Abstract
The interaction between a confined electron and the nuclei of an optically active quantum dot provides a uniquely rich manifestation of the central spin problem. Coherent qubit control combines with an ultrafast spin–photon interface to make these confined spins attractive candidates for quantum optical networks. Reaching the full potential of spin coherence has been hindered by the lack of knowledge of the key irreversible environment dynamics. Through alloptical Hahn echo decoupling we now recover the intrinsic coherence time set by the interaction with the inhomogeneously strained nuclear bath. The highfrequency nuclear dynamics are directly imprinted on the electron spin coherence, resulting in a dramatic jump of coherence times from few tens of nanoseconds to the microsecond regime between 2 and 3 T magnetic field and an exponential decay of coherence at high fields. These results reveal spin coherence can be improved by applying large magnetic fields and reducing strain inhomogeneity.
Introduction
Decoupling techniques, such as Hahn echo, protect quantum states by filtering the effects of correlated environment noise^{1,2}. They are essential to realising quantum information, computation and simulation protocols by extending the usable coherence time of quantum states, but also provide a powerful spectroscopic tool to understand environmental dynamics^{3,4}. A particularly rich environment is experienced by electrons confined to selfassembled indium–gallium–arsenide (InGaAs) quantum dots: the electron spin couples via the contact hyperfine interaction to the dense nuclear spin bath of the ∼10^{4}−10^{5} atoms which constitute the quantum dot. This nuclear spin bath is subject to inhomogeneous electric field gradients within the crystal lattice, which arise from the straindriven selfassembly during epitaxial growth. Local field gradients are present in addition as a consequence of the random alloying of atomic species in these systems. The nuclear spins in the quantum dot couple to these gradients via quadrupolar moments, resulting in spatiallydependent shifts of their energy levels, which has recently been shown to provide a natural isolation from dipolar interactions^{5}. The effect of the hyperfine interaction is a loss of electron spin coherence within a few nanoseconds in timeaveraged measurements^{6,7}. While Hahn echo decoupling has been shown to enable the recovery of spin coherence for up to 3 μs^{8}, identifying the irreversible mechanism preventing coherence from reaching closer to the millisecond spin relaxation time^{9,10} is an unresolved issue.
Recent studies of electronspin dynamics have provided a comprehensive understanding at zero and very low (mT) magnetic field^{11,12}, but dephasing mechanisms and the limits of spin coherence in selfassembled quantum dots remain unclear at larger fields (few T) where quantumdot spins are promising quantum bit candidates. A key challenge to understand electronnuclear dynamics at nonzero external magnetic fields in these structures is presented by dynamic nuclear spin polarization. This nonlinear measurementinduced process, often observed as the dragging of resonance frequencies by detuning resonant optical probes^{13}, affects both the evolution of electron spin states and the detuning of resonant spin readout to the extent that it can limit state retrieval and obscure the mechanisms influencing the electronspin state^{14}.
Here, we present an alloptical solution to access the quantum dot spin state dynamics free from polarizing the nuclear environment. We then use this method to implement Hahn echo decoupling^{1} without perturbing the nuclear bath, and find the coherence to be governed by the quadrupolar coupling of the nuclear bath to the inhomogeneous strain fields in the quantum dot. In this way we recover the intrinsic limit to coherence for the central spin, indicating that extension beyond this fewmicrosecond bound necessitates the reduction of straininduced quadrupolar broadening in these materials.
Results
Coherent electron spin control and electronnuclear dynamics
InGaAs quantum dots possess strong, coherent optical transitions^{15,16}, forming an ultrafast interface between the Zeemansplit ground states and welldefined optical modes^{17,18,19}. The level structure allowing optical access to the spin state of an electron resident in the InGaAs quantum dot is depicted in Fig. 1a. An inplane magnetic field permits four equalstrength nearinfrared transitions to the excited trion states, at 969 nm in our case, and splits the electronspin ground states by Δ_{e}, while the excited states are split by Δ_{h}. Resonant addressing of an optical transition provides spindependent optical readout and prepares a welldefined ground state by optical pumping^{20}. Application of a circularly polarized and spectrally detuned pulse rotates the electron spin via the ACStark shift^{21}. This can be observed in the fluorescence rate as powerdependent spinRabi oscillations, plotted in Fig. 1b, when the rotation pulses are accompanied by a resonant drive for spin readout and repump. In our experiments we achieve electron spin rotation with error rates of ≤3% using 2 pslong pulses.
