Application of the Landau-Zener-Stückelberg-Majorana dynamics to the electrically driven flip of a hole spin

An idea of employing the Landau-Zener-Stückelberg-Majorana dynamics to flip a spin of a single ground state hole is introduced and explored by a time-dependent simulation. This configuration interaction study considers a hole confined in a quantum molecule formed in an InSb $\langle 111\rangle$ quantum wire by application of an electrostatic potential. An up-down spin-mixing avoided crossing is formed by nonaxial terms in the Kohn-Luttinger Hamiltonian and the Dresselhaus spin-orbit one. Manipulation of the system is possible by the dynamic change of an external vertical electric field, which enables the consecutive driving of the hole through two anticrossings. Moreover, a simple model of the power-law-type noise that impedes precise electric control of the system is included in the form of random telegraph noise to estimate the limitations of the working conditions. We show that in principle the process is possible, but it requires precise control of the parameters of the driving impulse.

Figure 1

An illustrative scheme of the transfer process idea. The spin-dependent energy-level structure is shown with the desired spin-flipping process utilizing two avoided crossings.

Figure 2

(a) Solid line, left axis: The confinement potential along the $z$ axis. Dashed line, right axis: The $zs\left(z\right)$ shape function of the electric field $F_z$. The arrows correspond to the “sewing” points in Eq. (10). (b) Shape of the dots. The $z$ boundary is shown for $V_z\left(z\right)=\frac{V_0}{2}$.

Figure 3

(a) Hole energy spectrum of the axially symmetric system. (b) Hole energy spectrum of the system with non-axially-symmetric terms included and the scheme of the hole transfer. Arrows indicate the successive steps of the process. $F_A$ is the field range used for the Landau-Zener transfer in the $\beta$ step. The dashed $\zeta$ arrow denotes an alternative EDSR approach (see Sec. 4c). The inset is a fragment of the spectrum in magnification.

Figure 4

The hole one-band densities integrated over the $\varphi$ coordinate, for $F_z=4$ kV/m: (a) $P_{1,k}(\rho ,z)$ for the ground state, and (b) $P_{2,k}(\rho ,z)$ for the first excited level.

Figure 5

The evolution process for the complete transition. Upper part, right axis: The driving field $F_z\left(t\right)$ function. Lower part, left axis: The ${\left|d_j\left(t\right)\right|}^2$ projections for the time-dependent state $\mathrm{\Phi }(\vec{r},t)$ on the $j\mathrm{th}$ time-independent state ${\mathrm{\Lambda }}_j\left(\vec{r}\right)$, respectively. For $j=4$ the projection is not shown as it is negligible at every moment of the evolution.

Figure 6

The evolution process for the EDSR of uncoupled levels. The ${\left|d_j\left(t\right)\right|}^2$ projections for the time-dependent state $\mathrm{\Phi }(\vec{r},t)$ on the $j\mathrm{th}$ time-independent state ${\mathrm{\Lambda }}_j\left(\vec{r}\right)$, respectively. For $j\in \{3,4\}$ the projections are not shown as they are negligible at every moment of the evolution.

Figure 7

The final ${\left|d_2\right|}^2$ projection for the RTN $F_z$ noise simulation (the fast-variable noise regime). $F_N$ is the amplitude of the noise and $f_c$ is the characteristic frequency of the noise. The square in the upper left-hand corner marks the region where an analog of the motionary narrowing occurs (see Appendix pp6).

Figure 8

The final ${\left|d_2\right|}^2$ projection for the static $F_z$ offset (the slowly variable noise regime) (solid curve). The results for the maximal $f_c$ considered in the RTN model are shown for comparison. The points are simulation data and the dotted line is a $1/\left(1+c{F}_N^2\right)$ function fitted to the data.

Figure 9

Evolution for $\beta$ transition; $p_s=1.16$ kV/m, $p_b=-1.98$ kV/m, and $T_P=295$ ps. Left axis: $|d_j{\left(t\right)|}^2$ projections for the time-dependent state $\mathrm{\Phi }(\vec{r},t)$ on the time-independent states ${\mathrm{\Lambda }}_j\left(\vec{r}\right)$. For $j=4$ the projection is not shown as it is negligible for all $t\in \left(0,T_P\right)$. Right axis: $F_z\left(t\right)$ electric field drive.

Figure 10

Evolution for $\delta$ transition; $p_s=1.23$ kV/m, $p_f=2.56$ kV/m, and $T_P=240$ ps. Left axis: $|d_j{\left(t\right)|}^2$ projections for the time-dependent state $\mathrm{\Phi }(\vec{r},t)$ on the time-independent states ${\mathrm{\Lambda }}_j\left(\vec{r}\right)$. For $j\in \{1,4\}$ the projections are not shown as they are negligible for all $t\in \left(0,T_P\right)$. Right axis: $F_z\left(t\right)$ electric field drive.

Figure 11

Optimization of the $T_P$ parameter for $\alpha , \gamma$, and $\epsilon$ transitions. The ${\left|d_j\left(T_P\right)\right|}^2$ projection for the time-dependent state $\mathrm{\Phi }(\vec{r},t)$ on the $j\mathrm{th}$ time-independent state ${\mathrm{\Lambda }}_j\left(\vec{r}\right)$ as a function of evolution time $T_P$, where $j=1$ for $\alpha$ and $j=2$ for $\gamma$ and $\epsilon$.

Figure 12

(a) The confinement potential along the $z$ axis with no asymmetry (solid line), $1\%$ asymmetry (dashed line), and $5\%$ asymmetry (dotted line). (b), (c) (Bottom $F_z$ scale, left energy scale, solid line) Hole energy spectrum of the system with non-axially-symmetric terms included with (b) $1\%$ asymmetry, (c) $5\%$ asymmetry; (top $F_z$ scale, right energy scale, dotted line) hole energy spectrum of the system with non-axially-symmetric terms included without the $z$-axis asymmetry, for comparison.

Figure 13

The loss in final ${\left|d_2\right|}^2$ projection for the RTN noise simulation as a function of the $\frac{F_N^2}{f_c}$ argument. The line is a linear fit to the data.