The first part of this article attempts to highlight in chronological order some prominent papers that have direct connexion with the Born-Oppenheimer approximation. This timeline successively points to the role of level crossing, the notions of gauge field and geometrical or Berry phase, early experiments that were accounted for with that concept, and its major role in understanding anomalous transport in solids. The second part analyzes the time evolution of states under the adiabatic approximation when the states are spin degenerate. The geometrical phase thus found entails an electron spin evolution. Then, we examine the case of moleclar dynamics. Starting from the Born-Oppenheimer approximation, we obtain a spin evolution by replacing the classical nuclear positions by their quantum mechanical counterparts in the terms that describe electrostatic interactions between nuclei and electrons.

We present the quantum master equations for midsize molecules in the presence of an external magnetic field. The Hamiltonian describing the dynamics of a molecule accounts for the molecular deformation and orientation properties, as well as for the electronic properties. In order to establish the master equations governing the relaxation of free-standing molecules, we have to split the molecule into two weakly interacting parts, a bath and a bathed system. The adequate choice of these systems depends on the specific physical system under consideration. Here we consider a first system consisting of the molecular deformation and orientation properties and the electronic spin properties and a second system composed of the remaining electronic spatial properties. If the characteristic time scale associated with the second system is small with respect to that of the first, the second may be considered as a bath for the first. Assuming that both systems are weakly coupled and initially weakly correlated, we obtain the corresponding master equations. They describe notably the relaxation of magnetic properties of midsize molecules, where the change of the statistical properties of the electronic orbitals is expected to be slow with respect to the evolution time scale of the bathed system.

Since their discovery in 1927, the Heisenberg Inequalities have become an icon of quantum mechanics. Often inappropriately referred to as the Uncertainty Principle, these inequalities relating the standard deviations of the position and momentum observables to Planck's constant are one of the cornerstones of the quantum formalism even if the physical interpretation of quantum mechanics remains still open to controversy nowadays. The Heisenberg Inequalities governing translational motion are well understood. However, the corresponding inequalities pertaining to rotational motion have not been established so far. To fill this gap, we present here the Rotational Heisenberg Inequalities relating the standard deviations of the orientation axis and orbital angular momentum observables of an isolated molecule. The reason for choosing this system is that a molecule separated from its environment corresponds to a bound system preserving the orbital angular momentum.

A rigorous quantum description of molecular dynamics with a particular emphasis on internal observables is developed accounting explicitly for kinetic couplings between nuclei and electrons. Rotational modes are treated in a genuinely quantum framework by defining a molecular orientation operator. Canonical rotational commutation relations are established explicitly. Moreover, physical constraints are imposed on the observables in order to define the state of a molecular system located in the neighborhood of the ground state defined by the equilibrium condition.

The symmetry breaking due to a magnetic field applied on a hydrogen molecule H2 generates an electric polarization. This magnetoelectric effect occurs for electrons in a triplet state provided the magnetic induction field is not aligned with the symmetry axis of the molecule.

We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained N-particle density matrix, we obtain the equations, which describe the evolution of the n-particle density operators (D) over bar ((n)) for n < N. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.

Starting from the quantum-Boltzmann equation derived in a previous paper, we study the irreversible evolution of an electron gas in the one-particle phase space. The connection with phase space is established by expressing one-electron states in terms of the overcomplete and nonorthogonal generating system of coherent states. By using the generalized closure relation for coherent states, as well as the fact that a one-particle operator is completely determined by the ensemble of expectation values for all coherent states, we obtain the master equations in a form that allows us to follow the evolution in phase space. This form of the master equations provides a direct link between the quantum-statistical approach and the semi-classical Boltzmann equation. The latter is obtained after a coarse-graining procedure in the one-particle phase space and by using the fact that the electron-electron interaction, as well as the interactions between the electron gas and the bath subsystems provided by phonons or photons, are local in real space.