An electron spin superposition stored in the quantum dot evolves at the spin splitting Δ_{e}. This frequency, principally determined by the external quantization field B_{ext}, is influenced by the contact hyperfine interaction with the nuclear spins within the electronic wavefunction. This interaction can be represented semiclassically by an effective magnetic field, the Overhauser field B_{OH}, as depicted in Fig. 1c. In the limit of B_{ext}≫B_{OH}, Overhauser field components parallel to the electron quantization couple linearly to the spin splitting, while perpendicular terms perturb the splitting quadratically^{22,23}. The highfrequency dynamics of the Overhauser field are determined by the nuclear bath precessing in the external field, together with the coupling to nuclear sitedependent electric field gradients via quadrupolar moments, represented in Fig. 1d for an individual nuclear spin (I_{i}) as an effective field B_{Q}^{24}. The sum effect of the two is such that the resulting Overhauser field fluctuations contain components both parallel and perpendicular to the external field direction. The frequency chart in Fig. 1e outlines the relevant timescales for each of these processes in a fewTesla external field. At these fields, the fastest process is electron spin precession at g_{e}μ_{e}∼6 GHz T^{−1}, broadened by the Overhauser field width g_{e}μ_{e}B_{OH}∼100 MHz^{25,26}. Atomicspecies dependent nuclear Zeeman splitting of g_{N}μ_{N}∼10 MHz T^{−1} is supplemented by the quadrupolar coupling g_{N}μ_{N}B_{OH}∼1–10 MHz^{27}. The effect of other dynamics, such as dipolar interaction between nuclear spins^{5}, electronmediated nuclear flip flops^{28} or the precession of nuclei due to hyperfine coupling with the electron^{29,30} can be neglected in this regime due to their weaker coupling strengths. For reference, we include the Rabi frequency of our spin control at ∼500 GHz, which is at least an order of magnitude faster than any other process.
Suppressing dynamic nuclear spin polarization
The nuclear dynamics introduced in Fig. 1d affect the storage and retrieval of arbitrary spin states from the quantum dot. To investigate the extent of their effect we measure the Ramsey interference of spin states for varying delay, τ between two π/2 spin rotations. The subpanels of Fig. 2a, labelled (i–iv), display the resonant spin readout signal when we probe each of the four available transitions (see level schemes). All data sets show a striking departure from the expected sinusoidal beating associated with the precession of the electron spin. When probing the spindown projection (iii & iv), a hysteretic sawtooth behaviour emerges^{14}, while for probing spinup projection (i & ii) the readout signal is suppressed for delays beyond 200 ps^{17}. This behaviour emerges from the dynamic polarization of the nuclear spins due to resonant driving of the optical transitions, producing a feedback loop between bath polarization and the measured electron state^{13,31}. As can be seen in Fig. 2a, it is the groundstate electron projection of the readout transition which dictates the feedback dynamics we observe, consistent with mechanisms based on the noncollinear hyperfine interaction^{32}. The effect of this statedependent polarization is to frustrate timeaveraged measurements of the electronnuclear system and to prevent access to the full timescale of spin coherence. To inhibit the polarization of the nuclear bath we repeat the same measurement as in Fig. 2a, however we invert the spin before every other repeat, as depicted in Fig. 2b. The two resulting measurements presented in Fig. 2c suppress the phase dependence in the average signal (grey circles). In this way we prevent the buildup of nuclear polarization and resolve the evolution of the unperturbed electronnuclear system, as shown in Fig. 2d for one half of the measurement pair. The symmetric, Gaussianenvelope decays with which is testament to the large amplitude, yet quasistatic environmental noise, consistent with an Overhauser field standard deviation of 33 mT^{11,30} (Supplementary Fig. 1 and Supplementary Note 1).
