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

The exceptional points 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 $\epsilon^{1/n}$ dependence of the energy level splitting on a perturbative parameter $\epsilon$ near an $n$-th order exceptional point stimulates the idea of metrology with arbitrarily high sensitivity, since the susceptibility $d\epsilon^{1/n}/d\epsilon$ diverges at the exceptional point. Here we theoretically study the sensitivity of parameter estimation near the exceptional points, using the exact formalism of quantum Fisher information. The quantum Fisher information formalism allows the highest sensitivity to be determined without specifying a specific measurement approach. We find that the exceptional point 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.

Around the nth order EP [30,31], where the coalescence of n levels occurs, the eigenenergy shows an ò 1/ n dependence on the perturbative parameter ò. This result stands in sharp contrast to the Hermitian degeneracy, where the eigenenergy has a linear or high-order dependence. That means the eigenenergies around EPs have diverging susceptibility on the parameter change since dò 1/ n /dò=ò 1/ n−1 diverges at ò=0. Based on this divergence, schemes of parameter estimation (or sensing) working around EPs were proposed for the purpose of beating the metrology limit of Hermitian systems [32,33]. Recently, this idea has been experimentally studied [34][35][36]. However, the diverging eigenvalue susceptibility does not necessarily lead to arbitrary high sensitivity. In parameter estimation the sensitivity is usually defined as minimum parameter change that can be determined above the noise level within a given data acquisition time. Thus defined sensitivity is more relevant than the eigenvalue susceptibility is to practical applications of parameter estimation. In Hermitian systems, the sensitivity is inversely proportional to the eigenvalue susceptibility, i.e. the larger the susceptibility, the higher the sensitivity. Such a relation is based on the fact that all the eigenstates are distinguishable and the transitions 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 simulations are presented; Conclusions and discussions are given in section 6.

Model
We consider two near resonance coupled cavities with the effective non-Hermitian Hamiltonian where ν a(b) is the cavity frequency and γ a(b) is the decay rate induced by the photon leakage of the cavity a(b), g is the coupling strength, and the Planck constant  is taken as unity throughout this paper. For the quadratic Hamiltonian in equation (1), the dynamics are captured by the coefficient matrix and ò=ν a −ν b are the average and detuning of the cavity frequencies, respectively, and are the average and difference in decay rates, respectively. In sensing experiments, the detuning 0   is a perturbation term and can be introduced, e.g. by a nanoparticle that changes the effective volume and hence the frequency of one of the cavities, say, cavity a [34].
The eigenvalues and the corresponding right eigenvectors are obtained by diagonalizing the coefficient where z  are the normalization factors such that 1 The eigenvectors R y  of the non-Hermitian M are in general non-orthogonal and coalescent at the EP

