Report

Suppressing Spin Qubit Dephasing by Nuclear State Preparation

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Science  08 Aug 2008:
Vol. 321, Issue 5890, pp. 817-821
DOI: 10.1126/science.1159221

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 (46) 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 T2* ∼ 15 ns, consistent with theoretical estimates (911) and previous measurements (4). The time T2* represents an inhomogeneous dephasing time and can be extended with the use of spin-echo methods (4). Nonetheless, extending T2* 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 (1215), and schemes for decoupling the bath dynamics from the coherent evolution of the electron spin through the use of control pulses (1618). 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 ms = 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 ms = 0 triplet (T0) states (4). Consequently, although the flip-flop cycle generates only a modest net nuclear polarization (<1%), the resulting nuclear state extends T2*of the S-T0 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 × 1015 m–2, mobility 20 m2 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 Vrf (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+, T0, T) split by a magnetic field B0 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.

Fig. 1.

(A) Schematic of the energy levels of the two-electron system in a magnetic field. Detuning, ϵ, from the (2,0)-(1,1) charge degeneracy is gate-controlled. (B) Gate-pulse sequence used to separately probe the longitudinal (ΔB n, green dashed line) and transverse (ΔB n, red dashed line) components of the Overhauser field difference, depending on the position of the separation point S. (C and D) Time-averaged charge-sensing signal Vrf from the rf-QPC as a function of gate voltages VL and VR, showing features corresponding to the singlet mixing with T0 [bracketed green triangle in (C)] and T+ [bracketed green line segment in (D)]. (E) Schematic view of the S-T+ anticrossing, illustrating the pumping cycle. With each iteration of this cycle, with period τC = 250 ns, a new singlet state is taken adiabatically through the S-T+ anticrossing in a time τA = 50 ns, then returned nonadiabatically to (2,0) in ∼1 ns, where the S state is then reloaded.

Each confined electron interacts with N ∼106 nuclei via hyperfine coupling, giving rise to a spatially and temporally fluctuating effective magnetic (Overhauser) field (911, 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 Bn = (B1n + Brn)/2, the Overhauser part of the total average field, Btot = B0 + Bn, into components along (B n) and transverse (B n) to Btot. The difference field, due only to Overhauser contributions, is given by ΔBn = (B1nBrn)/2, with components along (ΔB n) and transverse (ΔB n) to Btot. At large negative ϵ, where the two electrons are well separated and exchange J(ϵ) is negligible, ΔB n sets the precession rate between S and T0 states. At the value of detuning where J(ϵ) equals the Zeeman energy EZ = gμBBtot (where g is the electron g factor and μB is the Bohr magneton), precession between S and T+ states occurs at a rate set by ΔB 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 τS, then measures (M) the probability of return to (2,0). States that evolve into triplets during τS 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 Vrf as a function of constant offsets to gate biases VL and VR, with this pulse sequence running continuously. Setting the amplitude of the S-pulse to mix S with T0 at large detuning (green dashed line, Fig. 1B) yields the “readout triangle” indicated in Fig. 1C. Within the triangle, Vrf 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 B0 serves as a calibration, mapping the gate voltage VL (at fixed VR) into the total effective field Btot, including possible Overhauser fields (Fig. 2A). The charge-sensing signal Vrf is also calibrated using equilibrium (1,1) and (2,0) sensing values to give the probability 1 – PS that an initialized singlet will evolve into a triplet during the separation time τS (Fig. 2G). A fit to PSS) (Fig. 2C) yields (11, 22, 24) a dephasing time T2*=(h/2π)/gμB〈ΔB nrms ∼ 15 ns, where h is Planck's constant and the subscript rms denotes a root-mean-square time-ensemble average.

Fig. 2.

