Modified Quantum Wheatstone Bridge based on current circulation

TL;DR

The paper investigates a minimal four-site fermionic system with geometric asymmetry to estimate an unknown hopping rate $J_2$ by monitoring current circulation near an AEDP. Using the Non-Equilibrium Green Function framework, it derives low-temperature, low-bias conductance relations; at the AEDP (where $\xi=0$ with $\xi= J_1 J_3 - J_2 J_4$) the Wheatstone balance $\frac{J_1}{J_4}=\frac{J_2}{J_3}$ aligns to reverse current circulation, enabling direct determination of $J_2$ via $J_2=\frac{J_1}{J_4}J_3^0$. The study analyzes environmental effects through zero-current Büttiker probes (dephasing) and lossy channels, finding robustness up to moderate couplings, but eventual degradation under strong perturbations. Extending to finite voltage and temperature shows continued functionality with some signature shifts; Quantum Fisher Information reveals a coherence-driven enhancement near the AEDP, underscoring the role of geometry in metrological performance.

Abstract

We investigate a simple fermionic system designed to detect an unknown hopping rate between two sites by analyzing current circulation. The system exploits geometric asymmetry and utilizes the connection between the additional energy degeneracy point (AEDP) and current circulation for precise parameter detection. In the low-temperature, low-bias regime, with baths chemical potentials aligned near the degenerate energy, we find that a balanced Wheatstone bridge condition emerges when the direction of current circulation reverses, providing a direct means to determine the unknown hopping strength. We further examine the impact of environmental interactions, demonstrating that the device remains functional under moderately strong dephasing and particle losses, though extreme environmental effects eventually degrade performance. Extending the analysis to general operating conditions, we show that the device continues to function effectively at higher voltages and temperatures. Finally, an analysis of the quantum Fisher information qualitatively supports our findings, revealing a sharp increase in the coherence contribution and a corresponding decrease in the population contribution near the AEDP. Our results highlight geometric asymmetry as a robust and practical tool for quantum metrology.

Figures (5)

• Figure 1: Schematic diagram of the proposed model. The modified Wheatstone bridge is asymmetrically connected to the baths, such that the upper branch $(1\to2\to3\to4)$ has more number of fermionic sites than the lower branch ($1\to4$) between the baths. Also, the usual link between sites 2 and 4 is not present. The hopping rate $\textnormal{J}_1$ and $\textnormal{J}_4$ are fixed. We want to determine the hopping rates $\textnormal{J}_2$ by slowly varying the controllable parameter $\textnormal{J}_3$. We can determine $\textnormal{J}_2$ by using the balanced Wheatstone bridge condition, $\textnormal{J}_2=\frac{\textnormal{J}_1}{\textnormal{J}_4} \textnormal{J}^{0}_3$, where $\textnormal{J}^{0}_3$ is the point where the current circulation reverses direction.
• Figure 2: (a) Location of additional degeneracy in the energy spectrum of the single particle Hamiltonian. Variation of total $(\mathcal{G}_T)$, upper $(\mathcal{G}_U)$ and lower $(\mathcal{G}_D)$ branch conductance with the controllable hopping parameter $\textnormal{J}_3$ for (b) Weak system bath coupling with $\gamma=0.1$, $(\mathcal{G}_S)$ denotes the upper branch conductance when the baths are symmetrically connected to the model. (c) Strong system bath coupling with $\gamma=0.5$. For all the figures, the default value of parameters if not specifically defined are $\mu=1,\textnormal{J}_1=0.25, \textnormal{J}_4=0.5,\gamma=0.5$. The hopping rate to be determined is set at $\textnormal{J}_2=0.5$. All parameters are expressed relative to the on-site potential $\mu$, the actual value of which depends on the details of the specific experimental setup exp_same_PhysRevX.7.031001exp_same_2.
• Figure 3: Variation of total, upper and lower branch conductance with the unknown hopping parameter $\textnormal{J}_3$ for (a) system- dephasing Büttiker bath coupling parameter $\gamma_P=0.5$. Contour diagrams showing the positive and negative phases for the (b) Upper branch conductance and (c) Lower branch conductance for the dephasing Büttiker bath. The red region specifies the place where the conductance is $<-10^{-4}$ and the blue region specifies conductance value $>10^{-4}$. (d) Variation of conductance for lossy channel coupling $\lambda=0.3$. (e) Phase diagram for the Upper branch conductance of the system with lossy channels . (f) Variation of the device efficiency in percentage with the strength of environmental interaction $\gamma_P (\lambda).$ The Yellowish (Off- white) part in the phase diagrams corresponds to currents with magnitude $<10 ^{-4}$
• Figure 4: (a) Variation of upper branch particle current with voltage for the ideal case and the numerical solutions. Phases showing the negative and positive regions of the upper branch current for the (b) ideal case, (c) numerically exact case. Phase diagram for (d) lower branch current and (e) upper branch current at higher temperature $T=1$. (f) Variation of device efficiency in percentage with voltage for different temperature values. If not specified otherwise, $T=0.01$.
• Figure 5: (a) Energy dependent upper and lower branch current at V=0.1, $\textnormal{J}_3=0.98$. (b) Energy dependent upper branch current for different voltages at $\textnormal{J}_3=0.98$. (c) Contour plot showing the value of efficiency with variation in dephasing and voltage for temperature T=0.1. (d) Variation of different components of the quantum Fisher information (QFI) $\mathcal{F}$ with $J_3$. (e) Variation of total QFI with $J_3$ for various dephasing $\gamma_P$. (f) Contour plot showing the value of total QFI with $J_3$ Voltage. All the currents are scaled by a factor of $2\pi$. Unless otherwise specified, the parameters are $V=0.01,T=0.01$.