Quenching of dynamic nuclear polarization by spin–orbit coupling in GaAs quantum dots

The central-spin problem is a widely studied model of quantum decoherence. Dynamic nuclear polarization occurs in central-spin systems when electronic angular momentum is transferred to nuclear spins and is exploited in quantum information processing for coherent spin manipulation. However, the mechanisms limiting this process remain only partially understood. Here we show that spin–orbit coupling can quench dynamic nuclear polarization in a GaAs quantum dot, because spin conservation is violated in the electron–nuclear system, despite weak spin–orbit coupling in GaAs. Using Landau–Zener sweeps to measure static and dynamic properties of the electron spin–flip probability, we observe that the size of the spin–orbit and hyperfine interactions depends on the magnitude and direction of applied magnetic field. We find that dynamic nuclear polarization is quenched when the spin–orbit contribution exceeds the hyperfine, in agreement with a theoretical model. Our results shed light on the surprisingly strong effect of spin–orbit coupling in central-spin systems.


SUPPLEMENTARY NOTE 1. MEASURING σ ST
Here we describe the fitting procedure to extract σ ST . The experimentally measured quantity is the average triplet occupation probability P T , which we interpret as the average Landau-Zener (LZ) probability P LZ , at the end of a sweep. Here · · · indicates an average over the hyperfine distribution and charge fluctuations for the same nominal sweep parameters. We calibrate the rate β = d(E S − E T + )/dt using the spin-funnel technique [1] and assume a linear change in the S − T + splitting near the avoided crossing.
∆ HF (t) varies in time because of the nuclear Larmor precession and statistical fluctuations in the magnitude of the nuclear polarizations. We argue that both types of hyperfine fluctuations occur on time scales much longer than LZ transitions and can be treated as quasi-static. In typical experiments, the S − T + splitting is swept through approximately 5 GHz in less than 1 µs. For splittings of order 10 MHz, the total time spent near the avoided crossing is less than 10 ns, which is much faster than the nuclear Larmor period at 1 T, roughly 100 ns. Furthermore, during 1 µs, the nuclear polarization diffuses by approximately 7 kHz [2], which is 3 orders of magnitude smaller than σ HF . We therefore assume that the splitting is constant during a single sweep. Numerical simulations discussed below also support the hypothesis that nuclear Larmor precession does not significantly affect P T for the sweep rates used here ( Supplementary Fig. 1).
In the absence of hyperfine or charge fluctuations, the probability for a transition is given by the LZ formula: [3]. Neglecting high-frequency charge noise, the exact form of the LZ probability averaged over the hyperfine distribution can be computed. Let the total splitting be ∆ ST = ∆ HF + ∆ SO . We take ∆ SO to be the constant, real spin-orbit part and ∆ HF the complex hyperfine contribution. Assuming that the real and imaginary parts of ∆ HF (u and v, respectively) are Gaussian-distributed around zero such that the root-mean-square hyperfine splitting is σ HF , the probability distribution for the splitting to have magnitude ∆ = |∆ ST | is where I 0 is the zeroth-order modified Bessel function of the first kind. Note that when ∆ SO = 0, Supplementary Equation (2) reduces to the familiar distribution p(∆) = 2∆ Integrating the LZ probability over this distribution yields the average LZ probability P LZ : with Note that this result agrees with another derivation [5]. Note also that to leading order in The average triplet return probability P T may be modified due to effects of charge noise on the defining gates or in the two-dimensional electron gas itself. High-frequency charge noise in double quantum dots has recently been identified as a major source of decoherence [6]. In the current setting, corrections to P T should occur, because charge We observe that for high magnetic fields and slow sweeps, the maximum LZ probability falls to 0.5 as shown in Supplementary Fig. 1. It was previously noted that strong detuning noise can have such an effect [7]. To confirm that charge noise causes the probability Even in the presence of noise, however, the average LZ probability in the limit of fast sweeps is still 2π|∆ ST (t)| 2 / β, which is identical to the leading order behavior of the usual LZ formula, as shown in section 3.1 of ref. [7]. Replacing the LZ formula in Supplementary Equation (4) by its leading order behavior, and performing the integration over the quasi-static distribution gives P LZ ≈ 2π (∆ 2 SO + σ 2 HF ) / β. Such a result can be understood because the effect of detuning noise is reduced on short time scales. Supplementary Fig. 1 demonstrates this idea because the analytic curves deviate significantly from the data for P LZ 0.2, but for 0 < P LZ < 0.1, the analytic results agrees well with the data.
Based on additional simulations, we estimate the systematic error in the deduced value of Additionally, we note that the simulations in Supplementary Fig. 1, which include -dependent coupling, demonstrate that the fitting procedure described above allows an accurate measurement of σ ST . Finally, we have also performed additional simulations, taking into account the measured values of E( ), which deviate slightly from the values predicted by assuming a constant tunnel coupling, and we observe no significant change in our results.

