Weak Values and the Leggett-Garg Inequality in Solid-State Qubits

An implementation of weak values is investigated in solid-state qubits. We demonstrate that a weak value can be nonclassical if and only if a Leggett-Garg inequality can also be violated. Generalized weak values are described in which post-selection occurs on a range of weak measurement results. Imposing classical weak values permits the derivation of Leggett-Garg inequalities for bounded operators. Our analysis is presented in terms of kicked quantum nondemolition measurements on a quantum double-dot charge qubit.

Figure 1

Post-selected weak value ${}_1⟨{\mathcal{I}}_1{⟩}_{\psi }$ starting from an initial state $|\psi ⟩=(i|1⟩+|2⟩)/\sqrt{2}$ as a function of the unitary rotation angle between the measurements. SWVs exceed the classical bound on the system signal, shown with horizontal lines at $\mathcal{I}=\pm 1$. The weak value measurement is implemented by averaging the QPC detector current (${\mathcal{I}}_1$) over a subensemble where the later QPC current gives a particular value (here ${\mathcal{I}}_2=+1$). The different curves show how the weak value changes for different measurement strengths.

Figure 2

Post-selected weak value ${}_{[0,\infty )}⟨{\mathcal{I}}_1{⟩}_{\psi }^{\max }$ starting from an initial state $|\psi ⟩=(i|1⟩+|2⟩)/\sqrt{2}$ as a function of the strength ($S_1$ and $S_2$) of both measurements. SWVs exceed the classical bound on the system signal and are outside the black region. Weak values are labeled at the contour lines.