Fluctuations and noise-limited sensing near the exceptional point of $\mathcal{PT}$-symmetric resonator systems

We theoretically explore the role of mesoscopic fluctuations and noise on the spectral and temporal properties of systems of $\mathcal{PT}$-symmetric coupled gain-loss resonators operating near the exceptional point, where eigenvalues and eigenvectors coalesce. We show that the inevitable detuning in the frequencies of the uncoupled resonators leads to an unavoidable modification of the conditions for reaching the exceptional point, while, as this point is approached in ensembles of resonator pairs, statistical averaging significantly smears the spectral features. We also discuss how these fluctuations affect the sensitivity of sensors based on coupled $\mathcal{PT}$-symmetric resonators. Finally, we show that temporal fluctuations in the detuning and gain of these sensors lead to a quadratic growth of the optical power in time, thus implying that maintaining operation at the exceptional point over a long period can be rather challenging. Our theoretical analysis clarifies issues central to the realization of $\mathcal{PT}$-symmetric devices, and should facilitate future experimental work in the field.

Fluctuations and noise-limited sensing near the exceptional point of parity-time-symmetric resonator systems 1

. INTRODUCTION
Pronounced sample-to-sample fluctuations constitute a hallmark of mesoscopic physics [1], where the finite number of degrees of freedom limits the self-averaging common to macroscopic systems. In mesoscopic systems, the interactions of waves with disordered potentials lead to many fascinating phenomena [2], including Anderson localization [3], weak localization [4], and universal conductance fluctuations [5]. A typical playground for such effects is many-body electron physics, which is rich in mesoscopic fluctuations [6][7][8][9]. Another class of interesting systems can be found in optics, with intriguing examples including random lasers [10] and quantum optical entanglement in multiple-scattering media [11], as well as cavity-quantum electrodynamics [12] and nanolasing [13] with Anderson localized states. Traditionally, many mesoscopic wave-interference phenomena have been explored using the tight-binding model of condensedmatter physics [14], while its optical analog-coupled-mode theory (CMT) [15]-has fostered the exploration of systems consisting of coupled resonators, with an emphasis on long chains (waveguides) [16], and the rich interplay of slow-light phenomena with the presence of both loss and gain [17] as well as disorderinduced Anderson localization [18]. While the quantum dynamics is commonly governed by Hermitian equations of motion, the electrodynamics of optical systems is in general non-Hermitian due to the inevitable presence of material absorption, but also the possibility of introducing optical gain. However, recent years have witnessed not only efforts to realize loss compensation in optical metamaterials [19], but also the possibility to enable parity-time (PT )-symmetric systems [20,21], where eigenvalues can be real despite the non-Hermitian aspects of the governing equations [22].

MODEL
Here, we turn to finite PT -symmetric systems and illustrate interesting new mesoscopic fluctuations of the spectral properties near the exceptional point (EP) with coalescing eigenstates. For the transparency of our illustration, we consider a problem of two coupled resonators, as illustrated in Fig. 1. Within CMT, the dynamics is governed by a Schrödinger-like equation: where ω a , ω b , a, and b are the resonance frequencies of the uncoupled resonators and the amplitudes of their respective modes, κ is the coupling parameter (which can be chosen real-valued), and g characterizes the gain and damping of the two resonators.
For convenience, we have introduced symbols ψt and H for the state and the Hamiltonian, respectively. In the analysis of such a problem, it is customary to study stationary solutions (we will come back to the time evolution toward the end of this paper), i.e., the eigenvalue problem Hψ ωψ. For g 0, this constitutes a Hermitian problem and corresponds to the usual hybridization of two levels, i.e., with the bonding and anti-bonding states (notation inherited from molecular orbital bonding theory) having real eigenfrequencies and, in the ω a ω b case, being separated by an energy of 2κ.
In the presence of a finite g, the system becomes non-Hermitian, while PT -symmetry may still allow real-valued eigenfrequencies [22], depending on the strength of the gain g relative to the coupling κ. Perhaps the most notable characteristic of a PT -symmetric system is a PT -symmetry breaking transition that takes place around g∕2κ 1. In optical settings, this abrupt phase transition has been experimentally demonstrated in coupled waveguides and cavities, by measuring both the real and imaginary components of the eigenvalues, as well as by observing the evolution of the corresponding mode profile [23][24][25][26][27][28].
In order to analyze the influence of temporally fluctuating environments or sample-to-sample fluctuations associated with inevitable small variations in ω a and ω b , we shall in the following allow a small, but finite frequency detuning between the two coupled resonators. To ease our notation, we first define a normalized frequency Ω ω∕κ and center frequencyΩ ω a ω b ∕2κ, while the normalized detuning of the two resonances is denoted by Δ ω a − ω b ∕2κ. The eigenvalue problem Hψ ωψ now takes the form where G g∕2κ is the normalized parameter central to the analysis of EPs in this problem. By straightforward diagonalizing, we get with corresponding eigenvectors Obviously, the eigenfrequencies of the coupled system can in general be complex, i.e., Ω Ω 0 iΩ 00 , and we immediately see how detuning enters simply as an imaginary part of the gain parameter: G 2 → G iΔ 2 . Equations (2b) and (2c) nicely illustrate how both the eigenvalues and the eigenstates coalesce (Ω Ω − and ψ ψ − ) when the square root vanishes, forming an EP. If the two uncoupled resonators are perfectly aligned (Δ 0), this occurs for G 1, where the gain and loss are exactly balanced by an appropriate coupling constant. Under realistic experimental conditions, the inherent material loss can always be compensated by carefully adjusting the gain, e.g., through electrical pumping of one of the resonators [29]. However, no matter the efforts spent in fabricating resonators with similar resonance frequencies, there will always be some small, yet inevitable frequency detuning. Moreover, this detuning will vary from sample to sample. In a particular sample, the detuning is also likely to fluctuate over time due to unavoidable fluctuations in the environment. In this paper, we study the interplay of such sample-to-sample fluctuations and its behavior and magnitude near EPs. We also consider possible implications of fluctuating environments for the exploration of EPs in sensing [30]. While we here focus on fluctuations in Δ, it is clear from Eq. (2a) that fluctuations in G would have quite similar implications.

