Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics

The paper explores whether distinct non-Hermitian dynamical effects can enhance quantum metrology beyond classical limits and beyond previously studied few-mode exceptional-point strategies. Prior work largely examined small coupled-mode systems and classical-domain implementations. In extended non-Hermitian lattices, phenomena like the non-Hermitian skin effect (NHSE) imply extreme boundary sensitivity, suggesting potential sensing benefits. The authors investigate Hamiltonian parameter estimation in a multi-mode, one-dimensional asymmetric tunneling lattice (Hatano–Nelson-type), asking which non-Hermitian mechanisms provide genuine quantum sensing enhancements when resources (photon number) are properly accounted for.

The work situates itself among quantum metrology and sensing using entanglement/squeezing, and non-Hermitian sensing proposals centered on exceptional points (EPs). Experiments have demonstrated EP-related enhancements in classical regimes, while theory has debated quantum advantages and limits, including non-reciprocal approaches. Recent non-Hermitian lattice studies revealed the NHSE and associated topology/symmetry classifications (including Z2 classes). Prior non-reciprocal quantum sensing in zero-dimensional dimers did not exploit spatial propagation or scaling with system size. The paper distinguishes its approach from squeezing-based enhancement schemes, which can be limited by added noise of subsequent amplifiers, and emphasizes autonomous, dissipation-free realizations via parametric driving compatible with superconducting circuits and quantum optics.

- System: A one-dimensional chain of N coupled bosonic cavities with nearest-neighbor hopping w and nearest-neighbor two-photon (parametric) drive Δ (A in text), with open boundaries. The Hamiltonian in a pump-rotating frame (site resonance set to zero) is H_B with hopping and parametric drive on bonds; mapping to quadratures shows dynamics equivalent to two uncoupled Hatano–Nelson chains with opposite chiralities for x and p quadratures. - Effective model: In quadrature basis, equations of motion correspond to asymmetric hopping with imaginary vector potential ±A (A>0 gives rightward amplification for x and leftward for p), with effective hopping J = sqrt(w^2−4Δ^2) and amplification parameter related via 2A = ln[(w+Δ)/(w−Δ)]. The susceptibility (Green’s function) factorizes as χ(n,m;t) = e^{A(n−m)} χ_0(n,m;t), where χ_0 is the reciprocal Hermitian chain response. - Sensing task: Estimate a small dispersive perturbation ε coupling locally as V_N = n̂_N = a_N^ a_N on the last site (frequency shift), which breaks a Z2 symmetry (combined spatial inversion and pseudospin inversion) of the doubled-chain system. Contrast case: a nonlocal boundary-tunneling perturbation V_NHSE linking sites 1 and N (sensitive to NHSE) is analyzed and found not to offer true advantage when normalizing by total photon number. - Drive and readout: The first site couples to a waveguide with rate κ; a classical coherent tone of amplitude β drives via the waveguide. The output field b^{(out)} is measured by homodyne detection. In the large-|β| limit, the optimal measurement is a homodyne quadrature. The relevant temporal mode B_φ(N) integrates b^{(out)} over time τ. - Performance metric: Quantum Fisher information (QFI) of B_φ(N) with respect to ε in the large-|β| limit equals the optimized homodyne SNR squared normalized by ε^2, and scales as |β|^2. To identify genuine sensing advantage, QFI is normalized by the total average intracavity photon number η_tot (resource), dominated by the coherent displacement and proportional to |β|^2/κ. - Linear-response analysis: Choose real drive phase (x-quadrature drive) and measure orthogonal quadrature (φ = π/2, measuring p), and take odd N so that a zero-frequency mode exists (resonant enhancement). The dispersive perturbation couples x and p chains only at site N, enabling an x wave to propagate right with gain e^{A(N−1)}, scatter into p at site N (∝ε), then propagate left with gain e^{A(N−1)} back to the readout—yielding a signal ∝ ε e^{2A(N−1)} while total photon number scales with a single traversal gain. - Noise: In absence of ε, the two chains are decoupled; input noise experiences balanced amplification-deamplification over a round trip, so the output noise temperature equals the input’s (vacuum if at T=0). Thus, homodyne noise is N = sqrt(2n_th+1)/√2 for n_th thermal quanta, independent of amplification. - Non-Markovian analysis: For finite measurement times, include internal timescales: round-trip propagation τ_tr = N/J and escape time τ_esc = (N+1)/κ. Using low-frequency resonance approximations (dominant zero-frequency mode with width ≈1/τ_esc), derive finite-time SNR and measurement time τ_M behaviors. With finite J, impose Θ(τ−τ_tr) cutoff reflecting causality. - Beyond linear response: For finite ε_0 where amplification makes ε_0 e^{2A(N−1)}/κ non-negligible, compute exact zero-frequency input-output relations via series in ε_0 and a canonical squeezing transformation. Quadrature-changing processes acquire fixed amplification factors e^{±2A(N−1)} independent of ε_0; quadrature-preserving terms have no amplification. SNR_ε includes amplified signal and amplified noise in the denominator. Optimize amplification e^{4A(N−1)} to balance noise growth and achieve maximal distinguishability.

