One of the theorems of quantum mechanics states that physical observables are represented by Hermitian linear operators. This is in particular true for the Hamiltonian H^, which represents the energy and governs the time evolution of a given system via the Schrödinger equation (ℏ is set to unity) However, for open quantum systems, it can be justified to consider effective Hamiltonians that are non-Hermitian or non-self-adjoint, H^≠H^† [1–5]. Note that we here do not distinguish between Hermiticity and self-adjointness; for caveats, see Ref. [6]. Non-Hermitian Hamiltonians can also be used for open classical wave systems when a Schrödinger-like equation, as in Eq. (1), holds [7–9]. The non-Hermiticity of the Hamiltonian implies that its eigenvalues, the energy eigenvalues, are in general complex numbers. The negative imaginary part is a decay constant related to a finite spectral linewidth at the resonant energy given by the real part of the eigenvalue. In classical optical systems, where resonant frequencies and modes play the role of energy eigenvalues and eigenstates, non-Hermiticity can enter by absorption, gain, and radiation. A special class of non-Hermitian systems is PT-symmetric systems [8,10–13] which are invariant under a combination of parity (P) and time-reversal (T) operations. Such a system can possess an entirely real eigenvalue spectrum despite being non-Hermitian. Applications of PT symmetry are reviewed in Refs. [14,15].

If the Hamiltonian is not only non-Hermitian but also non-normal, i.e., [H^,H^†]≠0, then its (right) eigenstates are mutually nonorthogonal. This nonorthogonality becomes maximal at exceptional points (EPs) in parameter space [16–19], at which the eigenstates of the Hamiltonian fail to provide a basis for the Hilbert space. Approaching an EP of order n (short: EPn), not only n eigenvalues but also the corresponding n states coalesce. While in earlier stages EPs have been regarded as a mathematical curiosity, their existence has been meanwhile proven in experiments on microwave cavities [20–22], optical microcavities [12,23–26] for light confinement, and other systems. An overview on EPs in optics and photonics can be found in Refs. [27,28].

A crucial difference between EPs and conventional degeneracies, so-called diabolic points (DPs), is the strong response of the eigenvalues to external perturbations. When a DP is subjected to a perturbation of strength ε, then the resulting energy (or frequency) splittings are proportional to ε. As illustrated in Fig. 1, this is in strong contrast to the case of an EPn where the resulting energy splittings are generically proportional to the nth root of ε [16]. Hence, the absolute values of the splittings in the EP case are parametrically larger, meaning that for sufficiently small perturbations, the splitting is larger than in the DP case for exactly the same perturbation.

The splitting enhancement factor, defined as the ratio of the EP-enhanced splitting and the splitting at a corresponding DP, therefore scales with ε−(1−1/n) for small ε. Formally, this enhancement factor goes to infinity as ε→0. It is this strong enhancement at EPs that can be exploited for sensing applications based on the detection of frequency splittings [29]. Devices based on the detection of frequency splittings at DPs are commonly utilized in microcavity sensors for particle detection [30–32], optical gyroscopes [33,34], weak magnetic field sensors [35], nanomechanical mass sensors [36], and bending curvature sensors [37].

In this review paper, we summarize the recent developments in the field of EP-based sensors. The outline of the paper is as follows. In Section 2, the basic mechanism is explained. Theoretical proposals are discussed in Section 3 and experimental realizations are reviewed in Section 4. Section 5 deals with the influence and classical and quantum noise. Finally, a summary and outlook are given in Section 6.

We describe EP-based sensors in the formalism established in Refs. [29,38]. For an n-fold degeneracy, the sensor is described by the n×n system Hamiltonian H^0, with all eigenvalues being equal. The input quantity of interest is parametrized by ε and is considered as a perturbation with the n×n perturbation Hamiltonian H^1. The resulting eigenvalue splittings are the output quantities which can be measured by spectral means; see the illustration in Fig. 1(b). According to Kato [16], the splittings are proportional to the nth root of ε for an EPn, whereas for a DP, the splittings are proportional to ε.

For illustration purpose, we focus in this section on sensors based on a two-fold degeneracy. In the case of a DP, the 2×2-matrix H^0 can be diagonalized to with complex eigenvalue E0. Obviously, two linearly independent eigenvectors exist which can be chosen to be (1,0)T and (0,1)T. The superscript T marks the transpose of the vector. At an EP a diagonalization is not possible, but a transformation to is with complex eigenvalue E0 and off-diagonal element A0≠0. By scaling the matrix [Eq. (4)], the off-diagonal element A0 can be replaced by unity, which would give the Jordan normal form of the EP. In contrast to the matrix in Eq. (3), the matrix in Eq. (4) has only one eigenvector (up to proportionality), (1,0)T.

