Quantum Sensing at Exceptional Points
- Quantum sensing near exceptional points (EPs) is defined by non-Hermitian degeneracies that induce a nonlinear eigenvalue splitting, which can be exploited for improved parameter estimation.
- Utilizing coherent and NOON states with balanced homodyne detection or photon counting protocols enables enhancements in sensitivity, achieving up to fourfold or Heisenberg-limited scaling.
- Dynamic adjustment of the working point, even slightly off the exceptional surface, may further maximize the quantum Fisher information and pave the way for innovative high-sensitivity sensor designs.
Quantum sensing near exceptional points (EPs) explores the interplay between non-Hermitian degeneracies and the ultimate quantum limits of parameter estimation in open and driven-dissipative systems. At an EP, both eigenvalues and eigenvectors of a non-Hermitian Hamiltonian coalesce, leading to a characteristic nonlinear spectral response—typically an eigenvalue splitting that scales as a fractional power of a perturbation. This property can be leveraged for parameter estimation, provided the measurement protocol and noise sources are fully accounted for. The concept of an exceptional surface (ES) generalizes isolated EPs to continuous submanifolds in parameter space. Understanding how the quantum Fisher information (QFI)—the metric underpinning the quantum Cramér–Rao bound (QCRB)—behaves near EPs and ESs, and how it depends on input quantum states and measurement protocols, is essential for designing sensors that saturate quantum limits.
1. Non-Hermitian Microring Resonator Model: EPs and Exceptional Surface
The canonical platform for exploring these effects is a microring resonator supporting two counter-propagating modes (CW and CCW). The open system is made non-Hermitian by partially reflecting the output of one mode into the other (reflection coefficient ρ > 0), in addition to a perturbing symmetric coupling (parameterized by ε) induced by an impurity. The effective Hamiltonian is non-Hermitian when ρ > 0, leading to eigenvalues and eigenvectors that coalesce for ε = 0—forming an EP. Varying other system parameters (such as decay rate γ and beam splitter phase φ) generates a continuous set—an exceptional surface (ES)—where the system remains at an EP for a range of configurations (Cook et al., 12 Mar 2025).
| Parameter | Role in Model | Effect at ε = 0 |
|---|---|---|
| ε | impurity cross-coupling | Sets proximity to EP/ES |
| ρ | reflection coefficient | ρ > 0 yields non-Hermitian Hamiltonian |
| φ, γ, etc. | phase, decay, detuning | ES spans multi-dimensional parameter set |
On the ES, the system’s spectral response is maximally nonlinear with respect to ε, allowing for potentially enhanced sensitivity. The eigenvalue splitting near an m-th order EP obeys a Puiseux expansion, scaling as ε1/m.
2. Quantum Fisher Information: Sensitivity Bounds and Scaling
The quantum Fisher information (QFI) quantifies the ultimate sensitivity of quantum parameter estimation via the quantum Cramér–Rao bound (QCRB), MSE(ε) ≥ 1/(m * 𝓘Q(ε)), where m is the number of trials. For output quantum states ρε, the QFI can be equivalently defined via the symmetric logarithmic derivative (SLD) or the Bures distance:
- 𝓘Q(ε) = tr[ρε L_ε²] = 4 * lim_{δε→0} [d_B(ρε, ρ{ε+δε})² / δ_ε²]
Perturbations in ε generate the local Hermitian generator A_ε = –i (𝐊ε)⁺ (∂𝐊ε/∂ε), with the output state determined by a quantum transfer matrix 𝐊_ε. The QFI thus depends both on the input state and on the generator spectrum.
- For coherent state input |β⟩, 𝓘_β(0) ≤ 64(1 + ρ)² |β|²/γ². The QFI is enhanced by a factor of up to 4 on the ES (ρ = 1, EP) compared to a diabolic point (DP, ρ = 0).
- For a NOON state (N-photon entangled superposition), 64 N²/γ² ≤ 𝓘_N(0) ≤ 108 N²/γ². The QFI enhancement is up to 1.69×, with Heisenberg scaling (∝ N²).
