Precession of entangled spin and pseudospin in double quantum dots

Quantum dot spin valves are characterized by exchange fields which induce spin precession and generate current spin resonances even in the absence of spin splitting. Analogous effects have been studied in double quantum dots, in which the orbital degree of freedom, the pseudospin, replaces the spin in the valve configuration. We generalize, now, this setup to allow for arbitrary spin and orbital polarization of the leads, thus obtaining an even richer variety of current resonances, stemming from the precession dynamics of entangled spin and pseudospin. We observe for both vectors a delicate interplay of decoherence, pumping, and precession which can only be understood by also considering the dynamics of the spin-pseudospin correlators. The numerical results are obtained in the framework of a generalized master equation within the cotunneling approximation and are complemented by the analytics of a coherent sequential tunneling model.

Figure 1

Energy splitting of the two-particle states: the spin-triplet, pseudospin-singlet states are depicted in blue ($S_z=0,\pm 1$ and $T=0$). The pseudospin anisotropy splits the pseudospin-triplet, spin-singlet states (highlighted in orange; $T_z=0,\pm 1$ and $S=0$) energetically.

Figure 2

Interplay of spin and pseudospin channels determines the transport through the system: in the spin space, the polarization vectors of the leads are almost antiparallel ($\varphi \approx \pi$), which translates into a spin valve configuration. Through pseudoexchange fields (purple) one can rotate the spin of the system and thus lift the current suppression of the spin valve. In the pseudospin space, we consider parallel polarization of the leads. Since there is a preferential plane in the pseudospin, indicated by the red lines, one defines the polarization direction of the leads in respect to this plane ($\theta$). The pseudospin of the system can precess under the influence of the pseudoexchange field ${\vec{B}}_T$.

Figure 3

Current plot of a DQD in a $V_{\text{g}}\text{−}{V}_{\text{b}}$ map shows an intricate set of current resonances: the $N=1,2,3$-Coulomb diamonds are decorated with resonances which cut deep into the Coulomb blockade regions. In the central $N=2$ diamond a simple ground-state-to-ground-state transition is appearing. The parameters are the following: $U=2V, k_{\text{B}}T=0.05V, P_T=0.6, P_S=0.99, \theta =1.5, \varphi =0.95\pi , {\mathrm{\Gamma }}_0^{\text{L}}=1\times {10}^{-2}V=2{\mathrm{\Gamma }}_0^{\text{R}}$, and ${\varepsilon }_0=-2V$.

Figure 4

Current resonances modulated by the pseudospin polarization angle $\theta$ and strength $P_T$: (a) $\theta \text{−}{V}_{\text{b}}$ map and (b) $P_T\text{−}{V}_{\text{b}}$ map, both at $V_{\text{g}}=1.5V$, exhibit a strong dependence on the respective parameters. The white dashed lines indicate the parameter set of Fig. 3. The black and magenta dashed lines are the resonance conditions for the ${\vec{S}}_{-}$ and ${\vec{S}}_{+}$ channel, which are explicitly discussed in Sec. 6.

Figure 5

Current in the one-particle Coulomb diamond calculated with two different approaches: in panel (a) the full cotunneling calculation is presented. Panel (b) shows the corresponding result in the coherent sequential tunneling limit. Both currents are renormalized by the current $I_0$ expected for a spin valve in the high bias limit. The white dashed line should help for the comparison with Fig. 4.

Figure 6

Pseudospin depends strongly on the gate and bias voltages: the components (a) ${\vec{\mathrm{e}}}_y$, (b) ${\vec{\mathrm{e}}}_T$, (c) ${\vec{\mathrm{e}}}_{\perp }$ of $\vec{T}$ underline the vector character of the pseudospin. The (anti)alignment of the pseudospin (decreases) increases the current flow through the DQD. Focusing on the upper right corner, one observes a clear rotation of $\vec{T}$ in the $y$ direction. Same parameters as in Fig. 3.

Figure 7

Comparison of stationary spin variables ${\vec{S}}_{\pm }$: panels (a) and (c) are obtained from the full cotunneling calculation, while (b) and (d) refer to the reduced sequential tunneling model. Same parameters as in Fig. 3.

Figure 8

Spin dephasing as the main mechanism behind spin resonances: (a) the limiting case $\theta =\pi /2$ shows a splitting into the $\pm$ channels. The default situation is that the electrons occupy $P_{+}$ (red area) since the pumping is polarized in that direction. $P_{-}$ is prevailing on the resonance condition for the “$+$” channel (white dashed line) since there the $+$ electrons can leave the spin valve blockade and thus only “$-$” electrons remain. (b) On the resonance, the spin coherence decreases faster than the respective population. (c) Clear resonance condition for the $-$ channel is outside of this $V_{\text{g}}\text{−}{V}_{\text{b}}$ window; thus the coherence of this channel is maintained and yields a blockade. Parameters of Fig. 3 except for $\theta =\pi /2$.

Figure 9

Concurrence $C$ in dependence of gate and bias voltage: remarkably, only in a limited area entanglement between spin and pseudospin can be observed. These features are determined by the correlator vectors ${\vec{\mathrm{\Lambda }}}_y$ and ${\vec{\mathrm{\Lambda }}}_{\perp }$. Same parameters as in Fig. 3.