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Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$.

We prove that there is a function $f$ such that every graph with no $K$-fat $K_4$ minor is $f(K)$-quasi-isometric to a graph with no $K_4$ minor. This solves the $K_4$-case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective $K_4^-$-case, which was first established by Fujiwara and Papasoglu., Comment: 29 pages, 11 figures, SVG files available in TeX Source

We extend Robertson and Seymour's tangle-tree duality theorem to infinite graphs.

Kriz and Thomas showed that every (finite or infinite) graph of tree-width $k \in \mathbb{N}$ admits a lean tree-decomposition of width $k$. We discuss a number of counterexamples demonstrating the limits of possible generalisations of their result to arbitrary infinite tree-width. In particular, we construct a locally finite, planar, connected graph that has no lean tree-decomposition.

We prove that every graph which admits a tree-decomposition into finite parts has a rooted tree-decomposition into finite parts that is linked, tight and componental. As an application, we obtain that every graph without half-grid minor has a lean tree-decomposition into finite parts, strengthening the corresponding result by Kriz and Thomas for graphs of finitely bounded tree-width. In particular, it follows that every graph without half-grid minor has a tree-decomposition which efficiently distinguishes all ends and critical vertex sets, strengthening results by Carmesin and by Elm and Kurkofka for this graph class. As a second application of our main result, it follows that every graph which admits a tree-decomposition into finite parts has a tree-decomposition into finite parts that displays all the ends of $G$ and their combined degrees, resolving a question of Halin from 1977. This latter tree-decomposition yields short, unified proofs of the characterisations due to Robertson, Seymour and Thomas of graphs without half-grid minor, and of graphs without binary tree subdivision.

Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition $(T, \mathcal{V})$ of adhesion less than $k$ that efficiently distinguishes every two distinct $k$-profiles, and which has the further property that every separable $k$-block is equal to the unique part of $(T, \mathcal{V})$ in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than $k$. For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.

Chip-scale multimode optomechanical systems have unique benefits for sensing, metrology and quantum technologies relative to their single-mode counterparts. Slot-mode optomechanical crystals enable sideband resolution and large optomechanical couplings of a single optical cavity to two microwave-frequency mechanical modes. Still, previous implementations have been limited to nanobeam geometries, whose effective quantum cooperativity at ultralow temperatures is limited by their low thermal conductance. In this work, we design and experimentally demonstrate a two-dimensional mechanical-optical-mechanical (MOM) platform that dispersively couples a slow-light slot-guided photonic-crystal waveguide mode and two slow-sound $\sim 7$ GHz phononic wire modes localized in physically distinct regions. We first demonstrate optomechanical interactions in long waveguide sections, unveiling acoustic group velocities below 800 m/s, and then move on to mode-gap adiabatic heterostructure cavities with a tailored mechanical frequency difference. Through optomechanical spectroscopy, we demonstrate optical quality factors $Q \sim 10^5$, vacuum optomechanical coupling rates, $g_o/2\pi$, of 1.5 MHz and dynamical backaction effects beyond the single-mode picture. At larger power and adequate laser-cavity detuning, we demonstrate regenerative optomechanical oscillations involving a single mechanical mode, extending to both mechanical modes through modulation of the input laser drive at their frequency difference. This work constitutes an important advance towards engineering MOM systems with nearly degenerate mechanical modes as part of hybrid multipartite quantum systems.

Given an arbitrary class $\mathcal{H}$ of graphs, we investigate which graphs admit a decomposition modelled on a graph in $\mathcal{H}$ into parts of small radius. The $\mathcal{H}$-decompositions that we consider here generalise the notion of tree-decompositions. We identify obstructions to such $\mathcal{H}$-decompositions of small radial width, and we prove that these obstructions occur in every graph of sufficiently large radial $\mathcal{H}$-width for the classes $\mathcal{H}$ of paths, of cycles and of subdivided stars., Comment: 32 pages, 6 figures

Nanoelectromechanical systems offer unique functionalities in photonics: The ability to elastically and reversibly deform dielectric beams with subwavelength dimensions enable electrical control of the propagation of light with a power consumption orders of magnitude below that of competing technologies, such as thermo-optic tuning. We present a study of the design, fabrication, and characterization of compact electrostatic comb-drive actuators tailored for integrated nanoelectromechanical silicon photonic circuits. Our design has a footprint of $1.2 \times 10^{3} \mu$m$^{2}$ and is found to reach displacements beyond 50 nm at 5 V with a mechanical resonance above 200 kHz, or, using different spring constants and skeletonization, a mechanical resonance above 2.5 MHz with displacements beyond 50 nm at 28 V. This is sufficient to induce very large phase shifts and other optical effects in nanoelectromechanical reconfigurable photonic circuits., Comment: 7 pages, 4 figure, 1 supplementary animation

We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such that for every graph $G$ and $X,Y \subseteq V(G)$, either there exist two $X-Y$ paths $P_1$ and $P_2$ such that the distance between $P_1$ and $P_2$ is at least $d$, or there exists $v \in V(G)$ such that the ball of radius $f(d)$ centered at $v$ intersects every $X-Y$ path., Comment: 13 pages, 10 figures