Nuclear bath evolution dominating electron spin coherence
Having successfully decoupled nuclear bath polarization from our resonant optical spin readout, we implement a Hahn echo scheme to assess the extent to which we can protect electronspin coherence. The measurement sequence is displayed in Fig. 3a for a 91.2ns delay between the π/2 rotations, in which the central π rotation refocuses the dephased spin state. We sweep the central rotation by and record the visibility of the oscillatory readout signal^{8}. In the limit of perfect rotations the visibility is a direct measure of the spinstate coherence. Figure 3b displays the integrated readout from the pulse sequence in Fig. 3a. The two traces correspond to the echo signal with (blue circles) and without (red circles) initial inversions. As with the freeinduction decay in Fig. 2, decoupling the readout from bath polarization is essential to obtain a meaningful readout signal (Supplementary Fig. 2 and Supplementary Note 2).
Figure 3c shows the visibilities we recover when varying storage time, T, up to 1.3 μs for four external magnetic fields. The refocusing π pulse filters the quasistatic noise components that resulted in the 2ns timeaveraged dephasing, yet the electron state is still susceptible to highfrequency dynamics of the Overhauser field. The data reveal a marked, nonlinear dependence on the external magnetic field strength: at low fields (2 T, top panel) spin coherence decays almost completely within 20 ns, followed by a very weak revival after ∼125 ns. At higher magnetic fields coherence times are dramatically prolonged, featuring damped oscillatory behaviour at short times and an exponential tail for longer delays.
The quality of the experimental data enables us to employ the Hahn echo as a sensitive spectroscopic probe of the nuclear bath dynamics: in protecting the electron spin coherence it provides a tunable filter of the Overhauser field noise spectrum. We can fully explain the measured spin coherence by considering the hyperfine interaction with an ensemble of noninteracting Ga^{69}, Ga^{71}, As^{75} and In^{115} nuclear spins in the presence of an external magnetic field and a significant quadrupolar interaction. A calculation of the nuclear noise spectra, presented in Fig. 3d, allows us to simulate the Hahn echo decay function (grey curves in Fig. 3c) which are in good quantitative agreement with the data. In general, we find the partial collapse and revival of the spinecho curve to be due to linearly coupled Overhauser field components along the electronspin quantization direction. The secondorder coupling to the perpendicular Overhauser components produces a smooth envelope function responsible for the exponential tail observed at higher magnetic fields.
The left panel of Fig. 3d shows the linearly coupled nuclear noise spectra while the right panel shows the transverse quadratically coupled counterpart. The broad features result from integrating over a Gaussian distribution of quadrupolar energies and orientations corresponding to the inhomogeneous spread of electric field gradients (Supplementary Note 3). Our choice of physical parameters (Supplementary Tables 1 and 2) is based on an indium concentration of 0.5 with strain profiles motivated by atomistic calculations^{33,34}. Fitting the power spectrum amplitude to our echo modulation is consistent with an Overhauser field s.d. of 28 and 40 mT for the quadratically and linearly coupled components, respectively. By resolving the nuclear spin spectra into the constituent atomic species we find that indium plays the dominant role in the dephasing of the electron, owing to its large nuclear spin, while the gallium isotopes have a negligible effect (Supplementary Fig. 3).
