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Canadian Journal of Physics

Green's function equations of motion for driven two-level atoms

Haider R. Zaidi

Published Mar 1, 1985• 1 citation
Publisher

Abstract

Using a diagram technique, equations of motion for N coherently driven two-level atoms are obtained from Green's functions. The longitudinal (Coulomb) dipole–dipole interaction is neglected. Three approximations (Born approximation, extended Born approximation, and screened interaction approximation) are considered. The existing theories are shown to be equivalent to the extended Born approximation. For a small sample, the equations of motion exhibit S conservation in this case. The screened interaction model predicts a different behaviour; in particular, S-breaking terms appear in the equations of motion for a small sample.

DOI
10.1139/p85-049

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Using a diagram technique, equations of motion for N coherently driven two-level atoms are obtained from Green's functions. The longitudinal (Coulomb) dipole–dipole interaction is neglected. Three approximations (Born approximation, extended Born approximation, and screened interaction approximation) are considered. The existing theories are shown to be equivalent to the extended Born approximation. For a small sample, the equations of motion exhibit S conservation in this case. The screened interaction model predicts a different behaviour; in particular, S-breaking terms appear in the equations of motion for a small sample.
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A set of coupled non-Markovian equations of motion is derived for the Green's function of a driven two-level atom. These equations can be expressed in terms of the density-matrix elements only in the Markovian limit, in which case the set reduces to optical Bloch equations. The non-Markovian effect arises from correlations between successive photon emissions, an effect which is neglected in the existing equations of motion. The resulting correction is of the order of ${\ensuremath{\gamma}}_{0}$/${\ensuremath{\Omega}}_{0}$ in the strong-field region (${\ensuremath{\gamma}}_{0}$${\ensuremath{\Omega}}_{0}$) and it vanishes in the weak-field region, where ${\ensuremath{\gamma}}_{0}$ is the Einstein A coefficient and ${\ensuremath{\Omega}}_{0}$ is the Rabi frequency.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
In this paper the equations of motion for open systems, derived recently by one of the present authors, are applied to the case of spontaneous emission from a system of NNN identical two-level atoms. Three different observation levels of the atomic system are considered. First, exact closed integrodifferential equations for the expectation values of each observation level are obtained. Second, making the standard approximations---Born, Markov, and long-time approximations, we find that these equations reduce to closed differential equations for the expectation values. The quality of the Born approximation for different observation levels is also discussed. Moreover, in the obtained equations no kind of rotating-wave approximation was made.
Using a diagram technique, the equations of motion for the density matrix and the field inside the medium are derived for a system of driven two-level atoms. It is pointed out that these equations do not necessarily imply that the local driving field is perfectly coherent even if the external field is. In fact, by calculating the emission spectrum, it is shown that the coherent-scattering component acquires a small nonzero width in the strong-field region. It is concluded that the local driving field is only quasi-coherent. The physical origin of this effect lies in the peculiar nature of resonant interaction between two atoms. The results are compared with the previously obtained results for the foreign gas broadening type of collisions. It is found that there are qualitative as well as quantitative differences between the two results.
The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons and nuclei beyond the Born-Oppenheimer approximation. The case of crystalline solids is discussed in detail.