Monday, 12 October 2026
The wave counterparts of classical particle paths and geometrical-optics rays are families of trajectories – patterns of streamlines – modified by a ‘wave (e.g. quantum) potential’. Wave interference corresponds to undulations in these trajectories, as envisaged in optics by Isaac Newton. Streamline patterns are dominated by singularities at wave vortices and stagnation points. The local momentum (phase gradient of the wave), can exceed the values classically allowed. Regions of such ‘superoscillations’ are bounded by manifolds where the quantum potential is zero. Some classical ‘curl forces’ – not the gradient of a potential – are associated with Hamiltonians (dispersion relations) anisotropic in momentum components, with unusual group velocity field singularities, and eigenfunctions with unfamiliar classical counterparts. For simple dispersion relations, some singularities coincide; for general dispersion, they are separate.
Berry's contributions in the early days
I will discuss how various well-known models for classical active systems, such as single-particle active Brownian motion and many-particle Vicsek-type models, and, in general, non-reciprocal interactions, can be microscopically obtained in the semiclassical limits of monitored quantum systems. The latter are monitored via continuous weak measurements and quantum feedback.
Tuesday, 13 October 2026
We show that surface waves in a draining-bathtub vortex provide a hydrodynamic realization of both Aharonov-Bohm phase shifts and Lense-Thirring frame dragging within a single system. A static time transformation maps the flat (2+1)-dimensional wave equation onto the convected shallow-water equation, yielding an effective vector potential set by the background flow. In this geometry, the circulation defines a global phase holonomy that controls wave structure. Traveling waves exhibit wavefront dislocations characteristic of Aharonov-Bohm scattering, while standing-wave superpositions produce nodal patterns that rotate at an angular velocity fixed by the circulation, providing a direct analogue of frame dragging. For noninteger circulation, the problem is naturally defined on the universal cover, ensuring single-valued partial-wave solutions. Experiments on a controlled vortex confirm these predictions and establish a laboratory platform in which topological phase and inertial effects, central to gauge and gravitational physics, emerge from a measurable velocity field.
In the first part of the lecture I shall provide a brief review of the method:
Given a quantum Hamiltonian H as a Hermitian matrix of an arbitrary large but finite dimension $N$. A graph on N vertices is associated with the matrix. Graph theory guarantees that an initial quantum state expressed as a vector in the space of the graph directed edges, evolves in time by a unitary matrix U – the quantum analogue of a discrete classical Poincare' map. U is written in terms of H and it is a function of the energy. The intimate connection between H and U is evident from the known theorem that the spectrum of H is the set of E values where U has 1 as an eigenvalue (if and only if). The semi classical evolution is written in terms of a Markovian matrix with elements which are the absolute square of the corresponding elements of U. Ergodic theory provides an expression for the Lyapunov exponent per trajectory on the graph. The mean Lyapunov exponent (with respect to the distribution of all graph trajectories) and its variance follow.
In the second part I shall describe the application of the formalism to four random matrix ensembles :
The adjacency matrices of random d regular graphs, GOE , GUE and the tri diagonal Dumitriu-Edelman ensembles. We computed the mean values of the Lyapunov exponents, variances, and thermal averages both numerically and by deriving approximate expressions. The results of the two methods agree very well in most circumstances. We also computed the spectral distributions of the stochastic evolution matrices, and in particular their gap distributions.
I shall conclude with a summary and a list of open problems.
This lecture marks the fiftieth anniversary of Berry and Tabor’s classic 1977 paper on energy-level statistics of integrable quantum systems. Written in the autumn of 1976 during Michael Berry’s three-month stay at IISc Bangalore, the paper has made a foundational contributions to quantum chaos, with an influence extending well beyond the field. After providing some historical context, I will review the paper’s impact over the past five decades and explore its connection to long-standing open problems in number theory and dynamical systems.
Hyperbolic surfaces are amazing toy models that can be used to study chaos, thanks to their negative curvature and rich algebraic structure. The aim of this talk is to present recent breakthroughs in the study of hyperbolic surfaces, due to the introduction of probabilistic ideas and methods.
Resurgent asymptotics leads to remarkable new insights into the connection between perturbative physics and nonperturbative physics. Michael Berry has made profound contributions to the underlying ideas, as well as presenting some beautiful physical examples. I will discuss several illustrative examples of resurgent asymptotics in quantum mechanics, matrix models and quantum field theory.
