Parity measurement of one- and two-electron double well systems

We outline a scheme to accomplish measurements of a solid state double well system (DWS) with both one and two electrons in nonlocalized bases. We show that, for a single particle, measuring the local charge distribution at the midpoint of a DWS using a SET as a sensitive electrometer amounts to performing a projective measurement in the parity (symmetric/antisymmetric) eigenbasis. For two-electrons in a DWS, a similar configuration of SET results in close-to-projective measurement in the singlet/triplet basis. We analyze the sensitivity of the scheme to asymmetry in the SET position for some experimentally relevant parameter, and show that it is experimentally realizable.

Figure 1

Schematic of the physical system under consideration.

Figure 2

SET island energies relative to the lead Fermi level, depending on the state of DWS.

Figure 3

Geometry of an asymmetric system.

Figure 4

(a) Projection of the state of the DWS onto the singlet state, $\mid ⟨S\mid \psi \left(t_j\right)⟩\mid$, immediately after jump $j$ for $\delta =\Delta ∕2=10{\gamma }^{\prime }=U∕10$. (b) Histogram of waiting times between jumps with $n_s=0$ and (c) with $n_s=1$, for a simulation with ${10}^4$ jumps.

Figure 5

(a) Transition energies, ${\omega }_m$, relative to lead Fermi levels, for a symmetrically placed SET island, where ${\omega }_1=-{\omega }_5=-U$, ${\omega }_2=-{\omega }_4=-({\delta }^2+2\delta \Delta )∕U$ and ${\omega }_{3,6,7,8}=0$, and (b) for an asymmetrically placed SET island, where ${\omega }_1=-{\omega }_5=-U$, ${\omega }_2=-({\delta }^2+2\delta \Delta )∕U$, ${\omega }_3=-{\omega }_4=-2ϵ$ and ${\omega }_{6,7,8}=0$.