Recent advances in exciton based quantum information processing in quantum dot nanostructures

Recent experimental developments in the field of semiconductor quantum dot spectroscopy will be discussed. First we report about single quantum dot exciton two-level systems and their coherent properties in terms of single qubit manipulations. In the second part we report on coherent quantum coupling in a prototype"two-qubit"system consisting of a vertically stacked pair of quantum dots. The interaction can be tuned in such quantum dot molecule devices using an applied voltage as external parameter.


Introduction
The use of coherent phenomena for the implementation of quantum information technology is expected to provide significant scope for advanced developments in the future [1]. Semiconductor quantum dot (QDs) nanostructures are artificial atoms and hence suitable entities to implement arrays of qubits for solid state based quantum information processing. One possible approach is the use of excitonic excitations in the ground state of a QD as basis for a two-level system. Recently coherent population oscillations, so called Rabi oscillations [2], have been demonstrated in the exciton population of single QDs [3,4,5,6,7,8]. Low temperature dephasing times for excitons in self-assembled QDs have been shown to exceed several hundred ps, allowing for large numbers of coherent manipulations with ps pulses before decoherence occurs. [9,10] In the present paper, we summarise recent experimental and theoretical developments in the field of semiconductor based quantum information research. The paper is organized as follows: In the first section we report about single QD exciton two-level systems, their coherent properties in terms of a single qubit manipulations, and quantum interference. In the second section, we demonstrate coherent quantum coupling in a prototype "two-qubit" system consisting of a vertically stacked pair of QDs. Furthermore, we show that the qubit-qubit interaction can be tuned using an external parameter. Such controlled coupling is required to perform conditional quantum operations using an intrinsically scalable system and, as such, may be a vital resource for the future implementation of quantum algorithms on the basis of semiconductor nanostructures. Finally, we present a brief outlook in section 3.0.

Section 1. Single quantum dot photo diodes
The experimental results presented here have been obtained from the ground states of single self-assembled In 0.5 Ga 0.5 As QDs. Within this section we further concentrate on n-i-Schottky diodes grown by molecular beam epitaxy on a (100)-oriented n + -GaAs substrate. While based on a conventional diode structure, here a GaAs n-i-Schottky structure, the only optically active part is a single self-assembled In 0.5 Ga 0.5 As QD contained in the intrinsic layer of the diode (see Figure 1.1a). The QDs are embedded in a 360 nm thick intrinsic GaAs-layer, 40 nm above the n-doped GaAs back contact.
A semitransparent Schottky contact is provided by a 5-nm-thick titanium layer. The optical selection of a single QD is done by shadow masks with apertures from 100 to 500 nm, which are prepared by electron beam lithography from a 80-nm-thick aluminium layer (see reference [11]). For resonant excitation we use a tunable Ti:Sapphire laser, which is focused on the sample by a NA = 0.75 microscope objective. All experiments were carried out at 4.2 K. A QD is an artificial atom in a semiconductor, which acts as a protective container for quantized electrons and holes. A single QD photodiode as used here essentially is an exciton two-level systems with electric contacts. In addition to the fundamental optical processes (B 12 , B 21 , A 21 indicated in Figure 1.1b, substantial tunneling escape τ esc -1 appears at electric fields beyond about 35 kV/cm. The diode arrangement allows for photocurrent (PC) detection, a very sensitive and, as a matter of fact, quantitative way to determine the excitonic occupancy of the two-level system.

Ultra narrow linewidth
Recently PC experiments on QD ensembles have given insight into the mechanisms and time scales of carrier capture, redistribution and escape processes [12,13,14,15].

