led to simulated traces clearly demonstrating the phe- nomena of interest on field and time scales approximately comparable to those observed in experiments. 

In the table below, N$_{L}$ and N$_{R}$ are the numbers of nuclear spins in the left and right dots, W$_{cot}$ is the co- 

1 R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha and L. M. K. Vandersypen, Rev. Mod. Phys. 79 , 1217 (2007). 2 

A. Pfund, I. Shorubalko, K. Ensslin, R. Leturcq, Phys. Rev. Lett. 99 , 036801 (2007). 3 

H. O. H. Churchill et al. , Nat. Phys. 5, 321 (2009). 4 

K. Ono, D. G. Austing, Y. Tokura, S. Tarucha, Science 297 , 1313 (2002). 5 

K. Ono and S. Tarucha, Phys. Rev. Lett. 92 , 256803 (2004). 6 

J. Baugh, Y. Kitamura, K. Ono, S. Tarucha, Phys. Rev. Lett. 99 , 096804 (2007). 7 

F. H. L. Koppens et al. , Science 309 , 1346 (2005). 8 

D. J. Reilly et al. , Phys. Rev. Lett. 101 , 236803 (2008) 9 

D. G. Austing, C. Payette, G. Yu, and J. A. Gupta, Physica E 40 , 1118 (2008). 10 

M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99 , 036602 (2007). 11 

C. Lopez-Moniz, J. Inarrea, and G. Platero, New Journal Phys. 13 , 053010 (2011). 12 

J. Inarrea, G. Platero, and A. H. Macdonald, Phys. Rev. 

tunneling rate, t is the tunnel coupling, Δ is the detuning, γ is the decay rate of (0 , 2)$_{S}$ [see Eq.(1), main text], Γ$_{rel}$ is the phenomenological relaxation rate for nuclear po- larization within the double dot, Eq.(6) of the main text, and B is the magnetic field strength. 

B 76 , 085329 (2007). 

13 J. Danon, I.T. Vink, F.H.L. Koppens, K.C. Nowack, L.M.K. Vandersypen, Yu.V. Nazarov, Phys. Rev. Lett. 103 , 046601 (2009). 14 

M. Gullans, J.J. Krich, J.M. Taylor, H. Bluhm, B.I. Halperin, C.M. Marcus, M. Stopa, A. Yacoby, M.D. Lukin, Phys. Rev. Lett. 104 , 226807 (2010). 15 

J. C. Vassilicos, Intermittency in turbulent flows, (Cam- bridge, U.K.: Cambridge University Press, 2000). 16 

F. D. Stefani, J. P. Hoogenboom, and E. Barkai, Physics Today 62 , 34 (February 2009). 17 

O. N. Jouravlev and Yu. V. Nazarov, Phys. Rev. Lett. 96 , 176804 (2006). 18 

K. Saito, S. Okubo, M. Eto, Physica E 40 , 1149 (2008). 19 

A. B. Vorontsov, M. G. Vavilov, Phys. Rev. Lett. 101 , 226805 (2008). 20 

F. Qassemi, W. A. Coish, F. K. Wilhelm, Phys. Rev. Lett. 102 , 176806 (2009). 21 

S. Foletti, H. Bluhm, D. Mahalu, V. Umansky, A. Yacoby, Nature Physics 5 , 903 (2009). 

