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We review the design, theory, and applications of two dimensional periodic lattices hosting conical intersections in their energy-momentum spectrum. The best known example is the Dirac cone, where propagation is governed by an effective Dirac equation, with electron spin replaced by a "fermionic" half-integer pseudospin. However, in many systems such as metamaterials, modal symmetries result in the formation of higher order conical intersections with integer or "bosonic" pseudospin. The ability to engineer lattices with these qualitatively different singular dispersion relations opens up many applications, including superior slab lasers, generation of orbital angular momentum, zero-index metamaterials, and quantum simulation of exotic phases of relativistic matter.

An ab initio analytical model for Fano resonances in two-channel systems is presented. We find that the lineshape parameter q factors into a background contribution q_b that depends only on the uncoupled channels and an interaction contribution q_i that is affected by the coupling between the channels, revealing how the overall lineshape parameter q may be controlled. In particular, we show how conical intersections of the background phase shifts have an important role in the interplay between q_b and q_i. Finally, control of Fano transmission profiles through q_b and q_i is demonstrated for quantum billiards.

An ab initio analytical model for Fano resonances in two-channel systems is presented. We find that the lineshape parameter q factors into a background contribution q_b that depends only on the uncoupled channels and an interaction contribution q_i that is affected by the coupling between the channels, revealing how the overall lineshape parameter q may be controlled. In particular, we show how conical intersections of the background phase shifts have an important role in the interplay between q_b and q_i. Finally, control of Fano transmission profiles through q_b and q_i is demonstrated for quantum billiards.

An ab initio analytical model for Fano resonances in two-channel systems is presented. We find that the lineshape parameter q factors into a background contribution q_b that depends only on the uncoupled channels and an interaction contribution q_i that is affected by the coupling between the channels, revealing how the overall lineshape parameter q may be controlled. In particular, we show how conical intersections of the background phase shifts have an important role in the interplay between q_b and q_i. Finally, control of Fano transmission profiles through q_b and q_i is demonstrated for quantum billiards.

Nonadiabatic effects are ubiquitous in physics, chemistry and biology. They are strongly amplified by conical intersections (CIs) which are degeneracies between electronic states of triatomic or larger molecules. A few years ago it has been revealed that CIs in molecular systems can be formed by laser light even in diatomics. Due to the prevailing strong nonadiabatic couplings, the existence of such laser-induced conical intersections (LICIs) may considerably change the dynamical behavior of molecular systems. By analyzing the photodissociation process of the D2+ molecule carefully, we found a robust effect in the angular distribution of the photofragments which serves as a direct signature of the LICI providing undoubted evidence for its existence.

Conical intersections (CoIn) dominate the pathways and outcomes of virtually all photophysical and photochemical molecular processes. Despite extensive experimental and theoretical effort, CoIns have not been directly observed yet and the experimental evidence is being inferred from fast reaction rates and some vibrational signatures. We show that short X-ray (rather than optical) pulses can directly detect the passage through a CoIn with the adequate temporal and spectral sensitivity. The technique is based on a coherent Raman process that employs a composite femtosecond/attosecond X-ray pulse to detect the electronic coherences (rather than populations) that are generated as the system passes through the CoIn.

We observe energy-dependent angle-resolved diffraction patterns in protons from strong-field dissociation of the molecular hydrogen ion H$_2^+$. The interference is a characteristic of dissociation around a laser-induced conical intersection (LICI), which is a point of contact between two surfaces in the dressed 2-dimensional Born-Oppenheimer potential energy landscape of a diatomic molecule in a strong laser field. The interference magnitude and angular period depend strongly on the energy difference between the initial state and the LICI, consistent with coherent diffraction around a cone-shaped potential barrier whose width and thickness depend on the relative energy of the initial state and the cone apex. These findings are supported by numerical solutions of the time-dependent Schr\"{o}dinger equation for similar experimental conditions.

We find that energy surfaces of more than two atoms or molecules interacting via dipole-dipole po- tentials generically possess conical intersections (CIs). Typically only few atoms participate strongly in such an intersection. For the fundamental case, a circular trimer, we show how the CI affects adiabatic excitation transport via electronic decoherence or geometric phase interference. These phe- nomena may be experimentally accessible if the trimer is realized by light alkali atoms in a ring trap, whose dipole-dipole interactions are induced by off-resonant dressing with Rydberg states. Such a setup promises a direct probe of the full many-body density dynamics near a conical intersection.

We show that dressing of diatomic molecules by running laser waves gives rise to conical intersections (CIs). Due to presence of such CIs, the rovibronic molecular motions are strongly coupled. A pronounced impact of the CI on the spectrum of $Na_2$ molecule is demonstrated via numerical calculation for weak and moderate laser intensity, and an experiment is suggested on this basis. The position of the light induced CI and the strength of its non-adiabatic couplings can be chosen by changing the frequency and intensity of the used running laser wave. This offers new possibilities to control the photo-induced rovibronic molecular dynamics.

