Coherent states in miicrocrystals are important as potential means for approaching the fundamental limits on speed and efficiency in optical switching. Workers are only now beginning to master means of preparing these coherent states at excitation levels of one or a few quanta. We have identified a mechanism that could explain some aspects of coherent state preparation in rnicrocrystals having optical wavelength dimensions. Realization of this mechanism requires no unusual properties, other than satisfying certain precise dimensional requirements on the rnicrocrystal, means of exciting the state, and a constraint on dephasing rates. Many real systems can satisfy these constraints.

  We have explored a broad parameter space and provide plots characterizing the dynamics of these states. These plots should be useful to both theorists and experimentalists in evaluating our hypothesis. In particular, we make close contact with earlier and more recent experimental observations[ 1-51. We find a remarkable degree of agreement between predicted and observed behavior. We also recognize alternative and related mechanisms and discuss specific means of distinguishing between contributions from different mechanisms in different structures. [2-7]

  We base our model on an interaction of the radiation field and the near field components of a single quantum excitation. The aton-tic system in microcrystals of particular dimensions mediates this interaction. We consider a system of two level ions in a niicrocrystal of the order of a half optical wavelength on a side excited at the level of one, or a few, quanta. We use Maxwell-Bloch equations adapted from work by Steve Harris [6] and work by Marlan Scully and others [7]. Their work applied to three level systems. We restrict our examples to two level systems.

  We show in Fig. I plots that characterize the dynamics of the coherent state in the rnicrocrysw. Key findings are that the interaction of the atomic system and the field in the microcrystal can reach very large Rabi frequencies, Fig. I (a). The emission can also show a delayed, but still relatively rapid, onset of emission, Fig. I (b), a rapid rise to the emission peak, Fig. I (c), and an emission pulse Fig. I (d) that has a delay, duration, and shape closely resembling emission from EuO [I], ZnO [2], and GaAs microcavities [3,4]. Here g is a measure of the threshold for nonlinear dependence of the susceptibility on field intensity. We denote the reflectivity at the resonator surface in the direction of propagation by r.

  The important points to be noted are: (1) the coherent state in the microcrystal can lead to relatively high values of the optical field (the peak field strength associated with the single quantum excitation is more than sufficient to reach levels for conventional nonlinear processes to play a role); (2) the reflectivity at the niicrocrystal surface can be significantly enhanced and ftu-ther enhance of the density of states; (3) the time required for the onset of the pulse is in the picosecond time domain consistent with the experimental observations, (4) the time to the peak of the pulse is also in the picosecond time domain and consistent with the experimental observations; (5) the asymmetry in the emitted pulse is also consistent with the experimental observations. [1-4]

View Figure 1

The basic problem in etablishing the coherence in the absence of a special resonator [3,4] is that of coupling the near-field and the radiation field. The near field alone tends to induce a coherence, but does not couple well to the external radiation field. The radiation field in the microcrystal couples well to the external fields, but tends to lose energy to other ions and, in the absence specifically fabricated reflectors [3,4], fails to build to a significant intensity in the experimentally observed time.

  We also examine the coupled dynamics of both the near field and the radiation field. There we find the combined dynamics can, in some cases without a special resonator, result in both strong coherence and highly enhanced emission. This does require enhancement of the density of states afforded by the three dimensional resonant confinement, a pump mechanism, and sufficiently long dephasing times for the atomic system.

  Three additional conditions appear important: (1) the terms in (r)- I and -(r)-3 in the dipolar fields have opposite signs; (2) the term with the (r)-3 dependence dominates the radiation field term (r)- I over a large region of the inicrocrystal; and (3) there is an enhancement of order (N) 112 for the total atomic system moment. Here N is the number of ions in the microcrystal that interact strongly with the -(Pr)-3 term. This enhancement is more than sufficient to give the initial impetus to the emission process that can eventually lead to strongly enhanced overall emission.

We can relate these calculations and simulations to prior and recent experimental observations [1-5]. Key factors are the reflectivity r at the microcrystal surface, the Rabi frequency at early times, the microcrystal dimensions, and the dipolar dephasing rates. We note that features of anomalous emission from microcrystals containing Eu2+ [1] and from microcrystals of other semiconductors [2-4] can be remarkably well approximated by simulations obtained from this model. We discuss alternative explanations of these enhanced emissions and suggest experiments to further clarify the nature of the mechanism that establishes coherent states in microcrystals.
 
 

References
 
 

1. R.L. Fork and D.W. Taylor, "Unusual Emission from Microcrystals containing Eu 2+: Experiment", Phys. Rev. B 19 3365-3398 (1979).

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3. Joseph Jacobson, Stanley Pau, Hui Cao, Gunnar Bjork, and Yoshihisa Yamamoto, "Observation of exciton-polariton oscillating emission in a single-qumtum-well semiconductor microcavity", Physical Review A 51, 2542, (1995).

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