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The coupling of matter to the quantized electromagnetic field of a plasmonic or optical cavity can be harnessed to modify and control chemical and physical properties of molecules. In optical cavities, a term known as the dipole self-energy (DSE) appears in the Hamiltonian to assure gauge invariance. The aim of this work is twofold. First, we introduce a method, which has its own merits and complements existing methods, to compute the DSE. Second, we study the impact of the DSE on cavity-induced nonadiabatic dynamics in a realistic system. For that purpose, various matrix elements of the DSE are computed as functions of the nuclear coordinates and the dynamics of the system after laser excitation is investigated. The cavity is known to induce conical intersections between polaritons, which gives rise to substantial nonadiabatic effects. The DSE is shown to slightly affect these light-induced conical intersections and, in particular, break their symmetry.

Placing an atom and a molecule in a cavity opens the door to initialize molecular dynamics by exciting a level of the atom. This approach enlarges the range of choosing the light source to trigger molecular dynamics substantially. The interplay of the atomic, molecular and photonic populations gives rise to rich dynamics. The cavity photon plays the role of a mediator between the atom and the molecule and it is found that the photonic population is rather low throughout and its evolution follows that of the molecule. Cavities are known to be subject to losses. In spite of the losses it is demonstrated that the presence of the atom gives rise to a long-lived dynamics which should be of relevance for experimental investigations. The presence of more atoms and molecules is expected to further enrich the dynamics.

The exchange of energy between electronic and nuclear motion is the origin of non-adiabaticity and plays an important role in many molecular phenomena and processes. Conical intersections (CIs) of different electronic potential energy surfaces lead to the most singular non-adiabaticity and have been intensely investigated. The coupling of light and matter induces conical intersections which are termed light-induced conical intersections (LICIs). There are two kinds of LICIs, those induced by classical (laser) light and those by quantum light like that provided by a cavity. The present work reviews the subject of LICIs, discussing the achievements made so far. Particular attention is paid to comparing classical and quantum LICIs, their similarities and differences and their relationship to naturally occurring CIs. In contrast to natural CIs, the properties of which are dictated by nature, the properties of their light-induced counterparts are controllable by choosing the frequency and intensity (or coupling to the cavity) of the external light source. This opens the door to inducing and manipulating various kinds of non-adiabatic effects. Several examples of diatomic and polyatomic molecules are presented covering both dynamics and spectroscopy. The computational methods employed are discussed as well. To our opinion, the young field of LICIs and their impact show much future potential.

The use of cavities to impact molecular structure and dynamics has become popular. As cavities, in particular plasmonic nanocavities, are lossy and the lifetime of their modes can be very short, their lossy nature must be incorporated into the calculations. The Lindblad master equation is commonly considered as an appropriate tool to describe this lossy nature. This approach requires the dynamics of the density operator and is thus substantially more costly than approaches employing the Schr\"odinger equation for the quantum wave function when several or many nuclear degrees of freedom are involved. In this work we compare numerically the Lindblad and Schr\"odinger descriptions discussed in the literature for a molecular example where the cavity is pumped by a laser. The laser and cavity properties are varied over a range of parameters. It is found that the Schr\"odinger description adequately describes the dynamics of the polaritons and emission signal as long as the laser intensity is moderate and the pump time is not much longer than the lifetime of the cavity mode. Otherwise, it is demonstrated that the Schr\"odinger description gradually fails. We also show that the failure of the Schr\"odinger description can often be remedied by renormalizing the wave function at every step of the time propagation. The results are discussed and analyzed.

It has been known that electronic conical intersections in a molecular system can also be created by laser light even in diatomics. The direct consequence of these light-induced degeneracies is the appearance of a strong mixing between the electronic and vibrational motions, which has a strong fingerprint on the ultrafast nuclear dynamics. In the present work, pump and probe numerical simulations have been performed with the NaH molecule involving the first three singlet electronic states (X1{\Sigma}+(X), A1{\Sigma}+(A) and B1{\Pi}(B)) and several light-induced degeneracies in the numerical description. To demonstrate the impact of the multiple light-induced non-adiabatic effects together with the molecular rotation on the dynamical properties of the molecule, the dissociation probabilities, kinetic energy release spectra (KER) and the angular distributions of the photofragments were calculated by discussing the role of the permanent dipole moment as well.

