Abstract
Qubits, the quantum mechanical bits required for quantum computing, must retain their quantum states for times long enough to allow the information contained in them to be processed. In many types of electronspin qubits, the primary source of information loss is decoherence due to the interaction with nuclear spins of the host lattice. For electrons in gatedefined GaAs quantum dots, spinecho measurements have revealed coherence times of about 1 μs at magnetic fields below 100 mT (refs 1, 2). Here, we show that coherence in such devices can survive much longer, and provide a detailed understanding of the measured nuclearspininduced decoherence. At fields above a few hundred millitesla, the coherence time measured using a singlepulse spin echo is 30 μs. At lower fields, the echo first collapses, but then revives at times determined by the relative Larmor precession of different nuclear species. This behaviour was recently predicted^{3,4}, and can, as we show, be quantitatively accounted for by a semiclassical model for the dynamics of electron and nuclear spins. Using a multiplepulse Carr–Purcell–Meiboom–Gillecho sequence, the decoherence time can be extended to more than 200 μs, an improvement by two orders of magnitude compared with previous measurements^{1,2,5}.
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The promise of quantumdot spin qubits as a solidstate approach to quantum computing is demonstrated by the successful realization of initialization, control and singleshot readout of electronspin qubits in GaAs quantum dots using optical^{6}, magnetic^{7} and fully electrical^{8,9,10} techniques. To further advance spinbased quantum computing, it is vital to mitigate decoherence due to the interaction of the electron spin with the spins of nuclei of the host material. Understanding the dynamics of this system is also of great fundamental interest^{11,12}.
Through the hyperfine interaction, an electron spin in a GaAs quantum dot is subjected to an effective magnetic field produced by the nuclear spins. Under typical experimental conditions, this socalled ‘Overhauser field’ has a random magnitude and direction. Typically, measurements of the coherent electronspin precession involve averaging over many experimental runs, and thus over many Overhauser field configurations. As a result, the coherence signal is suppressed for evolution times τ≳T_{2}^{*}≈10 ns (refs 1, 2). However, the nuclear spins evolve much more slowly than the electron spins, so that the Overhauser field is nearly static over sufficiently short time intervals. Therefore, one can partially eliminate the effect of the random nuclear field by flipping the electron spin halfway through an interval of free precession, a procedure known as Hahn echo. The random contributions of the Overhauser field to the electronspin precession before and after the spin reversal then approximately cancel out. For longer evolution times, the effective field acting on the electron spin generally changes over the precession interval. This change leads to an eventual loss of coherence on a timescale determined by the details of the nuclear spin dynamics.
Previous Hahnecho experiments on lateral GaAs quantum dots have demonstrated spindephasing times of around 1 μs at relatively low magnetic fields up to 100 mT for microwavecontrolled^{2} singleelectron spins and electrically controlled^{1} twoelectronspin qubits. For optically controlled selfassembled quantum dots, coherence times of 3 μs at 6 T were found^{5}. Recent theoretical studies of decoherence due to the hyperfine interaction^{3,4,13} are generally consistent with these experimental results, but predict revivals of the echo signal after several microseconds, similar to those seen in nitrogen–vacancy centres in diamond^{14}. This prediction already indicates that the initial decay of the echo does not reflect irreversible decoherence, but is a consequence of the coherent Larmor precession of the nuclei. Theoretical work also predicted much longer coherence times at higher external magnetic fields^{13,15} or when using more advanced pulse sequences^{16,17}. The classic example is the Carr–Purcell–Meiboom–Gill (CPMG) sequence^{1,18}, but several alternatives have recently been developed^{19,20} and demonstrated^{21,22}. Their performance is expected to improve as more control pulses are added^{17}. Here, we provide direct experimental confirmations for all of the above predictions.
The spin qubit studied in this work consists of two isolated electrons confined in a double quantum dot, created by applying negative voltages to metallic gates that locally deplete a twodimensional electron gas 90 nm below the wafer surface (see Fig. 1a). The Hilbert space of our logical qubit is spanned by the states ↑↓〉 and ↓↑〉, that is, the m=0 subspace of two separated spins. The arrows represent the alignment of the electron spins in each of the dots relative to an external magnetic field, B_{ext}, which lies in the plane of the twodimensional electron gas and defines the z direction. The remaining two states, T_{+}≡↑↑〉 and T_{−}≡↓↓〉, are energetically separated by the Zeeman energy E_{Z}=g^{*}μ_{B}B_{ext} (g^{*}=−0.44 is the gfactor in GaAs) and are not used for information storage. Tunnel coupling to the leads is used for initialization and interdot tunnelling allows spin exchange between the dots. This exchange interaction is modulated by means of the detuning ɛ, which is the difference between the electrostatic potentials in the two dots. This parameter is controlled by means of rapid, antisymmetric changes of the voltages on gates GL and GR (see Fig. 1) applied through highfrequency coaxial lines, which enables fast electrical control of the qubit^{1,8,23}.
