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Journal of Physics A Mathematical and General

Quantum electrodynamic approach to the Volkov-Coulomb problem

Dong‐Sheng Guo, Richard R. Freeman, Yong-Shi Wu

Published Oct 27, 2000• 4 citations
Publisher

Abstract

We have previously presented a nonperturbative quantum electrodynamic approach to the atom-light system, and solved it by nonperturbative treatment limited to the electron-photon interactions only. In this paper, we solve the equation of motion, a time-independent Schrödinger-like equation for the combined atom-photon-mode system, nonperturbatively for both the atomic Coulomb potential and the photon mode, by employing a Born-Oppenheimer approximation in momentum space. The solution obtained is the direct product of a photonic part, representing a stimulated photon cloud dressing the electron, and an electronic part, as if the atomic potential is distorted. For atoms with known spectra and wavefunctions, the integral equation for the distorted atomic wavefunctions is reduced to a solvable set of linear algebraic equations with explicitly determined coefficients. Applications of this result are also discussed.

DOI
10.1088/0305-4470/33/44/308

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We have previously presented a nonperturbative quantum electrodynamic approach to the atom-light system, and solved it by nonperturbative treatment limited to the electron-photon interactions only. In this paper, we solve the equation of motion, a time-independent Schrödinger-like equation for the combined atom-photon-mode system, nonperturbatively for both the atomic Coulomb potential and the photon mode, by employing a Born-Oppenheimer approximation in momentum space. The solution obtained is the direct product of a photonic part, representing a stimulated photon cloud dressing the electron, and an electronic part, as if the atomic potential is distorted. For atoms with known spectra and wavefunctions, the integral equation for the distorted atomic wavefunctions is reduced to a solvable set of linear algebraic equations with explicitly determined coefficients. Applications of this result are also discussed.
Many nonlinear quantum optical physics phenomena need more accurate wave functions and corresponding energy or quasienergy levels to account for. An analytic expression of wave functions with corresponding energy levels for an atomic electron interacting with a photon field is presented as an exact solution to the Schrödinger-like equation involved with both atomic Coulomb interaction and electron-photon interaction. The solution is a natural generalization of the quantum-field Volkov states for an otherwise free electron interacting with a photon field. The solution shows that an N level atom in light form stationary states without extra energy splitting in addition to the Floquet mechanism. The treatment developed here with computing codes can be conveniently transferred to quantum optics in classical-field version as research tools to benefit the whole physics community.
Introduction . In the previous chapter we have obtained the form of the coupling between matter and the electromagnetic field. We are thus in a position to treat electrodynamics in the presence of charges and currents. We start by obtaining the Euler-Lagrange equations of the matter-electromagnetic field system. These turn out to be the Maxwell-Lorentz equations, which is a set of coupled equations of motion in which the matter field acts as a source for the electromagnetic field and vice versa. In Section 4.2 we derive the Hamiltonian of the complete system in the Coulomb gauge, and this leads to the so-called minimal coupling Hamiltonian, containing both the electromagnetic potential and the matter-field amplitude. Specializing the nonrelativistic matter field to the case of a neutral atom, consisting of the field of electrons in a static nuclear potential, in Section 4.3 we obtain the atom-photon Hamiltonian in the minimal coupling scheme. Some of the basic processes induced by the atom-photon interaction part of the Hamiltonian are also discussed in this section. This gives us the possibility of introducing at this stage a fundamental simplification of the interaction Hamiltonian, namely the electric dipole approximation. The minimal coupling scheme, however, is not the only possible atom-field coupling.
Nonperturbative analytical and numerical methods are presented for the solution of the coupled nonlinear Maxwell-Schr\"odinger equations. The theory has been derived within the Hamiltonian or canonical formalism. The canonical approach to dynamics, starting from the Maxwell and Schr\"odinger Lagrangians with a Lorenz gauge fixing term, yields a set of first order Hamilton equations. They form a well-defined initial value problem. The Maxwell-Schr\"odinger equations of motion are then represented in a spatial basis of Gaussian functions. In the limit of a complete basis this representation is exact. For any choice of finite basis it provides an approximate system of dynamical equations that can be integrated in time and made systematically more accurate by enriching the basis. The basis form of the theory has been implemented numerically and is used to investigate the dynamics of a single nonrelativistic spinless one electron atom interacting with the electromagnetic modes of a cavity.
A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the g factor of the electron within nonrelativistic quantum electrodynamics.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.