Quantum Fisher information
In general, the sensing can be viewed as a scattering process. The input state ρ in after scattering with the sensing system yields an output state ρ(ò), which depends on the parameter ò that is to be estimated. Certain measurement of the output state ρ(ò) determines the parameter ò. The sensitivity is defined as the minimum detectable parameter change above the noise level within a given acquisition time where δò min is the minimum detectable parameter change for a detection time T [44]. In general, the sensitivity depends on the specific measurement scheme, which, in optics, is usually the measurement of the phase, the intensity, or various quadratures. However, there is a theoretical lower bound for all kinds of measurement, which is known as the quantum Cramér-Rao bound [45] F n T 1 . 5 Here F ò is the QFI of the output state ρ(ò) and n/T is the number of experiment repetitions per unit time.
Formally, it has an expression ] is the fidelity between the states ρ and ρ′. A particular advantage of the QFI is that it is independent of the specific measurement scheme. In the following, we use the QFI to characterize the sensitivity of a non-Hermitian system. According to the definition of QFI in equations (6) and (7), the highest sensitivity is determined by the change of the state ρ(ò) in response to the variation of the parameter ò.
The output state ρ(ò) and the input state ρ in are connected via the scattering process [49,50]. The input νfrequency photon c in n after scattering by the sensing system gives the output photon c out g is the strength of the coupling between the input photons and jth mode of the sensing system, and M lk is the coefficient matrix of different modes. For example, the coupled system shown in figure 1 has o , and M lk in equation (2). The cases considered in sections 4 and 5 are two specific examples with different M lk and o l in n ( ). Taking the interaction V as a perturbation and expanding it to the second order, we obtain is the density matrix element of the input state. Here we assume that the input state is a product state of different frequency modes. A small disturbance δò of the sensing system changes the output state to The QFI, with the expansion in equation (10) kept to the leading order of δò, becomes The output state ρ(ò) and its differential ∂ ò ρ(ò), as functions of M lj Note that such a singular condition is independent of the EP. For example, the coefficient matrix in equation (2) shows no divergence at the EP as ]¯¯for all frequencies. Therefore, the QFI for a sensing system with well defined M ν shows no ò singularity at the EP.
For the completeness of discussion, we briefly comment on sensing systems with In such a case, ρ(ò) and its differential ∂ ò ρ(ò), in general, are singular because the divergence of M lj ) makes the output state ρ(ò) sensitive to the parameter ò. A small change of ò can make an abrupt change of ρ(ò). In physics, the abrupt change of the output state indicates a non-equilibrium phase transition. An explicit example is the lasing transition of a gain cavity system [51]. By embedding a gain medium into cavity b and applying optical pumping, the effective decay rate is effectively reduced to b g ¢ and even change its sign (see figure 3). That yields the lasing Above the threshold, the system is in lasing phase.
The singular point is in general not related to the EP that occurs at g 2 4 ) , unless the nonequilibrium phase transition coincides with the EP. An example is the  phase transition that occurs at g=γ a /2 and b a g g ¢ = -. But even for such coincidence, the divergence of QFI is caused by the phase transition rather than the EP. This is evidenced by the fact that F ò , as a function of M lj ) | near the transition point, where α is the critical exponent. For example, α=2 for the lasing transition (see appendix D for details). The above discussions are based on a linear theory in which the dynamics of the sensing system are captured by a linear matrix M lk . However, near the non-equilibrium transition, the critical fluctuations diverge and their effects become nonlinear. The diverging critical fluctuations may prevent the singular behavior of the QFI. A systematic study on the competition between the critical fluctuation and the abrupt change of output near the non-equilibrium phase transition is needed before a conclusion can be made on whether the phase transition can dramatically enhance the sensitivity of parameter estimation, which, however, is beyond the scope of this paper.
Physically, the lack of divergence of QFI at the EP is due to the coalescence of the eigenvectors (quasinormal modes). The coefficient matrix in equation (2) can be diagonalized as M V D V 1 = n n -, where D ν is a diagonal matrix of eigenvalues (ν−ν ± ) and V is the matrix composed of the eigenvectors R y  . The differential is ⎞ ⎠ vanishes at the EP due to the eigenvector coalescence, canceling the Δ −1 susceptibility divergence near the EP. This conclusion can be generalized to higher order EPs, since the differential is a smooth function at any EPs. The divergency of the eigenvalues part V V Note that in [32][33][34][35][36], the eigenenergy susceptibility, i.e. dD v /dò is taken as the inverse sensitivity, which is justified for Hermitian systems but invalid for non-Hermitian ones where the state coalescence counteracts the divergent eigenenergy susceptibility.
The analysis based on equation (10) is applicable for a coefficient matrix M n of any dimensions and hence an EP of arbitrary order. Therefore, the QFI shows no divergence at the EP in general.

Input-output theory
We consider the configuration of a coupled cavity system with input and output channels (as shown in figure 1). The QFI is extracted from the output for the parameter estimation (e.g. estimation of the frequency of a cavity). In addition to the waveguide input and output, we also include the realistic leakage into the free space, with rates γ a/b for cavity a/b.
where a a ex g g g ¢ = + and the definitions of a t in ( ) and b t in ( ) are similar to that of c t in ( ). The input-output relation is found to be is the dressed propagator of cavity a, and G b (ˆˆ) † and hence the waveguide output state c out r (see appendix B for details). Now to be specific we consider a Gaussian input state, which is the most commonly used in experiments. The output state must also be Gaussian since the scattering is a linear transform. The Gaussian state enables the exact calculation of the QFI. We assume that the free space and the waveguide are in the thermal equilibrium state with temperature 1/β (Boltzmann constant k B taken as unity) and the input signal is in the coherent state.
gives a smooth, linear perturbation dependence.
Therefore the QFI F ò shows no divergence at the EP. Note that equation (18) holds for any single-mode Gaussian noises, e.g. thermal noises or single-mode squeezed noise. Different noises may induce different signal-to-noise ratio, but would not change the scattering amplitude, and therefore would not change the conclusion that F ò is smooth at the EPs. Figure 2 presents the numerical results of the QFI and the energy splitting susceptibility as functions of the coupling strength g. Here, the input coherent state is assumed to have a spectrum . In the calculation, zero temperature is considered, i.e. n 0 = n , and the two cavities are tuned to resonance, i.e. ν a =ν b . The results reveal that F ò is a smooth function of g even at the EP (indicated by vertical dashed lines in figure 2). The decreasing of the QFI with increasing the coupling g is understood as follows. When the coupling g approaches to zero (i.e. enters the weak coupling regime), the sensing scheme becomes a 'single-cavity' sensing one, in which cavity b acts as a noise source of the sensorcavity a. Decreasing g in this regime means decreasing decoherence due to the extra environment (cavity b) of cavity a, and therefore increases the QFI and the sensitivity.
To show that the absence of divergence of QFI at the EP is related to the state coalescence, we expand the QFI as  figure 2(b)). It reflects that the eigenstates are indistinguishable at the EP. Combining F Δ with the divergent susceptibility χ 2 (see figure 2(c)), we find that the susceptibility divergence is exactly counteracted by the vanishing QFI F Δ . Similar arguments apply to the second term in the expression of F ò shown in equation (20). Thus the QFI F ò is a smooth function around the EP.

