of the emitted photons are tunable via the qubit frequencies. Using quadrature amplitude detection, we reconstruct the moments and correlations of the photonic modes and demonstrate state preparation fidelities of $84%$. Our results provide a path towards realizing quantum communication and teleportation protocols using itinerant photons generated by quantum interference within a waveguide quantum electrodynamics architecture.

Waveguide Quantum Electrodynamics with Giant Superconducting Artificial Atoms

Models of light-matter interactions typically invoke the dipole approximation, within which atoms are treated as point-like objects when compared to the wavelength of the electromagnetic modes that they interact with. However, when the ratio between the size of the atom and the mode wavelength is increased, the dipole approximation no longer holds and the atom is referred to as a giant atom. Thus far, experimental studies with solid-state devices in the giant-atom regime have been limited to superconducting qubits that couple to short-wavelength surface acoustic waves, only probing the properties of the atom at a single frequency. Here we employ an alternative architecture that realizes a giant atom by coupling small atoms to a waveguide at multiple, but well separated, discrete locations. Our realization of giant atoms enables tunable atom-waveguide couplings with large on-off ratios and a coupling spectrum that can be engineered by device design. We also demonstrate decoherence-free interactions between multiple giant atoms that are mediated by the quasi-continuous spectrum of modes in the waveguide-- an effect that is not possible to achieve with small atoms. These features allow qubits in this architecture to switch between protected and emissive configurations in situ while retaining qubit-qubit interactions, opening new possibilities for high-fidelity quantum simulations and non-classical itinerant photon generation.