Abstract
Quantum control of solidstate spin qubits typically involves pulses in the microwave domain, drawing from the welldeveloped toolbox of magnetic resonance spectroscopy. Driving a solidstate spin by optical means offers a highspeed alternative, which in the presence of limited spin coherence makes it the preferred approach for highfidelity quantum control. Bringing the full versatility of magnetic spin resonance to the optical domain requires full phase and amplitude control of the optical fields. Here, we imprint a programmable microwave sequence onto a laser field and perform electron spin resonance in a semiconductor quantum dot via a twophoton Raman process. We show that this approach yields full SU(2) spin control with over \(98 \%\)\(\pi\)rotation fidelity. We then demonstrate its versatility by implementing a particular multiaxis control sequence, known as spin locking. Combined with electronnuclear Hartmann–Hahn resonances which we also report in this work, this sequence will enable efficient coherent transfer of a quantum state from the electron spin to the mesoscopic nuclear ensemble.
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Introduction
The existence of strong electric dipole transitions enables coherent optical control of matter qubits that is both fast and local.^{1,2,3} The optical techniques developed to address central spin systems in solids, such as colour centres in diamond and confined spins in semiconductors, typically fall into two categories: the first makes use of ultrashort, broadband, fardetuned pulses to induce quasiinstantaneous qubit rotations in the laboratory frame.^{4,5,6,7} Achieving complete quantum control with this technique further requires precisely timed free qubit precession accompanying the optical pulses. The second technique is based on spectrally selective control via a resonantly driven twophoton Raman process,^{8,9,10,11,12} and allows full control exclusively through tailoring of the drive field, echoing the versatility of magnetic spin resonance. Despite this attractive flexibility, achieving highfidelity control using the latter approach has proved challenging due to decoherence induced by the involvement of an excited state for colour centres in diamond,^{8,10,11} and due to nucleiinduced groundstate decoherence for optically active semiconductor quantum dots (QDs).^{9} In the case of QDs, the limitation of groundstate coherence can be suppressed by preparing the nuclei in a reducedfluctuation state.^{13,14,15,16,17} In this Letter, we achieve highfidelity SU(2) control on a nucleiprepared QD spin using a tailored waveform imprinted onto an optical field. We then demonstrate the protection of a known quantumstate via an alignedaxis continuous drive, a technique known as spin locking. Finally, by tuning the effective spinRabi frequency, we access the electronnuclear Hartmann–Hahn resonances, which holds promise for proxy control of nuclear states.
Results
Optical electron spin resonance
Our device is an indium gallium arsenide QD, embedded in an ntype Schottky heterostructure and housed in a liquidHelium cryostat at 4.2 K; Fig. 1a depicts this arrangement. The QD is charged deterministically with a single electron, and a magnetic field of 3.3 T perpendicular to the growth and optical axes creates an \({\omega }_{{\mathrm {e}}}=24.5\ {\rm{GHz}}\) Zeeman splitting of the electron spin states which form \(\Lambda\) systems with the two excited trion states. Using an electrooptic modulator (EOM), we access these \(\Lambda\) systems by tailoring a circularly polarised singlefrequency laser, of frequency \({\omega }_{{\mathrm {L}}}\) and detuned from the excited states by \({\Delta }_{{\mathrm {L}}}\approx 700\ {\rm{GHz}}\). The EOM is driven by an arbitrary waveform generator (AWG) output with amplitude \({V}_{0}\), frequency \({\omega }_{\mathrm {{\mu {w}}}}\) and phase \({\phi }_{\mathrm {{\mu {w}}}}\). Operating the EOM in the regime where the microwave field linearly modulates the input optical field, a signal \({V}_{0}\cos ({\omega }_{\mathrm {{\mu {w}}}}t+\Delta {\phi }_{\mathrm {{\mu {w}}}})\) produces a control field consisting of two frequencies at \({\omega }_{{\mathrm {L}}}\pm {\omega }_{\mathrm {{\mu {w}}}}\) with a relative phaseoffset of \(2\Delta {\phi }_{\mathrm {{\mu {w}}}}\). This bichromatic field of amplitude \({\Omega }_{{\mathrm {L}}}\) drives the twophoton Raman transitions with a Rabi coupling strength \(\Omega ={\Omega }_{{\mathrm {L}}}^{2}/{\Delta }_{{\mathrm {L}}}\) between the electron spin states (see Supplementary Note 2) in the limit \({({\Omega }_{{\mathrm {L}}}/{\Delta }_{{\mathrm {L}}})}^{2}\ll 1\). The Hamiltonian evolution is given by
where \({\hat{S}}_{i}\) are the spin operators in the electron rotating frame, \(\delta\) the twophoton detuning and \(\phi\) the relative phaseoffset of the Raman beams. The effect of this Hamiltonian is described geometrically by a precession of the Bloch vector around the Rabi vector [\(\Omega \cos (\phi ),\Omega \sin (\phi ),\delta\)]. We have full SU(2) control over the Rabi vector through the microwave waveform, via the Rabi frequency \(\Omega \propto {V}_{0}^{2}\), its phase \(\phi =2\Delta {\phi }_{\mu {\rm{w}}}\), and the twophoton detuning \(\delta ={\omega }_{{\mathrm {e}}}2{\omega }_{\mathrm {{\mu {w}}}}\). An additional resonant optical field of \(100\)\({\rm{ns}}\) duration performs spin initialisation and readout. Finally, prior to the whole protocol, we implement the recently developed nuclearspin narrowing scheme,^{17} which conveniently requires no additional laser or microwave source, in order to enhance groundstate coherence and so maximise control fidelity.
