Starting from first principles, I derive a time-dependent diagrammatic many-body formalism that can be used for strong harmonic fields. My aim has been to treat many-electron effects in an atom interacting with an intense laser field. The formalism derived describes the atom in terms of a damped many-body Floquet state. The diagrammatic expansion is similar to the ordinary time-independent atomic many-body theory, but some important changes have been made: The energy denominators contain time-dependent self-energies and the exact ground-state shift; the intermediate levels are dressed atomic states, for which each harmonic of a given atomic state is counted as a separate level; and the ground-state energy shift is given by a time-dependent Brillouin–Wigner-type energy equation, which in general must be solved numerically. The formalism is not linked but contains no divergences in the adiabatic limit of the electron–electron interaction. For weak fields the formalism reduces to the ordinary linked expansion. For a two-level system all diagrams can be summed, and the result is equivalent to the usual continued-fraction solution. I also derive the exact diagrammatic solution for a single level interacting with a continuum.
Starting from first principles, I derive a time-dependent diagrammatic many-body formalism that can be used for strong harmonic fields. My aim has been to treat many-electron effects in an atom interacting with an intense laser field. The formalism derived describes the atom in terms of a damped many-body Floquet state. The diagrammatic expansion is similar to the ordinary time-independent atomic many-body theory, but some important changes have been made: The energy denominators contain time-dependent self-energies and the exact ground-state shift; the intermediate levels are dressed atomic states, for which each harmonic of a given atomic state is counted as a separate level; and the ground-state energy shift is given by a time-dependent Brillouin–Wigner-type energy equation, which in general must be solved numerically. The formalism is not linked but contains no divergences in the adiabatic limit of the electron–electron interaction. For weak fields the formalism reduces to the ordinary linked expansion. For a two-level system all diagrams can be summed, and the result is equivalent to the usual continued-fraction solution. I also derive the exact diagrammatic solution for a single level interacting with a continuum.
Diagrammatic many-body theory has been used extensively in the treatment of electron–electron correlations in atomic photoionization at low intensities. We discuss how it can be used at high intensities also. The standard diagrammatic formalism is derived for time-independent interactions by using adiabatic switching. This formalism can be formally derived for harmonic fields also but leads to unphysical results for strong fields. If the amplitude function of the laser field is an exponential function, however, the standard derivation still applies. If the adiabatic limit is not taken, this time-dependent formalism can be used for any intensity. In practice one is still limited to slow turn-on, but, assuming slow turn-on, we show that for a discrete system the linked expansion, when summed to infinite order, gives the Floquet state that is adiabatically connected to the ground state. When continua are included, we find that qualitatively correct results are obtained only if one uses an expansion of the time derivative of the wave function instead of an expansion of the wave function itself. We also show that the imaginary part of the self-energies must be excluded from the energy denominators, since they otherwise lead to unphysical results. These results are found by treating model systems for which the exact solutions are known and for which the diagrams can be evaluated to any order.
The diagrammatic solution to a (sequentially coupled) one-electron three-level system in an intense laser field is derived. The solution is in the form of a continued-fraction expansion and is equal to the Floquet state that is adiabatically connected to the ground state. The formalism used is a recently presented diagrammatic many-body theory by Jonsson that can be used for non-perturbative calculations in strong fields. The purpose of the calculation is to demonstrate how a non-trivial one-electron problem can be solved exactly within the formalism.
The multiple-linear-time-scale method is used to construct a perturbation theory for N-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2), the approximate solution is valid for many thousands of Rabi cycles.
In this chapter,we shall analyze the particular case of an atom interacting with a laser pulse whose duration is sufficiently long, so that the evolution of the atom in the laser field is adiabatic. When this condition is fulfilled, the atom can be considered to interact with a monochromatic laser field. As a consequence, the Hamiltonian of the system is periodic in time, and the Floquet theory [1] can be used to solve the time-dependent Schrödinger equation (TDSE) non-perturbatively. We begin in Section 4.1 by considering the Hermitian Floquet theory. We first derive the Floquet theorem for a monochromatic, spatially homogeneous laser field and show that the solutions of the TDSE correspond to dressed states having real quasi-energies , which can be obtained by solving an infinite system of time-independent coupled equations. We then generalize the Floquet theory to multicolor laser fields and to “non-dipole” laser fields which are not spatially homogeneous. In Section 4.2, the Floquet theory is applied to study the dynamics of a model atom having M discrete levels interacting with a monochromatic laser field. In this case, the coupling between the bound and continuum atomic states is neglected.We analyze the relationship between the Floquet theory and the rotating wave approximation, and examine the perturbative limit of the Floquet theory. We also consider the population transfer between Floquet dressed states.