With the general 2×2 perturbation Hamiltonian it is straightforward to show that the eigenvalue splitting of H^ for the DP is proportional to ε, For the EP, the splitting is which is proportional to ε for small |ε| provided that B1≠0. In this generic case, we can conclude that for sufficiently small perturbation strength, the splitting in the EP case is larger than in the DP case; see Fig. 1(a). The splitting in the EP case strongly depends on |A0| which, in Ref. [39], is called the strength of the EP and is compared metaphorically to the tension of a crossbow. The larger the tension, the larger the enhancement of the effect of the perturbation, which is the pulling of the trigger. The DP has zero strength (A0=0), corresponding to an untensioned crossbow.

Note that for non-Hermitian Hamiltonians, the splittings in Eqs. (6) and (7) are in general complex valued. It is therefore appropriate to measure both the real frequency splitting ReΔE and the linewidth splitting −2ImΔE which can be done; see, for example, Refs. [30,40]. If only the real part of the splitting is experimentally accessible, then one should arrange a situation where the complex splitting is not dominated by the imaginary part.

It is mentioned that the EP enhancement also applies to the individual shift of (the real part of) the frequencies, but not to the mean value of the two frequencies. The measurement of frequency shifts is the more common sensing scheme for microcavity sensors. However, independent of the EP enhancement, one considerable advantage of measuring the splitting is that it is intrinsically self-referenced [30], i.e., there is no need for an external reference to suppress or eliminate thermal drifts, etc.

For passive systems, i.e., systems without gain, at an EP2 there is an upper bound of the strength of the EP in the Hamiltonian [Eq. (4)] [38] (for caveats see Ref. [41]): As the frequency splitting [Eq. (7)] has to overcome the individual linewidths in order to separate the modes spectrally [Fig. 1(b)] by standard means, one has to come close to this bound [Eq. (8)]. For active systems, including PT-symmetric systems, a bound as in Eq. (8) does not exist. Hence, one can introduce gain to narrow the spectral linewidth, thereby improving the resolvability of the peaks [31,42]. Further methods to efficiently resolve spectral peaks are the interferometric detection of frequency splittings [43] and the harmonic inversion technique [44].

For an EP3, the Jordan normal form [cf. Eq. (4) with A0=1] of the 3×3 system Hamiltonian reads The matrix in Eq. (9) has only one eigenvector (1,0,0)T, again up to proportionality, with complex eigenvalue E0. This degeneracy is lifted by a generic 3×3 perturbation Hamiltonian H^1 [as in Eq. (2)], leading to three eigenvalues. For small |ε|, the eigenvalue splittings scale with ε3, which gives an even stronger enhancement than in the EP2 case.

In the literature, the enhanced eigenvalue splitting has often been called enhanced sensitivity. However, the term “sensitivity” might be misleading [45], as it is used in two different ways: (i) for the transduction coefficient of the sensor from the input quantity of interest (here ε) to the output quantity (here ΔE), or (ii) for the smallest measurable change of the input quantity. The latter is the precision of the sensor, and it is characterized by the signal-to-noise ratio. Because of its ambiguity, the term “sensitivity” is avoided henceforth.

There have been a number of proposals for EP-based sensors in various different physical settings; the first was on whispering-gallery microcavities for single- or few-particle detection [29,38]; see Fig. 2(a). Whispering-gallery microcavities confine clockwise (CW) and counterclockwise (CCW) propagating light waves with ultralow losses by total internal reflection; see Refs. [46,47]. Microcavity sensors for label-free particle detection are commonly used; see, Refs. [30,48]. In the context of EP-enhanced sensing, the Hamiltonian in Eq. (4) appears naturally in a two-mode approximation in the basis of CCW and CW propagating waves, with state vector (ψCCW,ψCW)T. The diagonal elements are the frequencies of the waves, which are equal (here E0), as required by reciprocity [38]. The off-diagonal elements are backscattering coefficients, which can be different due to asymmetric backscattering at deformed boundaries and external or internal scatterers (e.g., air holes) [9,25,38,39,49–52]. At an EP2, one of the backscattering coefficients is zero, and the other one, here A0, is nonzero. Physically, this means that there is coherent backscattering from the CW to the CCW propagating direction but not the other way around. The backscattering is said to be fully asymmetric. As a result, only one eigenvector (1,0)T exists, which corresponds to a purely CCW propagating mode. This unidirectional propagation reflects the chirality of the EP [50,53–55]. A target particle approaching the evanescent field of the microcavity corresponds to the perturbation in Eq. (5); it leads to a square-root frequency splitting as in Eq. (7).

