Nonlocal quantum state engineering with the Cooper pair splitter beyond the Coulomb blockade regime

A Cooper pair splitter consists of two quantum dots side-coupled to a conventional superconductor. Usually, the quantum dots are assumed to have a large charging energy compared to the superconducting gap, in order to suppress processes other than the coherent splitting of Cooper pairs. In this work, in contrast, we investigate the limit in which the charging energy is smaller than the superconducting gap. This allows us, in particular, to study the effect of a Zeeman field comparable to the charging energy. We find analytically that in this parameter regime the superconductor mediates an interdot tunneling term with a spin symmetry determined by the Zeeman field. Together with electrostatically tunable quantum dots, we show that this makes it possible to engineer a spin triplet state shared between the quantum dots. Compared to previous works, we thus extend the capabilities of the Cooper pair splitter to create entangled nonlocal electron pairs.

Figure 1

Schematics of the Cooper pair splitter system. Such a system can, for example, be realized by deposition of a superconductor [wide (purple) structure on the top] on top of a patterned 2DEG at the interface of a semiconductor heterostructure. The 2DEG is electrostatically depleted underneath the superconductor and underneath the gates defining the quantum dots (yellow structures). Further gates (elongated thin gray structures) can be used to electrostatically control the potential of the quantum dots.

Figure 2

Schematics of the dominant processes for generating the nonlocal triplet state $|T\rangle =({|+\rangle }_L{|-\rangle }_R+{|-\rangle }_L{|+\rangle }_R)/\sqrt{2}$. Panel (a) shows the initial state right after switching the potential of the right QD to ${\mu }_R=-U/2$. Both QDs are unoccupied and electrons form CPs in the superconductor. A magnetic field lifts the spin degeneracy of the QD levels by ${\mathrm{\Delta }}_Z$. Panel (b) shows the state where a CP has been transferred to the right QD at time $T_1$. The rate for this process is given by ${\mathrm{\Gamma }}_P$, Eq. (13). Panel (c) shows the state right after time $T_1$ when the potential of the right QD has been switched to $-U$. Taking into account the chemical potential and the charging energy, we see that now the energy levels are shifted by $-U$. Panel (d) shows the state after an electron from the right QD (here the down-spin electron) has been transferred to the left QD at time $T_1+T_2$. This process is driven by the interdot tunneling term. If $U\ll {\mathrm{\Delta }}_Z/2$, the latter is dominated by the spin antisymmetric term with rate ${\mathrm{\Gamma }}_{-}$, Eq. (16), as compared with the spin symmetric term with rate ${\mathrm{\Gamma }}_{+}$, Eq. (15). The same processes but with the spin states interchanged are equally likely and their amplitudes add coherently resulting in the generation of a triplet state.

Figure 3

Dynamics of the triplet state generation for $U/\mathrm{\Delta }=0.12,{\mathrm{\Delta }}_Z/\mathrm{\Delta }=0.76$, and ${\mathrm{\Gamma }}_0/\mathrm{\Delta }=0.004$. The two shaded areas of the graph correspond to the two stages of the protocol described in the text and are separated by the switching of the potential of the right QD from $-U/2$ to $-U$ (see also Fig. 2). In the first stage, population is transferred from the vacuum [solid (blue) line] to the doubly occupied state of the right QD [dotted (black) line]. In the second stage, population is transferred from the doubly occupied state to the nonlocal triplet [solid thick (red) line]. A maximal triplet fidelity of $97\%$ is reached at time $T_1+T_2$. Note also the presence of a small oscillatory population of the nonlocal singlet state [dashed (green) line].

Figure 4

The contour plot shows the fidelity of the triplet state generation as a function of onsite interaction strength $U$ and Zeeman field ${\mathrm{\Delta }}_Z$ for ${\mathrm{\Gamma }}_0/\mathrm{\Delta }=0.004$. The (black) cross indicates the parameters used in the plot of Fig. 3. The solid (red) line in the upper panel shows the fidelity for a fixed value of ${\mathrm{\Delta }}_Z=0.56$ indicated by the solid (red) line in the contour plot. In the regime where ${\mathrm{\Gamma }}_0\ll U\ll {\mathrm{\Delta }}_Z\ll \mathrm{\Delta }$, the triplet fidelity is well approximated by the analytic expression ${\mathcal{F}}_T\approx 1-{(U/{\mathrm{\Delta }}_Z)}^2-8{({\mathrm{\Gamma }}_0/U)}^2$ [dotted (red) line]. Weak oscillations of the fidelity as a function of $U$ are clearly visible.

Figure 5

Analytical prediction for the dynamics of the triplet state generation for $U/\mathrm{\Delta }=0.12,{\mathrm{\Delta }}_Z/\mathrm{\Delta }=0.76$, and ${\mathrm{\Gamma }}_0/\mathrm{\Delta }=0.004$. This result agrees qualitatively with the full numerics shown in Fig. 3.