SUPPLEMENTARY NOTE 2. DERIVATION OF NUCLEAR POLARIZATION CHANGE δm
Here we derive equations (4) and (5) in the main text. Let ∆ ST = ∆ SO + ∆ HF where ∆ SO is real and where I + j is the raising operator for the j th nuclear spin, and the λ j are individual coupling constants. We assume that there are many nuclear spins, so that each coupling constant is small. Also, where I = 3 2 is the spin of the nuclei, and the angular brackets refer to an average over the distribution of nuclear spins.
We pick one of the nuclear spins, j, and we wish to compute δm j , the mean value of the change in I z j after one sweep. Let P LZ (∆ ST ) be the probability of an S − T + transition for a fixed value of ∆ HF . Clearly, P LZ depends on |∆ ST |. We calculate δm j as follows. Write where a includes the contributions of spin orbit and of all nuclei other than the nucleus j, and the second term represents the contribution (of order λ j ) from nucleus j. According to equation (31) of Ref. [4], the value of δm j for this configuration should be given by where ϕ = arctan(Im(∆ ST )/Re(∆ ST )) specifies the orientation of ∆ ST in the complex plane.
Without loss of generality, we may suppose that a is real. Then we have, ignoring terms that are higher order in b/a, P LZ (∆ ST ) = P LZ (a) + bP LZ (a) cos θ j (12) where P LZ (a) is the derivative of P LZ (a) with respect to a. Averaging over nuclear configurations, we obtain with b 2 = (5/2)λ 2 j . In the case of no charge noise, we have so and Finally, we sum over all nuclear spins and make the replacement a ≈ |∆ ST |, obtaining The collapse demonstrated in Fig. 4 In Supplementary Equation (19), δm → 0 for both β → 0 and β → ∞. In practice however, experiments necessarily average over the hyperfine distribution. Thus, using Supplementary Equation (4) (4) was then solved using the calculated parameters to find the rate β such that P LZ = 0.4. In order to compare with data on the dDNP rate, the theoretical curves for δm were multiplied by fitting constants C, which are different for the two curves. As explained in the main text, we expect the ratio between the dDNP rate and δm to depend on the magnetic field but to be independent of the sweep rate.
Data taken at fixed sweep rate β also show a suppression of DNP, as shown in Supplementary Fig. 4(a). In this case, P LZ increases with |φ| because of spin-orbit coupling [ Supplementary Fig. 4 Supplementary Fig. 5, and it does not significantly deviate from the case without noise, at least at the level of the experimental accuracy.

SUPPLEMENTARY NOTE 3. EXPECTED DNP RATE
In this section we give a simple calculation to explain the value of the peak (φ = 0 • ) dDNP rate, as shown in Fig. 4 of the main text. Additional measurements were carried out to measure the pumping rate of the sum hyperfine field, (B r + B l )/2, where B r and B l denote the longitudinal hyperfine fields in the right and left dots. This rate was determined by measuring the location of ST before and after a series of LZ sweeps to polarize the nuclei at B = 0.2 T. We observe that the sum field is pumped roughly twice as efficiently as the difference field, δB z = B r − B l . Setting (Ḃ r +Ḃ l )/2 = 2(Ḃ r −Ḃ l ), whereḂ l(r) indicates the pumping rate of the left(right) dot, we haveḂ l = (3/5)Ḃ r , meaning that the left dot is pumped 3/5 as often as the right dot. Under these conditions, the average gradient builds up at a rate (per electron spin flip) of (Ḃ r −Ḃ l )/(Ḃ r +Ḃ l ) that is only 1/4 the rate that would occur if nuclear spin flips occurred in only one dot.
To determine the expected change in δB z , we require the approximate number of spins overlapped by the electronic wave function in the double dot. We have measured the inhomogeneous dephasing time of electronic oscillations around δB z and find T * 2 = 18 ns [8]. This dephasing time corresponds to a rms value of the gradient σ δBz ≡ |δB z | 2 = h/ |g * |µ B √ 2πT * 2 = 2 mT, where h is Planck's constant. The total number of spins overlapped by the wavefunction is N = (h 1 /σ δBz ) 2 ≈ 3 × 10 6 , where h 1 = 4.0 T [9]. If all nuclear spins were fully polarized, then the dots would experience a hyperfine field of h 0 = 5.3 T [9], and if the nuclear spins in the two dots were fully polarized in opposite directions, the gradient would be 2h 0 . Therefore, the expected change in the gradient per electron spin flip, corresponding to a change in nuclear angular momentum of , is 2π × 2|g * |µ B h 0 2I(N/2) = 12 kHz, where I = 3/2 is the nuclear spin. The average dDNP under actual conditions is 1/4 of this value, or 3 kHz, in reasonable agreement with our observations. In addition, we note the reasonable agreement between the measured value of σ δBz = 2 mT and the root-mean-square