Anti-PT-Symmetric Optical Gyroscope at the Transmission Peak Degeneracy With Enhanced Signal-to-Noise Ratio

Non-Hermitian sensors have been widely studied for the enhanced response of their eigenvalues in the proximity of exceptional points. However, it has been shown that noise is enhanced in the same way as sensitivity, thus vanishing the advantages of the exceptional point. Recently, the idea of measuring the frequency splitting of transmission peaks of a non-lasing PT-symmetric sensor has shown the possibility of increasing the signal-to-noise ratio of a sensor close to the so called “transmission peak degeneracy”, with respect to an operating condition away from it. Here we analyze a non-Hermitian optical gyroscope, demonstrating that also anti-PT-symmetric Hamiltonians show an enhanced transmission peaks frequency splitting in the proximity of the transmission peak degeneracy. We also perform an analysis on noise and uncertainties and introduce new figures of merit to compare the proposed anti-PT-symmetric sensor with a classical resonant optical gyroscope. When the uncertainties due to fluctuations of parameters are negligible with respect to the uncertainties due to amplitude noise, the non-Hermitian sensor working at the transmission peak degeneracy is demonstrated to operate better than a classical resonant gyroscope in terms of signal-to-noise ratio.

In 2 × 2 NH Hamiltonians working at the EP, the eigenvalue splitting depends on the square root of the applied perturbation, thus showing an enhanced sensitivity for small values of perturbations.However, recently it has been shown that the simultaneous divergence of the noise and of the sensitivity of the eigenfrequency splitting at the EP cancels the advantage of non-Hermitian sensing [28], [29], [30], [31].
To overcome this problem, a non-lasing PT-symmetric electronic accelerometer coupled to a transmission line has been realized [32].Using the frequency splitting of the transmission peaks as the output of the sensor, rather than eigenfrequency splitting, has been shown to enhance the signal-to-noise ratio.In particular, the transmission peak frequencies collapse at a design condition called "transmission peak degeneracy" (TPD), where the sensitivity diverges.In this way the authors were able to separate the condition of the divergence of the sensitivity (corresponding to the TPD) from the condition of divergence of the noise (corresponding to the EP).In this work we apply the concept of the TPD to anti-PT-symmetric Hamiltonian for angular velocity sensing.We show that also anti-PT-symmetric systems show a TPD distinct from the EP, thus making it possible to enhance the signal-to-noise ratio.Moreover, we demonstrate that it is possible to set the TPD (and consequently the maximum sensitivity) in condition of null perturbation (zero angular velocity).Then, we propose a new approach for comparing the TPD-based sensor to a classical configuration of the same sensor.In our analysis, we demonstrate that for certain operating conditions a non-Hermitian gyroscope shows better performance than a classical resonant gyroscope in terms of signal-to-noise ratio.

A. Non-Hermitian Hamiltonian
In [32] it has been experimentally demonstrated that working next to a TPD of a non-Hermitian Hamiltonian can increase the signal-to-noise ratio of a sensor.In particular, a quasi-parity-time symmetric effective Hamiltonian has been set up, in order to obtain a transmission peak degeneracy distinct from the exceptional point (EP).Here we want to demonstrate the feasibility of an anti-PT-symmetric gyroscope with the architecture proposed in Fig. 1.In order to simultaneously consider the signal and noise inputs, the device is modelled as an open system with two optical resonators and five total ports (three physical buses and two phantom buses connecting the resonators with the gain/loss reservoirs).The amplitude vector: represents the amplitudes inside the optical cavity, normalized such that |a 1(2) | 2 corresponds to the energy stored in the first (second) resonators.Using the temporal coupled-mode theory formalism for optical resonators, we can write [22], [32], [33]: with H 0 the effective Hamiltonian, K the 5 × 2 coupling matrix, C a 5 × 5 scattering matrix, x the exciting incoming waves, y the outgoing waves.In particular, H 0 , x and y can be written as (in the linear hypothesis, i.e., much below the saturation conditions) [32], [33]: In particular, ω 1(2) is the isolated resonance of the first (second) resonator, γ o represents the intrinsic loss rate (with the "+" sign) or intrinsic gain rate (with the "-" sign) of each resonator, whereas γ e is the loss rate due to the coupling with each of the two buses adjacent to each resonator.Then, s in is the input amplitude wave (normalized such that |s in | 2 represents the input power), and n i1 , n i2 and n ic are the input noise amplitudes arising from the coupling with external physical buses, whereas n γ1 and n γ2 are the input amplitudes of the noise arising from the coupling with the gain/loss reservoirs (Fig. 1).Finally, y 1 , y 2 , y c are the output amplitudes at the physical buses, and y γ1 and y γ2 are the output amplitudes at the gain/loss reservoirs.
According to the constraint CK * = -K [33], we set C and K as [32]: with (2γ e ) 1/2 the mutual coupling between each physical bus and each resonator next to it and (2γ o ) 1/2 the mutual coupling between each gain/loss reservoir and the corresponding resonator.By setting γ t = 2γ e ± γ 0 , κ 0 = γ e , ω 0 = (ω 1 + ω 2 )/2 and Δ = (ω 1 -ω 2 )/2, we obtain: In (2) H 0 plays the role of an effective Hamiltonian in unperturbed conditions (at rest), and it is anti-parity-time-symmetric (PTH 0 (PT) -1 = -H 0 ).To take into account the rotation, we add an opposite perturbation ϵ 1(2) on the resonance frequencies [17] (due to the Sagnac effect).For simplicity we will assume the two resonators to have almost the same area and perimeter, so the perturbation is considered to be the same on the two resonators (ϵ = ϵ 1 ≈ ϵ 2 ).The perturbed Hamiltonian will be expressed as: In this case, H represents the effective Hamiltonian in the perturbed condition (during rotation).

