The spin of an electron is a natural two-level system for realizing a quantum bit in the solid state1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. For an electron trapped in a semiconductor quantum dot, strong quantum confinement highly suppresses the detrimental effect of phonon-related spin relaxation1,2,3,4,5,6,7. However, this advantage is offset by the hyperfine interaction between the electron spin and the 104 to 106 spins of the host nuclei in the quantum dot. Random fluctuations in the nuclear spin ensemble lead to fast spin decoherence in about ten nanoseconds8,9,10,11,12,13,14. Spin-echo techniques have been used to mitigate the hyperfine interaction14,15, but completely cancelling the effect is more attractive. In principle, polarizing all the nuclear spins can achieve this16,17 but is very difficult to realize in practice12,18,19. Exploring materials with zero-spin nuclei is another option, and carbon nanotubes20, graphene quantum dots21 and silicon have been proposed. An alternative is to use a semiconductor hole. Unlike an electron, a valence hole in a quantum dot has an atomic p orbital which conveniently goes to zero at the location of all the nuclei, massively suppressing the interaction with the nuclear spins. Furthermore, in a quantum dot with strong strain and strong quantization, the heavy hole with spin-3/2 behaves as a spin-1/2 system and spin decoherence mechanisms are weak22,23. We demonstrate here high fidelity (about 99 per cent) initialization of a single hole spin confined to a self-assembled quantum dot by optical pumping. Our scheme works even at zero magnetic field, demonstrating a negligible hole spin hyperfine interaction. We determine a hole spin relaxation time at low field of about one millisecond. These results suggest a route to the realization of solid-state quantum networks24 that can intra-convert the spin state with the polarization of a photon.
Our scheme to initialize a single hole spin is presented in Fig. 1. The quantum dot contains a single hole. The strong in-built strain in an InAs quantum dot shifts the valence light hole states with spin J = 3/2, Jz = ±1/2 away from the fundamental gap such that the uppermost valence states have heavy hole character with spin J = 3/2, Jz = ±3/2. The corresponding hole spin states are represented as and. A -polarized laser drives the hole to an exciton state with spin Sz = -1/2, , containing a spin-up, spin-down hole pair and a spin-down electron. Unlike the hole spin, the electron spin interacts with the nuclear spins through the contact hyperfine interaction. The electron spin experiences a small magnetic field, ∼20 mT (refs 8–12), as a result of the incomplete cancellation of the random nuclear spins in the quantum dot. The component of the magnetic field in the plane, , causes the electron spin in the excited state to precess with a period of ∼1 ns. The coherence of the precession is destroyed by spontaneous emission with a characteristic time of ∼1 ns and results in the shelving of hole spin in the state, which does not couple to the laser field, according to the spin selection rule. This process initializes the hole spin with high fidelity provided that the hole spin relaxation time is sufficiently large. Rather than a hindrance, the hyperfine interaction between the electron spin and nuclear spins is beneficial in this scheme because it enables fast hole spin initialization. The scheme works even when the spin states are degenerate. This is advantageous because hole spin flip processes involving a single phonon are turned off in the limit of zero magnetic field22. The overall scheme depends on stringent requirements: a very clean optical selection rule, slow hole spin relaxation, and negligible interaction between the hole spin and the nuclear spins. We report here successful implementation of this concept with very high fidelity even at zero applied magnetic field where the hole spin is shielded from the laser only by its spin and not by an energy detuning. We confirm our interpretation by applying a magnetic field along the quantization axis z which slows down the precession, gradually turning off the hole spin initialization process.