Figure 1b shows the evolution of the population of the \(\left\downarrow \right\rangle\) state for increasing durations of the Raman drive, taken at three different Raman powers. The Raman drive induces coherent Rabi oscillations within the groundstate manifold. The dependence of the fitted Rabi frequency on power is linear within the power range experimentally available as shown in Fig. 1c. This linearity is the result of modest optical power (~10 μW) and a sufficiently large singlephoton detuning \({\Delta }_{{\mathrm {L}}}\approx 700\ {\rm{GHz}}\), allowing us to work in the adiabatic limit where excitedstate population is negligible during the rotations. Even in this limit, we reach Rabi frequencies up to \(154\ {\rm{MHz}}\), exceeding that achieved by extrinsic spin–electric coupling^{18,19} and two orders of magnitude faster than direct magnetic control of gatedefined spin qubits.^{20} While rotations driven by ultrafast (few ps), modelockedlaser pulses naturally circumvent groundstate dephasing, the high visibility of the Rabi oscillations achieved here suggests that our electron spin resonance (ESR) yields equally coherent rotations with the added spectral selectivity and flexibility of microwave control.
Coherence of optical rotations
We characterise the coherence of the rotations with the quality factor \(Q\), which measures the number of \(\pi\) rotations before the Rabioscillation visibility falls below \(1/e\) of its initial value. Figure 2a summarises the dependence of the \(Q\) factor and decay of the Rabi envelope on the ESR drive strength \(\Omega\) and sheds light on three distinct regimes, which are dominated by one of three competing decoherence processes included in the model curve of Fig. 2: (i) inhomogeneous broadening of variance \(\sigma =4.8\ {\rm{MHz}}\), (ii) electronmediated nuclear spin–flipping transitions arising from the presence of strain, and (iii) a spin decay proportional to the laser power, which for simplicity we cast as \({\Gamma }_{1}=\alpha  \Omega \) with \(\alpha =2.7\times 1{0}^{2}\). In the lowpower regime, where \(\Omega\, < \,18\ {\rm{MHz}}\), the fidelity is affected by nucleiinduced shottoshot detuning errors, which in our model are fixed according to an independent Ramsey measurement. This inhomogeneous broadening induces a nonexponential decay of Rabi oscillation visibility^{3} (see also Supplementary Note 3). We shield the system from this effect by increasing the Rabi frequency, yielding an increase in \(Q\) factor. The intermediatepower regime, where \(\Omega =1880\ {\rm{MHz}}\), exhibits a dramatic decrease in \(Q\) and increase in decay rate. In this regime, the coherent spectrally selective drive induces electronmediated nuclear spinflips through a Hartmann–Hahn resonance,^{21} as we depict in the inset to Fig. 2a. Splitting the dressed electron states \(\left\tilde{\uparrow }\right\rangle ,\left\tilde{\downarrow }\right\rangle\) by an energy \(\hslash \Omega\) causes the dressed electronnuclear states to become degenerate, removing the energy cost associated to a single nuclear spin–flip \({\sim} {\hslash} {\omega }_{{\rm{nuc}}}^{z}\). The presence of intrinsic strain, which perturbs the nuclear quantisation axis set by the external magnetic field, allows coupling between these nowdegenerate states. The decay of electronic coherence is related to the nuclear spectral density shown in Fig. 2b, which captures the strength of the strainenabled nuclear transitions over a nuclear ensemble of \(N\approx 74,000\) nuclei inhomogeneously broadened by variation of the local strain fields across the QD (see Supplementary Note 4). As an intuitive semiclassical picture, one can think of the Knight field—the electronspin polarisation (\({S}_{z}\)) felt by the nuclei—acting as an effective radiofrequency field of frequency \(\Omega\) along the external magnetic field. Strain, tilted from this external field, is a perturbation that allows this Knight field to induce singlenucleus transitions between the eigenstates \(\tilde{m}\) and \(\tilde{m}^{\prime}\) to first order (inset of Fig. 2b) provided \(\Omega\) is close to \({\omega }_{{\rm{nuc}}}^{z}\) or \(2{\omega }_{{\rm{nuc}}}^{z}\). In the highpower regime (\(\Omega\, > \,80\ {\rm{MHz}}\)), we decouple from both inhomogeneous nuclearspin fluctuations and Hartmann–Hahn transitions, and consequently observe the highest \(Q\) factors (\(Q=47.6\pm 1.7\) over the four highest Rabi frequencies). Here, the decay envelope is dominated by \({\Gamma }_{1}\), an optically induced relaxation between the electron states proportional to power, and independent of the singlephoton detuning \(\Delta\) (see Supplementary Note 3). The nonresonant and nonradiative nature of this process is consistent with electronspin relaxation induced by photoactivated charges appearing in our device as Stark shifts of the resonance at the highest Raman power. This mechanism, extrinsic to the QD, will vary depending on device structure^{22,23} and quality. This process causes an exponential decay of the Rabi oscillations, bounding the \(Q\) factor to \(4/(3\alpha )\) and the \(\pi\)rotation fidelity to \({f}_{\pi }=\frac{1}{2}\times (1+{{\mathrm {e}}}^{1/Q})\) (see Supplementary Note 3). Our model allows us to evaluate the nonMarkovian corrections caused by the nuclear inhomogeneities and Hartmann–Hahn resonances within the spectral width \(1/{t}_{\pi }=2\Omega\) of the \(\pi\) pulse. At \(\Omega =154\ {\rm{MHz}}\), a correction of \(0.001\) applied to the experimentally measured \(Q\) factor yields a \(\pi\)pulse fidelity \({f}_{\pi }=0.9886(4)\).
Multiaxis control
Figure 3a shows Rabi oscillations taken while varying the detuning \(\delta\). With increasing \(\left\delta \right\), the frequency of the Rabi oscillations \(\Omega ^{\prime} =\sqrt{{\Omega }^{2}+{\delta }^{2}}\) increases, while the amplitude \({\Omega}^{2}/{\Omega ^{\prime} }^{2}\) decreases, as the spin precession follows smaller circles on the Bloch sphere. This confirms that we control the polar angle \(\theta\) of the Rabi vector through detuning of the microwave field. At the high Rabi frequencies used for taking these data (Fig. 3a), we observe that the Rabi drive weakly perturbs the nuclearspin polarisation, causing detuning fluctuations which become more visible as the pulse length increases.
In Fig. 3b, we demonstrate control over the azimuthal angle of the rotation axis by stepping the phase \(\phi\) between two consecutive \(\frac{\pi }{2}\) rotations. The \(\left\downarrow \right\rangle\)state population evolves sinusoidally with the phase shift between the two \(\frac{\pi }{2}\) pulses. For example, at \(\phi =0\), the two rotations add resulting in a \(\pi\) rotation and maximum readout signal, whilst for \(\phi =\pi\), the two pulses exactly cancel, returning the electron spin to its starting state and giving a minimum readout signal. Defining the measurement as the \(({\frac{\pi }{2}})_{\phi }\) pulse combined with the \(\left\downarrow \right\rangle\)state readout, the phase dependence shown here demonstrates our ability to perform \({S}_{\pm x}\) and \({S}_{\pm y}\) measurements, corresponding to twoaxis tomography.
Figure 3c displays Ramsey interferometry performed in the rotating frame, which allows us to further characterise our ESR control. We create a spin superposition using a resonant \(\frac{\pi }{2}\) pulse that evolves for a time \(\tau\) before measuring the state using a second \(\frac{\pi }{2}\) pulse with a relative phase \(\phi =0\) (\(\phi =\pi\)), performing an \({S}_{y}\) (\({S}_{y}\)) measurement. Within this observation window, there are no oscillations modulating the dephasinginduced decay (\({T}_{2}^{* }\)), confirming that the measurement basis is phaselocked to the rotating frame to below our resolution, set by the inhomogeneous nuclear broadening. Under these optimum nuclearspin narrowing conditions^{17} (see also the section “Methods”), the spin coherence decays according to \({T}_{2}^{* }=47.2\pm 0.2\ {\rm{ns}}\); this corresponds to a standard deviation of the spin splitting of \(\sigma =4.77\pm 0.02\ {\rm{MHz}}\) due to the hyperfine fluctuations.
