Coherent spin rotations in open driven double quantum dots

We analyze coherent spin rotations in a dc biased double quantum dot driven by crossed dc and ac magnetic fields. In this configuration, spatial delocalization due to interdot tunneling competes with intradot spin rotations induced by the time dependent magnetic field, giving rise to a complicated time dependent behavior of the tunneling current. When the Zeeman splitting has the same value in both dots and spin flip is negligible, the electrons remain in the triplet subspace performing coherent spin rotations and current does not flow. This electronic trapping is removed either by finite spin relaxation or when the Zeeman splitting is different in each quantum dot. In the last case, we will show that by applying a resonant bichromatic magnetic field, the electrons become trapped in a coherent superposition of states and electronic transport is blocked. Then, manipulating ac magnetic fields allows one to drive electrons to perform coherent spin rotations which can be unambiguously detected by direct measurement of the tunneling current.

Figure 1

Schematic diagram of the DQD in the presence of crossed dc and ac magnetic fields.

Figure 2

(a) $I\left(t\right)$ for initial state $\mid \uparrow ,\downarrow ⟩$ in the absence of spin relaxation for ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}=\Delta$ and ${\Omega }_{\mathrm{ac}}={\Omega }_{\mathrm{T}}∕2$. (b) The corresponding occupation probabilities: $\mid \uparrow ,\downarrow ⟩$ (solid line), $\mid \downarrow ,\uparrow ⟩$ (dash-dotted line), $\mid 0,\uparrow \downarrow ⟩$ (dotted line), and $\mid \uparrow ,\uparrow ⟩$ and $\mid \downarrow ,\downarrow ⟩$ (dashed line). Parameters $(e=\hslash =1)$: ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma ={10}^{-3}\mathrm{meV}$, $T_{1\left(2\right)}^{-1}=0$, ${\Omega }_{\mathrm{T}}=11.2\mathrm{GHz}$ and holding for the rest of the plots (in meV): ${\epsilon }_{\mathrm{L}}=1.5$, ${\epsilon }_{\mathrm{R}}=0.45$, $\Delta =0.026$ $(B_{\mathrm{dc}}\sim 1T)$, $U_{\mathrm{L}}=1$, $U_{\mathrm{R}}=1.45$, $V=0.4$, ${\mu }_{\mathrm{L}}=2$, and ${\mu }_{\mathrm{R}}=1.1$.

Figure 3

Effect of (a) finite spin relaxation rates, $T_1^{-1}$ and (b) the Zeeman inhomogenity, ${\Delta }_{\mathrm{L}}∕{\Delta }_{\mathrm{R}}$, on the stationary current when tuning the frequency of the magnetic field. In (a), ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}$; in (b), $T_1=0$. (Same parameters as in Fig. 2 but $\Gamma ={10}^{-2}\mathrm{meV}$).

Figure 4

(a) $I\left(t\right)$ for different spin-flip times (in $\mathrm{\mu }\mathrm{s}$), with ${\Omega }_{\mathrm{ac}}={\Omega }_{\mathrm{T}}=11.2\mathrm{GHz}$ and ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}=\Delta$. The initial state here is $\mid \uparrow ,\uparrow ⟩$, then, for $T_1^{-1}=0$, there is no mixing of the triplet and singlet subspaces and therefore no current flows through the system. Spin relaxation processes contribute to populate the singlet, producing a finite current. (b) Stationary current as a function of spin relaxation time. For long $T_1$, electrons remain in the dark space. As $T_1$ decreases, $I$ begins to flow, being again suppressed for short enough $T_1$, as discussed in the text. (c) $I\left(t\right)$ for different ratios between the ac field intensity and the interdot hopping, i.e., between ${\Omega }_{\mathrm{ac}}$ and ${\Omega }_{\mathrm{T}}$, with $T_1\sim 0.1\mathrm{\mu }\mathrm{s}$. (Same parameters as in Fig. 3.)

Figure 5

(a) Dependence of the stationary current on the dc field inhomogeneity ${\Delta }_{\mathrm{L}}-{\Delta }_{\mathrm{R}}$ for different relaxation times (in $\mathrm{\mu }\mathrm{s}$). The quenching of the current for ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}$ is lifted by spin relaxation. (b) $I\left(t\right)$ for different values of ${\Delta }_{\mathrm{L}}∕{\Delta }_{\mathrm{R}}$ when the electron in the right QD is kept in resonance, in the absence of relaxation. A crossover to the one electron spin resonance is observed by increasing the difference between ${\Delta }_{\mathrm{L}}$ and ${\Delta }_{\mathrm{R}}=\Delta$. (c) Dependence of the current oscillations on $B_{\mathrm{ac}}$ for ${\Delta }_{\mathrm{L}}=0.99{\Delta }_{\mathrm{R}}$ and $T_1^{-1}\sim 0.1\mathrm{\mu }\mathrm{s}$. Same parameters as in Fig. 4.

Figure 6

(a) Transient current in the presence of a bichromatic $B$, when ${\omega }_{1\left(2\right)}={\Delta }_{\mathrm{L}\left(\mathrm{R}\right)}$ for ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}∕2$ (solid) and ${\Delta }_{\mathrm{L}}=0.9{\Delta }_{\mathrm{R}}$ (dotted) and $T_1^{-1}$=0. Left inset: detail of the current oscillation. In the case where ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}∕2$, $I$ oscillates with ${\Omega }_{\mathrm{T}}$ and it presents faster oscillations overimposed (more important for ${\Delta }_{\mathrm{L}}=0.9{\Delta }_{\mathrm{R}}$) coming from the effect of each frequency on its off-resonance electron. Right inset: occupation probabilities for ${\Delta }_{\mathrm{L}}={\Delta }_{\mathrm{R}}∕2$: $\mid \uparrow ,\uparrow ⟩$ (solid line), $\mid \downarrow ,\downarrow ⟩$ (dotted line), $\mid \uparrow ,\downarrow ⟩\sim \mid \downarrow ,\uparrow ⟩$ (dashed line), and $\mid 0,\uparrow \downarrow ⟩$ (dash-dotted line, remaining very close to zero). The occupation of $\mid 0,\uparrow \downarrow ⟩$ drops to zero, and therefore $I$ drops as well. At long times the electrons fall in a coherent superposition of $\mid \uparrow ,\uparrow ⟩$ and $\mid \downarrow ,\downarrow ⟩$. (b) Stationary current as a function of ${\omega }_1$ when ${\omega }_2={\Delta }_{\mathrm{R}}$, for different relaxation rates, $T_1^{-1}$. $I$ drops at ${\omega }_1$ =${\Delta }_{\mathrm{L}}$. (${\Delta }_{\mathrm{R}}=\Delta$, $\Gamma ={10}^{-3}\mathrm{meV}$, and ${\Omega }_{\mathrm{T}}=1.12\mathrm{GHz}$.)