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The Astronomical Journal

The Two-Level Atom Revisited.

John Waddell

Published Oct 1, 1964• 0 citations
Publisher

Abstract

The two-level atom problem formulated in energy space by Warwick (Astrophys. J. 121,190,1955) and Thomas~ (Astrophys. J. 125, 260,1957) may be demonstrated to be incapable of exact formulation. Three difficulties of principle are encountered: (1) the inability to distinguish between photon numbers and photon energy; (2) the introduction of normalized Einstein coefficients which in turn violate the Einstein relations; and (3) an erroneous treatment of stimulated emission, which when properly accounted for, does not permit an angular dependence in the scattering term. The basic difficulty arises from the fact that non-LTE is unable, in principle, to account for finite linewidths. Some of these difficulties appear to be eliminated if the non-LTE problem is formulated in momentum space rather than the customary energy space.

DOI
10.1086/109331

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The two-level atom problem formulated in energy space by Warwick (Astrophys. J. 121,190,1955) and Thomas~ (Astrophys. J. 125, 260,1957) may be demonstrated to be incapable of exact formulation. Three difficulties of principle are encountered: (1) the inability to distinguish between photon numbers and photon energy; (2) the introduction of normalized Einstein coefficients which in turn violate the Einstein relations; and (3) an erroneous treatment of stimulated emission, which when properly accounted for, does not permit an angular dependence in the scattering term. The basic difficulty arises from the fact that non-LTE is unable, in principle, to account for finite linewidths. Some of these difficulties appear to be eliminated if the non-LTE problem is formulated in momentum space rather than the customary energy space.
Stimulated emission can be defined as the process when an incoming photon stimulates an additional quantum of energy from an atom into the same electromagnetic mode as the impinging photon. Hence, the two outgoing photons are identical. In a waveguide or free space, this intuition is typically found through Fermi's Golden rule, however, this does not properly account for the wave-like nature of the photons. Here, I present an exact solution to stimulated emission from a quantum two-level atom that properly accounts for the incoming and outgoing electromagnetic modes. This result is valid over a huge range of incident photon numbers. For a single incident photon, it shows how the photon must properly mode match the two-level system to cause stimulated emission. For a Fock state drive with large photon number, the exact solution shows how a two-level system Rabi oscillates with the traveling Fock mode as it passes by. I additionally use the same formalism to show that stimulated emission by a coherent pulse cannot be understood as an additional photon being emitted into the incident coherent state because the two-level system's state factorizes with the electromagnetic field's coherent state. Recent advances in superconducting circuits make them an ideal platform to test these predictions.
Light is both radiated and absorbed by atoms, and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics. However, real atoms are complicated systems, and even the simplest real atom, the hydrogen atom, has a non-trivial energy level structure. It is therefore often necessary or desirable to approximate the behavior of a real atom by that of a much simpler quantum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in many theoretical treatments to represent the atom by a quantum system with only two energy eigenstates. This is the most basic of all quantum systems, and it generally simplifies the treatment substantially. In a real atom the selection rules limit the allowed transitions between states, so that, in some cases, a certain state may couple to only one other. Moreover, optical pumping techniques have been developed that allow such preferred states to be prepared in the laboratory, and they have been successfully used in experiments (Abate, 1974). The two-level atom approximation is therefore close to the truth and not merely a mathematical convenience in some experimental situations. In the following we begin by developing the algebra for such a two-level atom. Dynamical variables for a two-level atom We consider an atomic quantum system with the two energy levels shown in Fig. 15.1.
Despite many years of research into Raman phenomena, the problem of how to include both spontaneous and stimulated Raman scattering into a unified set of partial differential equations persists. The issue is solved by formulating the quantum dynamics in the Heisenberg picture with a rigorous accounting for both time- and normal-ordering of the operators. It is shown how this can be done in a simple, straightforward way. Firstly, the technique is applied to a two-level Raman system, and comparison of analytical and numerical results verifies the approach. A connection to a fully time-dependent Langevin operator method is made for the spontaneous initiation of stimulated Raman scattering. Secondly, the technique is demonstrated for the much-studied two-level atom both in vacuum and in a lossy dielectric medium. It is shown to be fully consistent with accepted theories: using the rotating wave approximation, the Einstein A coefficient for the rate of spontaneous emission from a two-level atom can be derived in a manner parallel to the Weisskopf-Wigner approximation. The Lamb frequency shift is also calculated. It is shown throughout that field operators corresponding to spontaneous radiative terms do not commute with atomic/molecular operators. The approach may prove useful in many areas, including modeling the propagation of next-generation high-energy, high-intensity ultrafast laser pulses as well as spontaneous radiative processes in lossy media.
Usually it is assumed that the stimulated emission appears as a consequence of the Bose—Einstein statistics of photons and that to observe this effect at least two excitations have to be initially present in the atom—field system. That is, both the atom and the electromagnetic field have to be excited. In this paper we show that stimulated emission can appear exclusively as a consequence of quantum interference in a system with just a single excitation. Specifically, we consider a single two-level atom which is initially in its lower energy state and it interacts with a single-photon multi-mode wave packet. We show that for a proper choice of the photon-wave packet the atom can exhibit stimulated emission.