Quantum master equations for the superconductor–quantum dot entangler

The operation of a source of entangled electron spins, based on a superconductor and two quantum dots in parallel, is described in detail with the help of quantum master equations. These are derived including the main parasitic processes in a fully consistent and nonperturbative way, starting from a microscopic Hamiltonian. The average current is calculated, including the contribution of entangled and nonentangled pairs. The constraints on the operation of the device are illustrated by a calculation of the various charge state probabilities.

Figure 1

The Entangler setup: a superconductor injects electrons in quantum dots $D_1$ and $D_2$, whose energies in state $\mid 1⟩$ (i.e., one excess electron) are, respectively, $E_1$ and $E_2$. Electrons in the dots can subsequently tunnel into the normal reservoirs $L,R$.

Figure 2

Sequence of states for the crossed Andreev channel of the entangler. For instance, $b_{ik,\sigma }$ denotes the amplitude to have an electron in dot $i$ while a quasiparticle is created in the superconductor. First an electron is deposited in either dot, next the second electron tunnels and forms a singlet state in the pair of dots, next either electron is absorbed in the reservoir, and finally the two dots are empty.

Figure 3

A current channel sending a pair of electrons to the same reservoir. Andreev process towards one quantum dot can happen against strong Coulomb repulsion $U$.

Figure 4

Sequence corresponding to the tunneling of a singlet pair through one branch of the device. States $\mid a⟩$ and $\mid c⟩$ are coupled through two successive virtual states.

Figure 5

Cotunneling between the two dots. An electron from dot 1 tunnels towards the dot 2 via a virtual intermediate state containing a quasiparticle. Two contributions participate to the cotunneling depending on when the initial electron is transferred.

Figure 6

General operation including the three Andreev channels and $S$ cotunneling. States with three or four electron states are omitted for clarity. Real states are fully squared while virtual states are dashed squared. To make it simpler, spin is not represented. The $\Omega$’s correspond to transitions between two quantum states $\mid S⟩\otimes \mid \mathrm{dots}⟩\otimes \mid l,r⟩$, while $T_C$ indicates the resonant cotunneling process. Certain mixing process, such as the direct Andreev effect between states $\mid c⟩$ and state with one electron in dot 1 and two electrons in dot 2, are not presented for lack of space. However, such processes are included in the quantum master equations.

Figure 7

Charge states populations as a function of $U$ for ${\Delta }_S=9.5\mathrm{K}$, $E_1=-E_2=0.5\mathrm{K}$, ${\Gamma }_{L,R}={\Gamma }_{L,R}^{\prime }=T=0.1\mathrm{K}$, ${\gamma }_A$, ${\gamma }_C\sim 0.2$. States $\mid a⟩$, $\mid b⟩$, $\mid c⟩$, $\mid d⟩$, $\mid e⟩$, $\mid f⟩$ refer to Fig. 6. State $\mid k⟩$ refers to the triplet state shared between dots, and states $\mid g⟩$ and $\mid h⟩$ refer to three electrons states (see Appendix ). $\mid g⟩$, $\mid h⟩$, and $\mid k⟩$ populations correspond to the three lowest curves. The population of states containing doubly occupied dots vanishes when $U$ increases. For low values of $U(U\sim \mid E_1-E_2\mid )$, the asymmetry is introduced by energy difference between states $\mid e⟩$ (two electrons in dot 1) and $\mid f⟩$ (two electrons in dot 2).

Figure 8

Ratio between populations of state $\mid b⟩$ (singlet state shared between dots 1 and 2) and of state $\mid e⟩$ population (two electrons in dot 1) for ${\Delta }_S=9.5\mathrm{K}$, $E_1=-E_2=0.5\mathrm{K}$, ${\Gamma }_{L,R}={\Gamma }_{L,R}^{\prime }=T=0.1\mathrm{K}$, ${\gamma }_A$, ${\gamma }_C\sim 0.2$. It indicates the ratio between direct Andreev channel and crossed Andreev channel. The latter is strongly favored when $U$ increases $[p_b∕p_e(U∕\Gamma =90)=83.3]$.

Figure 9

Ratio between populations of state $\mid h⟩$ (one electron in dot 1, two electrons in dot 2) and of state $\mid c⟩$ (one electron in dot 1) for ${\Delta }_S=9.5\mathrm{K}$, $E_1=-E_2=0.05\mathrm{K}$, ${\Gamma }_{L,R}={\Gamma }_{L,R}^{\prime }$, $T=0.1\mathrm{K}$, $U=1\mathrm{K}$, ${\gamma }_A$, ${\gamma }_C\sim 0.2$. Increasing $\Gamma$ compared to the transition rate of direct Andreev and crossed Andreev processes allows us to favor the decay of single charge states before another Cooper pair tunnels to the free quantum dot. For $U=1\mathrm{K}=10\mathrm{T}$, $p_h∕p_e<1.5\%$.