Ionization phenomena are widely studied for decades. With the advent of cavity technology, the question arises how quantum light affects molecular ionization. As the ionization spectrum is recorded from the neutral ground state, it is usually possible to choose cavities which exert negligible effect on the neutral ground state, but have significant impact on the ion and the ionization spectrum. Particularly interesting are cases where the ion exhibits conical intersections between close-lying electronic states, which gives rise to substantial nonadiabatic effects. Assuming single-molecule strong coupling, we demonstrate that vibrational modes irrelevant in the absence of cavity play a decisive role when the molecule is in the cavity. Here, dynamical symmetry breaking is responsible for the ion-cavity coupling and high symmetry enables control of the coupling via molecular orientation relative to the cavity field polarization. Significant impact on the spectrum by the cavity is found and shown to even substantially increase for less symmetric molecules.

Polaritons - hybrid light-matter states formed in cavity - strongly change the properties of the underlying matter. In optical or plasmonic nanocavities, polaritons decay by radiative emission of the cavity, which is accessible experimentally. Due to the interaction of a molecule with the quantized radiation field, polaritons exhibit light-induced conical intersections (LICIs) which dramatically influence the nuclear dynamics of molecular polaritons. We show that ultrafast radiative emission from the lower polariton is controlled by the geometric phase imposed by the LICI. This finding provides insight into the process of emission and, furthermore, allows one to compute these signals by augmenting the Born-Oppenheimer approximation for polaritons with a geometric phase term.

The interaction of a molecule with the quantized electromagnetic field of a nano-cavity gives rise to light-induced conical intersections between polaritonic potential energy surfaces. We demonstrate for a realistic model of a polyatomic molecule that the time-resolved ultrafast radiative emission of the cavity enables to follow both nuclear wavepacket dynamics on and nonadiabatic population transfer between polaritonic surfaces without applying a probe pulse. The latter provides an unambiguous (and in principle experimentally accessible) dynamical fingerprint of light-induced conical intersections.

Nonadiabatic phenomena are investigated in the rovibrational motion of molecules confined in an infrared cavity. Conical intersections (CIs) between vibrational polaritons, similar to CIs between electronic polaritonic surfaces, are found. The spectral, topological, and dynamic properties of the vibrational polaritons show clear fingerprints of nonadiabatic couplings between molecular vibration, rotation and the cavity photonic mode. Furthermore, it is found that for the investigated system, composed of two rovibrating HCl molecules and the cavity mode, breaking the molecular permutational symmetry, by changing $^{35}$Cl to $^{37}$Cl in one of the HCl molecules, the polaritonic surfaces, nonadiabatic couplings, and related spectral, topological, and dynamic properties can deviate substantially. This implies that the natural occurrence of different molecular isotopologues needs to be considered when modeling realistic polaritonic systems.

The coupling of a molecule and a cavity induces non-adiabaticity in the molecule which makes the description of its dynamics complicated. For polyatomic molecules, reduced-dimensional models and the use of the Born-Oppenheimer approximation (BOA) may remedy the situation. It is demonstrated that contrary to expectation, BOA may even fail in a one-dimensional model and generally expected to fail in two- or more-dimensional models due to the appearance of conical intersections induced by the cavity.

Nonadiabaticity, i.e., the effect of mixing electronic states by nuclear motion, is a central phenomenon in molecular science. The strongest nonadiabatic effects arise due to the presence of conical intersections of electronic energy surfaces. These intersections are abundant in polyatomic molecules. Laser light can induce in a controlled manner new conical intersections, called light-induced conical intersections, which lead to strong nonadiabatic effects similar to those of the natural conical intersections. These effects are, however, controllable and may even compete with those of the natural intersections. In this work we show that the standard low-energy vibrational spectrum of the electronic ground state can change dramatically by inducing nonadiabaticity via a light-induced conical intersection. This generic effect is demonstrated for an explicit example by full-dimensional high-level quantum calculations using a pump-probe scheme with a moderate-intensity pump laser and a weak probe laser.

