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Physical Review B

Validity of semiclassical treatment of optical response in cavity systems

Hiroshi Ajiki

Published Mar 7, 2005• 2 citations
Publisher

Abstract

We have investigated the validity of the semiclassical treatment to calculate the third-order nonlinear spectra of cavity systems. Even if applied field is coherent, a quantized effect of a cavity-quasimode manifests itself in the nonlinear spectra in the weak- and strong-coupling regimes, and thus the semiclassical treatment is found to be invalid. One exception exists in the weak-coupling regime without a pure transverse damping.

DOI
10.1103/physrevb.71.125302

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We have investigated the validity of the semiclassical treatment to calculate the third-order nonlinear spectra of cavity systems. Even if applied field is coherent, a quantized effect of a cavity-quasimode manifests itself in the nonlinear spectra in the weak- and strong-coupling regimes, and thus the semiclassical treatment is found to be invalid. One exception exists in the weak-coupling regime without a pure transverse damping.
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We present a numerical study comparing semiclassical and quantum models of a damped, strongly interacting cavity QED system composed of a single two-level atom interacting with a single quantized cavity mode driven externally by a tunable monochromatic field. We compute the steady state transmission spectrum of the coupled system under each model and show that in the strong coupling regime, the two models yield starkly different results. The fully quantum mechanical model of the system correctly yields the expected multiphoton transmission spectra while the semiclassical approach results in a bistable spectrum.
The effect of phonons on a nonlinear optical response of a quantum dot-cavity system in quantum strong coupling regime can be accounted for by a fully analytical treatment, provided that the exciton-phonon dynamics is much faster than the exciton-cavity dynamics. Modern experiments involving semiconductor quantum dots embedded in optical microcavities typically meet this criterion. We find that, for a relatively small exciton-cavity coupling, the effect of phonons is concentrated mainly in the polaron shift of the exciton frequency and reduction of exciton-cavity coupling by the Huang-Rhys factor. We have generalized this result to an arbitrary optical nonlinearity and demonstrated a good agreement with the exact solution in a wide range of temperatures. This generalization provides access to higher rungs of the Jaynes-Cummings ladder, where exact numerical approaches are impractical or even impossible. At larger coupling strengths and low temperatures, our approximation is also in good agreement with the exact solution, which makes it a very useful tool for addressing the phonon contribution to the coherent dynamics of a nonlinear optical system. We demonstrate our results for third-order optical polarization with varying observation time and delay time between excitation pulses in the form of two-dimensional spectra. These spectra provide useful information about the coherent coupling between the exciton and the cavity modes. The presented analytical approach is also compared with another useful approximation, having the form of a matrix product, which is a special case of the asymptotically exact solution in the limit of a short phonon memory time.
We present detailed numerical simulations of semiclassical and quantum spectra of a cavity QED system consisting of a single three-level atom in Λ-configuration interacting with a quantized cavity mode through one of its transitions while its other transition is driven by a coherent classical field. We compute the semiclassical and quantum spectra of the system under various levels of external driving field strengths. In the semiclassical approach, we neglect the correlations between the cavity and atomic operators, whereas in the quantum case no such assumption is made. Our results show that the two approaches yield identical results under sufficiently weak driving field conditions. However, the two approaches yield starkly different results at stronger driving field intensities: the fully quantum approach yields a well-defined multiphoton spectrum whereas the semiclassical approach results in a bistable spectrum. Furthermore, our calculations reveal a complex Raman structure at the position of the dark-state transition.
A microscopic semi-classical theory for the self-consistent calculation of the nonlinear response field is applied to the exciton–microcavity coupled systems. Our results show that the respective spectral features of the degenerated four wave mixing (DFWM) signal in both the weak (Ω ≤ Γ) and the strong coupling (Ω ≫ Γ) regimes, where Ω is the Rabi splitting and Γ is the width of the modes, can be clearly described with a single framework. In the weak coupling regime, the nonlinear signal is strongest at the bare excitonic resonance, and its enhancement due to cavity effect is remarkable. In the strong coupling regime, our framework very well reproduces the polarization dependent DFWM spectra in a recent experiment.