Spin qubits in double quantum dots: Entanglement versus the Kondo effect

We investigate the competition between pair entanglement of two spin qubits in double quantum dots attached to leads with various topologies and the separate entanglement of each spin with nearby electrodes. Universal behavior of entanglement is demonstrated in dependence on the mutual interactions between the spin qubits, the coupling to their environment, temperature, and magnetic field. As a consequence of quantum phase transition an abrupt switch between fully entangled and unentangled states takes place when the dots are coupled in parallel.

Figure 1

(Color online) (a) Double-quantum-dot system. $t$ and $t_n$ are the matrix elements for tunneling between dots A and B and from the dots to the electrodes, respectively. (b) Entanglement between two spins on the quantum dots as a function of the interdot exchange coupling $J$: below $J_c$, the two spins are not entangled and the DQD is in some type of the Kondo regime. For elevated temperatures and magnetic field above $J_c$, the entanglement is zero if $J\lesssim \max (T,B)$.

Figure 2

(Color online) Top panels: spin-spin correlation function ${\mathbf{S}}_{\mathrm{A}}\bullet {\mathbf{S}}_{\mathrm{B}}$, charge fluctuations on one quantum dot, $\Delta n_{\mathrm{A}}^2$, and probabilities $P_{\uparrow \downarrow }$, $P_{\Vert }$ for $T⪡T_K$, $B\to 0$ and with $t^{\prime }=t_0∕\sqrt{20}$. Bottom left panel: concurrence $C$, corresponding to serially coupled dots, for a range of interactions $U∕\Gamma =40,32,24,16,8,4$ and calculated with both the NRG and GS methods (bullets), yielding the same $J_{1c}$, but due to the limited span of variational basis the GS method progressively overestimates $C$ for $U∕\Gamma \gtrsim 20$ regime. Bottom right two panels: the results for side-coupled and parallel configurations obtained from the NRG method. Temperatures range from the scale of the Coulomb repulsion parameter $U$, $T∕U=0.4$, to temperatures below the Kondo scale $T_K$; each consecutive curve corresponds to a temperature lowered by a factor of 4.

Figure 3

(Color online) In the phase diagram $(U∕\Gamma ,J∕T_K)$ a solid line separates $C>0$ and $C=0$ regions (line shaded), together with dotted line indicating $⟨{\mathbf{S}}_{\mathrm{A}}\bullet {\mathbf{S}}_{\mathrm{B}}⟩=-1∕4$ ($T\to 0$ and $B=0$). Both lines merge for $U∕\Gamma \gtrsim 12$, where charge fluctuations $\Delta n_{\mathrm{A}}^2$ progressively become negligible (contour plot).