Tomography of Many-Body Weak Values: Mach-Zehnder Interferometry

We propose and study a weak value (WV) protocol in the context of a solid state setup, specifically, an electronic Mach-Zehnder interferometer. This is the first specific proposal to measure both the real and imaginary part (i.e., complete tomography) of a WV. We also analyze the manifestation of many-body physics in the WV to be measured, including finite temperature and shot-noise-like contributions.

Figure 1

(a) Electronic MZI represented by edge states (full lines) in a Hall bar. Interedge tunneling (dashed lines) takes place at the quantum point contacts. (b) Scheme of a double electronic MZI. Electrons on edge states 2 and 3 interact via Coulomb interaction.

Figure 2

Real and imaginary part of the many-body WV as a function of the Aharanov-Bohm phase. Panels (a), (b) correspond to different plots for different values of temperature: $k_{B}T\Delta l/(\hslash v)=0$ (dotted), 0.05 (dash-dotted), 0.15 (dashed), 0.5 (long-dashed), 1.2 (full). $V=5\mu \mathrm{eV}$. Panels (c), (d) correspond to different plots for different voltage biases: $\mathrm{eV}\Delta l/(\hslash v)=0.2$ (dotted), 0.3 (dash-dotted), 0.5 (dashed), 1 (long-dashed), 2 (full). $T=3\mathrm{mK}$. In all plots $\hslash v/\Delta l=5\times {10}^{-6}\mathrm{eV}$, $\tau =2\times {10}^{-10}\mathrm{s}$, $t_A=r_A=1/\sqrt{2}$, $t_B=\sqrt{2}{r}_B=\sqrt{2}/\sqrt{3}$. The range for which $\mathrm{Re}[{}_{D2}⟨Q_2{⟩}_{S1}^{\mathrm{SP}}]$ is beyond the interval of standard (strong) values is shadowed.