Critical Behavior of Alternately Pumped Nuclear Spins in Quantum Dots

Nuclear spins in a spin-blocked quantum dot can be pumped and eventually polarized in either of two opposite directions that are selected by applying two different source-drain voltages. Applying a square pulse train as the source-drain voltage can continuously switch the pumping direction alternately. We propose and demonstrate a critical behavior in the polarization after alternate pumping, where the final polarization is sensitive to the initial polarization and pulse conditions. This sensitivity leads to stochastic behavior in the final polarization under nominally the same pumping conditions.

Figure 1

(a) Schematic energy diagram of two-electron energy for a DQD as a function of source-drain voltage $V_{\mathrm{SD}}$. The two crossing points, NPP and PPP, and the directions of the shift upon nuclear spin pumping, are indicated by arrows. The lengths of the arrows indicate the strength of pumping at each position. (b) Trajectories of the return map for alternate pumping. The two trajectories start from two initial states that are located very close to the repulsive fixed point.

Figure 2

(a) Plot of differential conductance $d{I}_{\mathrm{SD}}/d{V}_{\mathrm{SD}}$ at $B_{\text{ext}}=3.0\mathrm{T}$. The two gate voltages $V_{G1}$ and $V_{G2}$ change simultaneously, as shown on the left and right axes, respectively. The spin blockade region is indicated by dashed lines. Arrows indicate the lines of current peaks due to the NPP and PPP. (b) $V_{\mathrm{SD}}$ sequence used for alternate pumping measurement.

Figure 3

(a) Plot of the change in the nuclear field $\mathrm{\Delta }{B}_N$ after alternate pumping as a function of the high voltage of the square wave pulse train $V_{\mathrm{SD}}^H$ at $B_{\text{ext}}=3.0\mathrm{T}$. Each trace is obtained for a fixed low voltage $V_{\mathrm{SD}}^L$. The traces are offset for clarity. (b) Result of calculation using the rate equation. The horizontal axis is the high value ${\delta }^H$ of the interdot detuning pulse train. Each trace is calculated for different low values ${\delta }^L$ of the pulse train [32]. (c) Duty ratio dependence on $V_{\mathrm{SD}}^H$ with $V_{\mathrm{SD}}^L$ set at the NPP at $B_{\text{ext}}=3.0\mathrm{T}$. (d) Dependence of the frequency of the square pulse train on $V_{\mathrm{SD}}^H$ with $V_{\mathrm{SD}}^L$ set at the NPP.

Figure 4

(a) Results for $B_N$ in 100 trials under nominally the same alternate pumping conditions for $V_{\mathrm{SD}}^H=2.29\mathrm{mV}$ (top), 2.31 mV (middle), and 2.33 mV (bottom) with the same $V_{\mathrm{SD}}^L$ of 0.49 mV at $B_{\text{ext}}=3.0\mathrm{T}$. (b) Intensity plot of the count of $B_N$ as a function of $V_{\mathrm{SD}}^H$. (c) Switching probability as a function of $V_{\mathrm{SD}}^H$.