The experimental procedures follow those of ref. 1. We initialize the system at a large detuning, where the ground state is a spin singlet with both electrons residing in a single dot. As ɛ is swept to negative values, the electrons separate into different dots, thus preparing a singlet . For very large negative detunings, the electron spins in the two dots are decoupled, and each individually experiences a Zeeman field composed of the homogeneous external field and a fluctuating local hyperfine field. A difference ΔB_{nuc}^{z} between the zcomponents of the hyperfine fields in the two dots leads to an energy splitting between the basis states ↑↓〉 and ↓↑〉. This splitting causes precession between the singlet S and the triplet , and its fluctuations lead to dephasing of the qubit. We implement the echo πpulses by pulsing to small negative detunings, where interdot tunnelling leads to an exchange splitting between S and T_{0}. This splitting drives coherent oscillations between the states ↑↓〉 and ↓↑〉. The pulse profiles for the Hahn echo and CPMG sequence are shown in Fig. 1c,d.
Readout of the final qubit state is accomplished by switching to positive detuning, ɛ>0, where the state with both electrons sitting in the same dot is preferred for the spin singlet, but is energetically excluded for a spin triplet because of the Pauli exclusion principle. The two spin states thus acquire different charge densities. To sense this difference, we use a proximal quantum point contact (QPC), the conductance of which depends on the local electrostatic environment^{24}. After averaging over many identical pulse cycles, the mean QPC conductance, G_{QPC}, reflects the probability to find the qubit in the singlet state at the end of each cycle. The echo amplitudes presented below are normalized such that they are unity at short times (no decoherence) and eventually drop to zero for a fully randomized state. This normalization eliminates τindependent contrast losses (see Supplementary Information). Figure 2a shows the Hahnecho signals for different magnetic fields B_{ext}.
At high fields we find a decay of the Hahnecho signal approximately as exp(−(τ/30 μs)^{4}). As the magnetic field is reduced, the echo signal develops oscillations with a timescale of microseconds. For even lower fields, the oscillations evolve into full collapses of the signal, with revivals at later times on a tenmicrosecond timescale. These revivals were predicted in refs 3, 4 on the basis of a quantummechanical treatment of the hyperfine interaction between electron and nuclear spins. Below we outline a semiclassical model that can reproduce the lowestorder result of refs 3, 4 and accounts for additional effects that are essential for fitting our data. The derivation is provided in the Supplementary Information and will be discussed in more detail elsewhere. This model provides the theoretical echo signal, C(τ)≡2p(S)−1=−Re〈↑↓ρ(τ)↓↑〉, where p(S) is the probability of finding the electron in a singlet state and ρ(τ) is the density matrix of the qubit at the end of the evolution time. We have used this model to produce the quantitatively accurate fits shown in Fig. 2a.
For each electron spin, the Zeemanenergy splitting is proportional to the total magnetic field (Fig. 2b). Time dependence of the parallel and transverse nuclear components, B_{nuc}^{z} and , can lead to dephasing of the electron spin. Assuming statistical independence between B_{nuc}^{z} and , the theoretical echo signal can be written as a product, , where A_{z}(τ) and account for the contributions of B_{nuc}^{z} and to the electronspin precession. In the experimental range of the magnetic fields, the time dependence of B_{nuc}^{z} is mainly caused by spectral diffusion due to the dipole–dipole interaction between nuclear spins. The weak effect of this generally slow process is predicted to lead to a decay of the form A_{z}(τ)=exp(−(τ/T_{SD})^{4}) (refs 13, 15). As we will now discuss, (see Supplementary Equation S1) has a more interesting nonmonotonic structure that arises from the relative precession of nuclear spins in the external field with different Larmor frequencies.
The transverse hyperfine field, , is a vector sum of contributions from the three nuclear species ^{69}Ga, ^{71}Ga and ^{75}As. As a result of the different precession rates of these species (Fig. 2c), contains harmonics at the three relative Larmor frequencies (Fig. 2d) in addition to a constant term. The phase due to the constant term is eliminated by the echo pulse. For a general freeprecession period, the time dependence leads to a suppression of the echo signal. However, if τ/2 is a multiple of all three relative Larmor periods, the oscillatory components contribute no net phase to the electronspin evolution. As a result, the echo amplitude revives whenever the three commensurability conditions are approximately fulfilled, which is possible because of a fortuitous spacing of the Larmor frequencies. Averaging the singlet return probability over initial Overhauser field configurations leads to the collapseandrevival behaviour predicted in refs 3, 4.
For B_{ext}≲400 mT, the envelope of the echo revivals decays more quickly than at higher fields (see Fig. 2a). This field dependence can be accounted for by including a spread of the Larmor precession frequencies for each nuclear species. Such a variation is also manifest in the width of NMR lines and naturally arises from dipolar and other interactions between nuclei^{25}. We model it as a shift of the magnetic field acting on each individual nuclear spin by an amount B_{loc}, where B_{loc} is a Gaussian random variable with standard deviation δ B_{loc}. This spread of precession frequencies leads to an aperiodic time dependence of that cannot be removed by the electron spin echo.
