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Ultrafast Phenomena and Nanophotonics

Theoretical analysis of four-wave mixing on semiconductor quantum dot ensembles with quantum light

Hendrik Rose, Stefan Grisard, Artur V. Trifonov, Rilana Reichhardt, Matthias Reichelt, M. Bayer, Ilya A. Akimov, T. Meier

Published Mar 15, 2023• 0 citations
Publisher

Abstract

The nonlinear optical response of an ensemble of semiconductor quantum dots is analyzed by wave-mixing processes, where we focus on four-wave mixing with two incident pulses. Wave-mixing experiments are often described with semiclassical models, where the light is modeled classically and the material quantum mechanically. Here, however, we use a fully quantized model, where the light is given by a quantum state of light. Quantum light involves more degrees of freedom than classical light as e.g., its photon statistics and quantum correlations, which is a promising resource for quantum devices, such as quantum memories. The light-matter interaction is treated with a Jaynes-Cummings type model and the quantum field is given by a single mode since the quantum dots are embedded in a microcavity. We present numerical simulations of the four-wave-mixing response of a homogeneous system for pulse sequences and find a significant dependence of the result on the photon statistics of the incident pulses. The model constitutes a problem with a large state space which arises from the frequency distribution of the transition energies of the inhomogeneously broadened quantum dot ensemble that is coupled with a quantum light mode. Here we approximate the dynamics by summing over individual quantum dot-microcavity systems. Photon echoes arising from the excitation with different quantum states of light are simulated and compared.

