## Abstract

Coherent spin states in semiconductor quantum dots offer promise as electrically controllable quantum bits (qubits) with scalable fabrication. For few-electron quantum dots made from gallium arsenide (GaAs), fluctuating nuclear spins in the host lattice are the dominant source of spin decoherence. We report a method of preparing the nuclear spin environment that suppresses the relevant component of nuclear spin fluctuations below its equilibrium value by a factor of ∼70, extending the inhomogeneous dephasing time for the two-electron spin state beyond 1 microsecond. The nuclear state can be readily prepared by electrical gate manipulation and persists for more than 10 seconds.

Quantum information processing requires the realization of interconnected, controllable quantum two-level systems (qubits) that are sufficiently isolated from their environment that quantum coherence can be maintained for much longer than the characteristic operation time. Electron spins in quantum dots are an appealing candidate system for this application, as the spin of the electron is typically only weakly coupled to the environment relative to the charge degree of freedom (*1*). Logical qubits formed from pairs of spins provide additional immunity from collective dephasing, forming a dynamical decoherence-free subspace (*2*, *3*).

Implementing any spin-qubit architecture requires the manipulation (*4*–*6*) and detection (*7*, *8*) of few-electron spin states, as yet demonstrated only in III-V semiconductor heterostructure devices such as gallium arsenide (GaAs), which in all cases comprise atoms with nonzero nuclear spin. The nuclear spins of the host lattice couple to electrons via the hyperfine interaction and causes rapid electron spin dephasing. In the GaAs devices presented here, for instance, an ensemble of initialized spin pairs will retain their phase relationship for *T*_{2}* ∼ 15 ns, consistent with theoretical estimates (*9*–*11*) and previous measurements (*4*). The time *T*_{2}* represents an inhomogeneous dephasing time and can be extended with the use of spin-echo methods (*4*). Nonetheless, extending *T*_{2}* by nuclear state preparation considerably reduces the burden of using complex pulse sequences or large field gradients to overcome the influence of fluctuating hyperfine fields when controlling spin qubits.

Proposals to reduce dephasing by nuclear state preparation include complete nuclear polarization (*12*), state-narrowing of the nuclear distribution (*12*–*15*), and schemes for decoupling the bath dynamics from the coherent evolution of the electron spin through the use of control pulses (*16*–*18*). These approaches remain largely unexplored experimentally, although recent optical experiments (*19*) have demonstrated a suppression of nuclear fluctuations in ensembles of self-assembled quantum dots.

We demonstrate a nuclear state preparation scheme in a double quantum-dot system, using an electron-nuclear flip-flop pumping cycle controlled by voltages applied to electrostatic gates. Cyclic evolution of the two-electron state through the resonance between the singlet (*S*) and *m*_{s} = 1 triplet (*T*_{+}) (*20*), in the presence of a small (few mT) applied magnetic field, leads to a factor of 70 suppression of fluctuations below thermal equilibrium of the hyperfine field gradient between the dots along the total field direction. It is this component of the hyperfine field gradient that is responsible for dephasing of the two-electron spin qubit formed by *S* and *m*_{s} = 0 triplet (*T*_{0}) states (*4*). Consequently, although the flip-flop cycle generates only a modest net nuclear polarization (<1%), the resulting nuclear state extends *T*_{2}*of the *S*-*T*_{0} qubit from 15 ns to beyond 1 μs. Once prepared, this nonequilibrium nuclear state persists for ∼15 s, eventually recovering equilibrium fluctuations on the same time scale as the relaxation of the small induced nuclear polarization. This recovery time is longer than typical gate operation times by ∼9 to 10 orders of magnitude. We propose that occasional nuclear state preparation by these methods may provide a remedy to hyperfine-mediated spin dephasing in networks of interconnected spin qubits.

The double quantum dot is defined in a GaAs-AlGaAs heterostructure with a two-dimensional electron gas (2DEG) 100 nm below the wafer surface (density 2 × 10^{15} m^{–2}, mobility 20 m^{2} V^{–1} s^{–1}). Negative voltages applied to Ti-Au gates create a tunable double-well potential that is tunnel-coupled to adjacent electron reservoirs (fig. S1). A proximal radio-frequency quantum point contact (rf-QPC) senses the charge state of the double dot, measured in terms of the rectified sensor output voltage *V*_{rf} (*21*). Measurements were made in a dilution refrigerator at a base electron temperature of 120 mK.