Furthermore PC experiments on single self assembled QDs have been performed,
showing the discrete absorption characteristics of single QDs resulting in sharp spectral features [11]. By use of the quantum confined Stark effect the quantum dot ground state and hence the Eigenenergy of the two-level system can be nicely tuned in energy (see figure 1.2). If the electric field is increased the transition energies of the QD shows a red shift.
Within a limited range a PC spectrum can be obtained by a sweep of the bias voltage at fixed laser wavelength. Figure 1.2 shows a number of spectra all representing the same QD state, namely the ground state one exciton (1X) resonance. The excitation wavelength was slightly increased for each spectrum, resulting in a shift of the resonance towards higher bias voltages.
All spectra in figure 1.2 were taken at the same excitation power of approximately 65 nW. One immediately recognizes a notable increase in linewidth at higher bias voltage levels. In the regime of low bias voltages two sharp peaks are clearly distinguishable whereas at high voltages only one broad peak can be observed. A detailed measurement of the Stark effect gives us a conversion of voltage into energy scales with a relative uncertainty of less than 3%. Thus we are able to infer an increase in linewidth from 9 µeV at 0.4 V to about 150 µeV at 1.25 V. This increase corresponds to a higher tunneling probability and therefore shorter lifetime of the investigated 1X state with increasing electric fields. The doublet line structure visible at low voltages can be further investigated by control of the polarization of the excitation beam. On rotating the orientation of linear polarization each peak can be clearly suppressed with respect to the other. This can be explained by a slight shape asymmetry, present in almost any self assembled QDs, resulting in an energy splitting between wave functions oriented along or perpendicular to the elongation axis [16,17].We observe here an energy difference between both levels of about 30 µeV. The fact that this splitting is clearly visible underlines the high spectral resolution of our experiment. We actually expect the resolution to be only limited by the linewidth of the laser used for excitation. Resonant absorption spectroscopy therefore generally is a very capable method for line shape and fine structure analysis of QDs [18]. Due to improvements in sample design we were able to substantially reduce the linewidth of the QD resonance. The spacing between QDs and n + -back contact is 40nm in the present case as compared to 20 nm in the experiments reported in Ref. [19]. As a consequence, we think, Coulomb interactions between QD states and fluctuating background charges have been substantially reduced, which otherwise cause considerable dephasing and therefore an increase in linewidth. In Figure 1.3 we show our so far best experimental result with a directly measured linewidth below 3.5 µeV and a fine structure splitting of 11 µeV.

Non-linear saturation
Another feature of the spectra displayed in Figure 1  In addition a power dependent bleaching of the absorption also contributes to a reduction of PC peak height. This behaviour has been investigated in detail earlier [19] and will be outlined here briefly in the following. For the analysis we chose a bias voltage range where the tunneling time is long enough to allow for narrow linewidths but short enough so that optical recombination only plays a minor role.
Within this range a series of PC spectra has been recorded for varying laser intensities. Each spectrum has been fitted by two Lorentzian lines where data points and fit typically show a correlation of 99.95%. For further analysis the arithmetic mean of the two peak heights is used in order to reduce complexity and to minimize random fluctuations. In figure 1.4 we show such an analysis of peak height versus excitation power for a bias voltage of 0.4 V. A clearly nonlinear power dependence is observed, resulting in a saturation of PC peak amplitude at high excitation. The saturation curve can be described by the following equation [19]: Here I denotes the photocurrent peak amplitude, I sat its saturation value and P corresponds to the normalized excitation power [20]. The physical content of the PC saturation value can be derived fairly easily: If the QD is already occupied by one exciton no further absorption can take place due to a renormalization of energy levels [21,22] caused by few particle interactions. This also holds true if the QD is occupied only by one carrier. Under the applied conditions the first tunneling process happens fairly fast whereas the tunneling time τ slow of the slower carrier can even exceed the radiative lifetime. The observed saturation value then is given by I sat = e/2τ slow [20], where e is the elementary charge. The evaluation of the measured data gives a PC saturation value of I sat = 15 pA and a tunneling time of τ slow ≈ 5.3 ns, respectively.

