Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system

Abstract

The exceptional points (EPs) of non-Hermitian systems, where n different energy eigenstates merge into an identical one, have many intriguing properties that have no counterparts in Hermitian systems. In particular, the n 1  dependence of the energy level splitting on a perturbative parameter ò near an nth order EP stimulates the idea of metrology with arbitrarily high sensitivity, since the susceptibility dò 1/ n /dò diverges at the EP. Here we theoretically study the sensitivity of parameter estimation near the EPs, using the exact formalism of quantum Fisher information (QFI). The QFI formalism allows the highest sensitivity to be determined without specifying a specific measurement approach. We find that the EP bears no dramatic enhancement of the sensitivity. Instead, the coalescence of the eigenstates exactly counteracts the eigenvalue susceptibility divergence and makes the sensitivity a smooth function of the perturbative parameter. between these eigenenergies can be excited to measure the eigenvalue susceptibility. However, non-Hermitian systems are fundamentally different. Because different eigenstates of non-Hermitian systems are in general nonorthogonal and even become identical at the EP, exciting the transitions between different eigenstates near the EP to measure the eigenvalue susceptibility is infeasible.In this paper, we study the sensitivity around the EP of a coupled cavity system for its immediate relevance to recent experimental studies [34,35]. Nonetheless, the theoretical formalism and the main conclusion-no dramatic sensitivity enhancement at the EP-are applicable to a broad range of systems, such as magnon-cavity systems [37,38] and opto-mechanical systems [39,40]. We use the exact formalism of quantum Fisher information (QFI) [41] to characterize the sensitivity of parameter estimation. The QFI formalism enables us to evaluate the sensitivity without referring to a specific measurement scheme-be it phase, intensity, or any other complicated measurements of the output from the system. We find that no sensitivity boost exists at the EP. The reason boils down to the coalescence of the eigenstates around the EP. Due to the indistinguishability of different eigenstates around the EP, not one but all eigenstates are equally excited by an arbitrary detection field. The average of all eigenstates exactly cancels out the singularity in the susceptibility divergence of the eigenenergies and makes the sensitivity normal around the EP. The cancellation may also be understood by the divergent Pertermann excess-noise factor at the EP (or critical point) [42,43].The article is organized as follows: in section 2, the basic concept of EPs is introduced using a simple coupled-cavity model; in section 3, we introduce the definition of sensitivity. Sensitivity analysis for general linear systems is presented in this section; In sections 4 and 5, two specific experimentally relevant schemes, namely, coupled passive-passive cavities and coupled active-passive cavities, are considered and the numerical simul...

High-order exceptional points in supersymmetric arrays

“…In this regard, the EPs are useful for sensing in comparison with the diabolic points; this feature has been verified in optics, cavity optomechanics, cavity spintronics, and circuit quantum electrodynamics [34][35][36][37][38][39][40][41][42][43]. The sensing susceptibility is greatly enhanced near the EPs [44].…”

“…The non-Hermitian SUSY array at the high-order EP enhances the susceptibility in optical sensing, the frequency response near the high-order EP is greatly increased [30,31,44,71]. Near the high-order EP in the non-Hermitian SUSY array, a remarkable point is the enhanced frequency response to the detuning as well as the coupling when the array is subjected to the perturbation .…”

Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps

“…The advantages of EPs for applications remain a very active topic of research [16,[59][60][61][62][63][64][65][66][67][68][69][70][71][72] and correctly modelling noise and quantum jumps is fundamental to correctly adress the question of, e.g., EPs sensitivity. We believe that our work, showing explicitly the operational interpretation and the relation between classical and quantum EPs in terms of postselection and/or inefficient detectors, can stimulate more interest in experimental demonstrations of LEPs and their potential quantum applications, pointing out analogies and differences with respect to those studied for semiclassical HEPs.…”

Quantum state tomography across the exceptional point in a single dissipative qubit

“…2c). In Figure 2d we plot the observed oscillation frequency Ω versus coupling rate J, which displays a square-root singularity that is associated with increased sensitivity near the EP [30,31,[37][38][39]. The solid curve displays a fit to Re δλ = 2Re J 2 − J 2 0 with J 0 as the sole free parameter.…”