## Abstract

Engineered non-Hermitian systems featuring exceptional points (EPs) can lead to a host of extraordinary phenomena in diverse fields ranging from photonics, acoustics, opto-mechanics, and electronics to atomic physics. In optics, non-Hermitian dynamics are typically realized using dissipation and phase-insensitive gain accompanied by unavoidable fluctuations. Here, we introduce non-Hermitian dynamics of coupled optical parametric oscillators (OPOs) arising from phase-sensitive amplification and de-amplification, and show their distinct advantages over conventional non-Hermitian systems relying on laser gain and loss. OPO-based non-Hermitian systems can benefit from the instantaneous nature of the parametric gain, noiseless phase-sensitive amplification, and rich quantum and classical nonlinear dynamics. We show that two coupled OPOs can exhibit spectral anti-parity-time (anti-PT) symmetry and a EP between its degenerate and nondegenerate operation regimes. To demonstrate the distinct potentials of the coupled OPO system compared to conventional non-Hermitian systems, we present higher-order EPs with two OPOs, tunable Floquet EPs in a reconfigurable dynamic non-Hermitian system, and the generation of a squeezed vacuum around EPs, all of which are not easy to realize in other non-Hermitian platforms. We believe our results show that coupled OPOs are an outstanding non-Hermitian setting with unprecedented opportunities to realize nonlinear dynamical systems for enhanced sensing and quantum information processing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Non-Hermitian systems with engineered gain and dissipation have attracted a lot of attention thanks to remarkable properties and functionalities that are absent in their counterparts based on closed Hermitian setups [1,2]. A plethora of phenomena have been spawned by judiciously manipulating these non-Hermitian physical systems; namely, spontaneous parity-time (PT) symmetry breaking [1], unidirectional invisibility [3], coherent perfect absorption and lasing [4,5], single-mode lasing [6], and generation of structured light with a controllable topological charge of the orbital angular momentum mode [7], to name a few.

Non-Hermitian systems are often characterized by the presence of an exceptional point (EP), where the eigenvalues and eigenvectors simultaneously coalesce (non-Hermitian degeneracies), and have been explored in the context of PT-symmetric systems with balanced gain/loss and even in purely dissipative arrangements. The presence of an EP leads to several counterintuitive phenomenon, including loss-induced lasing [8,9], a breakdown of adiabaticity [10,11], and lasing without inversion [12]. However, most non-Hermitian optical systems realize gain/dissipation by deploying laser gain, which limits its viability in certain spectral regions [13].

Here we use parametric amplification and de-amplification in coupled optical parametric oscillators (OPOs) to implement EP in parametric non-Hermitian systems [14–16], thereby presenting a system that can exhibit unique phenomena not observed in their laser-gain-based counterparts. Parametric non-Hermitian systems can extend beyond the spectral coverage of laser gain [17], and the instantaneous nature of parametric gain also enables the realization of tunable/reconfigurable non-Hermitian systems that are otherwise difficult to achieve in conventional optics-based non-Hermitian setups. We leverage this tunable aspect of parametric gain to realize interesting functionalities. Fundamentally, the presented OPO-based non-Hermitian system is in sharp contrast with conventional optical systems and can enable unique opportunities for sensing, non-Hermitian nonlinear dynamics, and quantum information processing.

