Wannier-Stark localization has been proven to be a resource for quantum-enhanced sensitivity for precise estimation of a gradient field. An extremely promising feature of such probes is their ability to showcase such enhanced scaling even dynamically with system size, on top of the quadratic scaling in time. In this paper, we address the issue of decoherence that occurs during time evolution and characterize how that affects the sensing performance. We determine the parameter domains in which the enhancement is sustained under dephasing dynamics. In addition, we consider an effective non-Hermitian description of the open quantum system dynamics for describing the effect of decoherence on the sensing performance of the probe. By investigating the static and dynamic properties of the non-Hermitian Hamiltonians, we show that quantum-enhanced sensitivity can indeed be sustained over a certain range of decoherence strength for Wannier-Stark probes. This is demonstrated with two examples of non-Hermitian systems with nonreciprocal couplings.
Wannier-Stark localization has been proven to be a resource for quantum-enhanced sensitivity for precise estimation of a gradient field. An extremely promising feature of such probes is their ability to showcase such enhanced scaling even dynamically with system size, on top of the quadratic scaling in time. In this paper, we address the issue of decoherence that occurs during time evolution and characterize how that affects the sensing performance. We determine the parameter domains in which the enhancement is sustained under dephasing dynamics. In addition, we consider an effective non-Hermitian description of the open quantum system dynamics for describing the effect of decoherence on the sensing performance of the probe. By investigating the static and dynamic properties of the non-Hermitian Hamiltonians, we show that quantum-enhanced sensitivity can indeed be sustained over a certain range of decoherence strength for Wannier-Stark probes. This is demonstrated with two examples of non-Hermitian systems with nonreciprocal couplings.
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.
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.
We investigate the quantum sensing performance of a dephasing qubit as a probe in Ohmic environments, characterized by the coupling strength η, the Ohmicity parameter s, and the cutoff frequency ωc to be estimated. The performance is quantified by the dimensionless quantum signal-to-noise ratio Q. We show that the evolution of Q with the scaled time ωct is independent of ωc, and peaks at an optimal time topt, yielding optimal sensitivity Qopt. We analyze how Qopt depends on η, s and the temperature T. Our results demonstrate that, for any Ohmic environment, provided that ωctopt≪1, Qopt always reaches the upper bound: Qmax=0.648 at zero temperature, and consistently attains Qmax/4 at high temperatures. Remarkably, we find that increasing the scaled temperature T/ωc can enhance Qopt by nearly two orders of magnitude compared to its zero-temperature counterpart for certain Ohmic environments. Our work reveals that temperature can serve as a resource to enhance sensing precision, as it accelerates the encoding of the cutoff frequency information into the probe state, thereby enabling optimal measurement within a short time window.
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.
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.
In non-equilibrium probes, quantum-enhanced sensitivity is quantified through super-linear scaling of precision with respect to time, as a central metrology resource. However, this requires optimal measurements, which are often complex and time-dependent, making it challenging in practice. Sub-optimal measurements typically fail to retain robust super-linear scaling and show oscillations. Here, we begin by establishing a universal temporal scaling law in a general multi-parameter non-equilibrium quantum sensing framework. Within this setting, we identify the universal behavior of the sensing precision with sub-optimal measurements which shows quadratic scaling modulated by an additional bounded oscillatory function. To remove the oscillatory part, we propose a protocol in which measurements are partitioned into different groups, each performed at different times. The collective precision obtained from this protocol stabilizes a robust quadratic scaling even for sub-optimal measurements in the multi-parameter regime. We validate our protocol through three distinct examples as well as Bayesian estimation. Our protocol is applicable to every informative measurement and takes a key step towards practical realization of quantum-enhanced sensitivity.