3 I. TRANSIENTS AND INTERMITTENCY IN THE TWO-POLARIZATION MODEL A typical behavior, often seen in the data, is a rela- tively slow transient buildup of current after which the system enters an intermittent state, characterized by al- ternation of quiet and noisy behavior. Here we discuss the physics of how such behavior can arise naturally from the two-polarization model. The key elements of the mechanism are summarized in schematic form in Fig.1b, which shows a phase portrait of DNP in the ( s$_{L}$, s$_{R}$ ) plane. In our analysis we assume that, via the hyper- fine interaction, dynamics are primarily controlled by two variables s$_{L}$ and s$_{R}$ that describe independent nuclear polarizations in the two dots. The trajectories shown in Fig.1b are obtained by applying the ideas of Refs. 5 and 10 to the case of two coupled polarizations while ignoring noise; the fixed points associated with these trajectories describe steady-state DNP. Due to blockade of the triplet state | T$_{0}$ 〉 , current is low in the gray stripe indicated along the main diagonal, s$_{L}$ = s$_{R}$ ; away from this line the finite polarization gradient Δ s = s$_{L}$ − s$_{R}$ mixes | T$_{0}$ 〉 with the singlet states [see Fig.3 and Eq.(1) below] and gives rise to enhanced current, which is then only weakly sensitive to DNP. Intermittency originates naturally as follows. Due to asymmetry of the dots, DNP initially moves from the un- polarized state into the region s$_{L}$ = s$_{R}$ where current is insensitive to polarization (pink curve in Fig.1b). This corresponds to the quiet build-up period in the current trace (see Fig.2). After approaching the nearly symmet- ric fixed point, DNP continues to fluctuate locally. Here, relatively small fluctuations of polarization take the sys- tem in and out of the low-current stripe s$_{L}$ ≈ s$_{R}$ , re- sulting in large amplitude fluctuations of current and its apparent ‘switching’ between high and low values. FIG. 3: a) Energy level diagram showing hybridization of S z Tot = 0 levels, see Eq.(1), and relevant transitions for spin- blockaded transport. b) Current as a function of nuclear po- larizations s L and s R on the left and right dots. Current is suppressed for s L ≈ s R due to the vanishing coupling between | T 0 〉 and | (1 , 1) S 〉 . Small roman numerals indicate bands of high current due to resonance between blocked and unblocked levels, shown in panel a). Primed labels indicate analogous configurations with | T ± 〉 reversed. c) Simulated trajectory of polarization arcing through the smooth region of the current diagram with steady-state near the sensitive region s L ≈ s R , similar to the schematic in Fig.1. Arrows indicate the direc- tion of the flow (6). Inset: distorted zoom of a narrow strip containing the diagonal, magnified in the transverse direction to display the arc-shaped trajectory and steady state fluctu- ations. This behavior arises whenever a DNP fixed point re- sides near the sensitive region s$_{L}$ ≈ s$_{R}$ , irrespective of the details of the dynamics near the fixed point. Using a realistic model of the stochastic dynamics of electron transport and nuclear polarization, we have generated current traces exhibiting both the slow quiet build-up and long-time intermittent fluctuations. The relationship between the intermittent behavior of current and fluctu- ating DNP is illustrated in Fig.1c, where corresponding regions of low current and s$_{L}$ ≈ s$_{R}$ are marked in gray. When parameters such as magnetic field or detuning are changed such that the DNP fixed point moves away from s$_{L}$ ≈ s$_{R}$ , intermittent fluctuations of current are abruptly suppressed (see red line in Fig.2b); this picture is consis- tent with experiment (see Fig.2a and Ref. 7). coupled by spin-conserving interdot tunneling with am- plitude t , exhibit an avoided level crossing as a function of detuning Δ. A uniform magnetic field B splits the (1 , 1) triplet into the states | T$_{0}$ 〉 , | T ± 〉 with total z -projection of electron spin m = 0 , ± 1, respectively. The hyperfine interaction gives rise to the Overhauser shift of the Zeeman energy, which is different on the left and right dots: H Z L ( R ) = ( gµ$_{B}$B + As$_{L}$$_{(}$$_{R}$$_{)}$ ) S $^{z}$. The ‘po- larization gradient’ Δ s = s$_{L}$ − s$_{R}$ couples the states | T$_{0}$ 〉 and | (1 , 1)$_{S}$ 〉 . Including this coupling, we find the energy levels ε$_{n}$ and eigenstates | n 〉 within the S z Tot = 0 sub- space (spanned by the states | T$_{0}$ 〉 , | (1 , 1)$_{S}$ 〉 , | (0 , 2)$_{S}$ 〉 ) by diagonalizing the Hamiltonian We now turn to a detailed description of the model used to generate Figs.1 and 2. The relevant energy lev- els are depicted in Fig.3. The states (1 , 1)$_{S}$ and (0 , 2)$_{S}$, where the energy of the unhybridized (0 , 2) singlet state, ε$_{(0}$$_{,}$$_{2)}$$_{S}$ = − Δ − i ℏ γ/ 2, includes an imaginary part that ˆ H$_{3}$ × 3 =   0 A Δ s/ 2 0 A Δ s/ 2 0 t 0 t − Δ − i ℏ γ/ 2   , (1) II. MATHEMATICAL FORMULATION