Charge Sensing Amplification via Weak Values Measurement

A protocol employing weak values (WVs) to obtain ultrasensitive amplification of weak signals in the context of a solid-state setup is proposed. We consider an Aharonov-Bohm interferometer where both the orbital and the spin degrees of freedom are weakly affected by the presence of an external charge to be detected. The interplay between the spin and the orbital WVs leads to a significant amplification even in the presence of finite temperature, voltage, and external noise.

Figure 1

(a) A sketch of the WV hypersensitive charge measurement setup. It consists of a half-metal open AB interferometer with Zeeman magnetic field $\mathbf{B}=B_z{\mathbf{e}}_z$. The half-metal left and right leads with spin orientations ${\widehat{\mathbf{n}}}_L$, ${\widehat{\mathbf{n}}}_R$, respectively, serve as a SV that measures the spin orientations of the electrons. The length of the upper (lower) arm is $L_1(L_2)$. The spin orientation ${\widehat{\mathbf{n}}}_L^{\prime \prime }$ exits the right junction of the interferometer. A small charge, $q$, is situated next to one of the arms of the interferometer, weakly changing the confining geometry of this arm. Consequently, the electron trajectory and momentum of electrons passing through this arm are modified, inducing additional electron spin rotation which depends on whether the upper or the lower interferometer’s arm is traversed (i.e., the spin is coupled to the “which-path coordinate”). The signal due to $q$ is stored in the spin state which is read in the current through the SV. (b) A sketch of the dispersion curve for $\widehat{\mathcal{H}}$, cf. Eq. (1).

Figure 2

Density plot of the amplification factor, ${\alpha }_{\mathrm{SV}+\mathrm{interf}}/{\alpha }_{\mathrm{SV}}$, in the case of ${}_f⟨\widehat{A}{⟩}_i\in \mathbb{C}$ for (a) $(\Delta p\Delta L/\hslash )=0\mathrm{mod}(2\pi )$ and (b) $(\Delta p\Delta L/\hslash )=2.6\times {10}^{-3}$. In both plots ${\mathbf{n}}_L={\mathbf{n}}_R={\sigma }_x$, $B_z=1\mathrm{T}$, $(\overline{p}/\hslash )\sim 7\times {10}^{-2}(1/\mathrm{nm})$, $(\Delta p/\hslash )\sim 1\times {10}^{-4}(1/\mathrm{nm})$, $\hslash e\eta \sim 20.7\mathrm{nm}$, $\delta \times e\sim 1.5\times {10}^{-2}$ [21].

Figure 3

(a) The amplification factor, ${\alpha }_{\mathrm{SV}+\mathrm{interf}}/{\alpha }_{\mathrm{SV}}$, as a function of the transmission through arm 1 for different values of the temperature (a) and voltage bias (b). In (a), $V=0.1\mathrm{meV}$, and $T=10\mathrm{mK}$ (solid black curve), $T=25\mathrm{K}$ (long-dashed red curve), $T=50\mathrm{K}$ (medium-dashed green curve), $T=100\mathrm{K}$ (short-dashed blue curve). In (b), $T=10\mathrm{mK}$, and $V=0.1\mathrm{meV}$ (solid black curve); $V=10\mathrm{meV}$ (long-dashed red curve), $V=25\mathrm{meV}$ (medium-dashed green curve), $V=50\mathrm{meV}$ (short-dashed blue curve). In all plots ${\Phi }_{\mathrm{AB}}=\pi$ and all other parameters are chosen as in Fig. 2.

Figure 4

The amplification factor, ${\alpha }_{\mathrm{SV}+\mathrm{interf}}/{\alpha }_{\mathrm{SV}}$, as a function of the transmission through arm 1 for different strength of the magnetic field fluctuations: $\Delta B=10\mathrm{mT}$ (solid black curve), $\Delta B=100\mathrm{mT}$ (long-dashed red curve), $\Delta B=500\mathrm{mT}$ (medium-dashed green curve), and $\Delta B=700\mathrm{mT}$ (short-dashed blue curve). All plots are for ${\Phi }_{\mathrm{AB}}=\pi$ and all the other parameters are chosen as in Fig. 2.