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 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.
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.
Theory relevant to the treatment of resonance collisions between two-level dressed atoms has been extended to the three-level case in a self-contained calculation of field-dependent collision rates. With the laser field tuned close to the \ensuremath{\Vert}e1〉\ensuremath{\leftrightarrows}\ensuremath{\Vert}e2〉 transition, dressing of the upper two-level system is achieved. We solve coupled equations of motion for two-atom states that describe one atom in the ground state and the other in one of two dressed states. Dressed-state collision rates are calculated within the impact approximation using S-matrix theory. The coupled equations of motion are simplified in the limit of intense field. We have applied the theory to describe approximately collisions between Ne atoms in fields intense enough to dress the collision complex.
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.
A general and computationally efficient method for averaging both the time-dependent and the steady-state atomic or molecular state populations over the phases, ${\ensuremath{\delta}}_{1}$ and ${\ensuremath{\delta}}_{2},$ of two continuous-wave laser fields involved in an excitation process is developed based on the Floquet formalism. Explicit calculations are presented for the coherent one- and three-photon electronic excitation of a two-level model molecule in order to illustrate the importance of phase averaging in situations where the relative phase difference between the two fields is fixed. While the explicit results involve electronic excitation, they are presented in reduced form so that they can be scaled to other regions of the electromagnetic spectrum and to other field strengths. The results have important implications in situations where the relative phase difference between two intense continuous-wave laser fields is used to control the excitation process.
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.