Power broadening
The saturation behaviour outlined before also has a direct effect on the linewidth of absorption peaks, independent of any other line broadening mechanisms.
At exact resonance, i.e. at the center of a PC peak, the absorption naturally comes nearest to its saturation value. Therefore with increasing excitation the increase in absorption or in our case the increase in PC signal is weakest at the center of a peak and comparatively stronger at its sides. This results in a broadening of the absorption line known in literature as power broadening [23,24]. If one has a homogeneously broadened absorption peak with a Lorentzian line shape of width Г 0 the power broadened peak again is Lorentzian but with an increased width Г according to In figure 1.5 we show an analysis of linewidth versus excitation power obtained from the same set of data as used in Figure 1 Figure 1.4 gives us a measure for the time it takes the system to go back to its initial state. In genuine two-level systems this would be the lifetime of the excited state, which is usually denoted by T 1 . In our case this is the escape time of the slower of both photo excited carriers, as discussed above. The linewidth Г 0 derived in Figure 1.5 corresponds to the dephasing time of the system, in the context of two-level systems usually denoted as T 2 . It is important to note that the correct dephasing time can only be derived by an extrapolation to zero power. We performed the according measurements at V bias = 0.4 V and infer a linewidth Г 0 as low as 4.3 µeV. This reflects a significant increase of the tunneling time towards lower bias voltages. On the other hand a long tunneling time also means that the system needs a long time to come back to its initial state. Saturation and power broadening therefore play an important role even at a comparatively low excitation power. Thus in Figure 1

Coherent manipulations of a qubit: Rabi oscillations
In the following section we want to focus on the coherent behaviour of the QD. Due to finite dephasing times we use ultrashort laser pulses for excitation. The fundamental experiment in the coherent regime is the observation of Rabi oscillations [2]. The occupancy of the upper level of a two-level system under coherent resonant excitation is given by ) 2 / ( sin 2 t ⋅ Ω [23], where the Rabi frequency Ω is proportional to the square root of the laser intensity and t corresponds to the pulse length. A πpulse thereby results in a complete inversion of the two-level system. In the context of quantum computing this represents a qubit rotation analogous to the classical NOT operation. We define the pulse area, i.e. the rotation angle t ⋅ Ω = Θ , by adjusting the excitation amplitude rather than the pulse length (see: Figure 1   in wetting layer tail states. As the background is linear in excitation power, it is clearly distinguishable from the oscillatory coherent signal. At π-pulse excitation the incoherent part is about 6 % of the total signal. We use here circular polarized light, in order to avoid biexciton generation not only by spectral separation but also by Pauli blocking [25]. The maximum rotation angle of 6 π as shown here (almost 9π have been demonstrated by us in most recent work) significantly exceeds that of any previously published data on excitonic Rabi oscillations (see for example references [3][4][5][6][7][8]. Even more important, the observed oscillations only slightly decrease towards high pulse areas, although the excitation power at 6 π is 36 times higher than at π. This is also an experimental proof that the generally observed strong damping at Θ > 1 π is of no principal nature, but usually is caused by sample specifics or by the measurement technique. The data displayed in Figure 1

