Suppression of Zeeman Gradients by Nuclear Polarization in Double Quantum Dots

We use electric dipole spin resonance to measure dynamic nuclear polarization in InAs nanowire quantum dots. The resonance shifts in frequency when the system transitions between metastable high and low current states, indicating the presence of nuclear polarization. We propose that the low and the high current states correspond to different total Zeeman energy gradients between the two quantum dots. In the low current state, dynamic nuclear polarization efficiently compensates the Zeeman gradient due to the $g$-factor mismatch, resulting in a suppressed total Zeeman gradient. We present a theoretical model of electron-nuclear feedback that demonstrates a fixed point in nuclear polarization for nearly equal Zeeman splittings in the two dots and predicts a narrowed hyperfine gradient distribution.

Figure 1

(a) Charge stability diagram near the $(3,1)\to (2,2)$ transition ($V_{\mathrm{dc}}=7\mathrm{mV}$, $B_{\mathrm{ext}}=0.79\mathrm{T}$). Low and high current states are labeled, the dashed arrow indicates the detuning axis $\epsilon$. (b) Dependence of the leakage current on the magnetic field and detuning. The detuning is swept along the line in panel (a), and the field is stepped after each sweep. For (a) and (b) the sweep direction is from left to right. (c) The magnetic field retrace at fixed detuning $\epsilon \approx 3.5\mathrm{meV}$, the sweep directions are indicated with arrows. The current becomes zero above 4 T since the double dot then shifts into a Coulomb blocked state due to the Zeeman shift of $T_{+}(1,1)$. The same device was previously studied in Ref. 19.

Figure 2

(a) Double dot current as a function of magnetic field (device 2, $V_{\mathrm{dc}}=7.5\mathrm{mV}$, and $\epsilon \approx 2.5\mathrm{meV}$). The gray and white background indicates the high and low current states respectively. (b) The frequency of the ac voltage on a local gate is swept, while the magnetic field is stepped. At each point, we measure 50 ms with ac excitation on the gate and then 1 s without. The difference in the two measured currents is plotted. This procedure also avoids dragging of the spin resonance. The inset shows a zoom-in on the spectral line at the transition from the high to low current state. Low ac power is used to minimize switches between metastable states induced by the driving. The horizontal lines at fixed frequency correspond to photon assisted tunneling enhanced by cavity modes in the fridge. This is the same device as studied in Ref. 3.

Figure 3

(a) Sketch of the spectrum of the five electronic states. The coupling between the different states, as well as the decay rate ${\Gamma }_{\mathrm{out}}$ of $S(0,2)$ are indicated. (b) Simulated current in the presence of an external Zeeman gradient of 1 mT (red trace) and 3 mT (blue trace). We used $\epsilon ={\Gamma }_{\mathrm{out}}=1.5\mathrm{meV}$, $t_s=100\mu \mathrm{eV}$, $t_{\mathrm{SO}}=25\mu \mathrm{eV}$, and $\overline{g}=9$. We averaged over ${10}^5$ random nuclear fields taken from a normal distribution with $\sqrt{⟨(B_N^{x,y,z}{)}^2⟩}=B_N^{\mathrm{rms}}=1\mathrm{mT}$ for both dots.

Figure 4

(a) Thick blue lines, left scale: Spectrum of the (1,1) states as a function of $\Delta E_Z$ for $E_Z=-20\mu \mathrm{eV}$, $t_s=60\mu \mathrm{eV}$, $t_{\mathrm{SO}}=40\mu \mathrm{eV}$, and $\epsilon ={\Gamma }_{\mathrm{out}}=1.5\mathrm{meV}$. The thickness of the lines indicates the occupation probability of the corresponding eigenstate ($4\mu \mathrm{eV}=1$). Brown dashed line, right scale: Current through the double dot as a function of $\Delta E_Z$ for the same parameters. Red dashed arrows: Preferred directions of nuclear spin flips close to $\Delta E_Z=0$. (b) $d(\Delta P)/dt$ as a function of $\Delta P$ for three different magnetic fields, negative $E_Z$ corresponds to a positive field. We used the same parameters as at (a), with $AI=0.7\mathrm{meV}$, $I=\frac{1}{2}$, $N={10}^5$, $1/\tau =3\times {10}^{-10}\mu \mathrm{eV}$, and $\Delta g/\overline{g}=0.05$. Stable (un)polarized states of the nuclear spin ensembles are indicated.