An immediate opportunity derived from multiaxis control is the realisation of an optical analogue of spin locking, a magnetic resonance sequence that preserves a known quantum state well beyond its dephasing time. In this sequence (Fig. 4a), a \(\frac{\pi }{2}\) rotation creates a coherent superposition state in the equatorial plane. The azimuthal angle of the rotation axis is then shifted by \(\frac{\pi }{2}\), bringing the Rabi vector into alignment with the system state; this places the electron into one of the dressed states. The drive creates an energy gap \({\hslash} {\Omega}\) between the two dressed states, which provides protection against environmental dynamics occurring at frequencies different from \(\Omega\). By setting \(\Omega \sim 10\ {\rm{MHz}}\), we successfully avoid nuclearspin resonances observed in Fig. 2. In Fig. 4b we implement the spinlocking sequence for up to \(0.6\,{\mathrm {\mu {s}}}\), tracking the population in the \(\left\{\left\tilde{\uparrow }\right\rangle ,\left\tilde{\downarrow }\right\rangle\right\}\) basis to give a measure of the performance of our quantumstate preservation. At these short delays, a small unlocked component of the Bloch vector undergoes Rabi oscillations resulting in smallamplitude oscillations. As confirmed with our Blochequation model (black curve in Fig. 4b), this arises from detuning errors of the locking pulse consistent with the measured 4.8MHz nuclearfield inhomogeneity. The decay of the locked component of the Bloch vector is significantly slower than under a Rabi drive of the same amplitude (\(\Omega =11\ {\rm{MHz}}\)) (grey model curve in Fig. 4b). Figure 4c shows the decay of the spinlocked state on longer timescales. After each locking at \(\Omega =16\ {\rm{MHz}}\), we measure the length of the Bloch vector by performing statetomography and obtaining the visibility as in Fig. 3b. An exponential fit (black curve in Fig. 4c) reveals a decay time of \(2.3\pm 0.2\ {\mathrm {\mu {s}}}\). The close agreement with the decay rate expected from our Fig. 2 model is evidence that spin locking is similarly limited by the photoactivated spin relaxation (\({\Gamma }_{1}\)). The quantum state is thus locked and preserved for a longer time than would be accessible via direct Rabi drive. This confirms our ability to implement alloptically a multiaxis spincontrol sequence that can protect the qubit effectively against the intrinsic nuclear hyperfine coupling.
Discussion
The highfidelity alloptical ESR we report here enables the generation of any quantum superposition spin state on the Bloch sphere using a single waveformtailored optical pulse. This full SU(2) control further allows the alloptical implementation of spin locking, traditionally an NMR technique, for quantumstate preservation via gapped protection from decoherenceinducing environmental dynamics. In the case of semiconductor QDs, where the nuclei form the dominant noise source, the same quantum control capability enables us to reveal directly the spectrum of nuclearspin dynamics. An immediate extension of this work will be to perform spin locking in the spectral window of nuclearspin resonances, i.e. the Hartmann–Hahn regime, to sculpt collective nuclearspin states,^{24,25} and also to tailor the electron–nuclear interaction^{26,27,28} to realise an ancilla qubit or a local quantum register based on the collective states of the nuclear ensemble.^{29}
Methods
Quantum dot device
Our QD device is the one used in ref. ^{30}. Selfassembled InGaAs QDs are grown by molecular beam epitaxy and integrated inside a Schottky diode structure, above a distributed Bragg reflector to maximise photonoutcoupling efficiency. There is a \(35\)\({\rm{nm}}\) tunnel barrier between the ndoped layer and the QDs, and a blocking barrier above the QD layer to prevent charge leakage. The Schottky diode structure is electrically contacted through Ohmic AuGeNi contacts to the ndoped layer and a semitransparent Ti gate (\(6\ {\rm{nm}}\)) is evaporated onto the surface of the sample. The photon collection is enhanced with a superhemispherical cubic zirconia solid immersion lens on the top Schottky contact of the device. We estimate a photonoutcoupling efficiency of 10% at the first lens for QDs with an emission wavelength around \(970\ {\rm{nm}}\). A homebuilt microscope with spectral and polarisation filtering^{17} is used for resonance fluorescence, with a QDtolaser counts ratio exceeding 100:1.