Double pulse experiments: Quantum interference
While the measurement of Rabi oscillations represents the occupancy of a two-level system, we have to perform quantum interference experiments to also gain access to the phase of coherent excitations (see Figure 1.8). First experiments of this kind have been done in the weak excitation regime, i.e. at pulse areas much less than 1 π [9,26]. In order to obtain relevant results on phase coherence with respect to quantum information processing, these experiments have to be extended to the strong excitation regime [4,6,27]. Figure 1.8. Schematic diagrams of a 2-pulse quantum interference experiment on a qubit. The first π/2 laser pulse brings the qubit from |0> into a superposition state and defines thereby also the reference phase in the system. Depending on the phase of the second π/2 pulse, the qubit performs a quantum interference between |0> (phase shift π) and |1> (no phase shift).
We have performed here experiments with π/2 pulses, representing a 1 qubit Hadamard transformation in context of quantum computing [28]. The first pulse thereby creates a coherent superposition of the |0> and |1> state of the QD two-level system. The second pulse then follows with a variable delay in the range of 0 to 1000 ps. The relative phase of the second pulse can be controlled via an additional fine delay with sub-fs resolution. If coherence is maintained, the superposition state is expected to be transferred into the pure |1> or |0> state, depending on whether the two pulses are of the same or opposite phase, respectively. When varying the phase continuously, we observe an oscillation of the PC at the same period as the optical interference at overlapping pulses. The amplitude of these oscillations versus delay time is displayed in Figure 1.9. The data represented has been obtained for linear polarized excitation, for which only one of the asymmetry split levels contributes. A fit to these data points reveals a purely exponential decay at delay times >10ps, corresponding to a dephasing times of T 2 = 320ps, 230ps, and 110ps for bias voltages of 0.4 V, 0.48 V, and 0.59V respectively. The analysis of the first few picoseconds is complicated by the fact that an overlap of both pulses to some degree influences the measurement results. We still are able to determine some initial dephasing, though, in the best measurements, this amounts to less than 4 %.
We further are able to compare dephasing times measured by quantum interference with those derived from the linewidth analysis. At low bias and accordingly long tunneling times, however, saturation results in a broadening of the linewidth even at low excitation intensities [20]. We consequently performed a full power broadening analysis with an extrapolation to zero excitation for all measurements up to 0.6 V. At higher bias voltages the PC saturation value is high enough so that the linewidth of single low power spectra will already yield the correct results. The linewidth Γ can be converted into a dephasing time T 2 via Γ = / 2 2 h T [23]. Both sets of data show excellent agreement up to a bias of 0.7 V. At still higher bias we observe quantum beats independent of the choice of polarization and it is therefore difficult to infer a dephasing time.
Resuming the comparison of coherent versus steady state measurements, we get an agreement in several aspects: The ground state linewidth nicely corresponds to the decay time of the quantum interference. The asymmetry induced splitting of energy levels is reflected in the period of quantum beats (not shown here). The polarization at which these effects are suppressed is the same in both measurements. . This is in good agreement with saturation measurements, even though the optical peak power typically differs by five orders of magnitude.
In summary we have performed a whole range of fundamental experiments with respect to two-level systems. All experimental results can be brought down to few basic properties of the investigated single QD, giving evidence for an almost ideal quantum system. We are able here to draw a comparison between complementary experimental methods, so that any indirectly derived parameter can be confirmed by a direct measurement. Furthermore many results mark a major advance in experimentally proven quality, encouraging further work on this kind of quantum system.
In the context of quantum computing the present work demonstrates excellent control over an exciton qubit in a semiconductor QD. In the current experimental setup the ratio of dephasing times versus excitation pulse length would allow for the order of 10 2 coherent operations. This could be increased on the excitation side by going to shorter laser pulses (see e.g. [29]). In addition any tunnelling related dephasing can be inhibited by applying sufficiently low bias voltage during the coherent manipulation.
Electrical readout then would be done by applying a short voltage pulse after the optical qubit rotations have been completed.

Section 2 Controllable Coupling in Quantum Dot Molecules
The observation of well resolved Rabi oscillations in the interband optical response of isolated quantum dots (QDs) clearly demonstrates their potential for the realisation of qubits based on electron-hole pair excitations (excitons). As clearly demonstrated in section 1 of the present paper, for the single exciton (=1e+1h) long coherence times approaching the radiative limit are generally observed, much longer than the picosecond duration optical pulses required for quantum state manipulation. [5] The combination of ultrafast optical gating with sensitive schemes for quantitative electrical readout makes such excitonic qubits highly attractive for the implementation of quantum information technologies based on solid state hardware.
A basic requirement for any realistic quantum hardware is the ability to perform two qubit operations that, when combined with single qubit rotations, would enable the implementation of arbitrary quantum algorithms. [30] Such conditional quantum operations have already been demonstrated in the restricted basis of one and two exciton states in individual "natural" quantum dots [31] formed from interface fluctuations in quantum wells. However, a major drawback of this approach is that it has little or no prospects for further scalability due to the rapidly diverging complexity of the optical response as the number of carriers in the dot increases. [32,33,34] In the interests of constructing more complex quantum processors using an intrinsically scalable hardware, we have to focus on multiple QD  The section is organised in the following way: the growth engineering methods employed to realise low density, self-assembled QD molecules with similar electronic structure are introduced in section 2.1. This is followed by a discussion of the excitonic spectrum of QD-molecules and the influence of electric field on the electronic structure (section 2.2). Finally, the devices investigated and results of the single molecule spectroscopy are presented in 2.3.

