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Physical Review A

Collective population evolution of two-level atoms based on mean-field theory

Hiroyuki Takeda

Published Feb 24, 2012• 3 citations
Publisher

Abstract

I theoretically demonstrate the collective population evolution of two-level atoms, based on the mean-field theory. Optical materials with abrupt changes of electromagnetic local densities of states enable steady-state population inversion of two-level atoms which is generally considered impossible. In previous studies, details of the population evolution of a single two-level atom were discussed. However, behaviors of the population evolution of collective two-level atoms are very different from those of a single two-level atom. In this paper, temporal behaviors of the collective population evolution for input electric fields with constant and time-dependent amplitudes are discussed in detail. Moreover, I develop the effective method to calculate the collective spontaneous emission for non-Markovian dynamics. Finally, I discuss the validity of the mean-field theory for a large number of two-level atoms in spontaneous emission using the Markovian approximation.

DOI
10.1103/physreva.85.023837

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I theoretically demonstrate the collective population evolution of two-level atoms, based on the mean-field theory. Optical materials with abrupt changes of electromagnetic local densities of states enable steady-state population inversion of two-level atoms which is generally considered impossible. In previous studies, details of the population evolution of a single two-level atom were discussed. However, behaviors of the population evolution of collective two-level atoms are very different from those of a single two-level atom. In this paper, temporal behaviors of the collective population evolution for input electric fields with constant and time-dependent amplitudes are discussed in detail. Moreover, I develop the effective method to calculate the collective spontaneous emission for non-Markovian dynamics. Finally, I discuss the validity of the mean-field theory for a large number of two-level atoms in spontaneous emission using the Markovian approximation.
I theoretically demonstrate the population inversion of collective two-level atoms using photonic crystals in three-dimensional (3D) systems by self-consistent solution of the semiclassical Maxwell-Bloch equations. In the semiclassical theory, while electrons are quantized to ground and excited states, electromagnetic fields are treated classically. For control of spontaneous emission and steady-state population inversion of two-level atoms driven by an external laser which is generally considered impossible, large contrasts of electromagnetic local densities of states (EM LDOS’s) are necessary. When a large number of two-level atoms are coherently excited (Dicke model), the above properties can be recaptured by the Maxwell-Bloch equations based on the first-principle calculation. In this paper, I focus on the realistic 1D PC’s with finite structures perpendicular to periodic directions in 3D systems. In such structures, there appear pseudo photonic band gaps (PBG’s) in which light leaks into air regions, unlike complete PBG’s. Nevertheless, these pseudo PBG’s provide large contrasts of EM LDOS’s in the vicinity of the upper photonic band edges. I show that the realistic 1D PC’s in 3D systems enable the control of spontaneous emission and population inversion of collective two-level atoms driven by an external laser. This finding facilitates experimental fabrication and realization.
Spontaneous emission from a two-level atom embedded in photonic crystals was investigated in this paper. Due to the photon localization and the quantum interference between the two dressed states, some changes in the radiation field take place: when the excited level moves from the gap to the band, the energy of the localized field is transferred to the propagating field through the diffusion field. If the localized field and the propagating field could coexist, the population evolution in the excited level would exhibit quasi-oscillation behavior. If they could not coexist, there would be no quasi-oscillation. The steady-state trapped atom population in the excited level is changed also. All these characteristics depend not only on the relative position δ of the excited level ω _1 with respect to the band-edge ω _c but also on the photon density of states ρ(ω_k) (or the band-edge smoothing parameter e ) near the band edge.
A two-level atom illuminated by a laser may be driven into a population-inverted steady state if it is coupled to a cavity. This is a result of the quantum nature of the electromagnetic field and is forbidden by the semiclassical theory. We numerically and analytically analyze this inversion. We find the maximum possible inversion and determine the rate of approach to the steady state. The quantized cavity modifies the electromagnetic vacuum seen by the atom; this connects our work to that on cavity-enhanced spontaneous emission and on dynamical line narrowing. Experimental signatures of the inversion and potential experimental difficulties are considered. For example, neither the presence of many atoms in the mode nor many modes in the cavity destroys the inversion.