Quantum metrology and sensing aim to surpass classical measurement precision using quantum phenomena like entanglement and squeezing. This research investigates whether non-Hermitian dynamics can enhance quantum sensors. Classical settings have shown that non-Hermitian mode degeneracies (exceptional points) can improve parametric sensing in small systems. However, the potential of non-Hermitian effects in truly quantum, multi-mode systems remains largely unexplored. The non-Hermitian skin effect, a phenomenon in spatially extended systems where eigenvalues and wavefunctions are highly sensitive to boundary conditions, is a potential resource. This work focuses on utilizing non-Hermitian lattice dynamics to create enhanced sensors, exploring Hamiltonian parameter estimation using a one-dimensional asymmetric tunneling model (similar to the Hatano-Nelson model). The study unexpectedly finds that the non-Hermitian skin effect does not provide a sensing advantage. Instead, another mechanism is identified that leads to an exponential scaling of quantum Fisher information per photon with system size, even accounting for limitations like finite propagation time. The mechanism leverages non-reciprocity and a specific type of symmetry breaking, demonstrating strong enhancements even in small systems (as few as three lattice sites) and remaining effective in non-Markovian and non-perturbative regimes.

The literature review focuses on existing quantum metrology and sensing techniques that exploit quantum phenomena like entanglement and squeezing for improved precision. It also highlights previous research on non-Hermitian sensing schemes, mostly limited to a few coupled modes and utilizing exceptional points for enhancement. The review covers experimental demonstrations of exceptional point based sensing in classical systems. It then positions the current work within the broader context of non-Hermitian systems and multi-mode dynamics, emphasizing the relatively unexplored potential of the non-Hermitian skin effect in quantum sensing.

The research uses a parametric driving approach to realize non-Hermitian dynamics without needing external dissipation or post-selection. The focus is on dispersive sensing, where the parameter of interest shifts the frequency of a resonant mode, a common strategy in various applications. The proposed sensor utilizes two Hatano-Nelson chains with opposite chirality, whose amplification factors are interpreted as directional gain and loss. The Hatano-Nelson model, a 1D tight-binding chain with asymmetric nearest-neighbor hoppings, is analyzed to understand directional amplification. The authors provide a physically intuitive explanation for the amplification effect and its connection to non-reciprocity. The model incorporates a symmetry-breaking perturbation whose magnitude is the parameter to be estimated. A physically realizable system using a driven array of bosonic cavities is presented, where the canonical quadratures represent the two Hatano-Nelson chains. The sensitivity of the system to a small change in Hamiltonian parameter is analyzed using the quantum Fisher information (QFI). The QFI, scaled by the average photon number, serves as the key performance metric, distinguishing true sensing enhancement from trivial effects caused by increased photon numbers. The analysis involves calculating the QFI using linear response theory in the large drive limit, with the optimal measurement being a standard homodyne measurement. The investigation then considers scenarios beyond linear response, analyzing the system's behavior when the perturbation is too large for the linear response to be valid. In the strong amplification limit, the system demonstrates a square root improvement in the signal-to-noise ratio compared to standard dispersive measurement schemes. This surpasses the linear response prediction which scales linearly with the perturbation. The effects of non-Markovian dynamics are also investigated. Numerical and analytical estimations are used to estimate the minimum measurement times, even for rapid measurements, with the goal of finding out how the measurement timescale depends on the lattice size.

The key findings demonstrate that a specific type of symmetry breaking in a system of two coupled Hatano-Nelson chains with opposite chirality yields an exponential enhancement in sensing capability. The quantum Fisher information (QFI) per photon increases exponentially with system size, even when considering a fixed total photon number. This advantage is not due to the non-Hermitian skin effect, which surprisingly does not enhance sensing. Rather, the exponential enhancement originates from non-reciprocal amplification. This enhancement is achieved even in the presence of non-Markovian effects, where the measurement time decreases exponentially until reaching the round-trip propagation time in the lattice, before increasing linearly at larger system sizes. Beyond the linear response regime, where noise amplification becomes significant, the optimized setup still provides a substantial advantage, increasing the signal-to-noise ratio by a factor proportional to the square root of the perturbation, significantly outperforming standard dispersive measurements. These enhancements are evident even in small systems with just three lattice sites. The study also explicitly shows that the exponential signal enhancement is not due to approaching a parametric instability nor to large changes in the Hamiltonian's spectrum. The mechanism relies on large changes in wavefunctions due to symmetry breaking. The optimal measurement strategy is a standard homodyne measurement, making the system experimentally feasible. The methodology used addresses limitations due to finite propagation times and non-Markovian effects.

The findings challenge the common belief that exceptional points are necessary for enhanced sensing in non-Hermitian systems. This work reveals a new mechanism for exponential enhancement based on chiral amplification and symmetry breaking. The exponential scaling of the QFI per photon, even with fixed total photon numbers, significantly improves sensitivity compared to conventional dispersive detectors. The robustness to non-Markovian effects and the practicality of homodyne measurements make this approach highly promising. The results provide a strong foundation for future advancements in quantum sensing, potentially leading to highly sensitive and efficient sensors for various applications.

This paper demonstrates a novel method for exponentially enhancing quantum sensing using non-Hermitian lattice dynamics. The approach exploits chiral amplification and symmetry breaking in a system of coupled Hatano-Nelson chains, resulting in exponential scaling of the QFI per photon with system size, even with fixed photon number. The method is robust against non-Markovian effects and experimentally accessible. Future research could explore other non-Hermitian features, such as topological phases or chiral mode switching, to further enhance quantum sensing capabilities.

The study mainly focuses on a specific model and realization of the non-Hermitian lattice sensor. While the model is physically realizable in current experimental setups, variations in experimental parameters and noise sources beyond those considered could affect the observed enhancement. Future work should explore the impact of imperfections and noise on the proposed sensing scheme. Additionally, the analysis utilizes approximations, such as the large-drive limit and approximations for the measurement time, that warrant further investigation for a more precise understanding.