B. Non-Hermitian Gyroscope
The eigenfrequencies are immediately obtained from the secular equation (H -ω eig I = 0, with I the identity matrix) as: An EP arises for Whereas the lasing threshold is obtained for Im{ω eig } = 0: with In order to calculate the transfer function of the system when is subject to a rotation, we calculate the Green's Matrix, whose elements are: The transfer function T 12 = |y 2 /s in | 2 is obtained as: We can easily calculate the peak frequencies (ω p ), i.e., the angular frequencies for which the transfer function T 12 has transmission peaks, for: After defining ξ = ( + Δ) 2 − κ 2 0 − γ 2 t , we obtain the peak frequencies as: The transmission peaks frequencies are easily measurable at the output of the device.Two transmission peak degeneracy points (TPDs) arise for: and the peak frequencies can be written as: In this case, it is evident that the peak frequencies show a square root dependence in the proximity of = T P D1 (2) .This condition corresponds to the degeneracy of the transmission peaks.There are two interesting conditions for the design of the device: 1) The case with ϵ EP = 0 with Δ 2 = κ 2 , has been usually considered [20], [22] for the design of EP-based anti-PT-symmetric gyroscopes.However, to guarantee the maximum sensitivity of the gyroscope in the proximity of zero angular velocity, we set ϵ TPD1 = 0.This means that our design condition becomes: From an experimental point of view, the tuneability of the parameter γ t is enough to ensure the possibility of achieving the TPD for T P D1(2) = 0.

C. Signal Enhancement
In order to evaluate the performance of a TPD, we define the signal scale factor as [32]: with The SSF TPD is equal to: The SSF TPD diverges at = T P D1 (2) .In the same way, we can define a scale factor evaluating the eigenfrequency splitting as signal: with: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
It results: Clearly, the SSF EP diverges at the EP.

D. Amplitude Noise
The properties of the non-Hermitian Hamiltonian, in the presence of noise, can be analyzed by considering the amplitude noise sources in the coupled mode theory [33] with a semiclassical approach.In particular, we will consider the five different input noises shown in ( 5): three of them (n i1 , n i2 and n ic ) arise from the coupling with external physical buses, whereas two of them (n γ1 and n γ2 ) arise from the coupling of the system with the gain/loss reservoirs.By using (2), (3) (with H instead of H 0 ) and ( 14) we obtain in the harmonic regime: Using ( 5) and (28) with s in = 0 and evaluating the output signal noise at the output of the second bus, n o = y 2 , we obtain: Recalling that the properties of the noise sources are described by their correlations [32], here we calculate the output noise spectral density due to amplitude noise, with the same approach used in [32].By knowing that independent sources of noise are uncorrelated, and assuming the same input noise spectral density for the three physical buses (S n i ) and the same input noise spectral density for the two gain/loss reservoir contributions (S n γ ), we can express the output noise spectral density (S n o ) as: with