We combine the two fundamental fixed-order tangle theorems of Robertson and Seymour into a single theorem that implies both, in a best possible way. We show that, for every $k \in \mathbb{N}$, every tree-decomposition of a graph $G$ which efficiently distinguishes all its $k$-tangles can be refined to a tree-decomposition whose parts are either too small to be home to a $k$-tangle, or as small as possible while being home to a $k$-tangle., Comment: arXiv admin note: text overlap with arXiv:2302.01808

Principal component analysis (PCA) is commonly used in genetics to infer and visualize population structure and admixture between populations. PCA is often interpreted in a way similar to inferred admixture proportions, where it is assumed that individuals belong to one of several possible populations or are admixed between these populations. We propose a new method to assess the statistical fit of PCA (interpreted as a model spanned by the top principal components) and to show that violations of the PCA assumptions affect the fit. Our method uses the chosen top principal components to predict the genotypes. By assessing the covariance (and the correlation) of the residuals (the differences between observed and predicted genotypes), we are able to detect violation of the model assumptions. Based on simulations and genome wide human data we show that our assessment of fit can be used to guide the interpretation of the data and to pinpoint individuals that are not well represented by the chosen principal components. Our method works equally on other similar models, such as the admixture model, where the mean of the data is represented by linear matrix decomposition.

Robertson and Seymour proved two fundamental theorems about tangles in graphs: the tree-of-tangles theorem, which says that every graph has a tree-decomposition such that distinguishable tangles live in different nodes of the tree, and the tangle-tree duality theorem, which says that graphs without a $k$-tangle have a tree-decomposition that witnesses the non-existence of such tangles, in that $k$-tangles would have to live in a node but no node is large enough to accommodate one. Erde combined these two fundamental theorems into one, by constructing a single tree-decomposition such that every node either accommodates a single $k$-tangle or is too small to accommodate one. Such a tree-decomposition thus shows at a glance how many $k$-tangles a graph has and where they are. The two fundamental theorems have since been extended to abstract separation systems, which support tangles in more general discrete structures. In this paper we extend Erde's unified theorem to such general systems., Comment: This is the journal version of this paper. An extended version containing additional examples is available in this ArXiv thread as v1

Phonons offer the possibility to connect the microwave and optical domains while being efficiently transduced with electronic and optical signals. Here, we present a multimodal optomechanical platform, consisting of a mechanical-optical-mechanical resonator configuration. The mechanical modes, with frequencies at 265 MHz and 6.8 GHz, can be simultaneously excited into a phonon lasing regime as supported by a stability analysis of the system. Both the MHz and the GHz modes enter a self-sustained oscillation regime, leading to the intermodulation of two frequency combs in the optical field. We characterize this platform experimentally, demonstrating previously unexplored dynamical regimes. These results suggest the possibility to control multiple mechanical degrees of freedom via a single optical mode, with implications in GHz phononic devices, signal processing, and optical comb sensing applications., Comment: 10 pages

The unique properties of light underpin the visions of photonic quantum technologies, optical interconnects, and a wide range of novel sensors, but a key limiting factor today is losses due to either absorption or backscattering on defects. Recent developments in topological photonics have fostered the vision of backscattering-protected waveguides made from topological interface modes, but, surprisingly, measurements of their propagation losses were so far missing. Here we report on measurements of losses in the slow-light regime of valley-Hall topological waveguides and find no indications of topological protection against backscattering on ubiquitous structural defects. We image the light scattered out from the topological waveguides and find that the propagation losses are due to Anderson localization. The only photonic topological waveguides proposed for materials without intrinsic absorption in the optical domain are quantum spin-Hall and valley-Hall interface states, but the former exhibits strong out-of-plane losses, and our work therefore raises fundamental questions about the real-world value of topological protection in reciprocal photonics., Comment: Accepted Manuscript

Nanophononics has the potential for information transfer, in an analogous manner to its photonic and electronic counterparts. The adoption of phononic systems has been limited, due to difficulties associated with the generation, manipulation, and detection of phonons, especially at GHz frequencies. Existing techniques often require piezoelectric materials with an external radiofrequency excitation that are not readily integrated into existing CMOS infrastructures, while non-piezoelectric demonstrations have been inefficient. In this work, we explore the optomechanical generation of coherent phonons in a suspended 2D silicon phononic crystal cavity with a guided mode around 6.8 GHz. By incorporating an air-slot into this cavity, we turn the phononic waveguide into an optomechanical platform that exploits localized photonic modes resulting from inherent fabrication imperfections for the transduction of mechanics. Such a platform exhibits very fine control of phonons using light, and is capable of coherent self-sustained phonon generation via mechanical lasing around 6.8 GHz. The ability to generate high frequency coherent mechanical vibrations within such a simple 2D CMOS-compatible system could be a first step towards the development of sources in phononic circuitry and the coherent manipulation of other solid-state properties., Comment: 13 pages, 4 main text figures, 6 appendix figures

