Physica E 12 (2002) 900 – 903
Entangled exciton states in quantum dot molecules M. Bayera;∗ , G. Ortnera , A. Larionova , V. Timofeeva , A. Forchela , P. Hawrylakb , K. Hinzerb , M. Korkusinskib , S. Fafardb , Z. Wasilewskib a Technische
Physik, Universitat Wurzburg, Am Hubland, D-97074 Wurzburg, Germany for Microstructural Sciences, NRC, Ottawa, Canada K1A 6N5
Abstract The exciton states in self-assembled InAs=GaAs quantum dot molecules have been studied by optical spectroscopy and the results have been compared to detailed theoretical calculations. We demonstrate that excitons in symmetric molecule structures form entangled quantum states with the degree of entanglement controlled by tunneling and Coulomb interaction. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.21.La; 78.67.He Keywords: Quantum dot molecules; Entangled exciton
1. Introduction Currently there is strong interest in quantum information processing (see, for example, Ref. ) in a solid state environment, in particular using semiconductor quantum dots (QDs). An essential building block of a quantum processor is a quantum gate which consists of two coupled quantum bits. Recently, we have proposed that a pair of vertically aligned QDs could be used as the optically driven quantum gate : The quantum bits are individual carriers either on dot zero or dot one. The diBerent dot indices play the same role as a “spin”, therefore we call them “isospin”. Quantum mechanical tunneling between the dots rotates the “isospin” and leads to superposition of two quantum dot states. The quantum gate is built when two diBerent particles, an electron and a hole, are created optically. The two particles form entangled isospin states. The entanglement can be controlled by ∗
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application of an electric Ield along the heterostructure growth direction. Here, we present spectroscopic studies of single quantum dot molecules (QDMs) with diBerent vertical separation between the dots that support the feasibility of this proposal. The comparison of the evolution of the excitonic recombination spectrum with the results of calculations allows us to demonstrate coherent tunneling of electrons and holes across the separating barrier and the formation of entangled exciton states. For a given barrier width, we Ind only small variations of the tunneling-induced splitting between the entangled states demonstrating a good homogeneity within a QDM ensemble.
2. Quantum dot molecules A key prerequisite for this demonstration of entanglement is the development of a growth technique allowing the fabrication of vertically aligned QDs
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 4 6 2 - 3
M. Bayer et al. / Physica E 12 (2002) 900 – 903
Fig. 1. Photoluminescence spectra recorded at T = 2 K on ensembles of quantum dots (left panel) and quantum dot molecules (right panel) for varying excitation powers (increasing from bottom to top). The barrier thickness in the quantum dot molecules was 5 nm.
with the same geometries. A promising technique in this respect is the Stransky–Krastanov epitaxy, for which the strain Ield of a QD in a Irst layer facilitates the growth of a second QD above it. This technique was supplemented by an indium-Kush procedure which allows for a shape engineering of the QDs: The Irst layer of dots having the form of lenses with a height of ∼4–6 nm is grown on a wetting layer of less than two monolayers of InAs on GaAs. Subsequently, the dots in this Irst layer are covered with 3 nm of GaAs, at which time the indium-Kush leaves all the dots with the same height. The dots are subsequently covered by a GaAs layer with a total thickness d measured from the wetting layer, and the same process is repeated for the second dot layer. The width of the GaAs barriers was varied from 4 to 8 nm. From transmission electron microscope images we Ind only small variations of the geometries of the two QDs in a molecule. The dot shapes can be well approximated by disks with heights between 1 and 2 nm and radii between 8 and 12 nm. For narrow barrier widths the vertical alignment of the dots is perfect. For wide barriers we Ind as well molecules with perfect alignment. However, we also Ind structures, for which a lateral displacement of the dots with respect to each other occurs. This is well conceivable, because for wide barriers the inKuence of the strain from the dot in the Irst layer will be reduced.
3. Spectroscopic data and interpretation From the coupling of two system with a discrete electronic shell structure we expect a splitting of each state into ‘bonding’ and an ‘anti-bonding’ orbitals, in analogy to the binding observed for molecules found in nature. Fig. 1shows state Illing spectroscopy performed on ensembles containing ∼106 QDs (left panel) and QDMs (right panel) where d was 5 nm. For the QDs at low excitation only emission from the s-shell is observed. With increasing excitation, higher dot shells are occupied by electron–hole pairs according to the Pauli principle and emission from the p-shell appears in the spectra. The energy splitting between these shells is about 50 meV. For the QDMs, already at very low excitation two features separated by ∼25 meV are observed, which can be attributed to the expected coupling-induced splitting
en erg y [eV] Fig. 2. Low excitation photoluminescence spectra recorded at T = 60 K on a single QD (upper trace) and on single QDMs (lower traces) with barrier widths of 5 and 8 nm .