A. Uncertainties Due to Amplitude Noise
In order to evaluate the uncertainty on the peak frequency splitting detection due to the amplitude noise modelled in the previous paragraph, we have used the same approach as in [32].The readout of the sensor is based on the measurement of the transmission spectrum and on the measurement of the difference between the frequencies of the transmission peaks.The frequency uncertainty on the measurement of each transmission peak frequency is proportional to the linewidth (Γ) of the transmission peaks [32].The proportionality term between the frequency uncertainty and the linewidth depends on several parameters of acquisition and on the noise-to-signal ratio.We have assumed, as in [32], that the proportionality factor between the squared frequency uncertainty (at the transmission peaks) and the squared linewidth is dominated by the noise-to-signal ratio, evaluated at the frequencies of the transmission peaks [32], as: where the sum of the variances of the uncorrelated peak frequencies holds.Hereinafter, the proportionality coefficient between the left and right terms of (32) will be M S .The linewidth (Γ) has been calculated as:

B. Uncertainties Due to Parameters Fluctuations
Another important source of uncertainty on the peak splitting measurement is the effect of the uncertainty of Δ, κ 0 and γ t on the peak splitting.We can calculate their relevant fluctuations scale factors: So, the uncertainty of the output splitting due to the fluctuations on the l-th (with l = Δ, κ 0 , γ t ) parameter will be: So, the total uncertainty will be given by: Considering for simplicity a design with κ 0 >> γ t , we can assume (σ κ 0 Δω ) 2 (σ γ t Δω ) 2 , so that: ) where σ F Δω represents the fraction of uncertainty related to the fluctuations of κ 0 and Δ.

C. Comparison With a Classical Resonant Sensor
In order to evaluate the performance of the TPD-based sensor, we consider a new device with the same effective Hamiltonian, but with null coupling strength, thus making the two resonators uncoupled.This can be physically imagined as separating the two resonators, duplicating the intermediate bus and reading the output at the drop port of each resonator (see Fig. 2).The uncoupled effective Hamiltonian becomes: In this case the two resonators can be studied separately, leading to two different peak frequencies (equal to the real part of the eigenfrequencies, in this case): with linewidth: and the elements of the Green's matrix: The input-output transmission at the drop ports of each resonator is found as (with x in the input signal and y c1 (2) the output signal at the drop port): We can then define a signal scale factor for the uncoupled Hamiltonian: with Δω u = ω ua − ω ub .The noise amplitudes at the drop port of each resonator become: and the noise spectral density: with Considering the two uncoupled rings, each of them is supplied with half of the power (0.5|s in | 2 ) of the anti-PT-symmetric version (see Fig. 2) for comparison reason (same total input power as in the anti-PT-symmetric version).
The squared uncertainty on the difference between ω ua and ω ub is: .
(47) The proportionality coefficient between the left and right terms is M S .Now we also include the effect of fluctuations of design parameters.In this case the only effect is due to the fluctuations of Δ.We define the fluctuations scale factor in the uncoupled system (FSF u ): Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I PARAMETERS USED IN SIMULATIONS
The uncertainty on the output splitting in the uncoupled case is: Finally, the total uncertainty in the uncoupled sensor is: Eventually we define two metrics to compare the TPD-based sensor and the sensor realized with uncoupled resonator: In particular χ Δω (χ σ ) represents the enhancement of the frequency peak splitting (uncertainties) of a TPD-based sensor with respect to the uncoupled case.If the ratio between them, χ Δω/ χ σ , is higher than 1, it means that setting up a non-Hermitian system for sensing shows a better signal-to-noise ratio than a classical (uncoupled) sensor.Hereinafter, we will refer to the uncoupled version as the "classical" sensor.