Electrical control of light in integrated photonics is central to a wide range of research and applications. It is conventionally achieved with thermo-optic tuning, but this suffers from high energy consumption and crosstalk. These challenges could be resolved with nanoelectromechanical photonics but scalable platforms were so far not demonstrated because of incompatibility with conventional multilayer metal architectures and because conventional approaches do not allow crossings of electrical wires and photonic waveguides. Here we use topology optimization to devise a single-layer electronic-photonic circuit crossing with up to 99.8% optical transmission across an isolation trench with excellent electrical isolation. We use the crossing to demonstrate a monolithic silicon nano-electro-mechanical add-drop switch in which the flow of photons, electrons, and mechanical motions are fully integrated within the same layer. Our work enables hitherto impossible opto-electro-mechanical topologies and may lead to new functionalities in nano-opto-electro-mechanical systems, optomechanics, and integrated quantum photonics., Comment: 32 pages, 20 figures

Controlling the vibrations in solids is crucial to tailor their mechanical properties and their interaction with light. Thermal vibrations represent a source of noise and dephasing for many physical processes at the quantum level. One strategy to avoid these vibrations is to structure a solid such that it possesses a phononic stop band, i.e., a frequency range over which there are no available mechanical modes. Here, we demonstrate the complete absence of mechanical vibrations at room temperature over a broad spectral window, with a 5.3 GHz wide band gap centered at 8.4 GHz in a patterned silicon nanostructure membrane measured using Brillouin light scattering spectroscopy. By constructing a line-defect waveguide, we directly measure GHz localized modes at room temperature. Our experimental results of thermally excited guided mechanical modes at GHz frequencies provides an eficient platform for photon-phonon integration with applications in optomechanics and signal processing transduction.

We present a theoretical study of dielectric bowtie cavities and show that they are governed by two essentially different confinement regimes. The first is confinement inside the bulk dielectric and the second is a local lightning-rod regime where the field is locally enhanced at sharp corners and may yield a vanishing mode volume without necessarily enhancing the mode inside the bulk dielectric. We show that while the bulk regime is reminiscent of the confinement in conventional nanocavities, the most commonly used definition of the mode volume gauges in fact the lightning-rod effect when applied to ultra-compact cavities, such as bowties. Distinguishing between these two regimes will be crucial for future research on nanocavities, and our insights show how to obtain strongly enhanced light-matter interaction over large bandwidths., Comment: 9 pages, 5 figures, 39 references

Confining photons in cavities enhances the interactions between light and matter. In cavity optomechanics, this enables a wealth of phenomena ranging from optomechanically induced transparency to macroscopic objects cooled to their motional ground state. Previous work in cavity optomechanics employed devices where ubiquitous structural disorder played no role beyond perturbing resonance frequencies and quality factors. More generally, the interplay between disorder, which must be described by statistical physics, and optomechanical effects has thus far been unexplored. Here, we demonstrate how sidewall roughness in air-slot photonic-crystal waveguides can induce sufficiently strong backscattering of slot-guided light to create Anderson-localized modes with quality factors as high as half a million and mode volumes that are below the diffraction limit. We observe how the interaction between these disorder-induced optical modes and in-plane mechanical modes of the slotted membrane is governed by a distribution of coupling rates, which can exceed $g_{\text{o}}/2\pi\sim 200$ kHz, leading to mechanical amplification up to self sustained oscillations via optomechanical backaction. Our work constitutes the first steps towards understanding optomechanics in the multiple-scattering regime and opens new perspectives for exploring complex systems with multitude mutually-coupled degrees of freedom.

Nanotechnology enables in principle a precise mapping from design to device but relied so far on human intuition and simple optimizations. In nanophotonics, a central question is how to make devices in which the light-matter interaction strength is limited only by materials and nanofabrication. Here, we integrate measured fabrication constraints into topology optimization, aiming for the strongest possible light-matter interaction in a compact silicon membrane, demonstrating an unprecedented photonic nanocavity with a mode volume of $V\sim3\times10^{-4}\,\lambda^3$, quality factor $Q\sim1100$, and footprint $4\,\lambda^2$ for telecom photons with a $\lambda\sim 1550$ nm wavelength. We fabricate the cavity, which confines photons inside 8 nm silicon bridges and use near-field optical measurements to perform the first experimental demonstration of photon confinement to a single hotspot well below the diffraction limit in dielectrics.

We combine experimental and theoretical approaches to explore excited rotational states of molecules embedded in helium nanodroplets using CS$_2$ and I$_2$ as examples. Laser-induced nonadiabatic molecular alignment is employed to measure spectral lines for rotational states extending beyond those initially populated at the 0.37 K droplet temperature. We construct a simple quantum mechanical model, based on a linear rotor coupled to a single-mode bosonic bath, to determine the rotational energy structure in its entirety. The calculated and measured spectral lines are in good agreement. We show that the effect of the surrounding superfluid on molecular rotation can be rationalized by a single quantity -- the angular momentum, transferred from the molecule to the droplet., Comment: 5 pages, 4 figures; 5 pages, 3 figures