A schematic energy-level diagram (Fig. 1A), with (*n,m*) indicating equilibrium charge occupancies of the left and right dots, shows the three (1,1) triplet states (*T*_{+}, *T*_{0}, *T*_{–}) split by a magnetic field **B**_{0} applied perpendicular to the 2DEG. The detuning, ϵ, from the (2,0)-(1,1) degeneracy is controlled by high-bandwidth gate voltage pulses. The ground state of (2,0) is a singlet, with the (2,0) triplet out of the energy range of the experiment.

Each confined electron interacts with *N* ∼10^{6} nuclei via hyperfine coupling, giving rise to a spatially and temporally fluctuating effective magnetic (Overhauser) field (*9*–*11*, *22*). In the separated (1,1) state, precession rates for the two electron spins depends on their local effective fields, which can be decomposed into an average field and a difference field. It is useful to resolve **B**_{n} = (**B**^{1}_{n} + **B**^{r}_{n})/2, the Overhauser part of the total average field, **B**_{tot} = **B**_{0} + **B**_{n}, into components along (*B* ^{∥}_{n}) and transverse (*B* ^{⊥}_{n}) to **B**_{tot}. The difference field, due only to Overhauser contributions, is given by Δ**B**_{n} = (**B**^{1}_{n} – **B**^{r}_{n})/2, with components along (Δ*B* ^{∥}_{n}) and transverse (Δ*B* ^{⊥}_{n}) to **B**_{tot}. At large negative ϵ, where the two electrons are well separated and exchange *J*(ϵ) is negligible, Δ*B* ^{∥}_{n} sets the precession rate between *S* and *T*_{0} states. At the value of detuning where *J*(ϵ) equals the Zeeman energy *E*_{Z} = *g*μ_{B}*B*_{tot} (where *g* is the electron *g* factor and μ_{B} is the Bohr magneton), precession between *S* and *T*_{+} states occurs at a rate set by Δ*B* ^{⊥}_{n}.

For measurement of the precession or dephasing of spin pairs in the two dots, a gate-pulse cycle (“probe cycle”) first prepares (P) a singlet state in (2,0), then separates (S) the two electrons into (1,1) for a duration τ_{S}, then measures (M) the probability of return to (2,0). States that evolve into triplets during τ_{S} remain trapped in (1,1) by the Pauli blockade and are detected as such by the rf-QPC charge sensor (*4*). Figure 1, C and D, shows the time-averaged charge-sensing signal *V*_{rf} as a function of constant offsets to gate biases *V*_{L} and *V*_{R}, with this pulse sequence running continuously. Setting the amplitude of the S-pulse to mix *S* with *T*_{0} at large detuning (green dashed line, Fig. 1B) yields the “readout triangle” indicated in Fig. 1C. Within the triangle, *V*_{rf} is between (2,0) and (1,1) sensing values, indicating that for some probe cycles the system becomes Pauli-blockaded in (1,1) after evolving to a triplet state. Outside this triangle, alternative spin-independent relaxation pathways circumvent the blockade (*23*). For a smaller-amplitude S-pulse (red dashed line, Fig. 1B), *S* mixes with *T*_{+}, also leading to partial Pauli blockade and giving the narrow resonance feature seen in Fig. 1D. The dependence of the *S*-*T*_{+} resonance position on applied field *B*_{0} serves as a calibration, mapping the gate voltage *V*_{L} (at fixed *V*_{R}) into the total effective field *B*_{tot}, including possible Overhauser fields (Fig. 2A). The charge-sensing signal *V*_{rf} is also calibrated using equilibrium (1,1) and (2,0) sensing values to give the probability 1 – *P*_{S} that an initialized singlet will evolve into a triplet during the separation time τ_{S} (Fig. 2G). A fit to *P*_{S} (τ_{S}) (Fig. 2C) yields (*11*, *22*, *24*) a dephasing time *T*_{2}*=(*h*/2π)/*g*μ_{B}〈Δ*B* ^{∥}_{n}〉_{rms} ∼ 15 ns, where *h* is Planck's constant and the subscript rms denotes a root-mean-square time-ensemble average.

