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Physical Review A

Two-photon nonlinearity by an excitonic system placed in a cavity

Kazuki Koshino, Hajime Ishihara

Published Jun 24, 2005• 11 citations
Publisher

Abstract

We have investigated the two-photon nonlinearity attained by a finite-sized excitonic system placed inside of a cavity, paying attention to the dependence on the system size $N$. The two-photon nonlinearity is evaluated by utilizing semiclassical optical responses. In the quantum-dot limit, the excitonic system behaves as a single two-level system with an enhanced coupling to the cavity mode. In the bulk limit, the nonlinear phase shift decreases as $N$ is increased, proportionally to ${N}^{\ensuremath{-}1∕2}$.

DOI
10.1103/physreva.71.063818

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IEICE Transactions on Electronics•Sep 1, 2005•0 citations
We have investigated the two-photon nonlinearity attained by a finite-sized excitonic system placed inside of a cavity, paying attention to the dependence on the system size NNN. The two-photon nonlinearity is evaluated by utilizing semiclassical optical responses. In the quantum-dot limit, the excitonic system behaves as a single two-level system with an enhanced coupling to the cavity mode. In the bulk limit, the nonlinear phase shift decreases as NNN is increased, proportionally to N\ensuremath−1∕2{N}^{\ensuremath{-}1∕2}N\ensuremath−1∕2.
A single atom in a cavity is the model system of cavity quantum electrodynamics (CQED). The strong coupling regime between the atom and cavity-confined photon corresponds to the reversible exchange of energy between the two modes, and underpins a wide range of CQED phenomena with applications in quantum information science, including for example as quantum logic gates and as sources of entangled states. An important advance was achieved recently when strong coupling between excitons and cavity photons was reported for the first time for localized quantum dots (QDs) in micron-size solid state cavities. This has significance in terms of scalability and integration with other optical devices, and could lead to the emergence of "quantum optics on a chip" technology. However the results presented so far for quantum dots are in the linear regime, corresponding to coupling to the vacuum field (vacuum Rabi splitting); they are not a true QED effect and can equally well be described by classical physics as the coupling between two oscillators. In this paper, we present evidence for a purely quantum phenomenon for the QD/cavity photon system, namely the increase in splitting of the levels when the mean number of photons in the cavity is increased. This corresponds to non-linearities on the single-photon scale: the presence of a single excitation in the cavity changes the level structure, affecting the emission energies for a second photon. Such results are a first step in demonstrating the promise of quantum dots for CQED applications.
We coherently probe a quantum dot that is strongly coupled to a photonic crystal nano-cavity by scattering of a resonant laser beam. The coupled system's response is highly nonlinear as the quantum dot saturates with nearly one photon per cavity lifetime. This system enables large amplitude and phase shifts of a signal beam via a control beam, both at single photon levels. We demonstrate photon-photon interactions with short pulses in a system that is promising for ultra-low power switches and two-qubit quantum gates.
Exploiting the field-amplification effect of a cavity, the possibility of optical nonlinearity by only two photons was indicated experimentally. In the present article, we review our recent analysis of the two-photon dynamics in a cavity quantum electrodynamics (QED) system. Since a cavity-QED system is highly dispersive around its resonances, the shapes of photonic pulses are significantly deformed through interaction with the system. Thus, the present analysis is based on a formalism beyond single-mode approximations. The external photon field is treated rigorously as a continuum, which enables us to handle the two-photon wavefunction in the space representation. The degree of optical nonlinearity in a two-photon state is quantified by comparing the output wavefunction with the linear output wavefunction. It is revealed that the semiclassical optical response theory can be applied for evaluation of the two-photon optical nonlinearity. The two-photon nonlinearity appears not purely as a phase shift in the output wavefunction. The degradation of the fidelity between the output wavefunction and the linear output wavefunction always occurs, which hinders the application of this nonlinear effect as a quantum phase gate. The optimum condition for maximizing the two-photon nonlinearity is clarified, suggesting that pulse shape control is more essential than the Q-value control of the cavity QED system.
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.
The effect of a cavity on the third-order optical nonlinearity, is studied for a two-level system with excitation frequency ω 0 , as a function of the Q factor, coupling constant g, and longitudinal (γ 1 ) and pure transverse (γ 2 ) damping constants. The largest enhancement is found in the strong-coupling regime with γ 1 + 2γ 2 = ω 0 /2Q. Large enhancement is also achieved in the weak-coupling regime satisfying the condition ω 0 /Q = (g 2 /γ 1 )[ 1 + 8γ 1 /(γ 1 + 2γ 2 ) - 1], and the intensity depends on damping constants only. The calculation is based on the cavity QED because the semiclassical treatment of the cavity quasimode leads to incorrect optical nonlinearity.