Two-dimensional quantum rings with oscillating spin-orbit interaction strength: A wave function picture

We determine the relevant spinor valued wave functions for a two-dimensional quantum ring in the presence of Rashba-type spin-orbit interaction (SOI). The case of constant SOI strength is considered first, then we investigate the physical consequences of time-dependent (oscillating) SOI strength. Floquet’s method is applied to find time-dependent eigenspinors, and it is shown that the Floquet quasienergies and thus their differences (the generalized Rabi frequencies) are determined by the radial boundary conditions. Time evolution of various initial states is calculated.

Figure 1

Two-dimensional quantum ring and the relevant coordinates.

Figure 2

Level scheme for two-dimensional rings with different parameters. (a) $r_0/r_1=0.6,\omega /\Omega =0.1$; (b) $r_0/r_1=0.6,\omega /\Omega =4.0$; (c) $r_0/r_1=0.8,\omega /\Omega =0.1$, and (d) $r_0/r_1=0.8,\omega /\Omega =4.0$. Energy is measured in units of $\hslash \Omega =\frac{{\hslash }^2}{2m^{\ast }{r}_1^2}$. The vertical line in panel (b) corresponds to $\kappa =9/2$, which is the value for which the wave functions are plotted in Fig. 3.

Figure 3

(Color online) Wave functions for $\kappa =9/2,r_0/r_1=0.6,\omega /\Omega =4.0$ [See the vertical line in Fig. 2b]. Probability densities in Eq. (18)—that do not depend on $\phi$ in the current cases—are shown in panel (a) while the position-dependent expectation value of $S_x,S_y$ and $S_z$ for the states indicated by the arrows can be seen in panel (b).

Figure 4

Floquet quasienergies $k^2$ (measured in units of $\hslash \Omega$) for a ring with $r_0/r_1=0.6$.

Figure 5

(Color online) Spin oscillations in a ring with $r_0/r_1=0.6,\nu =0.01$ as a function of time and coordinates $r$ and $\phi$. The amplitude of the SOI oscillations is $A=0.1$ for panel (a), where $⟨S_y⟩\left(\tau \right)$ is shown, and $A=3.0$ for panel (b), where we can see the time evolution of $⟨S_x⟩\left(\tau \right)$. The initial state is $\left({\Phi }_{1,15/2}^{+}+{\Phi }_{1,15/2}^{-}\right)/\sqrt{2}$ in both cases and rapid oscillations due to the difference of the quasienergies have been removed by averaging in time domain. Panel (c) corresponds to the same parameters and expectation value as (a) but it shows a circular section. (See the schematic rings on the right hand side that illustrate what slices are plotted in the different panels.)

Figure 6

Normalized magnitudes of the frequency amplitudes in the Fourier series expansion of the probability density corresponding to $\left({\Phi }_{1,15/2}^{+}+{\Phi }_{2,15/2}^{+}\right)/\sqrt{2}$ at $r=0.75,\phi =0$. Note that frequency is measured in units $1/\hslash \Omega$.

Figure 7

(Color online) Collapse and revival phenomenon in a ring with constant SOI ($\omega /\Omega =3.0,r_0/r=0.6$.) The initial state is a Gaussian wave packet, being multiplied by $\text{sin}\frac{\pi \left(r-r_0\right)}{r_1-r_0}$ to satisfy the boundary conditions.