Enhanced sensing of a weak Stark field under the influence of Aubry-André-Harper criticality

TL;DR

The paper addresses precise sensing of a weak Stark field by leveraging the interplay between Aubry–André–Harper criticality and Stark localization in a one-dimensional fermionic lattice. It uses ground-state, mid-spectrum, thermal, and half-filled probes to compute the quantum Fisher information and experimentally accessible observable Fisher information, demonstrating super-Heisenberg scaling such as ${F_Q\sim L^{6.7}}$ at AA H criticality and ${F_Q\sim L^{5.6}}$ in mid-spectrum extended regions, with corresponding OFIs saturating the QFI for suitable observables. The work also shows a universal high-temperature HL behavior ${F_Q(T,h)\sim f(h) T^{-2} L^{2}}$ while retaining strong low-temperature enhancements, and identifies experimentally accessible observables like ${\hat{O}_{H_2}}$ and ${\hat{O}_{cdw}}$ that realize these gains. Overall, the study provides a practical framework for quantum-enhanced metrology in localization-based sensors and highlights the potential for extending these ideas to other criticalities and multi-parameter sensing scenarios.

Abstract

The localization transition can be exploited as a resource for achieving quantum-enhanced sensitivity in parameter estimation. We demonstrate that by employing different classes of localization inducing potentials, one can significantly enhance the precision of parameter estimation. Specifically, we focus on the precision measurement of the Stark strength parameter encoded in the low- and high-energy eigenstates of a one-dimensional fermionic lattice under the influence of Aubry-André-Harper localization-delocalization transition. For the ground state, we consider the single-particle system, in addition to the system at half filling. Our work reveals that Quantum Fisher Information (QFI) offers superior scaling with respect to the system size compared to the pure Stark case, leading to a better parameter estimation. However, experimental measurement of the QFI based on fidelity in a multibody system is a significant challenge. To address this, we suggest experimentally relevant operators that can be utilized to achieve precision surpassing the Heisenberg Limit (HL) or can even saturate the QFI scaling. These operators, relevant for practical experimental setups, provide a feasible pathway to harness the advantages offered by the localization-delocalization transition by exploiting two distinct localizing potentials for quantum-enhanced parameter estimation.

Figures (6)