Our sample, grown by molecular beam epitaxy, consists of InGaAs quantum dots embedded in the intrinsic region of a metal–insulator–semiconductor field electron transistor (Fig. 2a), allowing deterministic charging. The quantum dots are separated by 25 nm from a hole Fermi sea. With an applied bias between a semi-transparent Schottky gate and the Fermi sea, the quantum dot energy can be controlled relative to the Fermi energy thus determining the charge state. Initial characterization is carried out using photoluminescence spectroscopy (Fig. 2b). Each discrete jump in the emission spectrum signals a charging event and we can identify the charge of each state by the characteristic fine structures25,26. The overlap in gate voltage for different charge states is a consequence of the relatively large hole tunnelling time, which we estimate to be ∼10 ns, compared to the radiative emission lifetime of 0.8 ns. This is crucial: it ensures that the cotunnelling process6,7,27 of hole spin relaxation through spin-swap with the Fermi sea is sufficiently weak to implement our scheme.
We drive the optical transitions with a highly coherent continuous-wave laser, detecting resonant scattering of the laser light from the quantum dot by measuring the transmission coefficient with very high signal-to-noise ratio26,28. Figure 2c displays transmission curves as the quantum dot transition energy is tuned through the laser energy via the Stark effect when the quantum dot is empty (X0 transition) and charged with a single hole (X1+ transition), in both cases using linear polarization. We observe lorentzian lineshapes with linewidths of ∼5 µeV. This is larger than both the radiation-broadened linewidth (0.8 µeV) and the linewidth observed in samples with n-doped back contacts (1–2 µeV)6,7,26,28, but small enough to record high-quality laser spectroscopy data. The increase in linewidth beyond the radiative limit is caused by a spectral fluctuation (see Supplementary Information).
Our main result is revealed in the transmission spectra taken at zero externally applied magnetic field, , when the two hole spin states are degenerate. We find that the contrast, defined as the size of the transmission dip at zero detuning, is immeasurably small when the exciting laser has circular polarization, either (Fig. 3a) or (Fig. 3b). This suggests that the hole is prohibited by its spin from scattering the laser light, that is, that optical spin pumping is taking place. We prove this by pumping with two lasers with identical wavelength and with the same total power, one with and one with polarization. In this case, a clear transmission dip appears (Fig. 3c). This arises because spin pumping with polarization is frustrated by the excitation, and vice versa, a repumping phenomenon6.
Hole spin pumping at small magnetic fields is also highly effective. In the Faraday configuration, the degeneracy of the two optical transitions is lifted by the sum of the electron and hole Zeeman energies. When the -polarized laser is tuned to the resonance, the contrast is very small, signifying spin pumping into the state. Contrast reappears in a two-colour experiment when the -polarized laser is tuned to the resonance. Repumping is explored more closely in Fig. 3d. Here the maximum contrast is measured as the laser with polarization is detuned relative to the transition. As expected, maximum contrast is observed when the laser is on resonance. However, an asymmetry is observed with the repumping signal persisting more at larger than smaller detunings. The spin pumping results demonstrate that the hole spin relaxation time must be much, much larger than the spin precession time. For the electron spin at fields 0.3 T, spin pumping is prohibited due to the hyperfine interaction6,7. This is not the case for holes: the high fidelity hole spin pumping even at = 0 demonstrates a negligible hole spin hyperfine interaction.
To investigate the mechanisms involved in the spin pumping process, we measure the contrast as a function of for one-colour excitation with both and polarizations (Fig. 4a). The first feature observed is that the contrast increases as increases. This is consistent with our proposed scheme. As increases, the total magnetic field experienced by the electron spin is tilted further from the (x, y)-plane, reducing the electron spin precession rate and hence the spin pumping rate. When , the total magnetic field lies close to the z-direction and spin pumping becomes very slow, eventually becoming comparable to the hole spin relaxation rate. In this regime at ∼3 T, the transmission signal is fully established. A second feature of the magnetic field dependence is the increasing contrast inequality for excitation with relative to polarization. This dependence arises as the system moves gradually towards thermal equilibrium. The thermal population of (or ) increases (or decreases) with increasing Zeeman splitting and the experimental contrast for (or ) polarization tends to increase (or decrease) at high magnetic fields.