A. Below the Exceptional Point
For low gain (G ≪ 1), we have to leading order in the detuning that the eigenvalues become In the ideal case (Δ 0), this regime is characterized by a realvalued spectrum, i.e., Ω Ω 1. However, for a small, but finite detuning, the imaginary part is finite despite the symmetric gain-loss arrangement. In other words, the finite detuning breaks the PT -symmetry associated with perfectly aligned resonators [20]. This is also immediately clear by noticing that the Hamiltonian does not equal its adjoint, i.e., H ≠ H † .

B. Fluctuations Near the Exceptional Point
In order to see that detuning changes the conditions for having an EP, we expand the exact expression [Eq. (2b)] around the EP G 1; to leading order in Δ, we get [31] The detuning lifts the degeneracy that would otherwise be associated with the EP of two perfectly aligned resonators (Δ 0). Away from the EP, systems are commonly affected linearly by perturbation. However, the fact that the splitting scales as ffiffiffi ffi Δ p is an interesting manifestation of the system being very susceptible to perturbations near the EPs [32,33]. Obviously, this can be used to our advantage in the context of optical sensors [30], but has the Fig. 1. Schematic illustration of a PT -symmetric dimer formed by two identical (no frequency detuning) coupled optical resonators, but with opposite values of the gain-loss parameter G≡g∕2κ. In the absence of gain-loss (G 0), the two resonators form common hybridized states with splitting by 2κ, while for the exceptional point (EP) at G 1, the system is degenerate, i.e., Ω Ω − , with coalescing eigenstates.
Research Article natural drawback that the system is also very sensitive to any undesired, yet practically inevitable degrees of freedom associated with fabrication imperfections or fluctuating environments (e.g., temperature shifts or noise in the gain parameter).
We now assume an ensemble of resonator pairs with a Gaussian distribution of the detuning parameter: This can be interpreted either as fabrication tolerance or as temporal fluctuations assuming that an ergodic approximation to the system dynamics is valid. In order to appreciate the dramatic effect this has on the spectrum especially near the EP, we show in Fig. 2 the distribution of the real and imaginary parts of the eigenvalues, where the variance σ 0.1 was chosen sufficiently small, so that the common regime with G 0 is only slightly broadened. However, in the vicinity of the EP (and beyond), we observe a very pronounced smearing of the spectral features.
Pursuing a deeper understanding of this numerical observation, we proceed with analytical calculations based on the leading-order correction in Eq. (4). The eigenvalue Ω is not a convenient quantity to study at an isolated point of degeneracy that is lifted by a statistical process. Instead, we shall focus on the splitting of the eigenvalues' real part: It should be noted that corresponding expressions for fluctuations in the gain coefficient and for the splitting of Ω 00 are very similar. Its ensemble average is Given the Gaussian distribution of the detuning, the distribution of eigenfrequencies at the EP can now be evaluated: where F x 8 π 1∕2 jxj exp− 1 2 x 4 . This approximate universal distribution shown in Fig. 3 illustrates an interesting ensemble-averaged broadening of levels inside the gap, i.e., a PΩ 0 ∝ jΩ 0 −Ωj for energies smaller than the detuning. Within the square root scaling law, PΩ 00 is distributed in the same manner; the cuts through the EP (solid blue curves) of the two panels in Fig. 2 are nearly identical, but not quite, due to the finite σ.

