Nuclear Spin Dynamics in Double Quantum Dots: Fixed Points, Transients and Intermittency M. S. Rudner $^{1}$, F. H. L. Koppens2 , $^{3}$, J. A. Folk2 , $^{4}$, L. M. K. Vandersypen$^{2}$, L. S. Levitov5 1 Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA 02138 2 Kavli Institute of NanoScience, TU Delft, PO Box 5046, 2600 GA, Delft, The Netherlands 3 ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain 4 Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada 5 Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139 Transport through spin-blockaded quantum dots provides a means for electrical control and de- tection of nuclear spin dynamics in the host material. Although such experiments have become increasingly popular in recent years, interpretation of their results in terms of the underlying nu- clear spin dynamics remains challenging. Here we examine nuclear polarization dynamics within a two-polarization model that supports a wide range of nonlinear phenomena. We point out a fundamental process in which nuclear spin dynamics can be driven by electron shot noise; fast electric current fluctuations generate much slower nuclear polarization dynamics, which in turn af- fect electron dynamics via the Overhauser field. The resulting intermittent, extremely slow current fluctuations account for a variety of observed phenomena that were not previously understood. The opportunity to study spin coherence and many- body dynamics in a controllable solid-state setting has inspired a wide range of experiments in a variety of materials such as GaAs vertically grown and gate- defined structures $^{1}$, InAs nanowires$^{2}$, and $^{13}$C-enriched carbon nanotubes $^{3}$. In particular, electron transport through spin-blockaded double quantum dots 4 consti- tutes a purely electrical means of probing and manip- ulating the dynamics of nuclear spins. Such experiments have revealed complex dynamical phenomena, including bistability and hysteresis $^{2,3,5,6}$, switching$^{3,7,8}$, slow tran- sient build-up of current $^{7}$, and slow oscillations$^{5,9}$. Despite wide interest in these phenomena and their im- portance for quantum information processing, progress in understanding them has been slow. While there is little doubt that nuclear spins in the host material play a cru- cial role, the lack of a direct probe of nuclear spin dynam- ics requires their behavior to be inferred from electronic transport measurements. To meet this challenge, theo- retical modeling must be used to complement analysis of relevent features in transport data. arXiv:1001.1735v3 [cond-mat.mes-hall] 10 Aug 2011 In previous work on spin dynamics in double quantum dots, simple models involving a single dynaical variable describing the total nuclear polarization have been used to explain the origin of feedback in this system $^{10,11}$. Al- though such models can succesfully account for feedback- driven nonlinear phenomena such as bistability and hys- teresis, the range of phenomena which they can describe is somewhat limited. Here we expand the phase space of the model, and describe nuclear spin dynamics in terms of two dynamical variables, s$_{L}$ and s$_{R}$ , corresponding to the independent nuclear polarizations in the left and right dots (see also e.g. Ref. 12), thereby extending the range of phenomena that can be analyzed. Time evolution is described by trajectories in a two-dimensional phase space ( s$_{L}$, s$_{R}$ ), which can exhibit complex dynamics in- cluding non-monotonic behavior, limit-cycles, or spirals, as illustrated in Fig.1 (also see Refs. 13 and 14 for addi- tional examples of complex phenomena arising from two- Nuclear polarization dynamics in spin-blockaded dots is driven by carriers passing through the system $^{4}$. Each electron passing through the dot can produce a spin flip of the nuclei due to hyperfine exchange with nuclear spins in the host lattice, see Fig.1a. Dynamic nuclear polariza- tion (DNP) arises when the up and down spin-flip rates are imbalanced, Γ$_{+}$ = Γ $^{$_{−}$5,10}$. Since the spin-flip rates in the two dots are in general different, their correspond- polarization dynamics in other contexts). FIG. 1: Nuclear dynamics and intermittent current fluctu- ations driven by shot noise of the spin-blockaded current through a double dot. a) Hyperfine spin-exchange with nuclei mediates transitions between two-electron triplet and singlet states, relieving blockade and producing dynamical nuclear polarizations (DNP) s L and s R in the left and right dots. b) A phase portrait of DNP trajectories, with a fixed point (DNP steady-state) positioned near the main diagonal, where current is low due to spin-blockade (see Fig.3). c) Typical sim- ulated current trace showing the effects of steady state DNP fluctuations (pink traces in b and c), see appendix. Switching between quiet and noisy regions results from excursions into the dark stripe s L ≈ s R , marked by the shaded regions in panel c).