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Optical micropillar Bragg cavities of different diameters and coupled by a small bridge have been realized experimentally by means of a focused ion beam system. The resonator modes in these coupled microcavities are either localized in one pillar or delocalized over the whole photonic structure, a fact that could be exploited to control the coupling between two spatially separated quantum dots, i.e. placed in different pillars, via the enhanced electromagnetic field in such a coupled microcavity. A simplified two dimensional simulation has been used to predict the resonant wavelengths and design the optical modes in these coupled Bragg cavities.
©2007 Optical Society of America
1. Introduction
In recent years, quantum dots (QDs) in high-quality (Q) cavities have been proposed as feasible candidates for the realization of quantum information processing using their electronic states as quantum bits (qubits)[1, 2]. Furthermore, quantum computing requires an independent initialization and preparation of single qubits in QDs as well as a controlled mutual interaction for two-qubit-operations. The essential interaction between two QDs can be mediated by the electromagnetic field in a microcavity, in particular, delocalized cavity modes combined with a method of tuning the QDs’ emission in and out of such a resonant mode could be exploited for a controlled optical interaction whereas localized modes might serve for read-out processes. In this paper we propose AlAs/GaAs Bragg cavities for an optical coupling of spatially separated QDs. Such microcavities have been intensively studied for years [3, 4] reaching high-Q-factors meanwhile [5, 6]. Furthermore, symmetrical coupled microcavities have already been demonstrated [7, 8]. Even chains of coupled microcavities with photonic defect states have been reported [9] implying localized defect modes. In the following we study micropillar cavities of different diameters coupled by a small bridge and resulting in modes which are either localized in one of the pillars or delocalized over the whole photonic structure.
In the first step we develop and verify the modeling of single pillars which gives us a viable tool to design coupled pillar structures. These coupled microcavities are engineered such that both localized as well as delocalized optical modes coexist. Finally, the designed structures have been realized and indeed, both predicted mode types are observed, thus opening the way for a controlled coupling of spatially separated QDs.
2. Pillar fabrication
Our planar layer structures are grown in a molecular-beam epitaxy (MBE) system. The GaAs λ cavities contain InGaAs QDs (emission wavelength ~ 950 nm) in their center. Top and bottom AlAs/GaAs distributed Bragg reflectors (DBRs) consist of 20 to 25 pairs. All photonic structures mentioned here are fabricated directly by a focused ion beam system using an ion current from 5 nA for a rough excavation of material down to 100 pA for polishing the resulting surface. In such a way, single pillars in the diameter range from 2.7μm to 15μm have been produced, see the scanning electron microscope (SEM) image in Fig. 1(a). The microcavities are investigated optically in a confocal micro-photoluminescence (μ-PL) set-up with the sample in a cryostat at around T=10 K. For wide pillars we achieve Q-factors exceeding 12000. The inset of Fig. 2 shows the μ-PL spectrum with several spectrally separated modes as an example for the results obtained.
Fig. 1. SEM images showing a single pillar with a diameter of 2.7μm (a) and a coupled pillar structure with pillar diameters of 6.3μm and 4μm (coupling type I-II), a center distance of 5.7μm, and a bridge width of 3μm (b).
Fig. 2. Simulated (solid line) and experimentally found (crosses) resonance wavelengths in single pillars in dependence of their diameters. The dashed lines mark different coupling types (I-II and II-III). The inset shows a typical μ-PL spectrum.
3. Study of single micropillars
In order to achieve localized and delocalized optical modes within the same photonic structure we have to design the coupled micropillar cavities correctly. For this purpose we have implemented a simple and versatile numerical simulation technique to model the mode structure of arbitrary Bragg cavities. First this simulation has been developed and tested with single pillars and confirmed by experimental results. The Bragg cavity is a three-dimensional problem but can be separated into two: (1) a one-dimensional λ cavity with high reflectivity on both sides referring to the optical confinement between the two DBRs (z-direction), (2) a two-dimensional problem concerning the lateral confinement in the GaAs cavity surrounded by air/vacuum (x-y plane). The first criterium implies a fixed component of the wavevector in z-direction, i.e.kz(d) = 2πn GaAs/d with the distance d between the DBRs and the refractive index n GaAs of GaAs, whereas the latter 2D problem is solved by a commercial software which treats the cavity as an infinitely extended step-index fiber using the material parameters of GaAs in the core and air/vacuum in the cladding and searches for eigenmodes. As a result of this calculation, which utilizes a finite-element method, we obtain the electric and magnetic field distribution for each supported mode including its vertical and lateral components of the wavevector. Thus, a dispersion relation for the energy E(kz) can be derived for each mode. Recomposing the solutions of the separated problems determines for all modes the energy E(kz(d)) by the GaAs layer thickness d. In practice, d is allowed to vary slightly to better fit the experimental data, but is fixed for all modes and photonic structures processed from the same sample.
Figure 2 displays the calculated resonant wavelengths for the first 40 modes in dependence of the pillar diameter as solid lines. The modes are typically twice degenerate because of rotational symmetry in the single pillar [10]. Following the wavelengths of the modes they spread for smaller diameters and shift to higher energies. In total, eight single pillars with different diameters on the same sample have been fabricated. μ-PL measurements not only confirm how the modes tend to shift with size but also give the wavelength positions with high accuracy (crosses in Fig. 2). Although no information about the theoretical Q-factor can be extracted from this model, it has a great advantage: arbitrary shapes of DBR cavities can be modeled easily.