Fabrication of low density, self-assembled QD-Molecules
The optimisation of self-assembled nanostructure growth techniques in the early to mid 1990 gave significant impetus to the field of single dot spectroscopy. For a recent review of the advances in fabrication and understanding of the physical properties of such self-assembled nanostructures the reader is directed to reference [55].
Experiments such as those discussed in section 1 of the present paper, which are capable of probing the unique quantum mechanical properties of isolated dots, only became possible due to the development of growth techniques capable of engineering the dot transition energy and surface density during growth. For single dot spectroscopy, the QD density has to be sufficiently low in order to use shadow masks with sufficiently high light extraction efficiencies. Furthermore, ground state transition energies higher than ~1240meV are needed to use sensitive silicon-based detectors for emission (photoluminescence -PL) and tuneable Ti-Sapphire lasers for absorption (e.g. photocurrent) spectroscopy. Thus, both the density and emission energy must be optimised to use self-assembled Ga (1-x) In x As for experiments in quantum information processing.
Almost all approaches employed until now to realise suitable Ga (1-x) In x As-GaAs QD material rely on adjusting the Ga (1-x) In x As coverage to be just above the critical threshold for self-assembly to obtain a low dot density. Furthermore, control of the In-content enables tuning of the emission energy. We now introduce a widely applied technique to obtain low surface density QDs by interrupting the rotation of the substrate the dot layer growth. This produces a material gradient across the wafer and a position where the coverage is insufficient for QD formation. We will demonstrate how this concept can be readily extended to double layer QD-molecules, which have a high stacking probability, a low surface density and suitable emission energies by carefully adjusting the relative amount of Ga (1-x) In x As deposited in the upper and lower dot layers.    We have presented a readily applicable technique to grow QD-molecules with high stacking probability and low surface density by tuning the Ga 0.5 In 0.5 As coverage in the upper and lower dot layers with otherwise fixed growth parameters. A reduction of the total Ga 0.5 In 0.5 As coverage from 8ML to 7ML for the growth of the upper layer is found to produce structurally similar dots, with comparable transition energies. The growth parameters obtained from this study were used to grow electrically active samples for single QD-molecule spectroscopy presented below in section 2.3.

Exciton states of QD-Molecules and Quantum Confined Stark Effect
Before presenting the single molecule spectroscopy data we discuss the nature of the single exciton states for our QD-molecules and the influence of static electric field perturbations applied along the QD growth axis. The geometry and composition of the modelled QDMs were based on the X-TEM microscopy investigations discussed above ( Figure 2.2(b)) and previous studies of the microscopic In:Ga composition profile of similar, nominally Ga 0.5 In 0.5 As dots embedded within GaAs. [59] A one band effective mass Hamiltonian was used to calculate the single particle states including a realistic treatment of strain and piezoelectric effects. The Coulomb interaction between electron and hole was included self-consistently using the local spin density approximation in order to obtain the single exciton transition energies [60]. character and would be expected to vanish in the optical spectrum. We demonstrate below that precisely this behaviour is observed in our single molecule spectroscopy experiments.