Further approaches to implementing EPs in microdisk cavities with the purpose of enhanced sensing have been introduced by appropriate boundary deformations generating EPs of fourth order [56], and by using a rotationally symmetric layered refractive-index profile [57]. The latter approach has been proposed for sensing the refractive index of a thin additional layer.

Another scheme is based on coupled cavities [58], as illustrated in Fig. 2(b). Two microrings in close proximity were coupled by their evanescent fields. They are supposed to be identical except that one cavity possesses gain and the other one exactly the same amount of loss. Hence, the system is PT-symmetric. In a two-mode approximation with state vector (ψ1,ψ2)T, the Hamiltonian is that of a PT-symmetric dimer Here, ω0 is the real-valued frequency, α≥0 is the gain/loss coefficient, and g≥0 is the coupling strength. The eigenvalues of the Hamiltonian in Eq. (10) are given by The eigenvalues degenerate if g=α. This degeneracy is a DP with eigenvectors (1,0)T and (0,1)T if α=0; otherwise, it is an EP with a single eigenvector (1,−i)T/2. Associated with the latter situation, there is the so-called PT-symmetric phase g≥α, where the eigenvalues are real valued. In the PT-broken phase g<α, the eigenvalues are complex conjugate of each other. As in the previous example, the proximity of a target particle is a perturbation to the system Hamiltonian, here in Eq. (10). Assuming that the target particle comes close only to the gain microring, the particle modifies the frequency of the associated mode of the gain ring by an amount ε. This leads to the perturbation Hamiltonian The resulting frequency splitting is iαε for |ε|≪4α. Here, the real and imaginary splittings are of the same size. Note that the validity of the two-mode approximation [Eq. (10)] is questionable as long-lived modes in optical microrings come in degenerate pairs, with one CW propagating mode and one CCW propagating mode. Hence, a four-mode approximation is more appropriate [59].

A related two-mode PT-symmetric system consisting of two coupled nanobeam cavities has been suggested [60]. A combination with optomechanics for enhanced metrology for displacement measurements has been presented in Ref. [61].

It has been proposed to apply the concept of EP-based sensing also to cavity-based optical gyroscopes [62,63] for the measurement of rotations. Optical gyroscopes are being widely used nowadays for navigation, e.g., in commercial aircraft. The principle of operation is the Sagnac effect. It is the opposite shift of the frequencies of counterpropagating optical waves in a rotating cavity, and scales linearly with the rotational velocity Ω. If the non-rotating system is prepared at an EP2, then the rotation-induced frequency splitting is proportional to Ω. For a PT-symmetric pair of coupled microrings such as in Fig. 2(b), it has been shown that the splitting is enhanced when the pair of rings is rotated. Remarkably, in the limit of maximum enhancement, the splitting is independent of the radii of the rings involved [62]. An improvement of more than six orders of magnitude has been estimated. A single-cavity approach based on a microring with PT-symmetric variation of the refractive index in the azimuthal direction has also been introduced [63]. Yet another approach to EP-enhanced measurements of rotations is to exploit the chirality of EPs [64].

Using a similar Hamiltonian as in Eq. (10), a bilayer structure composed of two different metamaterials with split-ring resonators has been proposed for refractive-index sensing in the terahertz regime [65], and a multi-layer structure based on the resonant optical tunneling effect has been suggested for biosensing [66]. Further proposals have been made for detecting dark-matter candidates [67], non-Newtonian effects in gravitation [68], and gravitational waves [69].

The advantage of the strong response at an EP is a disadvantage for fabrication since low tolerances are required. As a solution to this problem, the so-called exceptional surfaces have been introduced [70,71]. The idea is that certain perturbations do not move the system away from the EP. Examples of such perturbations are changing the values of A0 and E0 (equally in both diagonal elements) in the Hamiltonian in Eq. (4). Hence, the values of A0 and E0 define a two-parameter family of EPs, the exceptional surface. Provided that the system’s response is tailored such that most of the experimental uncertainties shift the operating point along the exceptional surface, the system can be robust and, at the same time, enhanced eigenvalue splittings are possible if the target perturbation pushes the system away from the surface.