Active-passive cavity system
By embedding a gain medium into cavity b, the decay rate γ b is effectively reduced and can even change the sign to realize an active cavity (see figure 3). Through this method, an effective active-passive coupled cavity system has been realized to study the  symmetry [8,55]. It is interesting to know whether the EP in the active-passive system can enhance the sensitivity. The gain can be realized, e.g. by stimulated emission of a medium with population inversion. However, there exists a threshold that limits the maximal achievable gain rate. Above the threshold, the system will be in the lasing phase (a self-adaptive region) in which the effective decay rate description becomes invalid. In this study, we constrain the gain rate below the lasing threshold. Below the threshold, the gain cavity works as an amplifier. The decay rate of the gain cavity due to the pumped gain medium becomes , where S z ≡N e −N g denotes the population inversion of the gain medium, g G is the cavity-gain medium coupling coefficient, κ is the effective decay rate of the gain medium, and    n n = - + - are the eigenvalues of the coefficient matrix of the active-passive coupled cavity system. Therefore, F  is a smooth function at the EP. Figure 4 presents the numerical results of the quasinormal mode frequencies (a) and the QFI (b) as functions of S z . The input state takes the same form as in figure 2. Figure 4(b) reveals that the QFI is a smooth function of the population inversion S z around the EP (indicated by the vertical dashed line). The enhancement of the QFI with increasing the population inversion is induced by the gain medium. The stronger the optical pumping is, the larger population inversion is induced, and the higher sensitivity is obtained.
In the linear theory, the QFI diverges at the lasing threshold, i.e. S z =S c . In figure 4(b), the divergent behavior is shown in the dotted line. However, the critical fluctuation neglected in linear description becomes important near the threshold, which may prevent the sensitivity from divergence. The discussion of the effects of the critical fluctuations is beyond the scope of this paper. Further increasing the pumping power, the coupled cavity system will exceed the threshold and enter the lasing phase. The EP, known as the  phase transition point, occurs at the point g 2 a = g¢ and ò=0 in the parametric space; when g 2 a g > ¢ , the system is in the  symmetric lasing phase, where both modes are lasing; whereas when g 2 a g < ¢ , the  symmetry breaks and the system is in the single mode lasing phase [9,[56][57][58]. In contrast to the cases below the lasing threshold, a nonequilibrium phase transition occurs at the EP. The conclusion revealed in equation (10), that the enhancement of the QFI at the lasing transition is not caused by the divergence of the energy splitting susceptibility but the phase transition, can be generalized to this case. We can also understand this conclusion from the coalescence of the different quasinormal modes counteracts the susceptibility divergence at the EP.

Conclusion and discussion
We show that the EP in a non-Hermitian sensing system does not dramatically enhance the sensitivity, since the coalescence of the different quasinormal modes counteracts the singular behavior of the mode splitting. This is verified in the passive-passive and active-passive coupled cavity systems through the exact calculation of the QFI. This conclusion is valid for high-order EPs and other sensing schemes.
Notes. After completion of this work, we came across to [59][60][61]. Reference [59] is based on a slightly less general theoretical framework, which, without invoking the QFI, depends on the specific sensing scheme, but its conclusions are consistent with ours. References [60,61] also adopt the QFI formalism. The sensitivity of the reciprocal sensor in [60] is bounded by the intracavity photon numbers. It shows that when the intracavity photon number is fixed, EP sensors have no sensitivity enhancement, which is consistent with our conclusion. Reference [61] considers the case of lasing transition coincident with the EP in coupled cavities (the so called  -lasing transition) and argues that the sensitivity is enhanced by the  -lasing transition. The analysis, however, is based on the linearization approximation, which could be problematic due to the divergent fluctuations at the  transition point (see discussions in section 3). Interestingly, it is shown in [60] that the non-reciprocal coupling in a non-Hermitian coupled system can, without involving the EP, covert the high frequency sensitivity of an ancillary system to high sensitivity of the other parameters (e.g. cavity-cavity coupling).
The sensing process by the linear optical system can be described by a scattering process. We define