A quantum system composed of a molecule and an atomic ensemble, confined in a microscopic cavity, is investigated theoretically. The indirect coupling between atoms and the molecule, realized by their interaction with the cavity radiation mode, leads to a coherent mixing of atomic and molecular states, and at strong enough cavity field strengths hybrid atom-molecule-photon polaritons are formed. It is shown for the Na$_2$ molecule that by changing the cavity wavelength and the atomic transition frequency, the potential energy landscape of the polaritonic states and the corresponding spectrum could be changed significantly. Moreover, an unforeseen intensity borrowing effect, which can be seen as a strong nonadiabatic fingerprint, is identified in the atomic transition peak, originating from the contamination of the atomic excited state with excited molecular rovibronic states.

The absorption spectra of the cold Na2 molecule dressed by a linearly polarized standing laser wave is investigated. In the studied scenario the rotational motion of the molecules is frozen while the vibrational and translational degrees of freedom are accounted for as dynamical variables. In such a situation a light-induced conical intersection (LICI) can be formed. To measure the spectra a weak field is used whose propagation direction is perpendicular to the direction of the dressing field but has identical polarization direction. Although LICIs are present in our model, the simulations demonstrate a very robust absorption spectrum, which is insensitive to the intensity and the wavelength of the dressing field and which does not reflect clear signatures of light-induced nonadiabatic phenomena related to the strong mixing between the electronic, vibration and translational motions. However, by widening artificially the very narrow translational energy level gaps, the fingerprint of the LICI appears to some extent in the spectrum.

The theory of rovibronic spectroscopy of light-dressed molecules is presented within the framework of quantum mechanically treated molecules interacting with classical light fields. Numerical applications are demonstrated for the homonuclear diatomic molecule Na$_2$, for which the general formulae can be simplified considerably and the physical processes leading to the light-dressed spectra can be understood straightforwardly. The physical origin of different peaks in the light-dressed spectrum of Na$_2$ is given and the light-dressed spectrum is investigated in terms of its dependence on the dressing field's intensity and wavelength, the turn-on time of the dressing field, and the temperature. The important implications of light-dressed spectroscopy on deriving field-free spectroscopic quantities are also discussed.

Coherent control experiments in molecules are often done with shaped laser fields. The electric field is described classically and control over the time evolution of the system is achieved by shaping the laser pulses in the time or frequency domain. Moving on from a classical to a quantum description of the light field allows to engineer the quantum state of light to steer chemical processes. The quantum field description of the photon mode allows to manipulate the light-matter interaction directly in phase-space. In this paper we will demonstrate the basic principle of coherent control with quantum light on the avoided crossing in lithium fluoride. Using a quantum description of light together with the nonadiabatic couplings and vibronic degrees of freedoms opens up new perspective on quantum control. We show the deviations from control with purely classical light field and how back-action of the light field becomes important in a few photon regime.

Nonadiabatic effects appear due to avoided crossings or conical intersections that are either intrinsic properties in field-free space or induced by a classical laser field in a molecule. It was demonstrated that avoided crossings in diatomics can also be created in an optical cavity. Here, the quantized radiation field mixes the nuclear and electronic degrees of freedom creating hybrid field-matter states called polaritons. In the present theoretical study we go further and create conical intersections in diatomics by means of a radiation field in the framework of cavity quantum electrodynamics (QED). By treating all degrees of freedom, that is the rotational, vibrational, electronic and photonic degrees of freedom on an equal footing we can control the nonadiabatic quantum light-induced dynamics by means of conical intersections. First, the pronounced difference between the the quantum light-induced avoided crossing and the conical intersection with respect to the nonadiabatic dynamics of the molecule is demonstrated. Second, we discuss the similarities and differences between the classical and the quantum field description of the light for the studied scenario.

A fundamental theoretical framework is formulated for the investigation of rovibronic spectra resulting from the coupling of molecules to one mode of the radiation field in an optical cavity. The approach involves the computation of (1) cavity-field-dressed rovibronic states, which are hybrid light-matter eigenstates of the `molecule + cavity radiation field' system, and (2) the transition amplitudes between these field-dressed states with respect to a weak probe pulse. The predictions of the theory are shown for the homonuclear Na$_2$ molecule. The field-dressed rovibronic spectrum demonstrates undoubtedly that the Born--Oppenheimer approximation breaks down in the presence of the cavity radiation field. A clear fingerprint of the strong nonadiabaticity is found, which can only emerge in the close vicinity of conical intersections. In this work, the conical intersection is induced by the quantized radiation field, and it is thus called a "light-induced conical intersection" (LICI). Dependence of the cavity-field-dressed spectrum on the cavity-mode wavelength as well as on the light-matter coupling strength is investigated. Essential changes are identified in the spectra from the weak to the ultrastrong coupling regimes.

