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Physical Review A

Nonlinear theory of index enhancement via quantum coherence and interference

Ulrich W. Rathe, Michael Fleischhauer, Shi-Yao Zhu, Theodor W. Hänsch, Marlan O. Scully

Published Jun 1, 1993• 108 citations
Publisher

Abstract

We study the nonlinear behavior of the electric susceptibility for several systems showing a high index of refraction without absorption due to quantum coherence and interference. Estimates of cw intensity limits for a given index of refraction are obtained. We calculate critical field intensities for self-focusing and investigate the stability of the absorption cancellation with regard to intensity fluctuations.

DOI
10.1103/physreva.47.4994

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Physical review. B, Condensed matter•Dec 15, 2000•65 citations
We study the nonlinear behavior of the electric susceptibility for several systems showing a high index of refraction without absorption due to quantum coherence and interference. Estimates of cw intensity limits for a given index of refraction are obtained. We calculate critical field intensities for self-focusing and investigate the stability of the absorption cancellation with regard to intensity fluctuations.
A theory for the refractive-index enhancement due to quantum coherence and interference has been developed in dispersive polaritonic and photonic band-gap materials doped with an ensemble of noninteracting three-level atoms. Quantum coherence is introduced by driving the atoms with a coherent monochromatic laser field. The real and the imaginary parts of the susceptibility have been calculated by using the master equation and the Laplace transform methods. It is found that the energy gap in these materials plays an important role in the refractive-index enhancement. Numerical simulations for the real and the imaginary parts of susceptibility are performed for SiC as a function of the probe laser frequency. It is found that there is a giant refractive-index enhancement with vanishing absorption when the resonance frequencies lie near the lower band edge.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.
Refractive index and absorption experienced by a probe field propagating through a three-level atomic medium can be effectively manipulated by the bistable behavior of a control field. The probe field couples the lower transition of the atom in a ladder configuration and experiences normal or anomalous dispersion depending on the control field being in the upper or lower bistable state, respectively. We also obtain nonlinear dynamical instability in the form of periodic self-pulsing as the lower bistable branch becomes unstable, quite unlike earlier demonstrations of an unstable regime in the upper branch. Consequently, the susceptibility experienced by the probe field varies periodically in time as dictated by the control field self-pulsing.
We propose coherent control and enhancement of the refractive index with zero absorption in an n-type asymmetric double quantum well using intersubband transitions. These effects are caused by quantum coherence and interference whereby a strong infrared laser mixes upper conduction subbands of an intersubband transition with an auxiliary subband. In the ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ system considered here, an approximately 10% enhancement of the refractive index accompanied with zero absorption occurs at 10.5 \ensuremath{\mu}m, when the system is driven by a 6.2-\ensuremath{\mu}m laser field with an intensity of \ensuremath{\sim}1 MW/cm2{\mathrm{M}\mathrm{W}/\mathrm{c}\mathrm{m}}^{2}MW/cm2. The interplay between quantum coherence and electron population dynamics not only causes an enhancement of refractive index similar to that in atomic systems, but can also induce both low positive and negative group velocities.