Stark systems in which a linear gradient field is applied across a many-body system have recently been proposed for quantum sensing. Here, we explore sensing capacity of Stark probes, in both single-particle and many-body interacting systems, for estimating nonlinear forms of the gradient fields. Our analysis reveals that, this estimation can achieve super-Heisenberg scaling precision that grows linearly by increasing the nonlinearity. Specifically, we find a universal algebraic relation between the scaling of the precision and the degree of the nonlinearity. This universal behavior remains valid in both single-particle and many-body interacting probes and reflects itself in the properties of the phase transition from an extended to a localized phase, obtained through establishing a comprehensive finite-size scaling analysis. Considering a parabolic gradient potential composed of both linear and nonlinear fields, we used multiparameter estimation methodology to estimate the components of the gradient potential. The phase diagram of the system is determined in terms of both linear and nonlinear gradient fields showing how the nonlocalized phase turns into a localized one as the Stark fields increase. The sensing precision of both linear and nonlinear Stark fields follows the same universal algebraic relation that was found for the case of single parameter sensing. We demonstrate that simple and experimentally available measurements can reach the theoretical precision bounds. Finally, we show that quantum enhanced sensitivity is still achievable even when we incorporate the preparation time of the probe into our resource analysis.
Stark systems in which a linear gradient field is applied across a many-body system have recently been proposed for quantum sensing. Here, we explore sensing capacity of Stark probes, in both single-particle and many-body interacting systems, for estimating nonlinear forms of the gradient fields. Our analysis reveals that, this estimation can achieve super-Heisenberg scaling precision that grows linearly by increasing the nonlinearity. Specifically, we find a universal algebraic relation between the scaling of the precision and the degree of the nonlinearity. This universal behavior remains valid in both single-particle and many-body interacting probes and reflects itself in the properties of the phase transition from an extended to a localized phase, obtained through establishing a comprehensive finite-size scaling analysis. Considering a parabolic gradient potential composed of both linear and nonlinear fields, we used multiparameter estimation methodology to estimate the components of the gradient potential. The phase diagram of the system is determined in terms of both linear and nonlinear gradient fields showing how the nonlocalized phase turns into a localized one as the Stark fields increase. The sensing precision of both linear and nonlinear Stark fields follows the same universal algebraic relation that was found for the case of single parameter sensing. We demonstrate that simple and experimentally available measurements can reach the theoretical precision bounds. Finally, we show that quantum enhanced sensitivity is still achievable even when we incorporate the preparation time of the probe into our resource analysis.
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.
Stark-localized quantum probes have recently been shown to enable quantum-enhanced weak-field sensing with polynomial or super-polynomial scaling. In this paper, we show that the spatial geography of the encoded field can elevate this advantage to a genuine exponential scaling. We study a one-dimensional Stark probe subject to an exponential gradient profile, Vj=eaj, and analyze its metrological performance in both equilibrium and non-equilibrium regimes, for single-particle and interacting many-body settings. In the equilibrium single-particle case, we derive an analytical lower bound showing that the quantum Fisher information grows exponentially with system size, and confirm numerically that this enhancement persists throughout the extended phase and at the localization transition. We further show that the same exponential scaling survives for mid-spectrum eigenstates and in the interacting many-body regime. This advantage remains intact under a fair resource analysis because the relevant preparation gap closes only algebraically, so the polynomial preparation overhead cannot offset the exponential gain in sensitivity. In the non-equilibrium regime, a simple product-state initialization followed by free evolution already retains exponential enhancement, eliminating the need for cooling, adiabatic preparation, or operation within a narrowly tuned sensing window. Finally, we outline a superconducting implementation based on flux-tunable transmon qubits with graded mutual inductive coupling to a common sensing bus. Our results identify exponentially graded Stark potentials as a distinct and experimentally plausible route to weak-field sensing with exponentially improving precision.
In contrast to interferometry-based quantum sensing, where interparticle interaction is detrimental, quantum many-body probes exploit such interactions to achieve quantum-enhanced sensitivity. In most of the studied quantum many-body probes, the interaction is considered to be short-ranged. Here, we investigate the impact of long-range interaction at various filling factors on the performance of Stark quantum probes for measuring a small gradient field. These probes harness the ground state Stark localization phase transition which happens at an infinitesimal gradient field as the system size increases. Our results show that while super-Heisenberg precision is always achievable in all ranges of interaction, the long-range interacting Stark probe reveals two distinct behaviors. First, by algebraically increasing the range of interaction, the localization power is enhanced and thus the sensitivity of the probe decreases. Second, as the interaction range becomes close to a fully connected graph its effective localization power disappears and thus the sensitivity of the probe starts to enhance again. The super-Heisenberg precision is achievable throughout the extended phase until the transition point and remains valid even when the state preparation time is incorporated in the resource analysis. As the probe enters the localized phase, the sensitivity decreases and its performance becomes size-independent, following a universal behavior. In addition, our analysis shows that lower filling factors lead to better precision for measuring weak gradient fields.
Exploiting quantum features allows for estimating external parameters with precision well beyond the capacity of classical sensors, a phenomenon known as quantum-enhanced precision. Quantum criticality has been identified as a resource for achieving such enhancements with respect to the probe size. However, such protocols demand complex probe preparation and measurement, and the achievable enhancement is ultimately restricted to narrow parameter regimes. On the other hand, non-equilibrium probes harness dynamics, enabling quantum-enhanced precision with respect to time over a wide range of parameters through simple probe initialization. Here, we unify these approaches through a Stark-Wannier localization platform, where competition between a linear gradient field and particle tunneling enables quantum-enhanced sensitivity across an extended parameter regime. The probe is implemented on a 9-qubit superconducting quantum device, in both single- and double-excitation subspaces, where we explore its performance in the extended phase, the critical point and the localized phase. Despite employing only computational-basis measurements, we have been able to achieve near-Heisenberg-limited precision by combining outcomes at distinct evolution times. In addition, we demonstrate that the performance of the probe in the entire extended phase significantly outperforms the performance in the localized regime. Our results highlight Stark-Wannier systems as versatile platforms for quantum sensing, where the combination of criticality and non-equilibrium dynamics enhances precision over a wide range of parameters without stringent measurement requirements.
Discrete time crystals are non-equilibrium phases of matter in periodically driven systems, characterized by robust subharmonic oscillations and broken discrete time-translation symmetry. Their long-lived coherent dynamics and resilience to imperfections make them promising resources for quantum sensing. A disorder-free discrete-time crystal probe can provide the quantum-enhanced estimation of the coupling parameter. Here, we extend this sensing mechanism to nonlinear interactions and show that this nonlinear profile strongly enhances the sensing precision by increasing the system-size scaling exponent of the quantum Fisher information. Our analytical discussion separates a rigorous seminorm upper bound from the physically relevant scaling realized by product-state probes in the time crystal regime. Numerically, we find that the quantum Fisher information retains its quadratic long-time growth with the number of Floquet cycles, while its system-size exponent increases approximately linearly with the nonlinearity exponent, identifying nonlinearity as a resource for quantum-enhanced sensitivity. We further show that stronger nonlinearities shrink the time crystal stability window, making the probe more sensitive to small deviations from the resonant condition. We also analyze the effect of imperfect pulses and show that such imperfections can enhance, rather than suppress, the information encoded in the evolved state. Finally, we discuss a digital implementation of the nonlinear DTC sensing protocol using superconducting qubits.
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.