DOI
10.1117/12.2647700

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The nonlinear optical response of an ensemble of semiconductor quantum dots is analyzed by wave-mixing processes, where we focus on four-wave mixing with two incident pulses. Wave-mixing experiments are often described with semiclassical models, where the light is modeled classically and the material quantum mechanically. Here, however, we use a fully quantized model, where the light is given by a quantum state of light. Quantum light involves more degrees of freedom than classical light as e.g., its photon statistics and quantum correlations, which is a promising resource for quantum devices, such as quantum memories. The light-matter interaction is treated with a Jaynes-Cummings type model and the quantum field is given by a single mode since the quantum dots are embedded in a microcavity. We present numerical simulations of the four-wave-mixing response of a homogeneous system for pulse sequences and find a significant dependence of the result on the photon statistics of the incident pulses. The model constitutes a problem with a large state space which arises from the frequency distribution of the transition energies of the inhomogeneously broadened quantum dot ensemble that is coupled with a quantum light mode. Here we approximate the dynamics by summing over individual quantum dot-microcavity systems. Photon echoes arising from the excitation with different quantum states of light are simulated and compared.
We present a cavity-QED theory for generating squeezed light from semiconductor quantum dots (QDs) integrated in microcavities. We formulate equations of motion for an inhomogeneously broadened QD ensemble that is incoherently pumped and simultaneously driven by a coherent seed field, solve for steady states, and compute the output-field quadrature variances. The analysis identifies operating conditions that yield amplitude-quadrature squeezing, with photon-number fluctuations reduced below the coherent-state limit and squeezing levels as large as 5 dB attainable with presently accessible QD and cavity parameters using only ~ 1 uW pump power. We further show that quantum correlations originating from four-wave mixing play a dual role: they both shape the gain spectrum and generate squeezing. These correlations constitute the quantum counterpart of the mean-field (semiclassical) mechanisms responsible for self-mode-locking in QD lasers and the ultra-narrow lasing linewidths achieved under self-injection locking.
We develop an exact analytical approach to four-wave mixing and higher-order optical nonlinearities of a quantum dot-microcavity system. This approach allows us \nto analyse the non-linear response in terms of coherent transitions between rungs \nof the Jaynes-Cummings ladder, finding their complex amplitudes and frequencies. \nIncreasing the pulse area of the excitation field, a transition from quantum strong \ncoupling to coherent classical regime is observed, which manifests itself in a formation of the famous Mollow triplet, with the observation in the three-level model \nof a quantum dot, including a biexciton, to predict two new effects: One is the \nexistence of a Mollow-like quintuplet, the other is a complete elimination of the \nfour-wave-mixing optical non-linearity under certain parameter constraints. We further develop a simple and accurate approximation in the low-damping limit of the \nJaynes-Cummings model of the quantum dot-microcavity system, for nonlinear response of any order in a closed analytic form, predicting the positions and widths \nof the spectral lines. For higher damping, we demonstrate a strong relaxation down \nthe ladder, shown by both numerical and analytic calculation.
Interaction of light with atomic-like solid-state systems, such as semiconductor quantum dots (QDs) with discrete energy levels, is key for the implementation of high-speed quantum gates and information processing in semiconductor-based devices. Of particular interest are degenerate multiwave-mixing processes under resonant conditions that establish strong light–matter interactions beyond the limits of perturbative nonlinear optics. Here, we exploit a multitude of picosecond optical pulses to manipulate the unitary evolution of excitons in (In,Ga)As QDs and address the role of resonance detuning in an inhomogeneously broadened ensemble. In particular, we use the two-pulse photon echo sequence to uncover the coherent dynamics of charged excitons, whereas the areas of two additional control pulses serve as tuning knobs for adjusting the magnitude and timing of the coherent emission in a wide range up to several hundred of picoseconds. Furthermore, we make use of the spin degeneracy of the ground and excited states of charged QDs to control the polarization state of the emitted light. We reveal that control pulses, whose durations are comparable to the dephasing time of the ensemble, lead to a temporal sorting of optical multiwave-mixing processes, which is in accordance with the developed theoretical model. Lifting the temporal degeneracy allows us to smoothly trace the transition from the perturbative to the regime of Rabi rotations and opens up new possibilities for the optical investigation of complex energy-level structures in so far unexplored material systems.
We theoretically study quantum-optical properties of one- and two-dimensional semiconductor nanostructures, where the electronic band structure is described by a two-band tight-binding model. Since the focus of our work is on analyzing quantum-optical effects of systems with a continuous band structure, many-body processes like electron-electron and electron-phonon interactions are neglected. The quantum-optical interband excitations are fully incorporated and described by a Jaynes-Cummings-type model. The exciting quantum light can be a state with arbitrary photon statistics. For simplicity, the results discussed here are limited to single-photon and two-photon Fock states. Our analytical approach is based on the eigenvalue problem and can be utilized to obtain explicit expressions for the steady-state properties of the system. Numerical simulations are based on solutions of the equations of motion and allow for a more precise analysis of the dynamics and nonresonant excitations of the system. We demonstrate that during the interaction process, a collective excitation of the conduction band is formed. For nonresonant excitations, this collective dynamics results in interesting steady states, in which the resonantly addressed eigenstates are occupied. The presented model provides a framework for microscopic simulations of quantum-optically excited extended semiconductor nanostructures.
The Jaynes-Cummings (JC) model is at the heart of cavity QED and suitable to describe many different systems. Here, the authors use it to investigate a quantum dot strongly coupled to a micropillar cavity. Supported by experiments, they apply four-wave mixing spectroscopy to study the system's dynamics. They push the limits of JC physics by injecting up to $\ensuremath{\sim}20$ photons using ultrashort laser pulses. Starting with complicated dynamics, the signal evolves into a single-frequency oscillation monitoring the relaxation down the JC ladder.
Abstract Four‐wave mixing (FWM) optical response from a sample containing a non‐homogeneously broadened ensemble of vertically stacked pairs of quantum dots is studied theoretically. The effects of collective interaction between the dots and their electromagnetic environment are analyzed. It is shown that the shape of the FWM echo is affected by the coupling between the two dots and, to some extent, by the statistics of the ensemble. In the time‐integrated signal, oscillations on picosecond delay time scale appear which lead to a reduction of the coherent response for coupled dots. For longer delays, the evolution is dominated by radiative decay, with the decay rate increasing from Γ to 2Γ(superradiance effect) as the coupling becomes stronger. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)