An interesting setup for enhanced sensing with EPs of higher order is based on photonic spin–orbit interaction [72]. The spin–orbit interaction couples chiral dipolar modes in different cavities via an adjacent waveguide in a unidirectional manner. This is analogous to the fully asymmetric backscattering of CW and CCW propagating waves in a perturbed whispering-gallery cavity, but the setup can be scaled to higher-order EPs. Note that this setup can be considered as an application of the just-discussed concept of exceptional surfaces. Numerical simulations have shown an ε1/10 behavior of the frequency splittings at an EP10. Microwave experiments have demonstrated evidence for unidirectional coupling, but there is no experimental demonstration of the enhanced sensing in this setup.

Based on the original proposal [29], various experimental realizations of EP-based sensors have been developed [73–81]. Figure 3 shows a non-exhaustive compilation.

The first demonstrations of the superior performance of such sensors were done for single-particle detection [73] and thermal sensing [74] (see also Ref. [82]). In Ref. [73] the authors investigated a whispering-gallery microtoroidal cavity made of silica. The EP2 is based on asymmetric backscattering established by two precisely adjusted nano-tips [see Fig. 3(a)], following closely the proposal in Ref. [29]; see Fig. 2(a). Figure 4 embodies the central result of Ref. [73]. In the absence of a target particle the transmission spectra for the DP-based sensor [Fig. 4(a)] and EP-based sensor [Fig. 4(c)], obtained by coupling light in and out via two waveguides, show no frequency splitting. If the target particle [an (additional) nano-tip or a single polystyrene nanoparticle with a radius of 200 nm] is approaching the cavity a clear splitting can be seen, which is by a factor of around 2 larger in the EP case; cf. Figs. 4(b) and 4(d). Varying the distance between the target particle and the surface of the microtoroid, the splitting enhancement factor as a function of the perturbation strength has been determined [Fig. 4(e)]. Both the real and imaginary parts of the frequency splitting have been measured. A moderate enhancement factor of maximal 2.5 can be observed for small perturbations. Moreover, it has been demonstrated that the resolvability of the splittings at the EP can be improved by optical gain generated by pumping erbium ions embedded into the microcavity.

Figure 3(b) shows the PT-symmetric ternary microring system for thermal sensing [74]. The side cavities experience balanced gain (iα) and loss (−iα), whereas the middle one is neutral. The desired gain/loss distribution is realized by an inhomogeneous optical pumping of InGaAsP quantum wells inside the microrings. The individual frequencies of the cavities are fine-tuned to the same value using three gold microheaters underneath each cavity. Within a three-mode approximation and a rotating frame, the system Hamiltonian is Adjusting the coupling constant κ to be α/2, an EP3 is generated. To check the performance for thermal sensing, one of the microheaters has been used to introduce a thermal perturbation of strength ε to the gain cavity, A six-mode approximation is not necessary, as the perturbation acts equally on CW and CCW propagation ring modes. The experiment has clearly demonstrated (real-valued) frequency splittings proportional to ε1/3, with an impressive maximal enhancement factor of 23. The imaginary splittings are strongly influenced by nonlinear gain saturation and laser dynamics, which can be anticipated from the fact that one of the frequencies of the linear system resulting from Eqs. (13) and (14) has a positive imaginary part. These nonlinear dynamic aspects have not been studied in the theory.

The concept of EP-based sensing has also been applied in a modified fashion to thermal mapping [75]. Zhao and coworkers coated a conventional microscope slide such that the corresponding scattering matrix is at an EP2; note that the definition of an EP of a scattering matrix is ambiguous [83]. Figure 3(e) shows the three-layer structure with a polymer (PMMA) layer sandwiched between two gold films deposited on a silica glass slide. The key idea is that ambient thermal perturbations lead to a local change of the thickness of the polymer layer, causing a strong response of the eigenvalues of the scattering matrix and consequently a drastic increase of the reflection. Through reflection measurements, the temperature distribution of the glass slide can be monitored with a roughly tenfold enhanced optical-thermal signal transduction and high spatial resolution.