The striking magnetic field dependence can be understood by considering the competition between the nuclear Zeeman and the quadrupolar interaction. The large strain inherent to InGaAs quantum dots results in a quadrupolar coupling strength equivalent to a ∼1T magnetic field. As depicted in Fig. 1d, a tilt between the effective quadrupolar field axis and the external field leads to parallel Overhauser field components in the hyperfine interaction, which dominate at low external magnetic fields. The strong inhomogeneous broadening of the quadrupolar coupling prevents any meaningful rephasing of this large amplitude term below a 2T magnetic field. Electron spin coherence is then effectively suppressed beyond ∼20 ns. At higher magnetic fields the Zeeman interaction starts to dominate nuclear dynamics, reducing the amplitude of linearly coupled Overhauser field components (Supplementary Fig. 4). Consequently, we observe a strong increase of spin coherence between external field values of 2 and 3T. At higher fields the oscillation amplitude in the Hahn echo signal dies away and spin coherence decays exponentially at a fielddependent rate extending to 1.7 μs at 5T. This decay stems from the difference frequency terms in the secondorder hyperfine coupling, which result in a broad lowfrequency shoulder, plotted in the right panel of Fig. 3d. The continuous spread of noise components between 0 and ∼10 MHz causes a loss of spin coherence without revival. In this way, the quadrupolar inhomogeneity bounds the electron spin coherence to this microsecond timescale, set by the square of the Overhauser field strength. These components couple to the electron splitting with a 1/B_{ext} dependence, which reduces the effective amplitude at higher fields. The data in Fig. 3c, and further data taken for a different quantum dot at a 7 T external field yielding (Supplementary Fig. 5), support this expectation.
Calculations of a detailed magnetic field dependence of the Hahn echo decay based on the nuclear bath parameters found from fitting the experimental data are presented in Fig. 4, which summarizes our results: A clear transition from suppressed spin coherence with dephasing times of 10–20 ns to microsecond spin coherence takes place between 2 and 3 T, highlighted in the figure subpanel. Following this transition electron spin coherence increases approximately linearly with magnetic field at a rate of ∼0.7 μs T^{−1}, exceeding 5 μs at 10T. We note this linear trend is in agreement with earlier results at intermediate magnetic fields^{8}, but differs for high fields (Supplementary Note 3).
Discussion
By inhibiting dynamic nuclear spin polarization during resonant optical access to spin states, we observe the quadrupolar interaction of the nuclear bath with inhomogeneous electric field gradients providing an intrinsic bound to spin coherence in InGaAs quantum dots. In other systems, such as electrostatically defined GaAs quantum dots, the nitrogenvacancy centre in diamond or phosphorus donors in silicon, multipulse decoupling schemes can be used to prolong the spin coherence beyond the Hahn echo limits^{23,35,36}. Such schemes rely on noise being correlated on timescales longer than the pulse separation^{7}. The broadband quadrupolecoupled nuclear dynamics in our selfassembled quantum dots present a corresponding timescale of ∼50 ns and only pulses spaced more closely than this could extend spin coherence beyond the times we observe. In agreement, at lowmagnetic fields we find we can extend coherence for pulse separations of 6.5 ns, where the Hahn echo limit of ∼20 ns can be pushed beyond 65 ns using 9 π pulses (Supplementary Fig. 6). Further decoupling is limited in our case by pulse imperfections. While these imperfections may be improved using composite pulses^{37}, the number of rotations possible will be limited by irreversible excitonic dephasing through coupling to acoustic phonons^{38,39}, such that an improvement over Hahn echo T_{2} times in the highmagnetic field regime would face challenges.
The coherence of electron spins in electrostatically defined GaAs quantum dots is limited by quadrupolar nuclear broadening under certain conditions^{40}, but the combination of a larger nuclear bath, the absence of inherent strain, and the lack of indium has allowed retention of coherence for times close to 1 ms^{41}. By suppressing internuclear dipole–dipole coupling, the strainbroadening in selfassembled quantum dots has been shown to preserve nuclear coherence beyond these unstrained systems^{5,42}, while however preventing rephasing of the Overhauser field.
A single heavyhole as a quantumdot spin qubit should allow for longer dephasing times due to a much weaker hyperfine interaction, although other mechanisms can limit its coherence^{43,44}. Alternatively, longer coherence times in selfassembled quantum dots could be realised either for systems without quadrupolar nuclear moments, as in II–VI quantum dots^{45}, or through strainfree growth methods^{46,47}. One particular example of a strainfree system is provided by optically active GaAs/AlGaAs quantum dots grown by droplet epitaxy. With over two orders of magnitude weaker local strain than the quantum dots used here, significantly different nuclear dynamics have been reported^{48}. With reduced quadrupolar broadening absolute coherence times that are competitive with other spin qubit realizations could be achieved, while still maintaining the ultrafast control rates and optical integration the material properties enable.
