Abstract
Optically active quantum dots are one of the promising candidates for fundamental building blocks in quantum technology. Many practical applications would comprise of multiple coupled quantum dots, each of which must be individually chargeable. However, the most advanced demonstrations are limited to devices with only a single dot, and individual charging has neither been demonstrated nor proposed for an array of optically active quantum dots. Here we propose and numerically demonstrate a method for controlled charging of multiple quantum dots and charge transport between the dots. We show that our method can be implemented in realistic structures with fidelities greater than 99.9%. The scheme is based on alloptical resonant manipulation of charges in an array of quantum dots formed by a typeII band alignment, such as crystalphase quantum dots in nanowires. Our work opens new practical avenues for realizations of advanced quantum photonic devices, for instance, solidstate quantum registers with a photonic interface.
Introduction
Optically active quantum dots (QDs), as opposed to gatedefined QDs, provide a spinphoton interface^{1,2,3,4} which can act as a fundamental unit in a quantum computer^{5} or quantum network^{6}, onto which quantum information can be stored, manipulated and readout. In particular, the spin states of a confined electron or hole in a QD make up a qubit which can be efficiently manipulated optically^{7,8}. Many steps towards the realization of a QDbased spinphotonic quantum network have already been demonstrated for isolated QDs. These include picosecond optical coherent single spin initialization^{8,9,10,11}, manipulation^{7,12} and readout^{13,14}, spinphoton entanglement^{2,3}, photontospin teleportation^{15} and distant spin–spin entanglement^{16}. However, to exploit the full potential of such a quantum network, multiqubit nodes consisting of several coupled QDs in an array are necessary. Such a QD array would constitute a quantum register in which advanced quantum operations, such as CNOTgates^{17}, can be performed, enabling universal quantum computing and communication of multiqubit states.
The fabrication of semiconductor QDs is now reaching the point where multiQD structures can be engineered and reliably realized with atomic precision in nanowires thanks to recent progress on crystal phase control during the growth^{18}. However, the fabrication alone does not enable it as a spinphotonic interface as the spincarrying charges first have to be individually loaded into the QDs in a controlled manner. Such charge loading can be reliably achieved in single QD systems^{19}. However, a scalable method allowing controlled sitespecific charging of an array of optically active QDs remains yet to be proposed.
Here, we present a novel method for controlled alloptical charging of multiple QDs exploiting the unique advantages of an array of QDs formed by a typeII band alignment. Such an array can be reliably achieved using crystalphase quantum structures in nanowires^{18,20,21,22,23,24,25,26} in, for example, InP or GaAs. In these structures, excited electrons and holes are confined within different spatial regions, as shown in Fig. 1a. Consequently, excitons are spatially indirect and can share a single charge state. A hole can, therefore, act as a link, connecting spatially separated electron states, which we exploit in this work (likewise, an electron can connect two spatially separated hole states). Even though the hole and electron of an exciton in a typeII bandstructure are spatially separated^{21,27}, the exciton oscillator strength can still be substantial in an array of typeII QDs^{24,28}, allowing for efficient optical exciton generation.
The structure we are considering is an array of typeII QDs in a nanowire, as illustrated in Fig. 1b. Our scheme uses two sequential stages: In Stage 1, charge pairs are generated in a “Charging region” and in Stage 2 single charges are transferred to opposite regions of the nanowire, labeled as the “Electron region” and “Hole region” in Fig. 1b. These two stages are repeated to obtain the desired charge configuration. If only a single type of charge is required, e.g. electrons, one can add a reservoir in one end of the nanowire where the unused charges, e.g. the holes, are dumped.
The charge generation in Stage 1 is realized using the two steps illustrated in Fig. 2a:

Step I: The initial state is the ground state in which no charges are excited in the structure. Two lasers then simultaneously resonantly excite two neighboring excitons, labeled X_{A} and X_{B}. The structure is designed such that X_{A} and X_{B} have different energies to allow for their individual optical addressing.

Step II: The electron of X_{A} and the hole of X_{B} now form a third exciton, labeled X_{C}, which can also spontaneously recombine. Spontaneous recombination of X_{C} leaves behind a longlived exciton, X_{D}.
The large spatial separation of the charges of X_{D} results in a considerably lower electron–hole overlap and so this final charge configuration can be thought of as stable, with a lifetime multiple orders of magnitude larger compared to an electron–hole pair in adjacent QDs (like X_{A}, X_{B}, X_{C}), as shown in the Supplementary Information Section S1. X_{D} is the final configuration of Stage 1 in Fig. 1b, where we have generated one electron and one hole in separate regions of the structure. The spin state can subsequently be initialized using wellknown spin pumping techniques^{8,9,10,11}.