C. Sensitivity of Fluctuating Sensors
It is not entirely surprising that statistical detuning leads to a nonzero average eigenvalue splitting. The natural next question is how this affects the performance of a sensor, i.e., how the average splitting hΣi reflects additional, non-fluctuating detuning. We now assume that this detuning parameter has two contributions: first a fluctuating detuning due to unintended noise, which is inevitably present in any realization of such systems, and second, the signal Δ 0 that is meant to be detected or sensed. For the detuning probability distribution the sensitivity of the time-averaged frequency splitting can now be written as

Research Article
Here, the integral can be approximated in small-and large-signal limits: and in Fig. 4, we show these asymptotic behaviors (dashed lines) along with a full numerical evaluation of the integral (red line). The integral is always smaller than unity, implying that the sensitivity is noise limited, i.e., ∂hΣi∕∂Δ 0 < σ −1∕2 . The sensitivity should be contrasted to the case in the absence of fluctuations, where there is a tremendous sensitivity to small signals, i.e., ∂Σ∕∂Δ 0 Δ −1∕2 0 . Indeed, from Eq. (11), we recover this result for σ → 0. On the other hand, it is quite clear how the sensitivity vanishes linearly in the low-signal limit, where the perturbation is dressed by the noise. While figures of merit are commonly adapted to quantify the performance of linearly responding sensors (see, e.g., [34,35]), the power-law response of EPs makes it impossible to directly adapt such a linear concept. To illustrate the constrained sensitivity due to noise, we here use the limiting case (σ → 0) to introduce the sensitivity-diminution factor (Δ 1∕2 0 ∂hΣi∕∂Δ 0 ) shown in Fig. 4 (filled curve). There is a clear diminution of the sensitivity to perturbations for Δ 0 < σ, while the noise-less sensing performance is recovered only for Δ 0 ≳ 2σ.

D. Time Evolution
The use of EPs in highly sensitive sensors is seriously hampered by the low-frequency tail of the temporal fluctuations of detuning and gain. This tail is called drift and must be compensated by a feedback loop, i.e., the sensor is kept at the EP by constantly adjusting detuning and pump power, and the actually measured quantities are the values of these feedback variables (e.g., pump power for the gain or heating currents for the detuning). For this, it seems necessary to keep the sensor at the EP over an extended period of time. Naively, this seems trivial, because the eigenstate has a real eigenvalue, and one would therefore expect the timeevolution to be stationary and neither growing nor decaying in time. In reality, this is not the case.
At the EP, the equation of motion [Eq. (1)] reads where τ κt is the dimensionless time variable. We can solve this formally using the time-evolution operator [36]: ψτ U 0 τψ0 exp−iH 0 τψ0: The matrix exponential can be simplified by decomposing H 0 ΩI A, where I is the unit matrix, and A −i 1 1 i .
Since I and A commute, we find because A 2 vanishes (A is nilpotent). This dynamics is highly reminiscent of the critically damped classical harmonic oscillator, whose time evolution is a superposition of h 1 t ≃ exp−γt and h 2 t ≃ t exp−γt. Indeed, the critically damped harmonic oscillator formulated as two coupled first-order differential equations results in an EP in the coupling matrix and constitutes a beautiful didactic example for this phenomenon [37]. Equation (14) has several important implications. First, it means that the overall optical power is not conserved when operating at an EP. Instead, the optical amplitudes in general grow linearly, and the optical energy therefore grows quadratically in time.
This makes it rather difficult to keep a sensor at an EP for an extended period of time, with dynamics being sensitive to initial excitation conditions [36]. Second, this demonstrates nicely that having a Hamiltonian with only real eigenvalues is not sufficient to ensure energy conservation [38]. To address the former issue, in situations in which high sensitivity is desired, one may encounter a daunting scenario of being in constant need for continuously monitoring and correcting the system so as to bring it back at the EP. In practice, however, this task can be carried out in a considerably less demanding fashion by using the phase transition associated with the EP as a means to eliminate the requirement for constant correction. For example, by modulating the magnitude of the gain around the nominal value for EP and by monitoring the output signal, one can determine the parameter range where the derivative of the response over time reaches its extremum [39,40].

DISCUSSION AND CONCLUSION
So far, we have discussed the classical electrodynamics at EPs of PT -symmetric systems, where the spectrum can be real despite the presence of both loss and gain. We have emphasized mesoscopic fluctuations of classical origin, while we speculate that also quantum optics and quantum fluctuations would experience dramatic enhancement near the EP. Indeed, quantum emitter dynamics in the presence of EPs is in itself interesting [31]. In the present context, we note that loss-compensated metamaterials do not necessarily exhibit the dynamics of ideal loss-less structures when probed with quantum optics [41], and as such, there might also be interesting quantum fluctuation properties to be explored in the vicinity of EPs. Focusing here on the role of mesoscopic fluctuations and noise on the spectral and temporal properties of systems of PT -symmetric coupled gain-loss resonators operating near the EP, we have shown that the inevitable detuning in the frequencies of the uncoupled resonators leads to modified conditions for reaching the EP. In ensembles of resonator pairs, statistical averaging significantly smears the spectral features, which leaves sensitivity of EP-based sensors noise limited. Finally, we have shown how Research Article temporal fluctuations in the detuning and gain of such sensors lead to a quadratic growth of the optical power in time, suggesting that maintaining operation at the EP over a long period might be a formidable task. Since acceptance of this paper, we have become aware of related work in Ref. [42].