Methods
Sample
We use a sample grown by molecular beam epitaxy containing a single layer of selfassembled InGaAs quantum dots in a GaAs matrix, embedded in a Schottky diode for charge state control. The Schottky diode structure comprises an n^{+}doped layer 35 nm below the quantum dots and a ∼6nm thick partially transparent titanium layer evaporated on top of the sample surface. This device structure allows for deterministic charging of the quantum dots and shifting of the exciton energy levels via the DC Stark effect. Twenty pairs of AlAs/AlGaAs layers form a distributed Bragg reflector below the quantum dot layer for increased collection efficiency in the spectral region between 960 and 980 nm. Spatial resolution and collection efficiency are enhanced by a zirconia solid immersion lens in Weierstrass geometry positioned on the top surface of the device. The device is cooled in a liquidhelium bath cryostat to 4.2 K and surrounded by a superconducting magnet.
Spin inversion to prevent nuclear spin polarization
The inversion necessary to cancel phase terms in the average readout signal and suppress nuclear polarization can be provided by either a coherent πrotation, or an incoherent repumping step. For Hahn echo the short spin evolution measured (∼200 ps) is such that an imperfect πrotation suffices; however for measurements of the timeaveraged dephasing, the longer delays measured between the π/2rotations and the corresponding enhanced sensitivity to nuclear backaction require complete spin inversion to cancel the timeaveraged phase. Due to this we supplement our imperfect spin rotation with a pumping step.
Pulse sequence and detection
Optical pulse sequences are constructed from a Ti:Sapphire pulsed laser in picosecondmode, detuned from the optical resonance by 3 nm and a resonant continuouswave diode laser. Both are modulated with fibrecoupled waveguide electrooptic modulators. The modulators are locked to the 76MHz repetition rate of the pulsed laser via a pulse delay generator with 8ps jitter. Additional suppression of the readout pulse is provided by an acoustooptical modulator, realising 6,000:1 suppression of readout lasers and >2,000:1 rotation pulse suppression. The readout laser is used at a power below optical saturation to avoid spinpumping when not reading the spin state. The readout fluorescence is filtered from the resonant laser by polarization mode rejection. Additional filtering of the detuned rotation pulses is provided by a holographic grating with a 30GHz full width at half maximum and the firstorder diffraction efficiency above 90%. The photon detection events are recorded with a timecorrelation unit and a single photon detector with a timing resolution of 350 ps.
Data availability
All relevant data are available from the authors on request.
Additional information
How to cite this article: Stockill, R. et al. Quantum dot spin coherence governed by a strained nuclear environment. Nat. Commun. 7:12745 doi: 10.1038/ncomms12745 (2016).
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Acknowledgements
We gratefully acknowledge financial support by the University of Cambridge and the European Research Council ERC Consolidator grant agreement no. 617985. C.M. acknowledges Clare College, Cambridge, for financial support through a Junior Research Fellowship. We thank H. Bluhm, T. Botzem, L. Cywinski, D. Kara, M. Stanley and J.M. Taylor for fruitful discussions and S. Topliss for technical assistance.
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R.S., C.L.G., C.M., L.H. and M.A. devised and designed the experiments. M.H. and E.C. grew the sample. C.M. processed the devices. R.S., C.L.G., C.M. and L.H. performed the measurements and analysed the data. C.M. and C.L.G. performed the theoretical modelling shown in Fig. 3. All authors contributed to the discussion of results and preparation of the manuscript.
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Supplementary Figures 16, Supplementary Tables 12, Supplementary Notes 13 and Supplementary References (PDF 1446 kb)
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Stockill, R., Le Gall, C., Matthiesen, C. et al. Quantum dot spin coherence governed by a strained nuclear environment. Nat Commun 7, 12745 (2016). https://doi.org/10.1038/ncomms12745
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