Single QD-Molecule Spectroscopy
In order to test the predictions of the calculations presented in the previous section we performed low temperature PL spectroscopy on single QDMs embedded in n-type Schottky photodiodes using techniques similar to those employed in section 1 of the paper. The epitaxial layer sequence was as follows: firstly, a highly doped n + -back  Typical PL-spectra recorded from a single QD-molecule are presented in the main panel of Figure 2.5a as a function of electric field (F) increasing from the upper to lower spectrum. The spectra presented were recorded using a very low excitation power density (P ex~2 .5Wcm -2 ) to ensure that single exciton species dominate the emission spectra. For the lowest electric fields investigated a number of lines are observed in the emission spectrum distributed over a ~5meV wide window around ~1289meV. Measurements of the intensities of each of these peaks as a function of excitation power density (not shown) demonstrate that the peaks labelled X(e L h L ) at 1289.2meV and X(e U h U ) at 1289.8 vary linearly with P ex . These features are, therefore, identified as the single, optically active exciton transitions in the QDM as discussed in the previous section. [33,34] In contrast, the other peaks labelled 2X and 2X´ are found to exhibit a superlinear power dependence suggesting that they arise from biexciton states in the QDMs. A discussion of these higher order exciton complexes will be presented elsewhere, here we focus only on the single exciton species for comparison with the calculations discussed in section 2.2. We begin by discussing the peak X(e U h U ). This feature shifts weakly with increasing field until, at a critical field of F~17.5kV/cm denoted by the red spectrum in Figure   2.5, the shift rate suddenly increases and the peak intensity quenches rapidly. Over the same range of electric field, another peak with single exciton power dependence, labelled X(e L h U ) is observed at higher energy. This feature shows precisely the opposite behaviour: it is weak and shifts rapidly for F<17.5kV/cm whereafter the shift rate suddenly decreases and the peak gains intensity. [50] These two features clearly anticross at F crit =17.5kV/cm as depicted by the dashed lines on Figure 2  shown in Figure 2.5a was obtained using h l =3.5 nm, h u =4 nm and d=12nm to produce fairly good quantitative agreement with the experimental data.
We now turn to the second peak observed in the low field spectra of Figure 2.5 with single exciton character -X(e L h L ). This feature shifts weakly over the entire range of electric field investigated and clearly does not show an anticrossing with X(e L h U ).
The observation of a weak shift rate identifies this peak as an excitonic state with direct character. Reference to the calculations presented in Figure 2.3b indicate that it arises from the direct exciton state in the lower quantum dot X(e L h L ). Reference to the calculations presented in Figure  The characteristic form and electric field dependence of the PL spectra discussed above was found to be very reproducible, similar results having been obtained from more than 25 different QD-molecules. We now present the statistical distribution of the measured coupling energy (2E e-e ) extracted from the observed anticrossing between X(e U h U ) and X(e L h U ), the "centre energy" between these states , and the critical electric field F crit at which the anticrossings are observed. These results of these investigations are summarised in   The statistical distribution of the quantum coupling strength extracted from the splitting between X B and X A is plotted in Figure 2.6b together with the normal distribution, from which the mean coupling strength is measured to be <2E e-e > =1.6±0.4meV. Also shown on the figure by the blue arrows is the calculated quantum coupling strength for a number of dot-dot separations ranging from d=13-10nm, using the model QD-molecule parameters employed to fit the data of figure 2.5. The best agreement with calculation is obtained for d=12±0.5nm, very close to the nominal dot-dot separation of d=10nm for this sample. Experimentally, it appears that the two QD-molecules are more weakly coupled than would be expected for a nominal 10nm separation, for which we calculate 2E e-e =4.6meV. This discrepancy may reflect, for example, the microscopic In-distribution throughout the QD-molecules which is not expected to be homogeneous. [61] However, it is likely that this minor discrepancy is due to the one-band model used for our electronic structure calculations. Further investigations are currently in progress to study the dependence of <2E e-e > on the nominal dot-dot separation and provide definitive experimental data for more detailed comparison with theory.
The statistical distribution of the field at which anticrossings are observed (F crit ) is presented in Figure 2.6c. The mean value of the critical field is <F crit >=14±4 kV/cm.
As discussed in section 2.2, F crit is most sensitive to the difference of the electronic structure of the upper and lower dots. Indeed, for the model QD-molecule parameters used to fit the data of Figure 2.5 we assumed slightly detuned QDs. Comparing the distribution of F crit with the associated spectral detuning ∆E of the QDs (arrows in Figure 2.6c) we find an <∆E>=0±8meV emphasising the good spectral overlap achieved between the excitonic states in two QD layers.

Outlook
We has been recently demonstrated that the electron spin degree of freedom is a particularly stable variable in QDs [62], which may even have coherence times approaching the microsecond range or even longer. [63] Furthermore, single spins can already be selectively generated [62] and detected [64] using optical techniques and, it may be possible to manipulate them over ultrafast timescales using pulsed laser sources. [65] Regardless of which qubit basis eventually emerges to have the most favourable properties, the ability to selectively initialize, control and readout the quantum state using optical means is likely to remain of paramount importance due to the unique selectivity that it provides. The rapid progress in the field over the past five years is a firm indication that viable systems for quantum information processing will emerge in the near future.