In order to properly describe the dynamics of a many-particle system in strong, time-dependent fields, a nonperturbative treatment of all constituents and of their correlation is needed. An ab initio solution of the time-dependent many-body Schrödinger equation is only feasible for simple systems. Hence, for many-electron systems in intense laser fields practicable methods for solving the quantum-mechanical many-body problem are required. An formally exact approach is the time-dependent density functional theory. In this work a mathematically rigorous formulation of the foundations of this theory is given. Further the non-locality in time of the exchange-correlation functionals is examined, and we formally define the notion of "quantum memory". We investigate the fundamental process of Rabi oscillations from a density-functional point of view and find the few-level approximation to be in conflict with the basis of time-dependent density functional theory. Finally, we apply the theory to calculate the electron dynamics of C60 in intense laser pulses. Although the laser light is far off-resonant with respect to the collective modes of the C60 the multi-electron dynamics strongly influences the harmonic spectra. The efficiency of this multi-particle recollision process with respect to the usual single active electron approximation of high-order harmonic generation is estimated by two analytical models.
When matter is exposed to strong fields, electrons are ionized and a number of processes such as high harmonic generation (HHG), above threshold ionization (ATI), non-sequential ionization happen. These processes are usually investigated semiclassically, i.e. matter is treated by quantum mechanics, and radiation is treated classically. In particular HHG which is the generation of ultrashort coherent extreme ultraviolet (XUV) pulses, has enabled a wealth of novel ultrashort spectroscopic methods in atomic, molecular, and condensed matter physics. Further, ionization has applications in micro-machining. Strong field physics continues to have numerous open questions. As an illustration, the mechanisms underlying dominant HHG pathways in solids remain insufficiently understood. It is not clear which mechanism dominates HHG in various materials. Possible pathways are the interband and intraband contributions. Furthermore, many of these processes in strong field physics have been described by single active electron (SAE) approximation. This approach relies on the assumption that the interaction between a single electron and the intense laser field dominates all other interactions. However, this is clearly not the case especially in solids where interactions with the lattice and other electrons can be important. In addition, quantum optical aspects of strong field physics have generated increasing interest, yet they remain partially explored. This chapter involves treating electromagnetic fields using the formalism of second quantization. Quantum optical properties, such as squeezing, entanglement and the negativity of the Wigner function, are of fundamental importance for the field of quantum information and quantum computation. The central theme of my thesis is the development of quantum optical and statistical theoretical frameworks for describing intense laser field processes such as ionization and HHG in atomic and molecular gases and in solids. The second chapter focuses on gaining a detailed understanding of the mathematical steps involved in the Keldysh theory of ionization to establish a solid theoretical foundation. Through this analysis we identified an additional factor of two compared to Keldysh’s original derivation of atomic ionization rates. The third chapter addresses the inadequately understood mechanisms that govern dominant high harmonic generation (HHG) pathways in solids. One approach to clarify these mechanisms involves introducing real (resonant) and virtual processes. We developed the strong field adiabatic following (SFAF) formalism which is based on Dyson expansion using the von-Neumann equation of density matrix. Using SFAF a diagnostic method is obtained to separate virtual and resonant channels. Through this separation, and by comparing with experimental results, we identified the need to incorporate many-body effects. The fourth chapter explores the fact that a solid is a complex many body system in which the SAE approximation is very crude. The electron interacts with holes, other electrons, collective excitations such as plasmons, and phonons. Our work is based on the idea that all theses effects can be treated as a quantum statistical heat bath of bosonic harmonic oscillators. We apply this to our SFAF formalism. In the ionization case, this work has settled a long standing issue which is the fact that simple phenomenological approaches such as the relaxation time approximation 𝑇2 result in nonphysical enhancement of ionization. In chapter five, we have investigated ways to transfer quantum optical properties on the otherwise classical high harmonic radiation. The goal is to use HHG to scale quantum sources to smaller wavelengths. Quantum properties can be imprinted by perturbing HHG with a quantum field, such as bright squeezed vacuum (BSV). In this chapter, the theory of quantum sideband high harmonic generation (QSHHG) in atoms and solids is derived to find ways by which to transfer quantum properties from the perturbative BSV to the harmonic sideband. The theoretical framework is a quantum generalization of the semi-classical Lewenstein model of HHG. It gives closed-form solutions for the HHG and QSHHG wavefunctions. Knowing the wavefunction, we can identify the quantum properties of QSHHG. The additional photons absorbed and emitted from the quantum perturbation, here BSV, create entanglement between individual harmonic sidebands and between the harmonic sidebands and the BSV. We show how this entanglement can be used to create a variety of non-classical states commonly used in quantum information science, such as high-purity single-photon states, Schrödinger cat states, and photon-added squeezed vacuum states. Some of these non-Gaussian states with negative Wigner functions have been shown to provide a quantum computational advantage over their Gaussian counterparts. Additionally, they play a significant role in quantum metrology, enhancing precision measurements beyond classical limits. This chapter has opened a path to quantum engineering of HHG.