Talks (titles and abstracts)
Simon Becker: Convergence rates for the classical Trotter formula and a Gaussian Solovay-Kitaev theorem
The Trotter product formula is a classical formula used in numerical analysis to decompose the time-evolution of a generator A=B+C into an interactive product, thus separating B from C. I will highlight a simple but universal method to obtain explicit (and often sharp) convergence rates which were previously not even known for the Coulomb Hamiltonian in quantum mechanics. I will then describe how similar ideas are used to establish a Solovay-Kitaev theorem which is considered to be a key ingredient of modern quantum computing.
This is joint work with N. Galke, R. Salzmann, L. v. Luijk as well as L .Lami, C. Rouze and N. Datta.
Oscar P. Bruno: Green-function and high-frequency asymptotic methods for remote sensing and design
We present novel Green-function and high-frequency asymptotic methods for solution of problems of propagation and scattering of time-harmonic and transient linear waves, with applications to remote sensing as well as simulation, optimization and design of electromagnetic structures.
Richard Craster: Non-reciprocity for the time-modulated wave equation and diffusion equation through the lens of high-order homogenization
Laminated media with material properties modulated in space and time in the form of travelling waves have long been known to exhibit non-reciprocity.
However, when using the method of low frequency homogenisation, it was so far only possible to obtain non-reciprocal effective media when both material properties are modulated in time, in the form of a Willis-coupling (or bi-anisotropy in electromagnetism) model. If only one of the two properties is modulated in time, while the other is kept constant, it was thought impossible for the method of homogenisation to recover the expected non-reciprocity since this Willis-coupling coefficient then vanishes. Contrary to this belief, we show that effective media with a single time-modulated parameter are non-reciprocal, provided homogenization is pushed to the second order. This is illustrated by numerical experiments (dispersion diagrams and time-domain simulations) for a bilayered modulated medium.
Bryn Davies: The dynamical systems view of waves in quasicrystals: super band gaps and limiting statistics.
Quasicrystals have exotic spectra that are challenging to understand and are the basis of several longstanding and famous problems in spectral analysis. There is also signi cant recent excitement about utilising their exotic spectra (which typically feature fractal spectra of many spectral gaps) for wave control applications. Many of the important spectral theoretic results for one-dimensional quasicrystals are based on reformulating the problem as a dynamical system. In this talk, we will explore this connection in the particular setting of materials generated by Fibonacci tilings and an associated non-linear dynamical system. This non-linear second-order recursion relation describes the Lyapunov functions associated to the sequence of periodic operators and has facilitated numerous breakthroughs. On the one hand, we can study how the spectral band gaps evolve under the Fibonacci tiling rule and prove the existence of "super band gaps" (band gaps that exist for all suciently large tilings in the sequence). This characterises how periodic approximants (supercells) of Fibonacci quasicrystalline materials faithfully reproduce the main spectral gaps.
Conversely, the density of states associated to the di erential operators can be used to predict the statistics of the non-linear recursion relation, in spite of the apparent lack of an invariant measure. In both cases, these problems were solved by exploiting the connection between the two elds.
Florian Feppon: Asymptotic expansions for Stokes flows in periodic porous media
Periodic porous media are the analogous of wave guides and photonic crystals for fluid flows. In this talk, I will present two recent works about the asymptotic analysis of Stokes flows in such media:
(i) the asymptotic expansion of a Stokes flow in a finite periodic channel, where we establish the exponential convergence of the fluid velocity towards the sum of a periodic (propagating) and an exponentially decaying (evanescent) mode;
(ii) the full asymptotic expansion of the permeability matrix associated to a periodic array (crystal) of small solid obstacles, which captures the effective property of the medium.
The analysis brings into play interesting mathematical concepts, such as boundary layers, layer potentials, Hadamard finite parts and polarization tensors.
Erik Hiltunen: Coupled harmonics in time-modulated scattering systems
We consider the resonance and scattering properties of a composite medium containing scatterers whose properties are modulated in time. The temporal modulation induces a coupling between wave harmonics whose frequencies differ by the modulation frequency, described as a folding of the frequency axis into a temporal Brillouin zone. We develop an integral-operator characterization of the resonances and band structure of time-dependent scatterers, and present small-volume asymptotic formulas analogous to the classical results for the static (unmodulated) case. As an effect of broken energy conservation, we show that the scattering coefficients may blow up when (complex) resonances cross the real axis. We also show how the integral-operator approach can be leveraged for efficient numerical calculation of the scattering properties of time-dependent materials.