EPs in non-Hermitian systems have been extensively studied for potentially enhanced sensing capabilities [18–20]. In spite of the underlying high sensitivity near an EP, this class of sensors relying on resonant frequency splitting are not capable of improving the signal-to-noise ratio (SNR) because of the non-orthogonality of the eigenvectors near an EP [21–23]. This leads to Petermann factor-limited sensing [24], where the noise is enhanced proportionally to the signal enhancement, thereby limiting the efficacy of this class of sensors for quantum-limited sensing [25]. Fluctuations accompanying the gain/dissipation in conventional non-Hermitian systems limits the achievable precision. In fact, it has been shown that any linear reciprocal sensor is bounded in terms of SNR performance, and the conventional EP-based sensing cannot surpass this limit [21]. Recently, a sensing protocol that does not measure the eigenfrequency splitting but rather measures the superposition of output quadratures using heterodyne detection, has shown the possibility to alleviate the problem of noise enhancement and realize EP enhanced sensing when operated near the lasing threshold [26]. The noiseless nature of phase-sensitive degenerate parametric amplification motivates studying non-Hermitian dynamics of coupled OPOs for sensing. In this regard, we explore the possibility of reduced uncertainty of fluctuations manifested in the form of squeezed noise in the vicinity of parametric EP to leverage the high sensitivity of EP in the pursuit to obtain a high SNR. It must be noted that phase-sensitive, parametric-gain-based systems are not bounded by the limit outlined in [21].

Non-Hermitian dynamics of coupled OPOs can be extended to the nonlinear regime, which can lead to several intriguing possibilities. It has been previously shown that the interplay of nonlinearity and gain/loss in conventional non-Hermitian systems can result in unidirectional transport [27], one parameter family of solitons [28] (in contrast to isolated attractor-based dissipative solitons) in PT-symmetric systems, and robust wireless power transfer [29]. Previous studies implementing parametric amplification to realize non-Hermitian systems have only focused on the linear dynamics [14–16]. We exploit rich nonlinear dynamics in our parametric non-Hermitian system (operating in the parametric oscillator regime) arising from the interplay of phase-sensitive gain and the gain saturation owing to the signal-to-pump back conversion.

The presented coupled OPO system is also an appealing platform to investigate quantum non-Hermitian physics. Previous studies of the quantum behavior in non-Hermitian systems have identified the criticality of information flow between the system and environment around the EP in a PT-symmetric system [30], shift of the position of the Hong–Ou–Mandel dip [31], and delay of entanglement sudden death (ESD) near an EP [32]. Opto-mechanical systems provide a versatile test bed to study non-Hermitian dynamics in the quantum regime [33]. We demonstrate nonclassical behavior, including quadrature squeezing and tunable squeezing of the parametric EP. These behaviors may also be extended to the non-Gaussian regime [34].

## 2. MODEL OF COUPLED OPOs

We consider a system of evanescently coupled degenerate OPOs, as illustrated in Fig. 1(a). The coupled-mode equations governing our system are given by

The OPOs considered are phase matched to oscillate around the half-harmonic frequency [35]. The continuous-wave (CW) pump is nonresonant and its dynamics are adiabatically eliminated. The signal field envelopes in the two resonators are designated by $a$ and $b$, respectively. The signal in the first resonator experiences a round-trip loss (${\rm intrinsic} + {\rm out} - {\rm coupling}$) of ${\gamma _1}$, a detuning of ${\Delta _1}$, and a parametric gain of $g$ provided by the nonresonant pump. The gain can be assumed constant for the frequency range of interest around the half-harmonic frequency. The parametric gain is phase sensitive, and the phase of the pump driving the first resonator is taken as a reference. The gain saturation term is denoted by ${g_{{s_1}}}$, which originates from the signal-to-pump back conversion due to second-harmonic generation, which is the reverse of the down-conversion process. The strength of the dispersive coupling is represented by $\kappa$. Similar terms that appear in Eq. (1b) describe the associated quantities in the second resonator. The pump driving the parametric interaction in the second resonator is phase shifted by $\phi$, as compared to the first pump. The parametric gain is proportional to the pump strength and is given by $f$. Both the pumps are at $2{\omega _0}$, where ${\omega _0}$ is the half-harmonic frequency. The time scale is normalized to the round-trip time.

We assume that the resonators are identical in terms of the loss ($\gamma$) and gain-saturation (${g_s}$) terms for simplicity. This can be achieved by accessing the two degrees of freedom of a single resonator; namely, the clockwise and counterclockwise propagation modes. In the absence of these assumptions, the results discussed in this work will still hold true, albeit with some quantitative differences.

