Current-based metrology with two-terminal mesoscopic conductors

Abstract

The traditional approach to quantum parameter estimation focuses on the quantum state, deriving fundamental bounds on precision through the quantum Fisher information. In most experimental settings, however, performing arbitrary quantum measurements is highly unfeasible. In open quantum systems, an alternative approach to metrology involves the measurement of stochastic currents flowing from the system to its environment. However, the present understanding of current-based metrology is mostly limited to Markovian master equations. Considering a parameter estimation problem in a two-terminal mesoscopic conductor, we identify the key elements that determine estimation precision within the Landauer-Büttiker formalism. Crucially, this approach allows us to address arbitrary coupling and temperature regimes. Furthermore, we obtain analytical results for the precision in linear-response and zero-temperature regimes. For the specific parameter estimation task that we consider, we demonstrate that the boxcar transmission function is optimal for current-based metrology in all parameter regimes.

Figures (4)

• Figure 1: A two-terminal mesoscopic conductor embedded between two fermionic leads. The energy bias $eV$ provided by the bias voltage pushes the system out of equilibrium. The transport properties depend on the transmission function $\mathcal{T}(\epsilon)$; we consider (a) Lorentzian, (b) sums of Lorentzians and (c) boxcar shapes.
• Figure 2: Precision rate $\gamma$ as a function of $\theta$, (a) single Lorentzian, $N=1$, (b) $N=21$ and (c) boxcar transmission function for $k_BT=0.1\,eV$ (top) and $k_BT=3\,eV$ (bottom), and three values of $\Gamma=0.01,0.1,0.5$$eV$. Solid curves have been drawn with full Landauer-Büttiker expressions and dashed with the linear-response expression Eq. \ref{['eq:preclin']}. In all bottom panels, the left inset shows the conductance as a function of $\theta$ and the right, relative sensitivity. $\delta=100\Gamma$ is set throughout the figure.
• Figure 3: The maximum precision $\gamma_{\text{max}}$ (optimized over $\theta$) obtained with the full Landauer-Büttiker expressions, as a function of the number of Lorenztians $N$ added within the energy window $[-\delta,\delta]$. The dashed lines are obtained with a boxcar transmission function over the same energy window. Parameters: $k_BT=3\,eV$, $\delta=100\,\Gamma$. The inset shows $\mathcal{T}^{\text{box}}$ (dashed, black), a $\mathcal{T}^{\text{lor}}$ (orange) and $\mathcal{T}_N$ with $N_d=201$ (green).
• Figure 4: $\mathcal{T}_N$ as a function of $\epsilon$ for (a) $\Gamma=0.1\delta$ and (b) $\Gamma=0.01\delta$. Other parameters: $\delta=1$ and $\theta=0$.