All-optical charging and charge transport in quantum dots

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 all-optical resonant manipulation of charges in an array of quantum dots formed by a type-II band alignment, such as crystal-phase quantum dots in nanowires. Our work opens new practical avenues for realizations of advanced quantum photonic devices, for instance, solid-state quantum registers with a photonic interface.

First, an electron-hole pair is generated in a long-lived configuration in a "Charging region". Secondly, the charges are transferred along the QD array, electrons to the "Electron region" and holes to the "Hole region". These two stages are repeated to initialize multiple charges within the structure. Step I: Two lasers, A and B, simultaneously resonantly excite two neighboring excitons, X A and X B .
Step II: The electron of exciton X A spontaneously recombines with the hole of exciton X B , leaving behind a spatially separated electron-hole pair forming a long-lived exciton, X D . ( www.nature.com/scientificreports/ 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 long-lived 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 well-known 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 γ D , arising from a non-zero 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 γ , and much quicker than the recombination of X D , i.e. γ ≫ γ D (these assumptions are justified in Supplementary Section S1). The steady-state 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 � A /γ , � B /γ ≫ 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 , γ D , decreases, i.e. when γ D /γ → 0 , as expected. As an example, for a realistic nanowire QD array, we numerically calculate γ D /γ = 10 −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 ≈ 4 µs , 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  www.nature.com/scientificreports/ 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 ultra-fast 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 γ and γ D are decreased with γ D decreasing more quickly such that the ratio γ D /γ 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 nm to L = 62 nm . The tradeoff is an increased pumping time required to reach steady-state, ~ 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 long-lived charges using two tunable lasers is given in the Supplementary Section S5. We note that the calculations of γ D /γ above were made in the absence of Coulomb interactions, pure dephasing, and phonon-assisted processes. The effect of Coulomb interactions will be to increase γ compared to γ 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 phonon-assisted pumping of transition |X D � ↔ |X A � |X B � . 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 � , to another, |e R � , via an intermediate spatially distributed negatively charged exciton, |X − � (see Fig. 4a). These three states form a -type system, where the ground states are the spatially separated electron states |e L � and |e R � . 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 � ↔ |X − � with a single laser, i.e. 1 = 0 , 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 − � → |e R � , eventually leaves the electron in the right QD. Once the electron is in state |e R � 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 well-known optical spin-pumping techniques 9 , but with location-states instead of spin-states. 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 � = 20γ 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 pulse-lengths. For realistic pulse intensities, we can obtain high fidelity coherent transfer within 0.1 μs, i.e. faster than coherence times in state-of-the-art systems utilizing, for instance, spin-echo 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  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 all-optical 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 crystal-phase switching in nanowires. Along with well-known methods for optical preparation, manipulation, and read-out of the spin states, our method enables the realization of advanced multi-qubit structures with a photonic interface such as, for instance, a scalable all-optically controlled QD-based 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.