4. Coupled micropillar cavities
In particular, we have modeled coupled pillars. In first studies the magnitude of mode coupling in structures with equal diameters have been simulated and fabricated [8], followed by structures with unequal diameters (Fig. 1(b)). The diameters are designed in such a way that, e.g., the second mode in the smaller pillar would be energetically equal to the third mode in the wider pillar (coupling type II-III). Thus, in the coupled structure the formerly degenerate modes split into two different modes (II-IIIa and II-IIIb). Figure 3 shows the calculated color-coded intensity distribution for the lowest five modes (coupling type II-III, see dashed lines in Fig. 2). As a result this coupled structure develops new types of modes delocalized over the whole photonic structure, namely II-IIIa and II-IIIb in Fig. 3. These two modes correspond to a symmetric/antisymmetric mode splitting (II-IIIa/II-IIIb) [7] with an antinode/node of the intensity in the barycenter of the coupled microcavity. On the other hand, since the fundamental and the second mode of the wider pillar and the fundamental mode of the smaller pillar have no counterpart on the other side of the coupled resonator they still can be found localized on their side (Fig. 3; 0-I, 0-II and I-0, respectively). Generally, dependent on the magnitude of coupling, which can be varied by the bridge width and length, the coupled modes (especially the symmetric ones) are shifted to lower energies; in the example with a bridge width of 2.9μm and a center distance of 5.7μm the symmetric coupled mode (Fig. 3, II-IIIa) is energetically even lower than the former second mode of the single wide pillar (Fig. 3; corresponding to 0-II).
Fig. 3. Color-coded intensity distribution of the lowest optical modes in a coupled pillar structure with diameters of 5.0 μm and 3.7μm (coupling type II-III), a center distance of 5.7μm, and a bridge width of 2.9μm, according to the simulation.
In experiment, the detection of the μ-PL signal with a charge-coupled device (CCD) camera allows a spatial resolution along the direction of the spectrometer slit. Hence, we can map out each detected mode spatially and thus identify it. Figure 4 displays two μ-PL spectra for the same coupled pillar structure as modeled in Fig. 3 (they only vary in the different position of the exciting laser spot). Below each CCD image the spatially integrated spectrum is plotted. Vertical lines in these spectra mark the resonant wavelength positions expected according to the simulation. The measured peaks in the spectra fit to the simulation with notable accuracy and allow their identification. The simulated modes are labelled with the same Roman numbers as in Fig. 3. Comparing the five modes in the detected, spatially resolved image spectra (Fig. 4) and the simulated intensity distributions (Fig. 3) also shows conspicuous correlations, i.e., for the detected modes corresponding to 0-I and 0-II only intensity in the wider pillar could be measured whereas I-0 is mainly localized in the smaller pillar and II-IIIa/II-IIIb are detected over the whole coupled structure, exactly the same as predicted by the simulation.
The laser excitation (825 nm) for the QDs is placed on the wider pillar for the measurement shown in Fig. 4(a). For that reason the fundamental mode of the wider pillar (0-I), corresponding to the fundamental mode of the whole structure, dominates the spectrum, simultaneously mode 0-II with similar localization is increased. However, in the spectrum of Fig. 4(b) mode I-0 shows the highest μ-PL intensity as the excitation is focused on the smaller pillar. The de-localized modes II-IIIa and II-IIIb show comparable intensity in both measurements, hardly depending on the excitation position.
Fig. 4. Spatially resolved color-coded μ-PL spectra for a coupled pillar structure with diameters of 5.0μm and 3.7 μm (coupling type II-III), a center distance of 5.7μm, and a bridge width of 2.9μm. The contour on the left indicates the detection position by a vertical dashed line as well as the position in the spatial resolution. The spectra below the images display the spatially integrated μ-PL intensity and the modeled resonance wavelengths (vertical lines). The Roman numbers refer to the same modes as in Fig. 3. The two spectra differ in the excitation position which is placed on the wider pillar (a) or the smaller pillar (b).
As a consequence of this coupled microcavity structure, QDs which are in resonance with a localized optical mode in one of the coupled pillars are not able to interact via the electromagnetic field with QDs located in the other pillar, whereas QDs which emit at wavelengths that correspond to a delocalized mode couple within the whole photonic structure mediated by the electromagnetic field of the delocalized modes. Extending this microcavity structure by the capability of tuning the QD emission into and out of the different cavity resonances, e.g. by applying an external electric field, should enable to control the coupling of QDs in such a device.
5. Conclusions
Based on simplified numerical simulations we designed coupled pillar structures with unequal diameters to achieve localized and delocalized resonator modes. The latter have been demonstrated experimentally through μ-PL measurements of coupled structures fabricated according to the design. QDs in resonance with delocalized modes couple within both interconnected pillars. At the same time, the interaction of QDs in resonance with localized modes is spatially restricted to one single pillar. This could be used to control the optical coupling between different spatially separated QD states by tuning their emission energy via an applied external field, an approach of potential interest within the context of a possible QD-based quantum information processing.
Acknowledgments
We would like to thank E. Müller for technical assistance at the FIB. This work has been performed within project A2 of the DFG Research Center for Functional Nanostructures (CFN). It has been further supported by a grant from the Ministry of Science, Research and the Arts of Baden-Württemberg (Az: 7713.14-300).
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