of the s-shell exciton. Note, that the higher lying line cannot be attributed to p-shell emission, because the splitting between p- and s-shell is mainly given by the lateral conInement and therefore is approximately the same for the QDMs: p-shell emission can be observed 50 meV above the s-shell when increasing the optical excitation. For it, also a splitting by ∼30 meV is observed which is slightly larger than the splitting in the s-shell. The larger splitting is well conceivable because of the larger penetration of the carrier wave
M. Bayer et al. / Physica E 12 (2002) 900 – 903 50
QDM - d=5nm
energy splitting [meV]
energy splitting [meV]
barrier width [nm]
Fig. 3. Left panel: Variation of the energy splitting between the two lowest entangled exciton states for diBerent QDMs with a barrier thickness of 5 nm. Right panel: Dependence of the splitting obtained by averaging the data from a large number of QDMs on barrier thickness.
functions into the GaAs barrier between the dots for higher lying states. Still detailed information about the electronic shell structure cannot be obtained from these ensemble spectra because of the inhomogeneous broadening. Therefore, we have isolated single QDMs in lithographically fabricated mesa structures with lateral sizes of about 100 nm. Fig. 2shows corresponding photoluminescence spectra for molecules with barriers of 5 and 8 nm width in comparison to the spectrum from a single dot. The sample temperature was 60 K, which allows for thermal excitation of carriers into higher lying shells. Therefore excitonic emission from the s- and the p-shells are observed for the single dot. For the QDMs a splitting of the s-shell emission is observed that increases strongly with decreasing barrier width. For a barrier width of 8 nm the splitting is ∼10 meV, for a 5 nm barrier it is 30 meV. Let us analyze the exciton states that can be formed in QDMs. In a symmetric QDM structure the tunneling causes a splitting of the single particle states into symmetric (bonding) and antisymmetric (anti-bonding) orbitals. Because we generate optically electron–hole pairs, excitonic eBects have to be included in the discussion. There are four diBerent ways to distribute electron and hole among the two coupled dots with isospin indices 0 (upper dot) and 1 (lower dot): |0; 0 and |1; 1 (optically active states) as well as |0; 1 and |1; 0 (optically inactive states). To take the symmetry of coupling to the light Ield into account, a transformation into a basis of entangled isospin
states is useful . These entangled basis states are: |a |0; 0 + |1; 1 |b 1 |0; 1 + |1; 0 |c = √2 |0; 1 − |1; 0 : |d |0; 0 − |1; 1 Obviously states |c and |d are antisymmetric and therefore optically inactive. It is the states |a and |b, which form the optically active states observed experimentally. They are the excitonic analogs of the symmetric and antisymmetric single particle states. The splitting between them is given by the tunneling matrix elements of the carriers, which are renormalized by the Coulomb interaction between electron and hole. The left panel in Fig. 3 shows the measured energy splitting of the s-shell exciton states for diBerent QDMs having a GaAs barrier thickness of 5 nm. For all structures a considerable splitting of 24 ± 6 meV is observed. Its rather small variation indicates a good sample homogeneity, in particular a good vertical alignment of the dot structures. We note however, that for large barrier widths we Ind also QDMs whose spectra give clear indications for an asymmetry of the QDMs . The right panel of Fig. 3 shows the dependence of the average energy splitting of the s-shell exciton on the barrier width d. With decreasing d a strong increase of the splitting is observed. Most importantly, for d ¡ 5 nm the splitting becomes larger than the thermal energy at room
M. Bayer et al. / Physica E 12 (2002) 900 – 903
temperature which makes it insensitive on thermal perturbations and the system of QDMs attractive for quantum information applications. Acknowledgements This work has been carried out under the Canadian European Research Initiative on Nanostructures supported by IMS NRC, NSERC, and EC. The WPurzburg group acknowledges Inancial support by the DFG, the DARPA and the State of Bavaria.
References  D. Bouwmeester, A. Ekert, A. Zeilinger (Eds.), The Physics of Quantum Information, Springer, Berlin, 2000.  P. Hawrylak, S. Fafard, Z.R. Wasilewski, Condens. Matter News 7 (1999) 16.  M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z.R. Wasilewski, O. Stern, A. Forchel, Science 291 (2001) 451.  G. Ortner, M. Bayer, A. Larionov, V.B. Timofeev, A. Forchel, P. Hawrylak, M. Korkusinski, S. Fafard, Z. Wasilewski, submitted for publication.