A theoretical analysis provides a quantum mechanical description of the optical pumping scheme and a means of extracting a fidelity for the hole spin initialization and the hole spin relaxation time, . The model takes a four-level basis with the coherent couplings shown in Fig. 3d. Incoherent processes, spontaneous emission, electron spin relaxation and hole spin relaxation, are included with a master equation for the density matrix. The density matrix is linked to the experiment by computing the quantum dot polarization: the differential transmission signal arises through a destructive interference of the laser and Rayleigh scattered fields. The model is explained in detail in the Supplementary Information. All the parameters for the model are known, apart from , which we determine by fitting the calculated signal strength to the experimental data (Fig. 4a).
The theory fit to the two data sets in Fig. 4a is excellent over the full two decades of contrast and yields = 1 ms for = 21mT. Values for for similar dots vary in the literature from 12 to 26 mT (refs 7, and 10–12). Within this range of we generate equally good fits to the data with 0.2 < <1 ms, defining our uncertainty in. The we determine is very similar to that determined from ensemble polarization decay measurements on similar quantum dots at high fields, >1.5 T (ref. 23), where any hyperfine interaction is suppressed by the mismatch in hole spin and nuclear spin Zeeman energies. In our case, our experiment is most sensitive to at low fields where the external magnetic field does not suppress the hyperfine interaction.
The theory perfectly reproduces the repumping experiment in Fig. 3d, in particular the asymmetry about the peak. The electron spin hyperfine interaction couples the two exciton states and , which are diagonalized with a quantization axis tilted from z. In the tilted basis, the exciton states become and , and there are two repumping transitions: at = 0.25 T, is strong and is weak. The asymmetry in the repumping spectrum arises because at large laser energies there is a small contribution from the weak repumping transition. The asymmetry therefore provides further evidence for the coupling of the two exciton states.
We define the fidelity of the hole spin initialization as for excitation with polarization and as for polarization where (or ) is the population of state (or ). The experiment is sensitive not to the populations but to the off-diagonal components of the density matrix. We use the theory to link the two. Under the experimental conditions, a slightly elliptically polarized excitation ( = 99.7%; = 0.3% in power), we deduce a fidelity of 99.0 ± 0.5% at = 0. The uncertainty arises through the uncertainty in . The main experimental limitation is at present the polarization: for pure polarization, the theory predicts an increase in fidelity to above 99.9%. Figure 4b shows how the fidelity falls as the spin pumping mechanism is gradually turned off with an external magnetic field. We note that these results are not restricted to this particular dot: we have demonstrated high fidelity hole spin preparation on about ten different quantum dots.
We turn to the limiting hole spin relaxation mechanism. Cotunnelling is one possibility. A single hole is trapped in the dot over an applied bias range. At the edges of this voltage plateau, the quantum dot and Fermi energies are nearly degenerate, enabling fast hole spin relaxation through cotunnelling6,7,27. Consistent with this, the transmission contrast increases at the plateau edges (Fig. 4b), because cotunnelling now dominates the hole spin relaxation. We can fit the data to determine a cotunnelling spin relaxation time at the plateau edge of ∼3 μs. We then predict the cotunnelling spin relaxation time at the plateau centre to be ∼100 ms. This time of 100 ms is much longer than the deduced from the data in Fig. 4a, and demonstrates that cotunnelling is not the fastest hole spin relaxation mechanism in the plateau centre. Instead, it is likely that a combination of spin–orbit and hole–phonon interactions ultimately limits the coherence of the hole spin22,23.
The high-fidelity hole spin initialization demonstrated here opens the way to the generation of arbitrary hole spin states either by electric field induced spin resonance29 or by an all-optical process, stimulated Raman adiabatic passage30. It is tantalizing to note that in the limit in which hole spin relaxation is limited by an interaction with the phonon bath, has been predicted4 to reach the maximum value of 2, which would correspond to a millisecond timeframe. In the context of quantum information processing, this would allow many quantum operations to be executed before the hole spin dephases.
This work was funded by the EPSRC (UK), the DFG and the EU (SANDiE). B.D.G. thanks the Royal Society of Edinburgh for financial support.
The file contains Supplementary Notes with Supplementary Equations and additional references.
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Physical Review Applied (2019)