4 spin flip, a nuclear spin is flipped from down to up (up to down). By adding the transition rates from | T ± 〉 to all three final states in the decaying S z Tot = 0 subspace, obtained using Fermi’s Golden Rule, we find the “bare” transition rates W $^{±}$,L ( R ) for hyperfine decay of | T ± 〉 as- sisted by a nuclear spin flip in the left (right) dot: accounts for its decay due to coupling to the continuum of states in the drain lead. Due to a large applied bias, we assume that the (1 , 1) state cannot decay directly to the continuum. After diagonalization, each of the states | n 〉 obtains a nonzero | (0 , 2)$_{S}$ 〉 component and decays with a rate γ$_{n}$ = − 2Im[ ε$_{n}$ ] / ℏ , see Fig.3a. Current through the system results from electron transmission through any of the 5 states {| n 〉 , | T ± 〉} that may be populated when the second electron is injected from the lead. The total current with the hybridized states | n 〉 = a$_{n}$ | T$_{0}$ 〉 + b$_{n}$ | (1 , 1)$_{S}$ 〉 + c$_{n}$ | (0 , 2)$_{S}$ 〉 obtained from (1), and ε ± = gµ$_{B}$B ± A ( s$_{L}$ + s$_{R}$ ) / 2. The states | n 〉 are characterized by a nonuniform electron spin density on the two dots, which introduces an asymmetry in the rates W $^{±}$,L ( R ) to flip nuclear spins on the left and right dots via the dependence on a$_{n}$ , b$_{n}$ . is determined by the inverse of the average of the lifetimes { τ$_{n,}$ ± } of these states, weighted by the probabilities p$_{n,}$ ± to load each of the states. The net spin flip rate in the left (right) dot is propor- tional to the total current I , Eq.(2), to the probability p ± of loading | T ± 〉 , and to the probability W $^{±}$,L ( R $_{)}$/ ( W ± $_{,L}$+ W ± $_{,R}$+ W$_{cot}$ ) that this state, when loaded, decays by hy- perfine spin flip in the left (right) dot, giving W $^{±}$,L = ∑ n | a$_{n}$ ∓ b$_{n}$ | 2 16 N$_{L}$ A $^{2}$(1 ∓ s$_{L}$ ) γ$_{n}$ ( ε ± − Re ε$_{n}$ ) 2 + ( ℏ γ$_{n}$/ 2) $_{2}$, (3) W $^{±}$,R = ∑ n | a$_{n}$ ± b$_{n}$ | 2 16 N$_{R}$ A $^{2}$(1 ∓ s$_{R}$ ) γ$_{n}$ ( ε ± − Re ε$_{n}$ ) 2 + ( ℏ γ$_{n}$/ 2) $_{2}$, (4) I ( s$_{L}$, s$_{R}$ ; t, Δ , B$_{0}$, . . . ) = ( ∑ n p$_{n}$τ$_{n}$ + p$_{+}$τ$_{+}$ + p − τ − ) − 1 (2) Γ $^{±}$,L ( R ) = p ± ( I/e ) W $^{±}$,L ( R ) W $^{±}$,L + W $^{±}$,R + W$_{cot}$ . (5) ˙ s$_{L}$ = 2(Γ$_{+}$$_{,L}$ − Γ − $_{,L}$) /N$_{L}$ − Γ$_{rel}$ s$_{L}$ ˙ s$_{R}$ = 2(Γ$_{+}$$_{,R}$ − Γ − $_{,R}$) /N$_{R}$ − Γ$_{rel}$ s$_{R}$, (6) Cotunneling or spin-exchange with the leads with rate W$_{cot}$ adds an additional decay channel, leading to the in- verse lifetimes τ − 1 n = γ$_{n}$ + W$_{cot}$ , see Refs.10,18–20. The inverse lifetimes of | T ± 〉 are determined by the rates W ± of resonant hyperfine flip-flop transitions to each of the states {| n 〉} , Eqs. (3) and (4) below, and of cotunnel- ing: τ − 1 ± = W ± + W$_{cot}$ . We neglect spin-orbit coupling, which is weak in GaAs, but do not expect any qualitative changes if it is included. Our separate treatment of | T ± 〉 and | T$_{0}$ 〉 is valid in nonzero field where the degeneracy of these states is lifted. Near zero field one should employ a 5 × 5 gen- eralization of Eq.(1) as in Ref.17. Finally, using Eq.(5), and including relaxation with rate Γ$_{rel}$, we arrive at the equations of motion for s$_{L}$ and s$_{R}$ : Figure 3b shows current as a function of polarizations s$_{L}$ and s$_{R}$ on the two dots for a fixed set of external conditions. In the dark stripe along the main diagonal, s$_{L}$ = s$_{R}$ , current is set by the cotunneling rate W$_{cot}$ ; away from this line the finite polarization gradient Δ s mixes | T$_{0}$ 〉 with the singlet states and gives rise to the large red plateau of enhanced current that spans most of the figure. The width of the dark stripe is set by the Overhauser field difference required to mix the | T$_{0}$ 〉 state and the | (1 , 1)$_{S}$ 〉 -like hybridized singlet state, and depends on tunnel coupling and detuning, as well as the cotunneling rate W$_{cot}$ , which controls the saturation of current. Sharp yellow bands of further enhanced current appear where the Overhauser shift brings either | T$_{+}$ 〉 or | T − 〉 into resonance with one of the states with S z Tot = 0. With the help of Fig.3a, each line can be identified with a specific resonant transition. where s$_{L}$$_{(}$$_{R}$$_{)}$ are defined as ( N$_{+}$ − N − ) /N in each dot. Equation (6) defines a flow, illustrated by the arrows in Fig.3c, that describes the smooth dynamics of mean polarization. However, polarization is actually stochasti- cally driven by a train of electrons passing through the system, and executes a directed random walk around the flow (6). Fluctuations arise from shot noise in the number of entering up and down spins, from the random compe- tition between hyperfine and cotunelling decay channels, and from nuclear spin diffusion/relaxation. We simu- late this random walk by stochastically loading electrons into each of the 5 transport channels and then generating a corresponding sequence of randomly distributed decay times and numbers of nuclear spin flips, with mean values given by Eqs.(1), (3), and (4). As seen in Fig.3b, the ( s$_{L}$, s$_{R}$ ) plane includes vast re- gions in which current is essentially independent of the value of polarization. In these regions, the mean value of polarization cannot be inferred and fluctuations about the mean do not induce fluctuations of current. In other regions, in particular near the line s$_{L}$ = s$_{R}$ , the deriva- tives ∂I/∂s$_{L,R}$ are large; here current is very sensitive to small fluctuations of polarization which can result in large-amplitude intermittent fluctuations of current, as discussed above. As discussed in Ref.10, DNP arises during spin- blockaded transport as the result of competition between hyperfine decay of | T ± 〉 , which changes the z-component of nuclear spin, and non-spin flip decay channels. Each time an electron in the state | T$_{+(}$ − ) 〉 decays by hyperfine From the simulation we obtain current and polariza- tion trajectories as shown in Figs.1c and 2b, see appendix III. SIMULATIONS AND RESULTS