Rovibronic spectra of the field-dressed homonuclear diatomic Na2 molecule is in- vestigated to identify direct signatures of the light-induced conical intersection (LICI) on the spectrum. The theoretical framework formulated allows the computation of the (1) field-dressed rovibronic states induced by a medium intensity continuous-wave laser light and the (2) transition amplitudes between these field-dressed states with respect to an additional weak probe pulse. The field-dressed spectrum features ab- sorption peaks resembling the field-free spectrum as well as stimulated emission peaks corresponding to transitions not visible in the field-free case. By investigating the dependence of the field-dressed spectra on the dressing-field wavelength, both in full- and reduced dimensional simulations, direct signatures of the LICI can be identified. These signatures include (1) the appearance of new peaks and the splitting of peaks for both absorption and stimulated emission, and (2) the manifestation of an intensity borrowing effect in the field-dressed spectrum.

Conical intersections are degeneracies between electronic states and are very common in nature. It has been found that they can also be created both by standing or by running laser waves. The latter are called light-induced conical intersections. It is well known that conical intersections are the sources for numerous topological effects which are manifested e.g. in the appearance of the geometric or Berry phase. In one of our former works by incorporating the diabatic-to-adiabatic transformation angle with the line-integral technique we have calculated the Berry-phase of the light-induced conical intersections. Here we demonstrate that by using the time dependent adiabatic approach suggested by Berry the geometric phase of the light-induced conical intersections can also be obtained and the results are very similar to those of the time-independent calculations.

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., Comment: 21 pages, 4 figures, accepted in Faraday Discussions

In the presence of linearly varying frequency chirped laser pulses the photodissociation dynamics of D$_2^+$ is studied theoretically after ionization of D$_{2}$ . As a completion of our recent work (J. Chem. Phys. 143, 014305 (2015)) a comprehensive dependence on the pulse duration and delay time is presented in terms of total dissociation probabilities. Our numerical analysis carried out in the recently introduced light-induced conical intersection (LICI) framework clearly shows the effects of the changing position of the LICI which is induced by the frequency modulation of the chirped laser pulses. This impact is presented for positively, negatively and zero chirped short pulses., Comment: 21 pages, 5 figures, accepted in J. Chem. Phys

A new quantum dynamical model has been developed to describe the dissociative ionization of deuterium molecular ions by intense laser pulses ($\tau=10$ fs, $\lambda=200$ nm and $I=3\times10^{13}$ W/cm$^{2}$). We calculated the ionization probability densities by solving the time-dependent Schr\"odinger equation numerically. Throughout the simulation the nuclear vibration was considered as a dynamic variable with fixed molecular axis orientation. Benchmark calculations were performed for the ionization of $\mathrm{HeH}^{++}$, for which accurate numerical results are available in the literature, in order to check the performance of this new restricted model., Comment: 26 pages, 11 figures

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.

Recently it has been recognized that electronic conical intersections in molecular systems can be induced by laser light even in diatomics. As is known a direct consequence of these accidental degeneracies is the appearence of nonadiabatic effects which has a strong impact on the nuclear quantum dynamics. Studying the photodissociation process of the $\mathrm{D}_{2}^{+}$ molecule, we report here some novel and observable quantum interference phenomena that arise from the topological singularity induced by a strong laser field.

A coherent superposition of two electronic states of ozone (ground and Hartley B) is prepared with a UV pump pulse. Using the multiconfiguration time-dependent Hartree approach, we calculate the subsequent time evolution of the two corresponding nuclear wave packets and the coherence between them. The resulting wave packet shows an oscillation between the two chemical bonds. Even more interesting, the coherence between the two electronics states reappears after the laser pulse is switched off, which could be observed experimentally with an attosecond probe pulse.

Characterizing and localizing electronic energy degeneracies is important for describ-ing and controlling electronic energy flow in molecules. We show, using topological phase considerations that the Renner effect in polyatomic molecules with more than 3 nuclei is necessarily accompanied by 'satellite' conical intersections. In these intersections the non-adiabatic coupling term is on the average half an integer. We present ab-inito results on the tetra-atomic radical cation C2H2+ to demonstrate the theory