## 3. LINEAR DYNAMICAL ANALYSIS

There are two regimes of parametric oscillation: namely the nondegenerate regime and the degenerate regime [36]. In the degenerate regime, the system oscillates at ${\omega _0}$, while in the nondegenerate regime, owing to energy conservation constraint, the system oscillates with symmetric sidebands centered around ${\omega _0}$. First, we consider that both the half-harmonic signals are on resonance (i.e., ${\Delta _1} = {\Delta _2} = 0$).

In the nondegenerate regime [under the scope of linearized analysis (i.e., ignoring gain saturation)], we can consider the following ansatz for the signal envelopes in the two resonators as

where $A$ and $B$ represent the complex envelopes for the symmetric primary sidebands for resonator 1, and $C$ and $D$ represent the same for resonator 2. Here, the real part of the eigenvalues (${\lambda _R}$) corresponds to the spectral splitting, while the imaginary part (${\lambda _I}$) is related to the growth/decay rate. $A,C$ can also be read as the signals and $B,D$ as the idlers. The eigenvalues can be obtained fromThe underlying Hamiltonian of the coupled OPO system exhibits spectral anti-PT symmetry [14]. The Hamiltonian governs the dynamics as $i\frac{{d\tilde V}}{{dt}} = H\tilde V$, where $\tilde V = {[{\tilde A,{{\tilde B}^*},\tilde C,{{\tilde D}^*}}]^T}$, $\tilde A = A{e^{({\lambda _I} - i{\lambda _R})t}}$, $\tilde B = B{e^{({\lambda _I} + i{\lambda _R})t}}$, $\tilde C = C{e^{({\lambda _I} - i{\lambda _R})t}}$, and $\tilde D = D{e^{({\lambda _I} + i{\lambda _R})t}}$. The discrete symmetry of the system can be expressed as ${P_1}{P_2}TH = - H{P_1}{P_2}T,$ where $T$ is the time reversal operator, and the parity operators action in the spectral domain is defined by ${P_1} = \{A \leftrightarrow {B^*}\}$ and ${P_2} = \{C \leftrightarrow {D^*}\}$. The system dynamics is also unitarily equivalent to a PT-symmetric system, where the unitary transformation is $\mathbb{U} = \frac{1}{{\sqrt 2}}\left[{\begin{array}{*{20}{c}}1&{- 1}&0&0\\0&0&1&{- 1}\\1&1&0&0\\0&0&1&1\end{array}}\right]$, such that ${H_{{\rm PT}}} = \mathbb{U}H{\mathbb{U}^\dagger}$. This mapping is shown schematically in Figs. 1(c) and 1(d). The signals of the two OPOs and the idlers are coupled by the evanescent linear coupling $\kappa$, and the signal and the idler within the same OPO are coupled nonlinearly by the nonlinear phase-sensitive coupling engendered by ${\chi ^{(2)}}$. Under the said unitary transformation ($\mathbb{U}$), this process can be mapped to a PT-symmetric system of coupled synthetic resonators with the positive superposition of the signal and the idler conjugated fields experiencing amplification, while the negative superposition of the signal and idler fields get de-amplified. It should be noted that due to the onset of nonlinearity arising from back conversion (${g_s}$), additional sidebands will appear in the complete nonlinear solution.

In the degenerate regime, the signals in both resonators are half-harmonics. Here we can express the signal evolution in terms of their quadrature components. We define ${X_1} = (a + {a^*}),{Y_1} = \frac{{a - {a^*}}}{i}$ and ${X_2} = (b + {b^*}),{Y_2} = \frac{{b - {b^*}}}{i}$. These quadrature components evolve as ${e^{{\lambda _I}t}}$. The eigenvalues can be obtained from the following evolution equation:

The transition from the nondegenerate oscillation regime to the degenerate oscillation regime is marked by the presence of an exceptional point. This point in the parameter space is characterized by the simultaneous collapse of eigenvectors and the coalescence of the eigenvalues. The disparity between the geometric and the algebraic multiplicity at the exceptional point is determined by the order of the exceptional point.