We now calculate the preparation fidelity of X_{D}. In principle, the presence of X_{D} can be heralded by the detection of the photon spontaneously emitted by X_{C}. However, with a low collection and detection efficiency, this heralding method will be inefficient. A more efficient method would be to track the resonance fluorescence from the recombination of X_{A} or X_{B}. Once exciton X_{C} recombines, the lasers can no longer excite X_{A} and X_{B} and thus resonance fluorescence from these excitons are quenched, heralding the preparation of exciton X_{D}. Such dynamical charge sensitivity was recently experimentally demonstrated^{29}. This method is further explained in Supplementary Section S5. Alternatively, one can just pump the QDs with the two lasers (A and B) for a sufficient time, since once the system reaches the end of Step II the lasers will have no further effect as they are resonant only with their respective exciton energies. A detailed energy diagram of the involved states in our scheme is shown in Fig. 2b. A fundamental limit to the success probability of this approach is the spontaneous recombination rate of exciton X_{D}, denoted \({\gamma }_{D},\) arising from a nonzero electron–hole overlap. Due to the similar sizes of the QDs, we assume the recombination rates of excitons X_{A}, X_{B}, and X_{C} to be equal and denoted \(\gamma ,\) and much quicker than the recombination of X_{D}, i.e. \(\gamma \gg {\gamma }_{D}\) (these assumptions are justified in Supplementary Section S1). The steadystate fidelity, F, i.e. the probability to observe only exciton X_{D} after sufficiently long pumping time, can be calculated using the Master equation as (see Supplementary Section S3):
where we have assumed sufficient laser powers such that \({\Omega }_{A}/\gamma ,{\Omega }_{B}/\gamma \gg 1\). This assumption ensures that both excitons are generated before the first exciton recombines and is experimentally feasible (see Supplementary Section S3). Equation (1) shows that the fidelity approaches unity as the recombination rate of exciton X_{D}, \({\gamma }_{D}\), decreases, i.e. when \({\gamma }_{D}/\gamma \to 0\), as expected. As an example, for a realistic nanowire QD array, we numerically calculate \({\gamma }_{D}/\gamma =1{0}^{4}\) resulting in \(F=99.96\%\). The dimensions of the QDs should be chosen to make each transition individually addressable via appropriate laser frequencies and to suppress tunneling effects. Figure 3a shows the fidelity as a function of pumping time. We see that the fidelity reaches its maximum after \(t \approx 4\;{\upmu }{\text{s}}\), as indicated by the dashed lines. Note that the charging time cannot be decreased by increasing the laser powers, as the limiting time factor is the spontaneous recombination rate of exciton X_{C}. This charging time is relatively long compared to typical qubit coherence times of QD defined qubits, but we emphasize that the purpose of the charge generation protocol is to place the charges in the desired positions, before any subsequent quantum operations, requiring qubit coherence, are carried out. Yet, if ultrafast charging is required, one can stimulate the recombination of exciton X_{C} with a third laser in pulsed mode, enabling charging on a ps timescale.
In Fig. 3b we show how the maximum fidelity and charging time depends on the size of the structure. In these calculations, the structure length is varied while keeping the relative QD sizes constant. By increasing the size, both \(\gamma\) and \({\gamma }_{D}\) are decreased with \({\gamma }_{D}\) decreasing more quickly such that the ratio \({\gamma }_{D}/\gamma\) decreases (see Supplementary Section S1), yielding a larger fidelity. For example, to increase the fidelity from \(F=99.9\%\) to \(F=99.99\%\) requires an increase of the structure size by about 14%, from \(L=54\;\mathrm{nm}\) to \(L=62\;\mathrm{nm}\). The tradeoff is an increased pumping time required to reach steadystate, ~ 2.4 times longer in this case. Thus by making the structure sufficiently large, very high fidelities can be reached with a pumping time on the order of μs. A proposal for a practical experimental scheme to implement and verify the generation of longlived charges using two tunable lasers is given in the Supplementary Section S5. We note that the calculations of \({\gamma }_{D}/\gamma\) above were made in the absence of Coulomb interactions, pure dephasing, and phononassisted processes. The effect of Coulomb interactions will be to increase \(\gamma\) compared to \({\gamma }_{D}\), and therefore our calculations are, in fact, underestimating the charging fidelity (see Supplementary Section S2). Pure dephasing has the effect of slightly reducing the fidelity while increasing the required pumping time, due to less efficient pumping of the system (see Supplementary Section S3). Phonon interactions might limit the fidelity due to phononassisted pumping of transition \(\left. {\left {X_{D} } \right.} \right\rangle \leftrightarrow \left. {\left {X_{A} } \right.} \right\rangle \left. {\left {X_{B} } \right.} \right\rangle\). Therefore the energy levels should be engineered to minimize this effect.
