Stark Localization as a Resource for Weak-Field Sensing with Super-Heisenberg Precision

Gradient fields can effectively suppress particle tunneling in a lattice and localize the wave function at all energy scales, a phenomenon known as Stark localization. Here, we show that Stark systems can be used as a probe for the precise measurement of gradient fields, particularly in the weak-field regime where most sensors do not operate optimally. In the extended phase, Stark probes achieve super-Heisenberg precision, which is well beyond most of the known quantum sensing schemes. In the localized phase, the precision drops in a universal way showing fast convergence to the thermodynamic limit. For single-particle probes, we show that quantum-enhanced sensitivity, with super-Heisenberg precision, can be achieved through a simple position measurement for all the eigenstates across the entire spectrum. For such probes, we have identified several critical exponents of the Stark localization transition and established their relationship. Thermal fluctuations, whose universal behavior is identified, reduce the precision from super-Heisenberg to Heisenberg, still outperforming classical sensors. Multiparticle interacting probes also achieve super-Heisenberg scaling in their extended phase, which shows even further enhancement near the transition point. Quantum-enhanced sensitivity is still achievable even when state preparation time is included in resource analysis.

Figure 1

The QFI versus $h/J$ when our probe with sizes $L$ is prepared in (a) the ground state and (b) the midspectrum eigenstate. The dashed lines in both panels are the best fit of ${\mathcal{F}}_Q$ in the localized phase, showing the behavior in the thermodynamic limit. (c) The transition point $h_{\max }/J$ versus energy $ϵ$, in various system sizes, indicating the emergence of mobility edges in finite systems. (d) The scaling of $h_{\max }/J$ with $L$ for various $ϵ$. Markers and solid lines represent numerical results and fitting function $h_{\max }=a{L}^{-b}$, respectively. Inset: $a$ and $b$ for different $ϵ$.

Figure 2

The maximum of QFI (markers) as a function of $L$ for (a) the ground state and (b) the midspectrum eigenstate. The solid red lines are fitting of the form ${\mathcal{F}}_Q(h_{\max })\propto L^{\beta }$ with $\beta =5.98$ for $ϵ=0$ and $\beta =4.11$ for $ϵ=0.5$. The inset of each panel represents the behavior of $\beta$ away from criticality. In the midspectrum (right panel), $\beta$ remains almost steady in the extended phase and drops in the localized phase.

Figure 3

Finite-size scaling analysis using Eq. (3) for (a) the ground state and (b) the midspectrum eigenstate. The optimal data collapse is obtained for the reported $(h_c,\alpha ,\nu )$.

Figure 4

The QFI of the thermal state as a function of temperature $T$ for various system sizes, when the probe is operating at (a) the transition point $h=h_{\max }={10}^{-8}J$ and (b) the localized phase with $h=0.05J$.

Figure 5

(a) The QFI of a many-body interacting system with $N=L/2$ particles versus $h/J$ when the probe of size $L$ is in the ground state. The dashed line is the best fitting function of $|h-h_{\max }|$ in the localized phase illustrating the system’s behavior in the thermodynamic limit. (b) The QFI at the extended and transition point, namely $h=h_{\max }$, as a function of $L$.