For two states of opposite parity that cross as a function of an external magnetic field, the addition of an electric field will break the symmetry and induce an avoided crossing. A suitable arrangement of fields may be used to create a conical intersection as a function of external spatial coordinates. We consider the effect of the resulting geometric phase for ultracold polar molecules. For a Bose-Einstein condensate in the mean-field approximation, the geometric phase effect induces stable states of persistent superfluid flow that are characterized by half-integer quantized angular momentum.

In this paper we present a constructive method to control the bilinear Schr\"odinger equation via two controls. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, and if the latter are conical (as it happens generically). We provide sharp estimates of the relation between the error and the controllability time.

Light-induced conical intersections (LICIs) can be formed both by standing or by running laser waves. The position of a LICI is determined by the laser frequency while the laser intensity controls the strength of the nonadiabatic coupling. Recently, it was shown within the LICI framework that linearly chirped laser pulses have an impact on the dissociation dynamics of the $D_{2}^{+}$ molecule (J. Chem. Phys. 143, 014305, (2015); ibid 144, 074309, (2016)). In this work we exploit this finding and perform calculations using chirped laser pulses in which the time dependence of the laser frequency is designed so as to force the LICI to move together with the field-free vibrational wave packet as much as possible. Since nonadiabaticity is strongest in the vicinity of the conical intersection, this is the first step towards controlling the dissociation process via the LICI. Our showcase example is again the $D_{2}^{+}$ molecular ion. To demonstrate the impact of the LICIs on the dynamical properties of diatomics, the total dissociation probabilities and the population of the different vibrational levels after the dissociation process are studied and discussed.

This book chapter is a general introduction to photo-dynamics. An outlined structure to the theoretical studies on classical/quantum/mixed dynamics were described from the literature point of view. Such methods get constrained for the practical applications due to parametrizations/computational cost. As an alternate TDDFT-based dynamics studies are promising. we have shown that the TDDFT calculations could be as accurate as high quality monte-carlo calculations and the reliability of the TDDFT calculations for locating the conical intersection through the simple ring-opening reaction of oxirane. Pros and cons of TDDFT for photo-dynamics calculations have been presented for the development of this fast evolving field.

We consider a fully quadratic vibronic model Hamiltonian for studying photoinduced electronic transitions through conical intersections. Using a second order perturbative approximation for diabatic couplings we derive an analytical expression for the time evolution of electronic populations at a given temperature. This formalism extends upon a previously developed perturbative technique for a linear vibronic coupling Hamiltonian. The advantage of the quadratic model Hamiltonian is that it allows one to use separate quadratic representations for potential energy surfaces of different electronic states and a more flexible representation of interstate couplings. We explore features introduced by the quadratic Hamiltonian in a series of 2D models, and then apply our formalism to the 2,6-bis(methylene) adamantyl cation, and its dimethyl derivative. The Hamiltonian parameters for the molecular systems have been obtained from electronic structure calculations followed by a diabatization procedure. The evolution of electronic populations in the molecular systems using the perturbative formalism shows a good agreement with that from variational quantum dynamics.

Adequate simulation of non-adiabatic dynamics through conical intersection requires account for a non-trivial geometric phase (GP) emerging in electronic and nuclear wave-functions in the adiabatic representation. Popular mixed quantum-classical (MQC) methods, surface hopping and Ehrenfest, do not carry a nuclear wave-function to be able to incorporate the GP into nuclear dynamics. Surprisingly, the MQC methods reproduce ultra-fast interstate crossing dynamics generated with the exact quantum propagation so well as if they contained information about the GP. Using two-dimensional linear vibronic coupling models we unravel how the MQC methods can effectively mimic the most significant dynamical GP effects: 1) compensation for repulsive diagonal second order non-adiabatic couplings and 2) transfer enhancement for a fully cylindrically symmetric component of a nuclear distribution.

It is shown that the Dirac fermion structures created in the middle of the Landau bands in the vortex-lattice state of a pure 2D strongly type-II superconductor at half-integer filling factors can be effectively controlled by the external magnetic field. The resulting field-induced modulation of the magneto-oscillations is shown to arise from Fermi-surface resonance scattering in the vortex core regions. Possible observation of the predicted effect in a quasi 2D organic superconductor is discussed.

We show that conical intersections can be created in laboratory coordinates by dressing a parabolic trap for ultracold atoms or molecules with a combination of optical and static magnetic fields. The resulting ring trap can support single-particle states with half-integer rotational quantization and many-particle states with persistent flow. Two well-separated atomic or molecular states are brought into near-resonance by an optical field and tuned across each other with an inhomogeneous magnetic field. Conical intersections occur at the nodes in the optical field.