- True sensing enhancement does not stem from the non-Hermitian skin effect: a boundary-coupling perturbation V_NHSE yields large signals but equally increases resource (photon number), so QFI/η_tot shows no advantage over conventional sensors. - Z2-symmetry-breaking local detuning on the last site (V_N = n̂_N) in a doubled Hatano–Nelson chain yields exponential enhancement: both SNR and QFI per photon scale as ∝ e^{2A(N−1)} at fixed total photon number. Specifically, SNR_N = Z(A) e^{2A(N−1)} SNR_1, with Z(A) ≈ O(1) and SNR_1 the single-mode dispersive detector SNR. - The measurement time in the linear-response, long-time limit decreases exponentially with N: τ_M^(N) = [1/(16 Z(A) η_tot κ)] e^{-2A(N−1)} (vacuum input), demonstrating exponential resource-normalized sensitivity improvement. - Non-Markovian effects: Including escape and propagation times, the exponential improvement persists until τ_M reaches the round-trip time τ_tr = N/J; overall τ_M(N) ≈ max(τ_M^∞(N), τ_tr(N)), so τ_M decreases exponentially with N, then becomes propagation-limited and grows linearly with N. - Beyond linear response: Noise amplification eventually limits indefinite exponential gains. Optimizing amplification such that e^{4A(N−1)} ≈ 1+R(ε_0)^2 ≈ κ^2/(8ε_0^2) yields a maximum SNR scaling as SNR_ε ∝ √(ε_0/κ) (square-root dependence), representing a strong advantage over standard linear ε-scaling. The enhancement is achievable with small systems (as few as N=3) by increasing A. - The scheme is dynamically stable for w>Δ and κ>0, does not require operation near an exceptional point, does not rely on output squeezing, and maintains input-equivalent noise temperature in the unperturbed system, easing practical amplification requirements. The mechanism crucially exploits directional, phase-dependent amplification and symmetry breaking between two opposite-chirality chains.

The study addresses whether multi-mode non-Hermitian dynamics can provide genuine quantum sensing advantages when resource usage is fixed. By identifying a Z2-symmetry-breaking local perturbation that couples two oppositely chiral non-Hermitian chains, the authors demonstrate exponential growth of QFI and SNR per photon with system size. This mechanism leverages directional amplification in one quadrature, conversion at the perturbation, and reverse-direction amplification in the conjugate quadrature, while leaving the output noise unamplified in the unperturbed system. The results remain robust when considering finite measurement times and internal dynamics, with performance limited only when propagation time becomes the bottleneck. Beyond linear response, the method still yields a substantial advantage, changing the fundamental scaling of SNR with the parameter from linear to square-root, reflecting differential amplification of signal versus noise. Overall, the findings show that extended non-Hermitian systems can outperform traditional sensors without requiring EPs or squeezing, and they suggest a new design paradigm based on non-reciprocity and symmetry considerations.

The paper introduces a non-Hermitian lattice-based sensor that achieves exponential enhancement of quantum Fisher information and homodyne SNR per photon by coupling two opposite-chirality Hatano–Nelson chains via a local, Z2-symmetry-breaking detuning at the chain end. The enhancement persists under non-Markovian conditions until limited by lattice round-trip time and remains advantageous beyond linear response, where optimal amplification yields SNR scaling as √(ε_0/κ). The approach is compatible with quantum optical and superconducting circuit platforms, requires only homodyne detection, and does not rely on exceptional points or output squeezing. Future work could explore leveraging other non-Hermitian features—such as exotic topological phases, chiral mode switching, or varied symmetry classes—and alternative coupling geometries (e.g., unequal chain lengths or varied boundary conditions) to broaden applicability and identify additional sensing benefits.

- The exponential improvement in measurement time and SNR per photon holds in linear response and for long integration times; as amplification and system size increase, finite round-trip propagation time (τ_tr = N/J) and escape time (τ_esc = (N+1)/κ) impose non-Markovian limits, ultimately capping gains when τ_M approaches τ_tr. - Beyond linear response, noise amplification limits indefinite exponential enhancement; optimal performance requires tuning amplification such that output noise increases modestly (by ~factor 2), yielding SNR ∝ √(ε_0/κ) rather than linear in ε_0. - The strongest resonant enhancement at zero frequency requires odd N; even N reduces susceptibilities by a factor ~κ/(2J). - Stability and mapping constraints require w>Δ and κ>0; practical implementations must manage parametric driving and potential internal loss. - The NHSE does not provide a resource-normalized sensing advantage in this architecture; benefits depend on Z2 symmetry breaking and quadrature conversion at the chain end.