## 4. RESULTS

#### A. Classical Mean-Field Regime

The threshold for parametric oscillation in the coupled OPO is determined by the linear eigenvalues (i.e., ${\lambda _I} = 0$), with oscillation occurring for ${\lambda _I} \gt 0$ (see Supplement 1). This extra caution is because of the possibility of occurrence of oscillation self-termination (see Supplement 1) analogous to laser self-termination [8,9]. Just above threshold, the system of coupled OPOs can oscillate either in nondegenerate [Figs. 2(a) and 2(b)] or in degenerate mode [Figs. 2(c) and 2(d)]. However, far above the threshold, the effect of nonlinearity becomes significant and the system is no longer governed by linearized dynamics [Eqs. (2) and (3)]. In this regime, nonlinearity can induce a phase transition from nondegeneracy to degeneracy as shown in Figs. 3(b) and 3(c), similar to laser systems [37,38]. This transition resembles a soft/supercritical bifurcation. Analytical results depicting this phenomenon for a representative case is shown in Supplement 1.

The phase-sensitive nature of parametric gain provides an additional tuning knob in the form of a phase difference between the two driving pumps ($\phi$) that do not exist in the conventional phase insensitive gain/loss-based non-Hermitian systems. Figure 3(a) illustrates the solution space as the phase difference is varied, identifying the degenerate and the nondegenerate oscillation regimes.

The order of the EP determines the rate of eigenvalues splitting in the presence of a perturbation away from the EP [1]. If the perturbation appears in the form of detuning ($\delta \Delta$), then the splitting depends as $\delta (Re(\lambda)) = (\delta \Delta {)^{\frac{1}{n}}}$, where $n$ is the order of the EP. This leads to enhanced sensitivity in the proximity of an EP, which is given by $\frac{{d(\delta (Re(\lambda)))}}{{d(\delta \Delta)}} \sim {(\delta \Delta)^{\frac{{1 - n}}{n}}}$. This sensitivity function diverges at the EP, which is the basis for the enhanced sensitivity of the EP-based sensors [18–20]. This scaling law arising due to the branch point singularity nature of nonHermitian degeneracies does not arise in the case of Hermitian degeneracies characterized by the diabolical points. We present the occurrence of both a second-order EP and higher-order EP (fourth order) in the coupled OPO system. The second-order EP is accompanied by the collapse of eigenvalues and eigenvectors in pairs, and is shown in Figs. 4(a) and 4(b) by considering $f = g$, ${\Delta _1} = {\Delta _2} = 0$. We identify a family of higher-order EPs (see Supplement 1), by biasing the coupled OPO’s at a suitable detuning. In Figs. 4(c) and 4(d), we considered $f = mg$, ${\Delta _2} = 0$, where $m$ is the parameter that describes the family of exceptional points that determines the critical $g$ and ${\Delta _1}$ for the occurrence of the fourth-order EP. In this case, four eigenvectors and eigenvalues coalesce, resulting in higher-order dependence of sensitivity. This enhanced sensitivity of the fourth-order EP is reflected in the slope of the log–log plot in Fig. 4(d), as compared to the case in Fig. 4(b) corresponding to a second-order EP.

The instantaneous nature of parametric gain and the ability to modulate the gain by applying phase/amplitude modulation to the pump opens unprecedented avenues to explore time modulated dynamic non-Hermitian systems in the coupled OPO arrangement. Time periodic Floquet non-Hermitian systems have been used to tailor the EP and realize reconfigurable non-Hermitian systems with an enriched phase space, depending on the amplitude and frequency of the modulation [39–41]. Previous demonstrations relied on periodically modulating the coupling to realize Floquet-driven systems. Parametric non-Hermitian systems enable us to modulate the gain instead of the coupling, by modulating the pump, and then realize a tunable Floquet EP. In Figs. 5(a) and 5(b), we explore Floquet control of the EP when the pump is amplitude-modulated as $g = {g_0} + F{\sin}(\omega t)$. The eigenvalues of the Floquet periodic system can be extracted by analyzing the associated Monodromy matrix. As shown in Fig. 5(b), with increasing values of the pump amplitude modulation parameter $F$, the EP is progressively shifted to higher values of ${g_0}$.