• Figure 1: Single party OFI. (a) presents OFI, $F_{cdw}$, for observable $\hat{O}_{cdw}$ with respect to $h$ for system size L = 144 (blue circle), 233 (red square), 377 (turquoise diamond), 610 (black triangular up), 987 (green triangular left). (b) presents OFI, $F_{H_2}$ for observable $\hat{O}_{H_2}$ with respect to $h$ for system size L = 144 (blue circle), 233 (red square), 377 (turquoise diamond), 610 (black triangular up), 987 (green triangular left). (c) illustrates the scaling of OFI corresponding to the operator $\hat{O}_{cdw}$, $F_{cdw}$ (circles) and the operator $\hat{O}_{H_2}$, $F_{H_2}$ (stars) for L = 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 . The straight lines are the best fits. Whereas $F_{cdw}$ scales as $F_{cdw} \sim L^{6.2}$, $F_{H_2}$ saturates the scaling of the QFI, i.e., $F_{H_2} \sim L^{6.7}$. A configuration averaging over $\phi$ is performed with 8000 random samples.
• Figure 2: Single-party mideigenstate. (a) QFI for the mideigenstate against $h$ for system size $L = 144$ (blue), 233 (red), 377 (turquoise), and 610 (black). (b) presents OFI, $F_{cdw}$, for observable $\hat{O}_{cdw}$ with respect to $h$ for system size $L = 144$ (blue), 233 (red ), 377 (turquoise), and 610 (black). (c) OFI, $F_{H_2}$ for observable $\hat{O}_{H_2}$ with respect to $h$ for system size $L = 144$ (blue), 233 (red ), 377 (turquoise), and 610 (black). (d) illustrates the scaling of QFI (circular) and OFI, $F_{cdw}$ (square) and $F_{H_2}$ (triangular up) for $L = 21$, 34, 55, 89, 144, 233, 377, 610, 987, and 1597. The straight lines are the best fits. Whereas $F_{cdw}$ scales as $F_{cdw} \sim L^{5}$ shown by the dotted magenta line, $F_{H_2}$ saturates the scaling of the QFI, i.e., $F_{H_2} \sim L^{5.6}$, shown by the dotted green line. A configuration averaging over $\phi$ is performed with 8000 random samples.
• Figure 3: Single party thermal state probe for QFI. (a) presents an analysis of the QFI for the thermal state, $\rho(T,h)$, where $T$ stands for the temperature. Setting $h$ at a low value ($h=10^{-8}$), the $F_Q$ exhibits distinct values for different system sizes at low temperatures. At sufficiently low temperature, the QFI remains flat and recovers the scaling results obtained from the ground state, i.e., $F_Q \sim L^{6.7}$ (see the inset). As the temperature increases, there is a transition in the behavior of $F_Q$, and it is marked with a monotonic decay with the temperature. There, the scaling relation turns out to be $F_Q\sim T^{-2}$. (b) depicts $F_{Q}$ with temperature $T$ when $h$ is set at a comparatively large value, $h=0.05$, where the considered finite-size systems are in the localized phase at the low temperatures. Here at sufficiently low temperatures, all $F_Q$ turns out to be system-size invariant in a similar fashion reported for the ground state. Beyond a transition temperature, $F_Q$ decreases with $T$, $F_Q \sim T^{-2}$. In both cases, the considered system sizes are $L = 55$ (solid), $89$ (dot), $144$ (dash), $233$ (dash-dot), $377$ (dash-dash-dot). A configurational averaing with 500 random realizations of $\phi$ is performed.
• Figure 4: Single party thermal state probe for OFI. (a) presents behavior of the OFI, $F_{H_2}$, corresponding to the operator, $\hat{O}_{H_2}$ in the thermal state $\rho(T,h)$, for $h = 10^{-8}$. OFI has a trend similar to QFI (See. Fig. 3), and the scaling relation here is $F_{H_2} \sim L^{6.7}$ (see inset) at sufficiently low-temperatures, similar to the behavior observed in the ground state. As the temperature increases, $F_{H_2}$ transitions to a decay regime, following the scaling relation $F_{H_2} \sim T^{-2}$. (b) Temperature dependence of $F_{H_2}$ in the localized phase ($h = 0.05$). At low temperatures, $F_{H_2}$ collapses across system sizes, resembling ground-state behavior. Beyond a certain temperature, $F_{H_2}$ decays with $T$, while maintaining the scaling, $F_{H_2} \sim T^{-2}$. (c) For $h = 10^{-8}$), $F_{\text{cdw}}$ also follows the ground-state scaling relation, $F_{\text{cdw}} \sim L^{6.2}$ (see inset), at sufficiently low temperatures as well. Above a certain temperature, however, $F_{\text{cdw}}$ decreases with increasing $T$. (d) In the localized phase ($h = 0.05$), the temperature dependence of $F_{\text{cdw}}$ shows a collapse across system sizes at low temperatures, consistent with ground-state behavior. At higher temperatures, $F_{\text{cdw}}$ decreases with increasing $T$. In all the cases, considered system sizes are $L = 55$ (solid), $89$ (dot), $144$ (dash), $233$ (dash-dot) and $377$ (dot-dash-dot). We have performed an average over 500 random values of $\phi$.
• Figure 5: Half-filled case QFI. (a1) QFI for half filled case with respect to $h$. We have taken system size $L (n_f)$, where $L$ is the system size and $n_f$ is the number of filling. This figure shows that in the flat region, the QFI is increasing with system sizes, $L$ = 55 (28) (blue dash-dash), 89 (44) (red dot-dash), 233 (116) (black dot-dot-dash), 377 (188) (green dash-dash-dot). After a certain value of $h$, the QFI decays with certain initial fluctuations, and again it shows steady behavior. (a2) In this plot, we have shown the scaling of QFI with system size $L (n_f)$. The circle point is numerical value of QFI for $h = 10^{-9}$ for $L$ = 21 (10), 55 (28), 89 (44), 233 (116), 377 (188), 987 (493), 1597(798) and the dotted line is the best fitting with fitting function $F_Q(h=10^{-9}) \sim L^{6.6}$.