Once a pair of charges has been generated, they can be subsequently transferred to their desired positions, as illustrated in Stage 2 of Fig. 1b. This charge transfer process is described in Fig. 4. A single electron can be moved from one location state, \({e}_{L}\rangle\), to another, \({e}_{R}\rangle\), via an intermediate spatially distributed negatively charged exciton, \({X}^{}\rangle\) (see Fig. 4a). These three states form a \(\Lambda\)type system, where the ground states are the spatially separated electron states \({e}_{L}\rangle\) and \({e}_{R}\rangle\). This system allows optical transfer of an electron between the QDs. The easiest way to realize such transfer is to resonantly pump the transition \({e}_{L}\rangle \leftrightarrow {X}^{}\rangle\) with a single laser, i.e. \({\Omega }_{1}\ne 0\), \({\Omega }_{2}=0\) in Fig. 4a. This corresponds to the generation of an exciton consisting of the middle hole and the right electron, in the presence of the left electron. Spontaneous recombination of the left electron with the hole, i.e. the transition \({X}^{}\rangle\) → \({e}_{R}\rangle\), eventually leaves the electron in the right QD. Once the electron is in state \({e}_{R}\rangle\) the laser pump has no further effect, and so by pumping for a sufficiently long time, the electron is effectively transferred from one location to another. This is analogous to wellknown optical spinpumping techniques^{9}, but with locationstates instead of spinstates. This pumping technique is incoherent, i.e. any quantum coherence of the electron is lost, which, however, is fine for initializing the positions of the charges. Instead, if one wishes to extend the scheme for a more general transport of qubits, coherent transfer is required. Such a coherent transfer can be realized in our system using e.g. the stimulated Raman adiabatic passage (STIRAP) scheme^{30} as proposed in literature^{17}. In Fig. 4b we calculate and compare the fidelities of coherent and incoherent transfers against transfer times, expressed as the duration of the involved laser pulses (see Supplementary Section S4 for details). We see that for laser powers of \(\Omega =20{\gamma }_{1}\) (the same power used in the charge generation stage), the performances of the two schemes are comparable, with the STIRAP scheme resulting in larger fidelities for the considered pulselengths. For realistic pulse intensities, we can obtain high fidelity coherent transfer within 0.1 μs, i.e. faster than coherence times in stateoftheart systems utilizing, for instance, spinecho techniques^{31}. With the STIRAP protocol the charge can thus be moved several steps before spin coherence is lost. Generally, high laser powers make coherent charge transfer the fastest option, as incoherent transfer is limited by the spontaneous recombination rate of the charged exciton (Supplementary Section S4). On the other hand, incoherent charge transfer is a much easier option, requiring only a single laser, and no accurate control of pulse timing and duration.
To conclude, we have presented a novel alloptical scheme for a scalable method for charging an array of QDs. In addition, we have shown how to transfer charges in such an array of QDs, both coherently and incoherently. The utilization of spatially indirect excitons for coupling of spatially separated charge states, is central for our scheme. The structures modeled in our work can be readily fabricated with current growth technology of crystalphase switching in nanowires. Along with wellknown methods for optical preparation, manipulation, and readout of the spin states, our method enables the realization of advanced multiqubit structures with a photonic interface such as, for instance, a scalable alloptically controlled QDbased quantum register, not possible with existing schemes.
Data availability
The data supporting the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We gratefully acknowledge the support of Villum Fonden (Project no. VKR023444). We thank Niels Gregersen and Masoomeh Taherkhani for useful discussions and help in the initial stages of this work. We further thank Niels Gregersen for his technical review of the manuscript.
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J.H. and N.A. conceived the original concept. J.H. carried out the simulations. J.H., L.L. and N.A. discussed the results and wrote the manuscript. N.A. coordinated the project.
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Hastrup, J., Leandro, L. & Akopian, N. Alloptical charging and charge transport in quantum dots. Sci Rep 10, 14911 (2020). https://doi.org/10.1038/s4159802071601x
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