IV. NUMERICAL SIMULATIONS
In this section the results of some numerical simulations are shown.In particular, we will consider, without losing any generality, the parameters in Table I.
The color plot in Fig. 3 shows the transmission T 12 as a function of the normalized frequency difference (ω-ω 0 )/ω 0 and of the perturbation ϵ applied to the resonators, due to rotation.It is evident that for ϵ > ϵ TPD = 0 the frequency peaks (red dashed lines) diverge as the square root of the perturbation.In the same graph also the eigenfrequencies are shown (black dash-dotted lines).As expected, they diverge between each other according to a square root dependence around ϵ = ϵ EP .
Fig. 4 shows the signal scale factor of the peak frequency splitting (SSF TPD ) and the signal scale factor of the eigenfrequencies (SSF EP ).As expected, the signal scale factor related to the eigenfrequencies diverges at the exceptional point.And it is evident that the SSF TPD diverges at a different value of the perturbation (ϵ TPD ), so explaining the advantage of using the frequency splitting between transmission peaks as the output of the sensor.Fig. 5 shows the output noise spectral density, divided by the sum of the power spectral density at the input bus and at the gain/loss reservoir (S n i + S n γ ), for two values of S n γ /S n i .It can be noted that the output noise at the transmission peak frequencies diverges at the lasing frequency.The curves in Fig. 5 for different ratios S n γ /S n i result to be superposed, because of the simulation hypothesis κ 0 >> γ t .
The bi-logarithmic graph in Fig. 6 shows the different uncertainties on the peak frequency splitting in the non-Hermitian gyroscope (solid lines) and in the classical one (dashed lines), as a function of the perturbation in the proximity of the TPD ( → TPD ).All the uncertainties are normalized to the central frequency ω 0 and the ones due to amplitude noise (σ   values for these relative uncertainties of the design parameters (σ κ0 /κ 0 and σ Δ /Δ) have been chosen according to literature: the measurements in [34] suggest a worst-case approximation of the relative uncertainty of the coupling strength of an InP-based coupler lower than 1%, while the measurements in [35] suggest a worst-case approximation for the relative uncertainty of the resonance of a silicon resonator (proportional to the relative uncertainty of resonance half difference Δ) of 005%.Thus, we can conclude that the analyzed range of relative uncertainties of parameters (from 0.05% to 5%) in Fig. 6 can be representative of experimental data of resonators in different mature technologies.In order to compare the effects of the uncertainty on the peak frequency splitting due to amplitude noise and the one due to parameters fluctuations, the ratio S n γ /S n i is set to 10 and |s in | 2 /(S n i + S n γ ) is set to 20 dB in all the simulations.From Fig. 6 it can be seen that for ϵ >> κ 0 the uncertainty on the peak frequency splitting due to amplitude noise in the NH case (σ N Δω ) increases for increasing values of ϵ.Intuitively, this trend depends on the fact that for ϵ >> κ 0 the transfer function T 12 (equal to |y 2 /s in | 2 ) reduces as the perturbation increases (see (16) and ( 32)).
It is worth noting that by considering y 1 as the output in both the configurations (see Fig.

Fig. 1 .
Fig. 1.Rendering of the non-Hermitian optical gyroscope with labels indicating the input and output signals.

Fig. 2 .
Fig. 2. Schematic of the anti-PT-symmetric gyroscope (a) and of the uncoupled architecture with two resonators (b), with labels of input signals and noises, and output signals.

Fig. 4 .
Fig. 4. SSF TPD and SSF EP as a function of the normalized perturbation.

Fig. 5 .Fig. 6 .
Fig.5.Output noise spectral density at the transmission peak frequencies with a semiclassical approach divided by the sum of the power spectral density at the input bus and at the gain/loss reservoir (S n i + S n γ ), for two values of the ratio S n γ /S n i .

Fig. 7 .
Fig. 7. Ratio between χ Δω and χ σ for different values of uncertainties due to fluctuations of parameters, for a ratio S n γ /S n i equal to 10 and |s in | 2 /(S n i + S n γ ) equal to 20 dB.Above unity (dashed line) the NH sensor has a better signal-to-noise ratio than a classical sensor.The simulations are performed for M S = 1.To estimate the linear dependence of χ Δω /χ σ from M S , for → TPD , (51) and (46) can be used.

2
(a) and (b)), the transfer function and the uncertainties on the peak frequency splitting tends to be equal for the NH and the uncoupled configurations when ϵ >> κ 0 , thus confirming the validity of the model.Finally, Fig. 7 shows the ratio between χ Δω and χ σ , assuming S n γ /S n i = 10 and |s in | 2 /(S n i + S n γ ) = 20 dB.The ratio χ Δω /χ σ is a useful way to compare the performance of a NH anti-PTsymmetric gyroscope with classical resonant gyroscopes.A ratio higher than unity demonstrates the clear advantage of the use of a NH gyroscope with respect to a classical one.In simulations, different values of the relative uncertainty on the parameter Δ have been considered for the classical ring resonator gyroscope.