Another interesting and potentially very relevant class of realizations is wireless EP-based sensors [76–79]. Applications of wireless sensors range from medical diagnosis to industrial and environmental monitoring. The principle of operation is the inductive coupling between the sensor and a reader. The quantity to be measured detunes capacitive or inductive elements of the sensor, which lead to frequency shifts measured externally by the reader. A particular fascinating experiment, depicted in Fig. 3(d), is on an EP2-based microsensor subcutaneously implanted in a rat abdomen [78]. The sensor can continuously and accurately track the rat’s breathing rate arising from the motion of the abdomen during respiration, which causes a modulation of the inductive coupling between the reader and the sensor. A splitting enhancement factor of 3.2 beyond the DP case limit that applies to all existing wireless readout schemes has been observed. The overall detection limit has been in total lowered by a factor of 26, including the help of the narrowing of spectral lines by the reader’s gain. A conceptional difference from previous wireless EP-sensors is here that the reader is a PT-symmetric two-mode system (frequency ω0, gain/loss coefficient α, and coupling constant g) locked to an EP2 (g=α), and the sensor is an additional single-mode system (frequency ωs and decay rate γs) whose sole function is to generate the perturbation of strength ε by the inductive coupling to the reader, In previous wireless EP-sensors [76,77], the reader and sensor together form a PT-symmetric two-mode system biased at an EP2, and the perturbation acts on the sensor mode. The new design of Ref. [78] shows a significantly improved performance. A passive wireless sensing system at an EP3 has been fabricated as well [79].

Ring laser gyroscopes with EP-based enhancement have also been successfully built and investigated [80,81]. Figure 3(c) illustrates the setup of Ref. [80]. It is a macroscopic equilateral cavity made of three highly reflective mirrors. A Faraday rotator, a half-wave plate, and two Brewster windows incorporated on both ends of a He–Ne gas tube are used to establish different gain/loss coefficients γ1 and γ2 for CCW and CW propagating waves. In addition, a weak conservative backscattering of counterpropagating waves is introduced by inserting an etalon into the cavity. With g being the strength of the backscattering and ω0 the frequency of both propagating waves, the Hamiltonian can be written in a two-mode approximation as in the basis of the pair of counterpropagating waves. The system has been biased at the EP2 with g=|γ1−γ2|/2>0 by fine-tuning the differential gain/loss |γ1−γ2|. Rotating the cavity with velocity Ω acts as a perturbation Hamiltonian with perturbation strength ε∝Ω. The resulting frequency splitting is proportional to Ω. The measured data [80] have clearly demonstrated this enhanced response of the gyroscope at the EP with an enhancement factor larger than 1 order of magnitude.

The laser gyroscope in Ref. [81] is of a very different type. The microscopic system is made of a high-quality silica microdisk; see Fig. 3(f). Stimulated Brillouin processes lead to lasing action of a chosen pair of counterpropagating waves, resulting in ultranarrow spectral linewidths. Moreover, the Brillouin processes modify the frequencies of the two involved modes ω1 and ω2 such that they exhibit a nonzero detuning even in the absence of rotation. This detuning can be precisely adjusted by properly choosing the pump frequencies ωp1 and ωp2. The presence of the waveguide that is used to pump the Brillouin laser modes induces scattering losses with the rate γ and backscattering of the counterpropagating waves with the rate κ. Assuming dissipative backscattering, only the system Hamiltonian is given by in the basis of the pair of counterpropagating waves. Adjusting the pump frequencies such that |ω2−ω1|=2κ>0 brings the sensor to an EP2. Rotating the cavity with velocity Ω acts as the perturbation Hamiltonian in Eq. (17) with strength ε∝Ω. The induced frequency splitting is again proportional to Ω. The experiment in Ref. [81] has confirmed this enhanced response of the gyroscope and reported a splitting enhancement factor of 4.

Recently, EP-based sensors have also been produced in the plasmonic domain [84]. Plasmonic systems are of particular interest for sensing, as electromagnetic fields can be greatly enhanced inside plasmonic hot spots. This opens the possibility to carry out sensing on the nanoscale. The EP-based sensor in Ref. [84] consists of a bilayer structure made of two arrays of dissimilar plasmonic resonators. A Hamiltonian is not provided, as the system has been discussed in terms of second-order differential equations for coupled oscillators. By functionalizing the surface of the sensor, the authors have successfully demonstrated that the sensor can selectively detect anti-immunoglobulin G, an important immunoglobulin in human serum, with enhanced frequency splitting.