(A) Position of the S-T+ resonance in left-dot gate voltage VL as a function of applied magnetic field amplitude B0, without prior pump cycle. (B) Evolution of the S-T+ position as a function of time. Vertical scale is converted from gate voltage VL to Btot via the resonance position in (A). Resonance position in gate voltage VL and converted to Btot via (A). (C) Singlet return probability PS as a function of τS at B0 = 100 mT. Gaussian fit gives an inhomogeneous dephasing time T2* = 15 ns. (D) Sensor output Vrf, as in Fig. 1C, showing triangle that yields PS. Vertical cut (black line) with tick marks shows location of slices in upper panels of (E) to (I). (E and F) After the pump cycle, repeated slices across this triangle at the position indicated by the black line in (D) allow a calibrated measure of PS as it evolves in time, at (E) B0 = 10 mT and (F) B0 = 100 mT. (G) Calibration between the sensing signal Vrf and singlet return probability PS. (H and I) Control experiments showing slices across the triangle as in (E) and (F), but without a prior pump cycle.

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 Btot 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 PSS) where exchange is small and the S and T0 states are mixed by ΔB n. We examined PSS) at fixed τS as a function of time after the end of the pump cycle by sampling Vrf while rastering VL across the readout triangle. The black line in Fig. 2D shows the value of VR, with the tick marks indicating the upper and lower limits of the rastering. Slicing through the readout triangle allows PS to be calibrated within each slice. Remarkably, we find that PSS = 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 τS = 25 ns exceeds by a factor of ∼2 the value of T2* measured when not preceded by the pump cycle (Fig. 2C). The time after which PS resumes its equilibrium behavior, with characteristic fluctuations (24) around an average value PSS = 25 ns) = 0.5, is found to correspond to the time for the small (∼1%) nuclear polarization to relax (Fig. 2B). Measurements of PSS = 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 PSS) as a function of τS shows that T2* 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 τS, 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 T2* ∼1 μs for 0 to 5 s after the pump cycle and T2* ∼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 T2* returning to ∼15 ns, as before the pump cycle.

Fig. 3.

(A) Singlet return probability PS as a function of separation time τS deep in (1,1) (green dashed line in Fig. 1A), where S mixes with T0. PS values are shown averaged within the 0- to 5-s interval (black), the 5- to 10-s interval (green), the 10- to 15-s interval (red), and 60 to 120 s after the pump cycle (blue), along with Gaussian fits. (B to E) PS as a function of time following the pump cycle (B0 = 10 mT) for fixed τS = 5 ns (B), τS = 0.2 μs (C), τS = 0.4 μs (D), and τS = 0.8 μs (E).

The root-mean-square amplitude of longitudinal Overhauser field difference, 〈ΔB nrms = (h/2π)/gμBT2*, is evaluated using T2* values within several time blocks following the pump cycle (using data from Fig. 3A). The observed increase in T2* following the pump cycle is thus recast in terms of a suppression of fluctuations of ΔB 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, 〈ΔB nrms. Figure 4B shows PS (tS = 25 ns) near the S-T+ resonance. Unlike S-T0 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 ΔB 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 〈ΔB nrms 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 Btot (25). Beyond these first few seconds, w is dominated by fluctuations of ΔB 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 〈ΔB nrms 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 ΔB n following the pump cycle, no corresponding suppression of 〈ΔB nrms is observed.

Fig. 4.

(A) Amplitude of fluctuating longitudinal Overhauser field, 〈ΔBnrms, extracted from T2* values at the 5-s intervals in Fig. 3A. (B) S-T+ resonance probed immediately after the pump cycle. Position of the resonance yields Btot and its intensity gives PS (B0 = 10 mT, τS = 25 ns). (C) Slice from (B) at position marked by white dashed line, averaged for 1 s. For each slice a Gaussian fit yields the mean position m and width w of the S-T+ resonance, given in units of magnetic field via Fig. 2A. (D) Resonance width w as a function of time after the pumping cycle. (E) Schematic of the S-T+ anticrossing, showing how fluctuations of E n due to ΔB n give the resonance a width. (F) Fluctuations in ΔB n in terms of 〈ΔB nrms in a 1-s slice.

Reducing the cycle rate by a factor of ∼10 reduces but does not eliminate the suppression of fluctuations of ΔB 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 T0, deep in (1,1), effectively performing multiple fast measurements of ΔB n, no subsequent effect on S-T0 mixing is observed. This demonstrates that transitions involving S and T+, rather than S and T0, 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 ΔB 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 104, 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

References and Notes

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