We study the angular correlation of single photons emitted from excited semiconductor quantum wells. The considered physical system is described in terms of two subsystems, the electronic part constituting the bath and the photonic part constituting the bathed subsystem, both being coupled by the light-matter interaction. From the master equations describing the coarse-grained Markovian evolution of the photonic subsystem, we derive the corresponding equations of motion for the photonic angular correlation functions. These equations are solved in the stationary, low-density limit. Experimentally, the angular correlations can be assessed by studying the interference of light emitted in different directions. In agreement with recent experimental results, we find that for ordered quantum wells angular correlations exist only in emission directions for which the projections of the photon momenta onto the plane of the quantum well are equal. This feature is a direct consequence of the Bloch character of the electronic states in an ordered quantum well. Thus the experimental study of the angular correlations of emitted photons may provide an interesting diagnostic tool to reveal the presence of disorder in semiconductor heterostructures and to characterize its influence on the electronic states near the band edges.

Coherent phonon scattering and transmission via a three-terminal junction in low-dimensional structures is investigated. The model system consists of two-dimensional crystalline wave-guides, constructed of double atomic chains that are joined together by a nanostructured T-shaped junction. The matching method theoretical approach is used to study the dynamics of the system. A calculation is presented for the coherent reflection and transmission scattering probabilities of the system phonons, as elements of a Landauer type scattering matrix. The scattering and transmission spectra via the T-shaped junction are analyzed as a function of the incident frequency per propagating mode of the perfect wave-guides. Our results show that the junction is an effective phonon splitter and suggest that its characteristics may be controlled by varying its nanometric parameters. Asymmetric Fano-like resonances are observed in the spectra which result from the coherent coupling between the localized modes of the T-shaped junction and the propagating modes of the wave-guides.

The method of angular photonic correlations of spontaneous emission is introduced as an experimental, purely optical scheme to characterize disorder in semiconductor nanostructures. The theoretical expression for the angular correlations is derived and numerically evaluated for a model system. The results demonstrate how the proposed experimental method yields direct information about the spatial distribution of the relevant states and thus on the disorder present in the system.

The influence of a special class of atomic nanostructures embedded on a waveguide is analyzed for the scattering and transmission of elastic waves in quasi-one-dimensional multicanal wave-guides. The quasi-one-dimensional waveguide is constructed of double chains of atoms, and the nanostructures consist of geometrical configurations, where the double chains are arranged to form several types of double L-shaped joints. Numerical results are presented for the three types of nanostructures, using the matching method. The theoretical approach allows us to calculate the reflection and the transmission probabilities as well as the average phonon conductance of the system along the waveguide. The results show that the transmission probabilities and the average conductance depend strongly on the type of geometrical joint nanostructure. The pronounced fluctuations in the transmission and conductance spectra as a function of the frequency can be understood as Fano resonances that result from the coherent coupling between the propagating modes and the localized vibrational modes induced by the nanostructures.

We consider N strongly interacting electrons in a one-dimensional circular loop that is pierced by a time-periodic magnetic flux a(t) = a(0) + a(1)(t) with the angular frequency omega. Similar to our previous work, where we have considered a static magnetic flux a(0), the electron positions are expressed in terms of collective and relative coordinates. Strong e-e interaction can then be treated in a harmonic approximation for the relative motion. The presently searched solutions of the time-dependent Schrodinger equation for a time-periodic flux are given by the Floquet states. The Floquet states for a spatially constant one-particle potential form a complete set of Floquet-basis states, which is used to study the influence of the one-particle potential on the electronic states. While for a spatially constant one-particle potential the time-averaged observables, such as the electronic energy and the electronic current or angular momentum, depend solely on the time-averaged magnetic flux a(0), a spatially varying one-particle potential leads to pronounced resonances. In the case of moderate electronic relaxation, the stationary properties are determined by the Floquet state with lowest time-averaged energy. For the associated persistent angular momentum we predict jumps with heights proportional to the number of electrons N. We further show that, already by measuring the locations of these jumps in the (a(0),omega) plane, one could determine the number of electrons N as well as the effective e-e interaction.

We present a detailed discussion of the evolution of a statistical ensemble of quantum mechanical systems coupled weakly to a bath. The Hilbert space of the full system is given by the tensor product between the Hilbert spaces associated with the bath and the bathed system. The statistical states of the ensemble are described in terms of density matrices. Supposing the bath to be held at some - not necessarily thermal - statistical equilibrium and tracing over the bath degrees of freedom, we obtain reduced density matrices defining the statistical states of the bathed system. The master equations describing the evolution of these reduced density matrices are derived under the most general conditions. On time scales that are large with respect to the bath correlation time \(\tau_{B}^{\mathrm{corr}}\) and with respect to the reciprocal transition frequencies of the bathed system, the resulting evolution of the reduced density matrix of the bathed system is of Markovian type. The detailed balance relations valid for a thermal equilibrium of the bath are derived and the conditions for the validity of the fluctuation-dissipation theorem are given. Based on the general approach, we investigate the non-linear response of the bathed subsystem to a time-periodic perturbation. Summing the perturbation series we obtain the coherences and the populations for arbitrary strengths of the perturbation.