The strong response of EPs to weak perturbations naturally raises the question of the influence of noise on the performance of EP-based sensors. Here we review the work done on classical, i.e., technical, noise related to EP-based sensors. Let us consider the total Hamiltonian with parametric noise where K≥1 is the number of statistically independent noise sources and the ξj are real-valued fluctuations with zero mean ξj(t)¯=0. The overline denotes an ensemble average over all possible realizations of the noise. As before, H^(ε) is the time-independent and non-Hermitian Hamiltonian describing the sensor and perturbation by virtue of Eq. (2). H^noise,j are time-independent and in general non-Hermitian operators describing the fluctuations of the matrix elements of the total Hamiltonian. Note that the noise in the total Hamiltonian in Eq. (19) enters the sensor as multiplicative noise in the Schrödinger equation [Eq. (1)]. It describes a noisy EP rather than a noisy input channel. The latter is relevant for the quantum noise discussed in Section 5.B.

We may classify the noise according to the correlation time. Infinite correlation time implies static disorder which has been studied in Ref. [85]. Not unexpectedly, static disorder blurs the spectral features after ensemble average. Fluctuations with limited correlation time, i.e., temporal noise, have been investigated in Refs. [86–88].

The authors in Ref. [86] have addressed the impact of a single noise source (K=1) on an EP-based sensor made of the PT-symmetric dimer in Eq. (10). The noisy element is a fluctuating on-site frequency detuning described by the noise Hamiltonian The spectrum of ξ1(t) has been required to be nonwhite, as it has a high-frequency cutoff and a gap around the frequency of the EP. The latter has been justified by the presence of an active stabilization circuit. It is difficult to judge to what extent the various approximations involved are fully consistent. The basic finding of Ref. [86] is that the system is dynamically unstable. Numerics show that the norm of any initial state grows asymptotically like exp(2γt−1) with a fitting constant γ. Nonlinear effects, which might counteract the instability, are ignored.

The general case of K noise sources is treated in Ref. [88]. It is assumed that the ξj describe stationary white Gaussian noise ξi(t)ξj(t′)¯=γjδijδ(t−t′) with noise strengths γj. For monochromatic excitation with frequency ω, the dynamics of the density operator ρ^(t)=|ψ(t)⟩⟨ψ(t)|¯ is described by an inhomogeneous Lindblad-type master equation with the superoperator L: and the effective Hamiltonian The pump operator P^(ω) in Eq. (21) has poles at the eigenvalues of H^eff. From the preceding equations it can be concluded that (i) the resolvability of frequency splittings at and near an EP is determined by H^eff, and (ii) the stability of the system is determined by the superoperator L. The latter can be discussed in Liouville space, where ρ^ and P^ are represented as n2-dimensional vectors and L is represented as an n2×n2 matrix (the Liouvillian). A stationary state ρ^ is in general only possible if all eigenvalues of the Liouvillian λl have nonpositive real part. A crucial result is that for a noiseless sensor, the Hamiltonian EPn carries over to a Liouvillian EP2n−1 [88,89], i.e., the latter is of higher order than the Hamiltonian one. Hence, the eigenvalues λl show an extreme response to noise, so one eigenvalue may easily cross the imaginary axis leading to a dynamical instability.

The instability, however, can be removed by a uniform damping of the sensor, i.e., by adding −iκ1 to H^0, with the rate κ above the critical rate κc=Reλ1/2, with λ1(γ1,…,γK) being the eigenvalue with the largest real part. Obviously, this reduces the resolvability because the linewidths broaden significantly. Hence, for EP-based sensors operating at or close to the real frequency axis, as for instance in the case of PT-symmetric systems, even small noise can indirectly, via the necessary stabilization, spoil the resolvability.

However, this does not mean that EP-based sensors are fundamentally limited by classical/technical noise. In principle, classical noise can be made arbitrarily small. For instance, it has been demonstrated theoretically for a class of electronic circuits that the extra technical noise originating from the gain in PT-symmetric arrangements can be made negligibly small [87].