Using the above model (see also Supplementary Equation S1), we have fitted all of the data in Fig. 2a with a single set of fieldindependent parameters that were chosen to obtain a good match with all data sets: the number of nuclei in each of the two dots, N, the spectral diffusion time constant, T_{SD}, and δ B_{loc}. In addition, the scale factor for each data set was allowed to vary to account for the imperfect normalization of the data. The value of N determines the depths of the dips between revivals. The best fit yields N=4.4×10^{6}, which is in good agreement with an independent determination from a measurement of giving N=4.9×10^{6} (see ref. 26 and Supplementary Information). From the fit we also obtain T_{SD}≈37 μs and δ B_{loc}=0.3 mT. The measured NMR linewidth in pure GaAs is about 0.1 mT (ref. 25). A possible origin for the larger field inhomogeneity found here is the quadrupole splitting arising from the presence of the two localized electrons^{27}. The inhomogeneity of the Knight shift is expected to have a similar but quantitatively negligible effect for our parameters. The value of T_{SD} is consistent with theoretical estimates^{15} (see Supplementary Information). Interestingly, the spread of nuclear Larmor frequencies, captured by δ B_{loc}, contributes significantly to the echo decay even at the highest fields investigated. We have also verified that the Hahnecho lifetime is not significantly affected by dynamic nuclear polarization, which can be used to increase T_{2}^{*} (see ref. 28 and Supplementary Information).
To measure the long Hahnecho decay times of up to 30 μs, it was necessary to systematically optimize the pulses (see Supplementary Information). Small differences in the gate voltages before and after the πpulse shift the electronic wavefunction relative to the inhomogeneously polarized nuclei. Such shifts cause the electrons to sample different Overhauser fields at different times, and thus lead to an imperfect echo. We have minimized this effect by compensating for a systematic drift of ɛ over the course of each pulse sequence (see Supplementary Information).
Substantially longer coherence times are expected for more elaborate decoupling sequences^{17}. We implemented the CPMG sequence^{18}, which consists of an nfold repetition of the Hahn echo, thus requiring n πpulses, as shown in Fig. 1d. Figure 3 shows data for n=6, 10 and 16. For n=16, the echo signal clearly persists for more than 200 μs. The field dependence for n=4 is reported in the Supplementary Information. To verify the interpretation of the data, we have measured the dependence of the echo on small changes in the final freeprecession time and the duration of the exchange pulses for n=10, τ=5 and 120 μs (Supplementary Information). As a result of the large number of potential tuning parameters,we have not optimized these CPMG pulses. We expect that with improved pulses the same extension of the coherence time could be achieved with fewer pulses. The initial linear decay of the signal in Fig. 3 is not well understood. The similar variation of the reference signal corresponding to a completely mixed state is suggestive of a singleelectron T_{1} process causing leakage into the T^{+} and T^{−} states (see Supplementary Information). The decay time constant sets a lower bound for the largest achievable coherence time.
Our measurements demonstrate coherence times of GaAs spin qubits of at least 200 μs, two orders of magnitude larger than previously shown. The duration of each of the πpulses could easily be reduced below the 6 ns used here. One may hope to achieve millisecondscale coherence times with improved decoupling sequences^{17} without adding complexity. Thus, 10^{5}–10^{6} operations could be carried out on one qubit while maintaining the state of another. However, more effort is required to realize pulses that use decoupling to improve the fidelity of short, nontrivial operations^{29,30}, which seems feasible at least to lowest order. The excellent agreement with the model for the field and time dependence of the Hahnecho revivals shows that many aspects of the dephasing of electron spins due to the nuclear hyperfine interaction are now well understood. The insight gained may also help pave the way towards probing macroscopic quantum effects in a mesoscopic ensemble of a few million nuclear spins.
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Acknowledgements
We thank D. J. Reilly for advice on implementing the radiofrequency readout system and J. R. Maze for discussions. We acknowledge financial support from ARO/IARPA, the Department of Defense and the National Science Foundation under award number 0653336. I.N. and M.R. were supported by NSF grant DMR0906475. This work was carried out in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN), which is supported by the National Science Foundation under NSF award no. ECS0335765.
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Electronbeam lithography and molecularbeamepitaxy growth were carried out by D.M. and V.U., respectively. H.B., S.F. and A.Y. fabricated the sample, planned and executed the experiment and analysed the data. I.N., M.R., H.B. and A.Y. developed the theoretical model. H.B., S.F., I.N., M.R. and A.Y. wrote the paper.
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Bluhm, H., Foletti, S., Neder, I. et al. Dephasing time of GaAs electronspin qubits coupled to a nuclear bath exceeding 200 μs. Nature Phys 7, 109–113 (2011). https://doi.org/10.1038/nphys1856
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DOI: https://doi.org/10.1038/nphys1856
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