Wednesday, 14 October 2026
Distinct phases of quantum systems, such as different symmetry-protected topological (SPT) phases, are separated by critical points. However, critical points, dubbed unnecessary quantum critical points (UQCPs), can also appear within a single phase and do not separate two distinct phases. In this work, we investigate the geometric structure of such UQCPs in the parameter space of quantum systems and their relation to SPT phases. We show, using free-fermionic systems in various dimensions, that the locus of UQCPs arises from the phase boundaries of SPT phases. The manifold of UQCPs in the parameter space forms a geometric ``pencil'' whose axis coincides with this SPT phase boundary. This is work done in collaboration with Basudeb Mondal (IISc).
Reflecting a polarized beam of light at a single or multilayer dielectric interface leads to a variety of electromagnetic-wave optical effects that depend intrinsically on the underlying geometric phase and conservation of angular momentum. The observed effects in this context include in- and perpendicular-to-the-plane-of-incidence beam shifts, spectral and temporal shifts, as well as a complicated polarization structure and complex orbital and spin current flows around polarization singularities. All of these wave-optical phenomena are understood to arise from the superposition of composite electromagnetic plane waves. The role of the geometric phase in generating optical beams with a twisted phase structure, arising from the spin-orbit interaction of light, is central to our exploration of this fundamental area of research. Theoretical simulations and experimental results that justify ‘complicated’ effects arising from a simple optical system will be explored in the presentation.
The geometry of quantum states has drawn increasing attention in condensed matter physics, particularly in nonlinear electronic and optical responses and in adiabatic state-preparation protocols. The state space of quantum mechanics is naturally endowed with a Riemannian metric, a symplectic structure, and a complex structure, which together underlie familiar objects such as the quantum geometric tensor — whose real and imaginary parts are the quantum metric and the Berry curvature. Rather than treating these as isolated quantities that surface in individual calculations, we develop a unified geometric treatment of the dynamics of an entire congruence of flows on the curved manifold of states, described through projectors. This is precisely the setting of the Raychaudhuri equation, which characterises the evolution of geodesic congruences in curved spacetime. We identify unavoidable quantum-mechanical consequences of such flows, including caustics and tidal anisotropies. Building on this, we show that Boltzmann transport and adiabatic dynamics generate flows embedded in the full state space, with the associated responses, such as berry curvature dipole, being intricately linked to the expansion, shear, and vorticity of the congruence
"Spontaneous parametric down-conversion (SPDC), the process of converting a pump photon into an entangled pair of lower-frequency photons inside a nonlinear crystal, is undeniably the workhorse of modern quantum optics. It serves as the main resource for breakthroughs in quantum communication, sensing, imaging, photonic computing, and testing foundational questions in quantum physics. Yet the apparent simplicity of this process hides a remarkably rich structure in the generated quantum states. While conservation laws dictate which photon pairs can be generated, understanding the precise states we actually create requires a much deeper look at the interplay between nonlinear optics and quantum correlations. Whether observing isolated photon pairs in the weak-pumping regime or complex multiphoton states at higher gains, SPDC offers access to incredibly diverse forms of entanglement. Capturing the full richness of these states, however, remains challenging. Common approximations simplify the description but can obscure important features, while characterising high-dimensional states is often difficult and time-consuming.
In this talk, I will share our recent progress in characterising biphoton entanglement using digital holography and phase retrieval techniques. I will discuss new findings that reveal unexpected complexities in SPDC-generated states, challenging some of our field’s most familiar assumptions about the photon pairs we routinely produce. Finally, I will explore three specific applications across quantum imaging, sensing, and communication, demonstrating how these high-dimensional complex states can be transformed from an experimental hurdle into a distinct, practical advantage. "
Thursday, 15 October 2026
We present a simple, visual proof of the spin-statistics theorem, based on the suggestion that histories of spinning particles should be represented not by worldlines but by ribbons. This suggestion is motivated and justified by the coadjoint orbit analysis of the symmetry group. Our argument applies whenever the symmetry group topologically retracts to the Euclidean group $E(3)$, implying that neither special relativity nor even Galileian relativity is needed to prove the theorem. We also recover the Berry–Robbins geometric phase connection from this more general topological viewpoint.
We present some results concerning chiral configurations of point particles in Euclidean space, including a principle bundle characterisation; geometry, dynamics and integrability; and a connection to geometric phases. This is joint work with Michael Berry.
The Event Horizon Telescope collaboration has presented remarkable images of polarised radiation from the immediate surroundings of the central black holes in M87 and in our own galaxy. These are constructed from the measured correlations between signals received by telescopes scattered over an entire hemisphere of the earth. Aberrations which modify the incoming electric fields in phase and amplitude are a major issue. One long standing solution is to construct 'closure quantities' which cancel the aberrations, using measured correlations between antenna pairs chosen to form a closed loop. The generalisation from complex scalar fields to the more realistic case of polarised EM fields is fairly recent and mathematically richer. This talk covers our joint work on this problem, which leads to a non-abelian GL(2,c) gauge theory on a graph. This has antennas as vertices and pairs with measured correlations as edges. . A complete set of gauge invariants which cancel the aberrations can be identified. The inclusion of measurement noise makes this a (Euclidean) quantum theory. Applications of this framework to real data are now emerging.