We investigate the ensemble averaged evolution of N-electron systems dynamically coupled to a statistical environment. The electrons are characterized by their spatial and by their spin properties. While the Hilbert space for single electrons is given by the tensor product of the Hilbert spaces associated with both properties, the corresponding Hilbert space for N-electron systems cannot be factorized. Consequently, quantum correlations between spatial and spin properties become extremely important. We assume that the evolution of the spin properties is controlled by spin-orbit interaction and that the spatial properties take the part of a bath held near some equilibrium. This description is appropriate for magnetic systems where the electronic states near the ground state correspond to different spin configurations, whereas electronic states with large excitation energies belong to different spatial-orbital configurations. In order to determine the coarse grained evolution of the spin properties, we have to know the evolution of the N-electron system over time intervals larger than the bath-correlation time. This is obtained from the first- and second-order contributions in the interaction picture. We show that, in spite of the strong quantum correlations between spin properties and spatial properties, the coarse grained statistical evolution of the electronic spin properties may be described by a set of coupled master equations.

We consider N electrons in a one-dimensional loop enclosing a static magnetic flux. The electronic positions are described in terms of collective and relative coordinates. The corresponding partition of the Hilbert space is particularly convenient to investigate the electronic spectrum in the range of large electron-electron interaction. The electronic current is determined by the collective motion of the electrons. Thus, in absence of backscattering by a one-particle potential, the electron-electron interaction does not affect the persistent current associated with the electronic ground state. This is different when backscattering is present, since then collective and relative motions are coupled. We present a detailed study of the interplay between potential backscattering and electron-electron interaction, which is revealed by the resulting ground-state currents as well as by the electronic excitation spectra.

A density-matrix theory for absorption changes in semiconductors induced by electron or hole occupations is outlined. Bound and unbound trions are included via four-point correlation functions representing electron-hole pair transitions in the presence of carrier populations. The spectra calculated for semiconductor nanorings show bleaching of the exciton resonance and induced absorption at energies corresponding to transitions to bound and unbound trion states. Without a magnetic field, induced absorption below the exciton line due to bound negatively (positively) charged trions appears when the two electrons (holes) of the trion are in different bands. A magnetic field introduces characteristic modifications of the spectra that can be attributed to the Aharonov-Bohm effect. It may lead to the formation of additional bound magneto-trion states.

: The coherent dynamics of magnetoexcitons in semiconductor nanorings following pulsed optical excitation is studied. The calculated temporal evolution of the excitonic dipole moment may be understood as a superposition of the relative motion of electrons and holes and a global circular motion associated with the magnetic-field splitting of these states. This dynamics of the electron-hole pairs can be generated either by local optical excitation of an ordered ring or, alternatively, by homogeneous excitation of rings with broken rotational symmetry due to disorder or band tilting.

The dynamics of optically generated electron-hole pairs is investigated in a disordered semiconductor nanowire. The particle pairs are generated by short laser pulses and their dynamics is followed using the Heisenberg equation of motion. Is is shown that Coulomb-correlation acts against localization in the case of the two-interacting particles (TIP) problem. Furthermore, currents are generated using a coherent combination of full-gap and half-gap pulses. The subsequent application of a full-gap pulse after time $τ$ produces an intraband echo phenomenon $2τ$ time later. The echo current is shown to depend on the mass ratio between the electrons and the holes.

We consider electronic relaxation in a small sample embedded in a host material that is held at thermal equilibrium. In our model description, the coupling of the sample electrons with the surrounding environment is mediated by the local vibrational modes in the sample region. Due to their coupling with the environment, these modes have a finite lifetime. The evolution of the electronic configuration with time is determined by studying the evolution of a density matrix, which is restricted to the electronic and vibrational degrees of freedom in the sample region. The thermalization of an excited electron is described using an iterative procedure. We present results for a single electron in a loop structure, where the electrons are coupled either to an ensemble of decoupled harmonic oscillators or to the acoustical phonons of the loop.