We now investigate effects of the electron-nuclear flip-flop cycle (“pump cycle”) (Fig. 1E). Each iteration of the pump cycle moves a singlet, prepared in (2,0), adiabatically through the *S*-*T*_{+} resonance, then returns nonadiabatically to (2,0), where the state is re-initialized to a singlet by exchanging an electron with the adjacent reservoir (*20*). In principle, with each iteration of this cycle, a change in the angular momentum of the electron state occurs, with a corresponding change to the nuclear system. Iterating the pump cycle at 4 MHz creates a modest nuclear polarization on the order of 1%, as seen previously (*20*). The pump cycle was always iterated for more than 1 s, and no dependence on pumping time beyond 1 s was observed. What limits the efficiency of the pumping cycle, keeping the polarization in the few-mT regime, is not understood.

Immediately after the pump cycle, the gate voltage pattern is switched to execute one of two types of probe cycles. The first type of probe cycle starts in (2,0) and makes a short excursion into (1,1) to locate the *S*-*T*_{+} resonance, allowing *B*_{tot} to be measured via Fig. 2A. Figure 2B shows that the nuclear polarization established by the pumping cycle relaxes over ∼15 s. The second type of probe cycle starts in (2,0) and makes a long excursion deep into (1,1) to measure *P*_{S} (τ_{S}) where exchange is small and the *S* and *T*_{0} states are mixed by Δ*B* ^{∥}_{n}. We examined *P*_{S} (τ_{S}) at fixed τ_{S} as a function of time after the end of the pump cycle by sampling *V*_{rf} while rastering *V*_{L} across the readout triangle. The black line in Fig. 2D shows the value of *V*_{R}, with the tick marks indicating the upper and lower limits of the rastering. Slicing through the readout triangle allows *P*_{S} to be calibrated within each slice. Remarkably, we find that *P*_{S} (τ_{S} = 25 ns) remains close to unity—that is, the prepared singlet remains in the singlet state after 25 ns of separation—for ∼15 s following the pump cycle (Fig. 2, E and F). Note that τ_{S} = 25 ns exceeds by a factor of ∼2 the value of *T*_{2}* measured when not preceded by the pump cycle (Fig. 2C). The time after which *P*_{S} resumes its equilibrium behavior, with characteristic fluctuations (*24*) around an average value *P*_{S} (τ_{S} = 25 ns) = 0.5, is found to correspond to the time for the small (∼1%) nuclear polarization to relax (Fig. 2B). Measurements of *P*_{S} (τ_{S} = 25 ns) using the same probe cycle without the preceding pump cycle (Fig. 2, H and I) do not show suppressed mixing of the separated singlet state.

Measurement of *P*_{S} (τ_{S}) as a function of τ_{S} shows that *T*_{2}* for the separated singlet can be extended from 15 ns to 1 μs, and that this enhancement lasts for several seconds following the pump cycle. These results are summarized in Fig. 3. Over a range of values of τ_{S}, slices through the readout triangle (as in Fig. 2E) are sampled as a function of time after pumping, calibrated using the out-of-triangle background, and averaged, giving traces such as those in Fig. 3, B to E. Gaussian fits yield *T*_{2}* ∼1 μs for 0 to 5 s after the pump cycle and *T*_{2}* ∼0.5 μs for 10 to 15 s after the pump cycle. After 60 s, no remnant effect of the pump cycle can be seen, with *T*_{2}* returning to ∼15 ns, as before the pump cycle.

