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Physical Review B

Interaction of a quantum-dot cavity system with acoustic phonons: Stronger light-matter coupling can reduce the visibility of strong coupling effects

Martin Glässl, L. Sörgel, Alexei Vagov, Mihail D Croitoru, T. Kühn, V. M. Axt

Published Jul 20, 2012• 25 citations
Publisher

Abstract

We present a numerically complete study of the combined dynamics of a quantum dot exciton coupled to a single quantized cavity mode and a continuum of acoustic phonons. We demonstrate that acoustic phonons have a pronounced impact on effects characteristic of the strong light-matter coupling regime, such as vacuum Rabi oscillations and collapse and revival scenarios. This impact is considerable already at zero temperature, where initially no phonons are present. Counterintuitively it is found that an increase of the light-matter coupling does not necessarily enhance the visibility of strong-coupling effects. In fact, for typical experimental situations, a stronger light-matter coupling will considerably reduce the visibility.

DOI
10.1103/physrevb.86.035319

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We present a numerically complete study of the combined dynamics of a quantum dot exciton coupled to a single quantized cavity mode and a continuum of acoustic phonons. We demonstrate that acoustic phonons have a pronounced impact on effects characteristic of the strong light-matter coupling regime, such as vacuum Rabi oscillations and collapse and revival scenarios. This impact is considerable already at zero temperature, where initially no phonons are present. Counterintuitively it is found that an increase of the light-matter coupling does not necessarily enhance the visibility of strong-coupling effects. In fact, for typical experimental situations, a stronger light-matter coupling will considerably reduce the visibility.
A microscopic theory to study the combined dynamics of a quantum dot exciton coupled to a single cavity mode as well as acoustic phonons is presented. By considering the phonon subsystem in the coherent state representation, the phonon’s impact on the emitted photons is investigated. The single photon emission probability and phonon effects on this quantity are studied. On the other hand, we illustrate that while a quantum dot interacts with a single cavity mode, the collapse and revival phenomenon will occur. It is demonstrated that acoustic phonons have a pronounced impact on the collapse and revival of the quantum dot exciton. Our finding reveals that at elevated temperatures the phonon interaction decreases the Rabi frequency. Moreover, the cavity mode possesses non-classical features such as sub-Poissonian statistics.
We study experimentally the influence of temperature on the emission characteristics of quantum dot-micropillars in the strong coupling regime of cavity quantum electrodynamics (cQED). In particular, we investigate its impact on the vacuum Rabi splitting (VRS) and we address the important question of the temperature stability of the coherent coupling regime in a semiconductor system, which is relevant in view of both fundamental study and future applications. To study the temperature dependence we investigate an unprecedentedly large number of strong coupling cases (89) in a wide temperature range from 10 up to 50 K, which constitutes a good basis for statistical analysis. The experiment indicates a statistically significant increase of the VRS with temperature in contrast to an expected decrease of the VRS due to the dephasing induced by acoustic phonons. From the theoretical point of view, the phonon-induced renormalization of the VRS is calculated using a real-time path-integral approach for strongly confined quantum dots (QDs), which allows for a numerical exact treatment of the coupling between the QD and a continuum of longitudinal acoustic phonons. The absence of the expected decrease of the VRS with temperature in our experimental data can be attributed to a unique optical property of laterally extended In0.4Ga0.6As\mathrm{I}{\mathrm{n}}_{0.4}\mathrm{G}{\mathrm{a}}_{0.6}\mathrm{As}In0.4​Ga0.6​As QDs used in this study. Their electronic structure facilitates an effective temperature-driven increase of the oscillator strength of the excitonic state by up to 40% in the given temperature range. This leads to enhanced light-matter interaction and overcompensates the phonon-related decrease of the VRS. The observed persistence of strong coupling in the presence of phonon-induced decoherence demonstrates the appealing possibility to counteract detrimental phonon effects in the cQED regime via engineering the electronic structure of QDs.
