Charge localization and dynamical spin locking in double quantum dots driven by ac magnetic fields

We investigate electron localization and dynamical spin locking induced by ac magnetic fields in double quantum dots. We demonstrate that by tuning the ac magnetic fields parameters, i.e., the field intensity, frequency, and the phase difference between the fields within each dot, coherent destruction of tunneling (and thus charge localization) can be achieved. We show that in contrast with ac electric fields, ac magnetic fields are able to induce spin locking, i.e., to freeze the electronic spin, at certain field parameters. We show how the symmetry of the Hamiltonian determines the quasienergy spectrum which presents degeneracies at certain field parameters, and how it is reflected in the charge and spin dynamics.

Figure 1

Double quantum dot with crossed ac and dc magnetic fields: $\mathbf{B}\left(t\right)=[B_{\mathrm{ac}}\left(t\right),0,B_z]$. The Zeeman splitting in the left and right dot is given by $B_{z,L}$ and $B_{z,R}$, respectively. The parameter $\tau$ describes the tunneling between the dots.

Figure 2

Top: Quasienergies vs $B_{\mathrm{ac}}/\omega$ for $\phi =\pi$. Note the crossings of all the quasienergies due to the existence of two generalized parity symmetries ${\mathbb{Z}}_2$ (GP and GSP) for $J_0\left(B_{\mathrm{ac}}/\omega \right)=0$. (bottom) Occupation probability vs time for the initial state $|{\Psi }_{x,+}^L\rangle =\left({|\uparrow \rangle }_L+{|\downarrow \rangle }_L\right)/\sqrt{2}$, at the first crossing of the quasienergies ($B_{\mathrm{ac}}/\omega \simeq 2.404$). The probability is almost constant in time, and the electron remains in a well-defined spin projection and is spatially localized in the left dot. In this case, both CDT (spatial localization) and dynamical spin locking occurs at the first zero of $J_0\left(B_{\mathrm{ac}}/\omega \right)$. Parameters: $B_{z,L}=B_{z,R}=0.7$, $\omega =8$, and $\tau =0.1$ in units ${\mu }_B=\hslash =1$. These parameters correspond to a $\omega =16\mu \mathit{eV}$, $B_z=1.4\mu \mathit{eV}$, and $\tau =0.2\mu \mathit{eV}$.

Figure 3

Quasienergies vs $B_{\mathrm{ac}}/\omega$ for $\phi =0$. The GP symmetry is broken. Crossings between quasienergies with opposite GSP appear, but the quasienergies with opposite parity do not cross due to a $2\tau$ splitting. Therefore, at the first zero of $J_0$, CDT does not happen, and only spin locking is achieved. $|{\Psi }_{x,+}^L\rangle$ is the initial state, being $|{\Psi }_{x,\pm }^{i=L,R}\rangle =\left({|\uparrow \rangle }_i\pm {|\downarrow \rangle }_i\right)/\sqrt{2}$. In this case, the electron performs Rabi oscillations between $|{\Psi }_{x,+}^L\rangle$ and $|{\Psi }_{x,+}^R\rangle$, remaining in the initial spin projection. Parameters: $B_{z,L}=B_{z,R}=0.7$, $\omega =8$, and $\tau =1/10$.

Figure 4

$P_{\min }^k$ vs the difference of phase for the first zero of $J_0(B_{\mathrm{ac}}/\omega )$, i.e., $B_{\mathrm{ac}}/\omega \simeq 2.404$. The dashed line shows the initial state freezing ($P_{\min }^1$, red/gray), the dotted line shows the spatial localization ($P_{\min }^2$, blue/dark gray), while the continuous line shows the spin locking ($P_{\min }^3$, brown/light gray). In this case $P_{\min }^1$ and $P_{\min }^2$ are coincident. As the phase $\phi$ moves away from $\pi$, $P_{\min }^{k=1,2}$ decreases, reducing and even destroying CDT. Spin localization ($P_{\min }^3$) holds for all $\phi$ (the state remains with a well-defined spin projection, but is oscillating within the dots), because, although GP symmetry is broken as $\phi$ varies from $\pi$, GSP remains. Parameters considered: $\omega =8$, $B_{z,L}=B_{z,R}=0.7$, and $\tau =1/10$.

Figure 5

Quasienergies vs $B_{\mathrm{ac}}/\omega$ for $\phi =3\pi /7$. The avoided crossings arise from the lack of GP symmetry produced by the difference of phase $\phi \ne \pi$. The right-hand figure shows a zoom of the framed area. Long-time spatial localization is obtained in such avoided crossings. At $B_{\mathrm{ac}}\simeq 5.5$ the crossing of quasienergies gives rise to dynamically induced spin locking. Also for $B_{\mathrm{ac}}\simeq 8.8$ a nearby degeneracy between all four quasienergies occurs, giving rise to charge and spin locking. This is in good agreement with Eq. (4) for $n=3$, where spatial and spin localizations occur. $B_{z,L}=B_{z,R}=0.7$, and $\tau =1/10$.