The root-mean-square amplitude of longitudinal Overhauser field difference, 〈Δ*B* ^{∥}_{n}〉_{rms} = (*h*/2π)/*g*μ_{B}*T*_{2}*, is evaluated using *T*_{2}* values within several time blocks following the pump cycle (using data from Fig. 3A). The observed increase in *T*_{2}* following the pump cycle is thus recast in terms of a suppression of fluctuations of Δ*B* ^{∥}_{n} (Fig. 4A). Similarly, the *S*-*T*_{+} mixing rate is used to infer the size of fluctuations of the transverse component of the Overhauser field, 〈Δ*B* ^{⊥}_{n}〉_{rms}. Figure 4B shows *P*_{S} *(*t_{S} = 25 ns) near the *S*-*T*_{+} resonance. Unlike *S*-*T*_{0} mixing, which is strongly suppressed by the pump cycle, the *S*-*T*_{+} resonance appears as strong as before the pump cycle. This suggests that the energy gap *E* ^{⊥}_{n} (Fig. 4E) is not closed by the pump cycle. Note that fluctuations in Δ*B* ^{⊥}_{n} produce fluctuations in *E* ^{⊥}_{n}, which give the *S*-*T*_{+} anticrossing a width in detuning ϵ (Fig. 4E). Converting to a width in magnetic field via Fig. 2A gives the fluctuation amplitude 〈Δ*B* ^{⊥}_{n}〉_{rms} following the pump cycle. Figure 4C shows a representative slice taken from Fig. 4B at the position indicated by the white dashed line. Gaussian fits to each 1-s slice yield mean positions *m* and widths *w* in the magnetic field, which fluctuate in time (Fig. 4D). The increase in *w* for short times (*t* < 10 s) reflects gate voltage noise amplified by the saturating conversion from gate voltage to effective field at large *B*_{tot} (*25*). Beyond these first few seconds, *w* is dominated by fluctuations of Δ*B* ^{⊥}_{n} but is also sensitive to fluctuations in *m* that result from fluctuations of *B* ^{∥}_{n} (Fig. 4E). [For *t* > 10 s, gate voltage noise makes a relatively small (<10%) contribution to the fluctuations.] Estimating and removing the contribution due to *B* ^{∥}_{n} (*25*) gives an estimate of 〈Δ*B* ^{⊥}_{n}〉_{rms} as a function of time following the pump cycle. These results are summarized by comparing Fig. 4A and Fig. 4F: In contrast to the strong suppression of fluctuations in Δ*B* ^{∥}_{n} following the pump cycle, no corresponding suppression of 〈Δ*B* ^{⊥}_{n}〉_{rms} is observed.

Reducing the cycle rate by a factor of ∼10 reduces but does not eliminate the suppression of fluctuations of Δ*B* ^{∥}_{n} [see (*25*) for a discussion of the dependence of polarization on pump cycle rate]. Also, when the pump cycle is substituted by a cycle that rapidly brings the singlet into resonance with *T*_{0}, deep in (1,1), effectively performing multiple fast measurements of Δ*B* ^{∥}_{n}, no subsequent effect on *S*-*T*_{0} mixing is observed. This demonstrates that transitions involving *S* and *T*_{+}, rather than *S* and *T*_{0}, lead to the suppression of nuclear field gradient fluctuations.

The observation that an adiabatic electron-nuclear flip-flop cycle will suppress fluctuations of the nuclear field gradient has been investigated theoretically (*26*, *27*). These models explain some, but not all, of the phenomenology described here, and it is fair to say that a complete physical picture of the effect has not yet emerged. Other nuclear preparation schemes arising from various hyperfine mechanisms, not directly related to the specific pump cycle investigated here, have also been addressed theoretically in the recent literature (*28*, *29*).

Control spin qubits in the presence of time-varying equilibrium Overhauser gradients require complex pulse sequences (*4*) or control of sizable magnetic field gradients (*2*, *30*). Suppressing fluctuations of Δ*B* ^{∥}_{n} by a factor of ∼100, as demonstrated here by means of nuclear state preparation, leads to an improvement in control fidelity on the order of 10^{4}, assuming typical control errors, which scale quadratically with the size of the fluctuating field at low-frequencies. We further anticipate generalizations of the present results using more than two confined spins, which would allow arbitrary gradients in nuclear fields to be created by active control of Overhauser coupling.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1159221/DC1

Materials and Methods

Figs. S1 to S3

References