Similarly, we can dynamically encircle the EP by periodically modulating the parametric gain. Dynamical encirclement involves adiabatically tracing a close path in the parameter space enclosing an EP, which has been used to realize robust and asymmetric switching [11], nonreciprocal energy transfer [42] and an omnipolarizer [10]. However, these promising results have only been demonstrated in lossy systems [11,43], due to the stringent requirement of non-Hermitian system tunability. Here, we propose that the tunable nature of the parametric gain provides a very promising platform to realize the chiral dynamics that are contingent to the topological structure of the EP. We perform adiabatic encirclement in the parametric space [Figs. 5(c) and 5(d)] of detuning and gain by undergoing the adiabatic evolution, $f = g = {g_0} + r{\cos}(\omega t)$ and ${\Delta _1} = r{\sin}(\omega t)$, where $r$ is the radius of encirclement, and ${g_0} = \kappa$ is the EP. Due to the breakdown of adiabaticity in a non-Hermitian system, we obtain an asymmetric/chiral behavior, where the final state at the end of the encirclement depends on the direction of the loop and is independent of the starting point. The distinct outcome by parametrically traversing a loop enclosing the EP counterclockwise [Fig. 5(c)] and clockwise [Fig. 5(d)] is shown.

#### B. Quantum Regime

OPOs have been the workhorses to generate quantum states of light for decades [44], and coupled OPOs have also been predicted to exhibit nonclassical properties [45]. When we approach the EP from below threshold, the vacuum fluctuations in the quadratures of the intracavity field can be squeezed below the standard noise limit. We assume the vacuum fluctuations entering the cavity from different open channels, to be delta-correlated white Gaussian noise and obtain the power spectral density of the output quadrature fields via a linearized analysis of the Langevin equations [46]. The formalism including the relevant noise operators is derived as shown in Supplement 1, Section 6. Figures 6(a) and 6(b) show that there exists a bandwidth where the intracavity field is squeezed as we approach the EP. The reduced noise in one quadrature is accompanied by increased uncertainty (antisqueezing) in the conjugate quadrature. Although the maximum squeezing attainable in the vicinity of EP is 3 dB (see Supplement 1), it can potentially allow the combination of the high sensitivity of the EP and the reduced uncertainty in the parametric EP to realize unparalleled sensing capabilities in an optimum sensing arrangement. The amount of squeezing depends on the OPO escape efficiency, which is the ratio between the outcoupling loss and the total round-trip loss, with higher squeezing attainable with a larger escape efficiency. More so, one can tune the squeezing response by changing $\kappa$ in the coupled OPO, as shown in Fig. 6(b), thereby operating at a frequency where external/technical noise of the sensing system is minimum. In response to a perturbation in the form of detuning, only the optimum quadrature for squeezing is rotated, still preserving the noise reduction property (see Supplement 1). In this regard, our parametric EP can pave the way for ultrasensitive detection with high SNR in shot-noise limited detection scenarios.

## 5. CONCLUSION

In summary, we have presented EP in parametrically driven coupled OPOs. We identified the presence of two distinct phases of oscillation, namely the degenerate and the nondegenerate, and have shown nonlinearity-induced phase transition. We discussed the potential benefits of a tunable parametric gain in realizing dynamically modulated non-Hermitian systems. The nonclassical behavior of the parametric EP is presented, and its implications in highly sensitive/high SNR detection are highlighted.