In contrast to classical noise, quantum noise may fundamentally limit EP-based sensing. All existing theoretical approaches [45,90–92] assume the so-called weak dispersive limit, where frequency shifts are much smaller than the spectral linewidths. It is worth emphasizing that none of the experiments reviewed in Section 4 are restricted to this regime. In the theoretical approaches, it is argued that the frequency splitting is not directly measured; instead, it is constructed from field measurements (sampled at various frequencies). Hence, quantum noise limits apply to the field rather than to frequency splittings. The authors in Ref. [45] have shown with elementary calculations based on the Schrödinger equation [Eq. (1)] that the field changes in lowest order are proportional to ε both for DPs and EPs. Hence, the response at the EP is not parametrically larger than at a DP (in absolute values, the response at the EP can be still larger [93]). Without performing any quantum-noise calculations, one can conclude that EP-based sensors do not have a superior scaling behavior in terms of the quantum-limited signal-to-noise ratio.

The full quantum calculations in Refs. [90–92] are based on Heisenberg–Langevin equations The operator a^j is a bosonic annihilation operator for the jth mode in the second-quantization formalism. Noise enters additively via stochastic Langevin terms describing the driving and coupling of the modes to separate baths. The presence of the latter is needed for a consistent quantum description of the dynamics based on the non-Hermitian Hamiltonian H^(ε). Using input–output theory, a signal is extracted from the linear system [Eq. (24)] for which an upper bound of the signal-to-noise ratio (or the measurement rate) is derived.

The upper bound in Ref. [90] for reciprocal systems is independent of whether the sensor is at an EP or not. The conclusion reached is that EP-based sensors do not have an advantage in terms of quantum-limited signal-to-noise ratio. The derived bound is, however, not valid for non-reciprocal systems. The authors of Ref. [90] therefore promote non-reciprocity as a resource for sensing. In contrast, the authors in Ref. [91] have concluded that EP-based sensing has a more favorable bound in the quantum regime when using an EP-based amplifier near its lasing threshold. However, their analysis is based on a linearization that might be problematic due to large fluctuations near the EP, as conjectured in Ref. [92].

The first experiment observing quantum limits of EP-based sensors was done by Wang and coworkers [94]. It is an extension of their work on Brillouin laser gyroscopes [81]; remember the discussion in Section 4 and Fig. 3(f). In strong contrast to the just-discussed theoretical approaches, the experiment went well beyond the weak dispersive limit. The key observation is that the diverging frequency-splitting enhancement of the gyroscope is precisely compensated by a diverging broadening of the laser linewidths; see Fig. 5. The origin of the latter is excess quantum noise due to the nonorthogonality of the modes. The linewidth broadening factor is called Petermann factor [95–98]. Near the EP, the nonorthogonality of the involved modes becomes extreme and the Petermann factor is therefore expected to diverge at the EP [99], even though it is difficult to imagine that this can be observed directly in an experiment. The experiment and theoretical analysis of Wang et al. have shown that the quantum-limited signal-to-noise ratio is not improved for EP-based sensors, at least in this kind of system.

I have reviewed the progress on exceptional point-based sensors since the proposal of this concept in 2014 [29]. Such a sensor exploits the enhanced response of the energy eigenvalues (frequencies) at an exceptional-point singularity on external perturbations. Several theoretical studies have broadened the field of possible applications. Eight proof-of-principle experiments have already demonstrated the potential of such sensors for particle detection, thermal sensing, thermal mapping, wireless sensing, and rotation sensing.

The deleterious influence of classical and quantum noise has been discussed. Classical noise can lead to a dynamical instability of exceptional point-based sensors operating at or close to the real frequency axis, which in turn can spoil the resolvability via the necessary stabilization [88]. The fundamental limitations due to quantum noise are still under debate. The experiment by Wang et al. has revealed that for laser gyroscopes, the enhanced splitting is compensated by laser linewidth broadening due to excess quantum noise [94].

For the future it would be important to have a deeper theoretical understanding of nonlinear processes in exceptional point-based sensors such as gain saturation and laser dynamics; a first step has been done in (the supplementary material of) Ref. [80]. This understanding will lead to further improvements of the performance of sensors based on exceptional points.

While the even more enhanced frequency splittings of third-order exceptional points have already been experimentally demonstrated [74], it remains to fabricate sensors with exceptional points of even higher order. This is obviously a challenging task because of the extreme sensitivity to fabrication tolerances. A promising approach to overcoming this problem is the concept of exceptional surfaces [70].

Even though most of the applications of exceptional point-based sensors will be in the classical domain, it can be expected that recent and future developments in the study of exceptional point physics in the quantum domain, see for example Refs. [89,100], will broaden the range of potential applications of exceptional point-based sensors.

Acknowledgment. I thank J. Kullig for discussions.