Quantum Wheatstone Bridge

We propose a quantum Wheatstone bridge as a fully quantum analog to the classical version. The bridge is a few-body boundary-driven spin chain exploiting quantum effects to gain an enhanced sensitivity to an unknown coupling. The sensitivity is explained by a drop in population of an entangled Bell state due to destructive interference as the controllable coupling approaches the unknown coupling. A simple criterion for the destructive interference is found, and an approximate expression for the width of the drop is derived. The sensitivity to the unknown coupling is quantified using the quantum Fisher information, and we show that the state of the bridge can be measured indirectly through the spin current. Our results are robust toward calibration errors and generic in the sense that several of the current state-of-the-art quantum platforms could be used as a means of realization. The quantum Wheatstone bridge may thus find use in fields such as sensing and metrology using near-term quantum devices.

Figure 1

(a) Classical Wheatstone bridge (WB) consisting of three resistors with known resistances $R$ and $R_C$ and one resistor with unknown resistance $R_x$. (b) Sketch of the voltage dependence used in the classical Wheatstone bridge. (c) Quantum Wheatstone bridge consisting of four spins interacting with three known coupling strengths $J_C,J$, and $J_{23}$ and one unknown coupling strength $J_x$. Spins 1 and 4 are interacting with two thermal baths of different temperature, and two magnetic fields of strength $h_1$ and $h_2$ are applied to spins 1 and 2, respectively. (d) Sketch of the spin current dependence used in the quantum Wheatstone bridge. (e) Numbering for the four spins used throughout the text.

Figure 2

(a) Population of the two states $|E_{\downarrow \downarrow }⟩$ and $|E_{-}⟩$ as a function of the unknown parameter $J_x$. In the inset, the gray area can be seen in more detail. (b) Population of $|E_{-}⟩$ as a function of both $J_x$ and $h_2$. The solid red line shows the points of destructive interference, Eq. (6), while the dashed lines denote the width of the population drop, Eq. (10). (c) Population of $|E_{-}⟩$ as a function of $J_x$ for both the full numerical solution (solid line) and the approximate solution (dashed line) given by Eq. (9). Here $J_C=J$ was used.

Figure 3

(a) Quantum Fisher information $\mathcal{F}$ as a function of the unknown parameter $J_x$ for different values of the magnetic field $h_2$ and $J_C=J$. (b) Maximum quantum Fisher information for both the full numerical solution and the approximate solution given in Eq. (13) and using $J_C=J$. (c) Spin current $\mathcal{J}$ as a function of $J_C$ for different values of the unknown parameter $J_x$. (d) $P_{\mathrm{SS}}(|{\mathrm{\Psi }}_{-}⟩)$ as a function of $J_C$ for 10 sets of random errors on all parameters except $h_2$ and $J_x$ (see text) where $h_2=0$.