These enhancements arise because the generator A_ε achieves its maximal eigenvalue spread at the ES. Nonclassical states provide a further advantage by maximizing susceptibility to Hamiltonian perturbations.
3. Input Quantum States and Achievable Bounds
Two classes of input states are analyzed:
- Semiclassical coherent state |β⟩: Achieves the shot-noise limit (QFI ∝ |β|²), with maximum QFI at the EP/ES as above. The optimal measurement protocol is balanced homodyne detection, where the output state is mixed with a strong local oscillator, yielding photon-count differences whose Fisher information saturates the QFI in the large local oscillator power limit.
- Highly nonclassical NOON state: (|ψ₁⊗N⟩ + |ψ₂⊗N⟩)/√2. The optimal protocol is photon counting after a frequency shift operation that renders the two constituent modes degenerate, followed by interference on a symmetric beam splitter. This scheme produces interference fringes with phase Nθ(ε) and achieves Heisenberg scaling (variance ∝ 1/N² for large N).
For both cases, the QCRB can be saturated by an experimentally feasible measurement procedure. Selecting the input state to be an eigenstate (or superposition thereof) of A_ε with maximal eigenvalue magnitude directly maximizes QFI and ensures the bound is tight.
| Input State | Measurement Scheme | QFI Scaling | Relative Enhancement |
|---|---|---|---|
| Coherent state | Balanced homodyne detection | ∝ | β |
| NOON state | Frequency-shifted photon counting | ∝ N² (Heisenberg) | Up to 1.69× at ES/EP |
4. Alternative Parameter Regimes and Optimization
Although operating exactly on the ES maximizes the nonlinear spectral response, the optimal QFI is not always achieved strictly at ε = 0. The maximal eigenvalue or spectral range of the generator A_ε can be larger in regions slightly away from the ES, such as near saddle points, resonances, or points of constructive interference. This suggests that “off-ES” parameter regimes may offer higher sensitivity for non-infinitesimal ε, motivating further exploration of system tailoring—including the use of coherent feedback or PT-symmetric Hamiltonians with real spectra—to engineer larger generator eigenvalues or enhanced gradients in the transfer function.
A plausible implication is that optimizing parameter estimation in practice may require dynamic or adaptive adjustment of the working point, rather than remaining strictly on the ES, in order to maximize the accessible QFI for a given measurement context.
5. Experimental Feasibility and Measurement Protocols
Balanced homodyne detection for coherent state inputs is well established and requires mixing the output with a strong local oscillator, allowing extraction of both quadratures. For NOON state inputs, frequency-shifting one of the constituent spectral modes into degeneracy, followed by mixing on a symmetric beam splitter and photon counting, offers an experimentally plausible protocol—assuming efficient dispersive mode separation and high-fidelity modulation are achievable.
The main technical challenges are:
- Maintaining phase coherence for both coherent and NOON-states (especially the beam splitter phase φ).
- Minimizing photon loss and excess noise, which directly lowers QFI.
- Ensuring the measurement regime stays linear in ε.
- Efficient and low-loss frequency shifting for NOON state protocols.
Within these constraints, both proposed protocols are capable of achieving the quantum Cramér–Rao bound as established for this system.
6. Summary and Implications for Quantum Sensing Design
The analysis demonstrates that quantum sensing performance near an exceptional surface is fundamentally determined by the structure of the non-Hermitian generator A_ε, the chosen input quantum state, and the optimal measurement protocol. Coherent state inputs benefit from a fourfold QFI enhancement at the ES/EP, while NOON states achieve Heisenberg scaling and a 1.69× gain. Feasible measurement schemes—balanced homodyne for coherent states and photon counting interferometry for NOON states—are identified to saturate these quantum bounds. Notably, the possibility that off-ES working points may provide even higher sensitivity suggests new directions for system engineering and adaptive sensing strategies. These insights set the stage for the design and realization of quantum sensors that explicitly leverage non-Hermitian degeneracies for enhanced parameter estimation (Cook et al., 12 Mar 2025).