Raman laser system
Sidebands are generated from the continuouswave laser by modulating a fibrebased EOSPACE EOM with a microwave derived from a Tektronix AWG 70002A. The output of the EOM depends on the voltage applied, with maximum and minimum transmissions for applied voltages \({V}_{\max }\) and \({V}_{\min }\), respectively, and the \(\pi\) voltage of the EOM, \({V}_{\pi }=\left{V}_{\max }{V}_{\min }\right\). Applying a voltage \(V={V}_{\min }+{V}_{0}(t)\), where \({V}_{0}(t)\) is a microwave field of small amplitude compared with \({V}_{\pi }\), the electric field at the EOM output \({E}_{{\rm{out}}}\) is described by \({E}_{{\rm{out}}}(t)\propto {V}_{0}(t)\times {E}_{{\rm{in}}}(t)\). In other words, we work with small amplitude around the minimum intensity transmission of the EOM to imprint the microwave amplitude onto the optical field.
Generation of the microwave signal \({V}_{{\rm{in}}}(t)\) is depicted in Fig. 5. We produce a digital signal with a sampling rate that is four times the microwave frequency (a factor 2 is obtained by setting the AWG sampling rate at \(2{\omega }_{\mathrm {{\mu {w}}}}\) and another factor 2 is obtained by combining two independently programmable AWG outputs with a splitter). We thus arrive at a digital signal containing four bits per period, the minimum required to carry phase information to the EOM. To generate the signal shown in Fig. 5, we add the two AWG outputs in quadrature, which we realise after characterisation of the relative delay between the two microwave lines arriving at the splitter. From each output, we generate a squarewave signal at \(12.25\ {\rm{GHz}}\). By tuning their relative amplitudes, we construct a sinusoidal signal at \(12.25\ {\rm{GHz}}\) whose phase \(\phi\) is determined by the relative amplitude \({A}_{1,2}\) of channels 1 and 2 according to tan (\(\phi\)) = \({A}_{1}/{A}_{2}\).
Experimental cycle
Nuclearspin preparation
Figure 6 shows our experimental cycle which involves narrowing the nuclearspin distribution before a spinmanipulation experiment. Nuclearspin preparation is done using the scheme detailed in ref. ^{17}, operating in a configuration analogous to Raman cooling in atomic systems. It involves driving the system continuously with the Raman laser, while pumping the \(\left\downarrow \right\rangle\) spin state optically. Optimum cooling, assessed using Ramsey interferometry, occurs for a Raman drive at \(\Omega =22\ {\rm{MHz}}\) and a resonant repump of \({\Omega }_{{\rm{res}}}=0.9{\Gamma }_{0}/\sqrt{2}\) for an excitedstate linewidth \({\Gamma }_{0}\), in agreement with the optimum conditions found in ref. ^{17}. These settings give an orderofmagnitude improvement in our electron spin inhomogeneous dephasing time \({T}_{2}^{* }\) (Fig. 2a).
Electron spin control
During spin control, we conserve the total Raman pulse area in our sequences by pairing pulses of increasing length with pulses of decreasing length (Fig. 6). This allows us to stabilise the Raman laser power using a PID loop and maintain relative fluctuations below a per cent. We operate with a duty cycle of around 50%, preparing the nuclearspin bath for a few μs before spending a similar amount of time performing electron spin control. The alternation on μs timescale of coherent manipulation and nuclearspin preparation is fast compared with the nuclearspin dynamics^{16} such that the nuclearspin distribution is at steady state.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The code used to produce the theoretical findings of this study is available from the corresponding authors upon reasonable request.
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Acknowledgements
This work was supported by the ERC PHOENICS grant (617985), the EPSRC Quantum Technology Hub NQIT (EP/M013243/1) and the Royal Society (RGF/EA/181068). D.A.G. acknowledges support from St John’s College Title A Fellowship. E.V.D. acknowledges funding from the Danish Council for Independent Research (Grant No. DFF418100416). C.L.G. acknowledges support from a Royal Society Dorothy Hodgkin Fellowship.
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J.H.B., R.S., D.A.G., G.É.M., D.M.J., C.L.G. and M.A. conceived the experiments. J.H.B., R.S. and C.L.G. acquired and analysed data. E.V.D., C.L.G. and J.H.B. developed the theory and performed simulations. E.C. and M.H. grew the sample. J.H.B., R.S., E.V.D., D.A.G., G.É.M., D.M.J., C.L.G. and M.A. prepared the manuscript.
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Bodey, J.H., Stockill, R., Denning, E.V. et al. Optical spin locking of a solidstate qubit. npj Quantum Inf 5, 95 (2019). https://doi.org/10.1038/s4153401902063
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DOI: https://doi.org/10.1038/s4153401902063
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