We survey exact sum rules for the spectral zeta functions of homogeneous 1D quantum anharmonic oscillators. Those are countably many numerical identities on the zeta values at positive integers, that mainly stem from the exact WKB method.
The Riemann Hypothesis has captivated mathematicians for more than a century, but its allure extends naturally to theoretical physics: the possibility that a deep arithmetic regularity may emerge from the simple logic of a physical system is hard to resist. Here we pursue this idea by recasting the Generalised Riemann Hypothesis in the language of random walks. The key ingredients are the strikingly aleatory behaviour of Möbius coefficients on square-free integers and of Dirichlet characters on prime numbers. Their statistical randomness can be tested with remarkable precision. From this viewpoint, the critical line Re s=1/2 acquires a natural physical meaning: it is the universal 1/2 scaling exponent characteristic of unbiased random walks.
Friday, 16 October 2026
The Wave and Particle nature of light is well understood in the context of double slit interference. Ascertaining the path of the photon, namely the particle nature of light, leads to the collapse of the wave function and hence resulting in no interference pattern. This wave particle duality can be precisely quantified in terms of the path predictability (P), visibility (V) and surprisingly a geometric measure of entanglement (E). In the general multislit interferometry a precise triality relationship emerges P^2 + V^2 +E^2 =1, which has found experimental verification. The geometric measure of entanglement, the so-called I-concurrence, is found to be related to the area between the constituent vectors of the Hilbert Space. This quantum mechanical treatment of the wave particle duality is in the presence of detectors to quantify the path predictability of the quanton. We illustrate the manifestation of the universal triality relation in various physical scenarios, both in optics and other multi-particle systems. Interestingly, it also manifests in Neutrino oscillations which may be verifiable in present day experience.
"The optical propagation of monochromatic light through simple linear optical components is fully modelled through plane-wave decomposition. As is familiar to any observation of a swimming pool, the ripples on the surface of the pool form highly aberrated lenses creating caustic patterns of partially focussed sunlight on the bottom of the pool. But what happens if the pool is much deeper?
For both monochromatic and polychromatic plane waves their propagation through random but smoothly varying phase masks is modelled and the transition from a uniform intensity cross section to other cross sections is examined. In the far-field, i.e. a very deep pool, monochromatic light gives rise to optical speckle with its characteristic exponential distribution of intensity values. However, in an interim plane, caustic-like patters are formed, and these patterns are distinctly non exponential in their intensity values and have positions of extreme intensity. These positions of extreme intensity resemble rare, rouge-wave type events. If rouge-waves are defined solely in terms of the creation and observation of extreme events, then here we seem to have rouge-wave formation in the absence of any non-linearity?
For polychromatic light these caustic-like, rouge-waves, persist but of course the far-field becomes more complicated. In the far-field the positions of the optical vortices is wavelength dependent giving rise to the subtle colours of darkness associated with complex white light fields."
The complexity e.g. due to presence of disorder or complicated interactions in many realistic physical situations can lead to hamiltonians that include randomness (in physically natural bases). The latter in turn manifest in the typical eigenstate of the Hamiltonian. Quantum properties such as the quantum metric tensor G for describing distances between parameterised families of quantum states or geometric curvature C governing the geometric phase of transported quantum states are then best described by a statistical approach. In this talk, I will discuss the probability distribution of the quantum metric as well as curvature in a natural minimal model i.e an ensemble of three parameters Hamiltonians represented by standard random matrix ensembles.
TBA
In this talk, I will present a quantitative theory for the emergence of quantum many-body chaos as integrability is broken via a tunable parameter. In a circuit model of free fermions, 'doped' with a tunable density of integrability-breaking gates, we uncover the microscopic mechanisms underpinning the crossover from early-time integrable behaviour to late-time chaos through the lens of the out-of-time-ordered correlators (OTOCs). The integrability-breaking gates act as local, in spacetime, hotspots which locally amplify the OTOCs such that an accumulation of them eventually leads to fully-developed chaos. We identify the explicit characteristic time and length scales governing this crossover, as well as the dependence of the chaotic OTOC characteristics -- such as the butterfly velocity and front broadening -- on the integrability-breaking parameter. [Based on joint work with S. Biswas and R. Moessner, arXiv:2607.02506].