Simon Horsley: Quantum and classical wave in time varying materials
When material properties vary in time, wave energy is no longer conserved. This simple fact not only gives us a novel means to amplify or absorb wave energy, but - when combined with spatial inhomogeneities - allows us to reshape electromagnetic modes in an arbitrary way. In this talk I will discuss first, the classical mathematical description of waves in dispersive time varying media; borrowing some methods from quantum mechanics. Secondly I will discuss some new quantum effects that arise in dispersive time varying media due to the quantum mechanics of absorbing media.
Bowen Li: Variation analysis for irreversible Lindblad dynamics
The use of Lindblad equations to prepare thermal and ground states has received an increasing amount of attention in quantum computing, yet there have been not many analytical results regarding its convergence rate. In this talk, we will discuss some recent progress in the analytical understanding of the equilibration mechanism of the Lindblad equation, drawing connections with the study of classical dynamics. We shall investigate the mixing properties of primitive hypocoercive Lindblad dynamics. By extending the variational framework originally developed for underdamped Langevin dynamics, we derive fully explicit and constructive exponential decay estimates for the convergence of these dynamics in the L2-norm. Our analysis relies on establishing a quantum analog of space-time Poincare inequalities. To complement these hypocoercive estimates, we also analyze the limiting behavior of the spectral gap for Lindblad dynamics with a large coherent contribution, providing sharper convergence rate estimates in this asymptotic regime.
Svitlana Mayboroda: tba
Graeme Milton: Space-Time Metamaterials: energy conserving temporal interfaces and new field patterns
It is now well known that a temporal interface between two media can act to time reverse waves, creating, after a delay, an image of the original source. However, this requires an input of energy into the wave at the temporal interface. We show how energy can be conserved at a temporal metasurface. If one thinks of lattices of masses and springs the basic idea is that the spring stiffnesses can be changed at the moments where they are unstretched, with no change in energy. The imaging nature of the interface is still preserved.
We also report on new field pattern materials. Introduced by Ornella Mattei and myself, these are space time microstructures such that when waves generated by a point source propagate through them, they form patterns - the field patterns. The work on energy conserving temporal interfaces is joint with Kshiteej Deshmukh, and the work on new field patterns is with my undergraduate class, Mellisa Carter, Randen Davis, Eden Heyen, Zev Katz, Owen Koppe, Kimball Schipaanboord, Kyle Smith, Tyler Trotter, Evelyn Van Den Akker, Yolotl Villarreal, and Aiden Wilde.
John Schotland: Quantum optics in random media
Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics. The goal is to understand the propagation of nonclassical states of light in systems consisting of many atoms. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.
Hai Zhang: Mathematics of in-gap interface modes in topological photonic/phononic structures
The developments of topological insulators have provided a new avenue of creating interface modes (or edge modes) in photonic/phononic structures. In this talk, we will report recent progress on establishing the existence of in-gap interface modes in various topological photonic/phononic structures.
Yi Zhu: Schrödinger operators with super honeycomb potentials: double Dirac cone and topological edge states
Honeycomb lattice plays an important role in the field of topological materials. Numerous efforts have been put to understand this delicate structure, such as the existence of conically degenerate spectrum point (Dirac points), topologically protected edge states, topology characterizations and so on. In this talk, I shall focus on recent works of waves in superhoneycomb structures. More specifically, we prove that the bulk structure admits double Dirac cones on its energy bands, and two branches of topological edge states are bifurcated under perturbations even with a PT symmetry.
Maciej Zworski: Overdamped quasinormal modes for Schwarzschild black holes
We prove a lower bound for the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild–de Sitter (SdS) black holes in discs (in fact we give asymptotics in conic neighbourhoods of the real axis). This shows that Jezequel’s recent global upper bound (in the case of SdS) is sharp. The argument is an application of spectral asymptotics results for non-self-adjoint operators which provide a finer description of QNM, explaining the emergence of a distorted lattice "deep in the complex" and generalizing a rougher lattice structure. The precise asymptotics are available in the region where numerics break down due to pseudospectral effects, a phenomenon which we will illustrate experimentally. The talk is based on joint work with M Hitrik.