The excitonic dynamics in a single quantum dot within a microcavity is investigated theoretically in terms of the perturbation treatment based on a unitary transformation. The excitonic Rabi oscillations with a damping rate are obtained explicitly. It is shown that the damping rate depends on exciton-phonon coupling constant and the vacuum Rabi frequency. Even at absolute zero temperature and in the vacuum cavity, the exciton-phonon interaction and exciton-photon interaction still cause excitonic decoherence. We also predict that under strong control fields the collapse and revival of excitonic Rabi oscillation may be observed for the quantum dots with small exciton-phonon coupling in the $T\ensuremath{\rightarrow}0$ limit.
Semiconductor quantum dots (QDs) in photonic nanocavities provide monolithic, robust platforms for both quantum information processing and cavity quantum electrodynamics (QED). An inherent feature of such solid-state cavity QED systems is the presence of electron-phonon interactions, which distinguishes these systems from conventional atomic cavity QED. Understanding the effects of electron-phonon interactions on these systems is indispensable for controlling and exploiting the rich physics that they exhibit. Here we investigate the effects of electron-phonon interactions on a QD-based cavity QED system. When the QD and the cavity are off-resonance, we observe phonon-assisted cavity mode emission that strongly depends on the temperature and cavity-detuning. When they are on-resonance, we observe an asymmetric vacuum Rabi doublet, the splitting of which narrows with increasing temperature. These experimental observations can be well reproduced using a cavity QED model that includes electron-acoustic-phonon interactions. Our work provides significant insight into the important but hitherto poorly understood mechanism of non-resonant QD-cavity coupling and into the physics of various cavity QED systems utilizing emitters coupled to phonons, such as nitrogen-vacancy centres in diamond and colloidal nanocrystals.
In the first part of this dissertation, a dissipative quantum dot (QD) -cavity system coupled to a longitudinal acoustic (LA) phonon reservoir is studied using a numerically exact real-time path-integral approach. Three distinct dynamical regimes of weak (WC), strong (SC), and coherent coupling (CC) are discussed and more accurate conditions identifying them are presented. Our results show that to have the CC regime, which is characterized by clear vacuum Rabi oscillation (VRO), vacuum Rabi splitting (VRS) should be larger than the sum of the widths of the corresponding peaks. In order to distinguish between contributions of population decay and impure dephasing, induced by LA phonons and the dissipations, on the quantum dynamics of the QD-cavity system, we propose a two-part phenomenological expression which fits the QD-cavity decay curves perfectly and is used to calculate the corresponding spectra. The emission rate increases from the carrier recombination rate to a maximum value, which is the mean of the QD and cavity dissipation rates, with QD-cavity coupling strength. We introduce a quantity that can be applied in determining the distinct coupling regimes; This quantity enables us to identify the onset of the SC regime as the point where the impure dephasing term begins to contribute to the central band of the spectrum significantly, as a result of the existence of VRO with a very small frequency (unclear VRO) at the corresponding decay curve. Its contribution to the width of the central band increases with the coupling strength up to the onset of the CC regime, then reduces as a result of the appearance of sidebands in the spectra. The effective population decay and impure dephasing rate contribute solely to the width—of the central and sideband peaks of the triplet spectra respectively—only beyond a very large coupling strength which is the same across the considered temperature range. In the second part, we demonstrate that all the available experimental data of temperature (T)-dependent shift of photoluminescence (PL) peak of In(Ga)As QD samiii ples can be modeled successfully by using a two-oscillator model if and only if the whole temperature interval (0–300 K) is divided into a few parts (at most four parts), depending on dispersion degree of the PL peak from a monotonic behavior. Analysis of the numerical results show that excitons mostly interact (inelastically) with acoustic (AC) or optical (OP) phonons separately. Increasing QDs uniformity, by using some improved growth techniques, results in decreasing or removing the sigmoidal behavior, enhancing total AC phonon contribution to the redshift of the PL peak. Elevation of the zero bandgap (ZBG) energy up to a critical value about 1.4 eV, for In(Ga)As QDs grown using molecularbeam epitaxy, results in enhancement of QD symmetry and total OP phonon contribution and reduction of QDs uniformity and total AC phonon contribution, while a rollover happens for further increase of the ZBG. Therefore we find that the highest QD symmetry and the lowest exciton fine structure splitting correspond to this critical value of ZBG, in accordance with previous experimental results.
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.