Recent developments in the realization of large-scale, time-multiplexed OPO networks [47] and integrated lithium-niobate-based devices [48] can be ideal candidates for experimental realization of the presented concept. Entanglement can be used as a resource to increase the sensor performance [25] based on parametric EP. An optimum sensing arrangement guided by quantum Fisher information calculations (Supplement 1, Section 8) must be designed to obtain high SNR sensing from parametric EP [21,26]. In addition, the enhancement provided by the higher-order parametric EP and the limits of sensors based on them, including their dynamic range, is worth exploring and will be subjects for future investigations. It will be interesting to extend it to the case of a lattice of parametric oscillators, where interesting nonequilibrium dynamics is expected [49].

## Funding

National Science Foundation (1846273, 1918549); Army Research Office (W911NF-18-1-0285); U.S. Department of Defense (N00014-17-1-3030).

## Acknowledgment

The authors wish to thank NTT Research for their financial and technical support.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

## REFERENCES

**1. **M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science **363**, eaar7709 (2019). [CrossRef]

**2. **Ş. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. **18**, 783–798 (2019). [CrossRef]

**3. **Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. **106**, 213901 (2011). [CrossRef]

**4. **S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A **82**, 031801 (2010). [CrossRef]

**5. **Z. J. Wong, Y.-L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics **10**, 796 (2016). [CrossRef]

**6. **L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science **346**, 972–975 (2014). [CrossRef]

**7. **P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science **353**, 464–467 (2016). [CrossRef]

**8. **B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science **346**, 328–332 (2014). [CrossRef]

**9. **R. El-Ganainy, M. Khajavikhan, and L. Ge, “Exceptional points and lasing self-termination in photonic molecules,” Phys. Rev. A **90**, 013802 (2014). [CrossRef]

**10. **A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, and D. N. Christodoulides, “Dynamically encircling exceptional points: exact evolution and polarization state conversion,” Phys. Rev. Lett. **118**, 093002 (2017). [CrossRef]

**11. **J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature **537**,76–79 (2016). [CrossRef]

**12. **I. Doronin, A. Zyablovsky, E. Andrianov, A. Pukhov, and A. Vinogradov, “Lasing without inversion due to parametric instability of the laser near the exceptional point,” Phys. Rev. A **100**, 021801 (2019). [CrossRef]

**13. **B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. **10**, 394 (2014). [CrossRef]

**14. **D. A. Antonosyan, A. S. Solntsev, and A. A. Sukhorukov, “Parity-time anti-symmetric parametric amplifier,” Opt. Lett. **40**, 4575–4578 (2015). [CrossRef]

**15. **R. El-Ganainy, J. I. Dadap, and R. M. Osgood, “Optical parametric amplification via non-Hermitian phase matching,” Opt. Lett. **40**, 5086–5089 (2015). [CrossRef]

**16. **Y.-X. Wang and A. Clerk, “Non-Hermitian dynamics without dissipation in quantum systems,” Phys. Rev. A **99**, 063834 (2019). [CrossRef]

**17. **M.-A. Miri and A. Alù, “Nonlinearity-induced PT-symmetry without material gain,” New J. Phys. **18**, 065001 (2016). [CrossRef]

**18. **H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature **548**, 187–191 (2017). [CrossRef]

**19. **Y. Lai, Y. Lu, M. Suh, Z. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature **576**, 65–69 (2019). [CrossRef]

**20. **W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature **548**, 192–196 (2017). [CrossRef]

**21. **H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing,” Nat. Commun. **9**, 4320 (2018). [CrossRef]

**22. **W. Langbein, “No exceptional precision of exceptional-point sensors,” Phys. Rev. A **98**, 023805 (2018). [CrossRef]

**23. **C. Chen, L. Jin, and R.-B. Liu, “Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system,” New J. Phys. **21**, 083002 (2019). [CrossRef]

**24. **H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Vahala, “Petermann-factor limited sensing near an exceptional point,” Nat. Commun. **11**, 1610 (2020). [CrossRef]

