Single-spin manipulation in a double quantum dot in the field of a micromagnet

The manipulation of single spins in double quantum dots by making use of the exchange interaction and a highly inhomogeneous magnetic field was discussed in Coish and Loss [Phys. Rev. B 75, 161302 (2007)]. However, such large inhomogeneity is difficult to achieve through the slanting field of a micromagnet in current designs of lateral double dots. Therefore, we examine an analogous spin manipulation scheme directly applicable to realistic GaAs double dot setups. We estimate that typical gate times, realized at the singlet-triplet anticrossing induced by the inhomogeneous micromagnet field, can be a few nanoseconds. We discuss the optimization of initialization, read-out, and single-spin gates through suitable choices of detuning pulses and an improved geometry. We also examine the effect of nuclear dephasing and charge noise. The latter induces fluctuations of both detuning and tunneling amplitude. Our results suggest that this scheme is a promising approach for the realization of fast single-spin operations.

Figure 1

Spectrum of the double quantum dot as function of detuning $\varepsilon$. The solid lines are computed from the full Hamiltonian Eq. (9). The dashed circle in the left panel highlights the anticrossing region around ${\varepsilon }_B$, shown in more detail in the right panel. The dashed curves are obtained from the effective two-level system of Eq. (13) and describe well the anticrossing region. We used $t=5\mu \mathrm{eV}$ and ${\mathbf{b}}_{1,2}$ obtained from the geometry of Fig. 3 with $B=0.5\mathrm{T}$ and $\phi =0$.

Figure 2

Detuning pulses to initialize the system from ${\varepsilon }_I$ into the logical subspace at ${\varepsilon }_A$ (first blue pulse), to execute single-spin rotations using the anticrossing at ${\varepsilon }_B$ (second red pulse), and to read out the logical states (third green pulse).

Figure 3

Geometries used in the simulations. The micromagnet extends vertically from $z=-0.2\mu \mathrm{m}$ to $z=0$, and is symmetric with respect to the $x=3\mu \mathrm{m}$ plane. Panel (a) indicates relevant dimensions of the micromagnet (in $\mu \mathrm{m})$. The coordinates of the two quantum dots are (a) ${\mathbf{x}}_1=(3.1,0.4,-0.3)\mu \mathrm{m},{\mathbf{x}}_2=(3.1,0.2,-0.3)\mu \mathrm{m}$, and (b) ${\mathbf{x}}_1=(3,0.2,-0.3)\mu \mathrm{m},{\mathbf{x}}_2=(3.2,0.2,-0.3)\mu \mathrm{m}$. We also indicate the angle $\phi$ defining the directions of $\mathbf{B}$ and $\mathbf{m}$; see Eq. (22).

Figure 4

Difference in direction $\theta$, see Eq. (5), and magnitudes, $\Delta b=b_1-b_2$, between the local magnetic fields at the two quantum dot locations, as functions of the direction of $\mathbf{B}$. Solid and dashed curves are for the setup of Fig. 3 and Fig. 3, respectively. Each family of curves (solid/dashed in the left/right panel) are for $B=0.5,1,1.5,2$ T from top to bottom, except the dashed curves of the right panel (where $B=0.5,1,1.5,2$ T from bottom to top).

Figure 5

Plot of the initialization fidelity $P_{-}\left({\tau }_I\right)={\left|\langle -|\psi \left({\tau }_I\right)\rangle \right|}^2$ for a linear ramp in detuning $\varepsilon \left(\tau \right)$ starting at ${\varepsilon }_I=-{\varepsilon }_A=200\mu \mathrm{eV}$ [with $\left|\psi \right(0)\rangle$ the ground state] and ending at $\varepsilon \left({\tau }_I\right)={\varepsilon }_A$. We used the geometry of Fig. 3 with $\phi =0$. The thin solid curves are for $t=10\mu \mathrm{eV}$ where the lower green (upper orange) curve is for $B=0.5$ T $(B=2$ T). The thick dashed red curve is the approximate formula $[1-exp(-\frac{\pi t^2{\tau }_I}{\hslash {\varepsilon }_A})]exp(-\frac{\pi \Delta E^2{\tau }_I}{2\hslash {\varepsilon }_A})$, obtained from the Landau-Zener probabilities, for $t=10\mu \mathrm{eV}$ and $B=0.5$ T. The thick solid blue curve is for $t=20\mu \mathrm{eV}$ and $B=1$ T.

Figure 6

Main panel: The solid red line is the return probability $P_S\left(2{\tau }_I\right)$ to the $\varepsilon (\tau =0)=200\mu \mathrm{eV}$ ground state after two linear detuning ramps of duration ${\tau }_I$: first to $\varepsilon \left({\tau }_I\right)=-200\mu \mathrm{eV}$ and then back to $\varepsilon \left(2{\tau }_I\right)=200\mu \mathrm{eV}$. We used $t=10\mu \mathrm{eV}$, and the geometry of Fig. 3 with $\phi =0,B=1$ T. The growing oscillations at large ${\tau }_I$ reflect the larger amplitude of the $|+\rangle$ state at the initialization time ${\tau }_I$. The dashed blue line shows $P_{-}\left({\tau }_I\right)$, reproduced from Fig. 5. The inset shows the return probability by introducing a waiting time ${\tau }_W$ between the two linear ramps. The thin blue, thick black, and dotted red curves correspond respectively to ${\tau }_I=1,2,4$ ns (see dots in the main panel). A smaller oscillation amplitude is obtained for ${\tau }_I=2$ ns.