**25. **C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Rev. Mod. Phys. **89**, 035002 (2017). [CrossRef]

**26. **M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. Stone, and L. Jiang, “Quantum noise theory of exceptional point amplifying sensors,” Phys. Rev. Lett. **123**, 180501 (2019). [CrossRef]

**27. **H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A **82**, 043803 (2010). [CrossRef]

**28. **F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A **83**, 041805 (2011). [CrossRef]

**29. **S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity–time-symmetric circuit,” Nature **546**, 387–390 (2017). [CrossRef]

**30. **K. Kawabata, Y. Ashida, and M. Ueda, “Information retrieval and criticality in parity-time-symmetric systems,” Phys. Rev. Lett. **119**, 190401 (2017). [CrossRef]

**31. **F. U. Klauck, L. Teuber, M. Ornigotti, M. Heinrich, S. Scheel, and A. Szameit, “Observation of PT-symmetric quantum interference,” Nat. Photonics **13**, 883–887 (2019). [CrossRef]

**32. **S. Chakraborty and A. K. Sarma, “Delayed sudden death of entanglement at exceptional points,” Phys. Rev. A **100**, 063846 (2019). [CrossRef]

**33. **Á. B. Jaramillo, C. Ventura-Velázquez, Y. N. Joglekar, and B. Rodríguez-Lara, “PT-symmetry from Lindblad dynamics in a linearized optomechanical system,” Sci. Rep. **10**, 1761 (2020). [CrossRef]

**34. **T. Onodera, E. Ng, N. Lörch, A. Yamamura, R. Hamerly, P. L. McMahon, A. Marandi, and H. Mabuchi, “Nonlinear quantum behavior of ultrashort-pulse optical parametric oscillators,” arXiv:1811.10583 (2018).

**35. **R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A **94**, 063809 (2016). [CrossRef]

**36. **A. Roy, S. Jahani, C. Langrock, M. Fejer, and A. Marandi, “Spectral phase transitions in optical parametric oscillators,” Nat. Commun. **12**, 835 (2021). [CrossRef]

**37. **Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. **111**, 263901 (2013). [CrossRef]

**38. **A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A **92**, 063807 (2015). [CrossRef]

**39. **X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar, and C. Lee, “Pseudo-parity-time symmetry in optical systems,” Phys. Rev. Lett. **110**, 243902 (2013). [CrossRef]

**40. **J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms,” Nat. Commun. **10**, 855 (2019). [CrossRef]

**41. **M. Chitsazi, H. Li, F. Ellis, and T. Kottos, “Experimental realization of Floquet PT-symmetric systems,” Phys. Rev. Lett. **119**, 093901 (2017). [CrossRef]

**42. **H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature **537**, 80–83 (2016). [CrossRef]

**43. **J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, “Time-asymmetric loop around an exceptional point over the full optical communications band,” Nature **562**, 86–90 (2018). [CrossRef]

**44. **L.-A. Wu, H. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. **57**, 2520 (1986). [CrossRef]

**45. **M. Olsen and P. Drummond, “Entanglement and the Einstein-Podolsky-Rosen paradox with coupled intracavity optical down-converters,” Phys. Rev. A **71**, 053803 (2005). [CrossRef]

**46. **Y. K. Chembo, “Quantum dynamics of Kerr optical frequency combs below and above threshold: Spontaneous four-wave mixing, entanglement, and squeezed states of light,” Phys. Rev. A **93**, 033820 (2016). [CrossRef]

**47. **A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics **8**, 937–942 (2014). [CrossRef]

**48. **C. Wang, M. Zhang, X. Chen, M. Bertrand, A. Shams-Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature **562**, 101–104 (2018). [CrossRef]

**49. **A. McDonald, T. Pereg-Barnea, and A. Clerk, “Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev-Majorana chain,” Phys. Rev. X **8**, 041031 (2018). [CrossRef]