Figure 7

Upper panels: the blue lower curves show $P_{+}\left({\tau }_G\right)$ as function of ${\tau }_W$ (i.e., the waiting time at ${\varepsilon }_B$ such that ${\tau }_G={\tau }_W+2{\tau }_R$; see Fig. 2). The left panel is for the setup in Fig. 3 with $\phi =5\pi /8$ and the right panel is for Fig. 3 with $\phi =0$. Lower panels: the blue lower curves show $F_H\left({\tau }_G\right)$ as defined in Eq. (24) for the same parameters as the upper panels [as above, left and right panels are for the setups of Figs. 3 and 3, respectively]. In all panels the upper red curves are $F\left({\tau }_G\right),t=25\mu \mathrm{eV},B=0.5$ T, ${\tau }_R=0.2$ ns, and the initial state is $|-\rangle$ at ${\varepsilon }_A=-200\mu \mathrm{eV}$.

Figure 8

Main panel: Maximum value of $P_{+}\left({\tau }_G\right)$ obtained with an optimum value ${\tau }_R^{*}$, the initial state $|-\rangle$ $\left({\varepsilon }_A=-200\mu \mathrm{eV}\right)$, and a waiting time ${\tau }_W={\tau }_{\pi }$ at ${\varepsilon }_B$; see Eq. (18). The plot illustrates the relation between the $\pi$-rotation fidelity $P_{+}\left({\tau }_G\right)$ and the corresponding total gate time ${\tau }_G={\tau }_{\pi }+2{\tau }_R^{*}$. The three curves are a guide for the eye through the numerical data (dots), computed for $B=0.5,1,1.5,...$ T from left to right. The dashed curve refers to the geometry of Fig. 3 with $\phi =5\pi /8$ and $t=10\mu \mathrm{eV}$. The other two curves are for Fig. 3 with $\phi =0$ and $t=10\mu \mathrm{eV}$ (solid) or $t=20\mu \mathrm{eV}$ (dot-dashed). The inset shows two examples on how to determine the optimum value ${\tau }_R^{*}$ from the maximum in $P_{+}\left({\tau }_G\right)$ as function of ${\tau }_R$. The slanting field is the same as in Fig. 3, $t=10\mu \mathrm{eV}$, and the thick (thin) curve is for $B=0.5\mathrm{T}$ $\left(1\mathrm{T}\right)$.

Figure 9

Plot of the fidelity $P_{+}\left({\tau }_G\right)={\left|\langle +|\psi \left({\tau }_G\right)\rangle \right|}^2$ for a rotation starting from $\left|\psi \right(0)\rangle =|-\rangle$ $\left({\varepsilon }_A=-200\mu \mathrm{eV}\right)$ as a function of the waiting time ${\tau }_W$ at ${\varepsilon }_B$. We used ${\tau }_R=2\mathrm{ns},t=10\mu \mathrm{eV}$, and the geometry of Fig. 3 with $B=1\mathrm{T}$ and $\phi =5\pi /8$. The thin solid line is obtained without nuclear dephasing, while the thick red curve includes nuclear fluctuations with $\sigma =2\mathrm{mT}$ (average over 400 runs). The thick black dashed line is Eq. (25) with ${\tau }_N$ given by Eq. (28).

Figure 10

Plot of the fidelity $P_{+}\left({\tau }_G\right)={\left|\langle +|\psi \left({\tau }_G\right)\rangle \right|}^2$ for a rotation of $|-\rangle$ $\left({\varepsilon }_A=-200\mu \mathrm{eV}\right)$, as function of the waiting time ${\tau }_W$ at ${\varepsilon }_B$. For both panels we assumed the slanting field of Fig. 3 with $\phi =0$. The solid curves are averages of 400 runs using ${\sigma }_{\varepsilon }=1\mu \mathrm{eV}$ for the thin blue curves and ${\sigma }_{\varepsilon }=3\mu \mathrm{eV}$ for the thick red curves. In the upper panel ${\tau }_R=2$ ns, $B=1$ T, and $t=10\mu \mathrm{eV}$, which give $\Delta E=1.4\mu \mathrm{eV}$. In the lower panel ${\tau }_R=0.1$ ns, $B=0.5$ T, and $t=20\mu \mathrm{eV}$, which give $\Delta E=2.8\mu \mathrm{eV}$. The dashed curves refer to the asymptotic expression Eq. (32). The ${\sigma }_{\varepsilon }=1\text{−}\mu \mathrm{eV}$ curve of the second panel has small decay and Eq. (32) is not plotted, as it is applicable only for ${\tau }_W\gg \hslash \Delta E/{{\sigma }_{\varepsilon }^{\prime }}^2\simeq 35$ ns [35].

Figure 11

Plot of the fidelity $P_{+}\left({\tau }_G\right)={\left|\langle +|\psi \left({\tau }_G\right)\rangle \right|}^2$ with the same parameters of the upper panel of Fig. 10, but including fluctuations in the tunnel amplitude $t$ instead of $\varepsilon$. The thick red (thin blue) solid curve is for ${\sigma }_t/t=5\%$ $\left({\sigma }_t/t=1\%\right)$. The dashed curve is the asymptotic formula for the ${\sigma }_t/t=5\%$ curve, calculated using Eq. (34).