Non-Hermitian physics in photonic systems

Lei Xiao; Kunkun Wang; Dengke Qu; Huixia Gao; Quan Lin; Zhihao Bian; Xiang Zhan; Peng Xue

doi:10.3788/PI.2025.R09

1 Introduction

In quantum mechanics, the observable energy is described by a Hamiltonian operator. According to the postulate of real-valued measurements and the conservation of probability for physical observables, Hermiticity (self-adjointness) is a crucial property of the Hamiltonian when describing a closed system. However, in real physical systems, uncontrolled flows of information, energy, or even particles to the environment lead to a breakdown of probability conservation^[1–3]. Generally, the Schrödinger equation with a Hermitian Hamiltonian, which describes the dynamics of a closed system, must be replaced by approaches such as the Lindblad quantum master equation for open systems^[4]. Although these methods are extremely powerful, their technical complexity greatly restricts the range of systems that can be efficiently handled. Therefore, the non-Hermitian Hamiltonian, which directly incorporates environmental influences, is proposed as a useful methodology for describing open systems^[5–7]. In fact, the non-Hermitian approach and the Lindblad quantum master equation are equivalent in some scenarios. In the short-time limit, typically defined by the inverse of the loss rate^[8], the quantum jump terms can be neglected. Effective non-Hermitian Hamiltonians derived from the master equation then provide a good approximation to the dynamics of open systems when post-selection is implemented^[9].

Serving as a simple and powerful method for studying open physical systems, non-Hermitian Hamiltonians possess complex spectra, which initially limited their use to purely mathematical tools in earlier research. Driven by curiosity, Bender and Boettcher introduced a broad class of non-Hermitian Hamiltonians exhibiting parity-time ( $P T$ ) symmetry. These Hamiltonians, characterized by balanced gain and loss, can possess entirely real eigenenergies^[10–12]. By abandoning the assumption of Hermiticity, $P T$ -symmetric non-Hermitian systems have been regarded as a fundamental modification of quantum physics. However, $P T$ symmetry is not always preserved across all parameter regimes. It can be easily broken by increasing gain and loss, where the eigenenergies transition into complex values. As a result, the properties of the systems differ drastically in the $P T$ -symmetry-preserving and $P T$ -symmetry-broken regimes^[1,13–17]. Striking examples include the fact that, in the $P T$ -preserving phases, information can be fully retrieved from the environment, whereas in the $P T$ -broken phases, retrieval becomes impossible^[18–20]. Additionally, under the quench dynamics, quantum dynamic phase transitions accompanied by the emergent momentum-time skyrmions^[21] occur in the $P T$ -preserving phases but vanish in the $P T$ -broken phases. These significant differences between $P T$ -preserving and $P T$ -broken phases have led to intriguing new applications and fascinating phenomena, generating substantial interest over the past decades^[22–29]. For instance, in $P T$ -preserving phases, non-Hermitian systems can enable superluminal information transmission^[30]. The exploration of $P T$ -symmetric non-Hermitian systems has also led to novel applications, such as invisibility^[22], non-reciprocal light propagation^[23,31], $P T$ -symmetric laser absorbers^[32], and $P T$ -symmetric phonon lasers^[33]. Beyond $P T$ symmetry, the concept of pseudo-Hermiticity has been proposed to ensure real eigenenergies, with proof that all $P T$ -symmetric Hamiltonians are indeed pseudo-Hermitian^[34,35].

The boundary between $P T$ -preserving and $P T$ -broken phases arises at the spectral degeneracies in non-Hermitian systems, known as exceptional points (EPs)^[3,35]. In contrast to the conventional energy degeneracy in Hermitian systems, EPs manifest as degeneracies in the spectrum of non-Hermitian Hamiltonian where the corresponding eigenvectors coalesce. This generally makes the system appear as though it suddenly loses the dimensionality of its eigenspace at these critical points^[36]. Therefore, while EPs signal $P T$ -phase transitions, they are also of intrinsic physical interest. A striking feature of EPs is their strong response of the degenerate eigenvalues to perturbations, which makes the system with EPs a good candidate for sensing applications^[37–39]. Another immediate consequence of EPs is the general presence of nodal non-Hermitian topological phases, where traditional band-touching points are replaced by these exceptional degeneracies^[40,41]. Similar to Weyl semimetals in the Hermitian case, EPs reveal intriguing topologically stable phenomena in non-Hermitian semimetals^[42–44]. Furthermore, carrying a nonzero topological charge, EPs exhibit unusual braiding topological and dynamical properties when encircling an EP in the parameter space^[45–49]. As one of the unique features of non-Hermitian systems, how to construct higher-order EPs or even exceptional structures is an important topic in this field^[50–53]. It has been shown that symmetry is not a necessary condition for the existence of EPs. An $n$ th-order EP (a point of n-fold degeneracy in eigenenergies and eigenstates) can be stable existing in a $2 n -$ two-dimensional non-Hermitian system without any symmetry. In addition, the required dimensionality for EPs can be reduced through various symmetries. Moreover, these symmetries impart a significantly broader and more conceptually rich range of properties to EPs, which are still being explored^[54–56].

Besides the aforementioned intriguing aspects of non-Hermitian systems, another fascinating area that has garnered extensive research is the impact of non-Hermiticity on the topological properties of lattice systems. In this domain, symmetries are well known to generally refine the topological classification by constraining the set of permissible physical systems. In Hermitian systems, tenfold symmetry classes are categorized based on the presence or absence of particle-hole symmetry, time-reversal symmetry, and chiral symmetry, as outlined in the comprehensive Altland–Zirnbauer symmetry classification^[57]. However, the problem is much more complicated when extending to a non-Hermitian system with the breaking of Hermiticity. In non-Hermitian systems, sublattice and chiral symmetries no longer align. Moreover, non-Hermitian systems possess unique symmetries, such as gain-loss-induced $P T$ symmetry or pseudo-Hermiticity, which are absent in conventional Hermitian systems. To address this problem, Bernard and LeClair (BLC) developed a 43-fold symmetry classification for non-Hermitian matrices^[58]. Building on this foundation, Kawabata et al.^[59] proposed a comprehensive theory of symmetry and topology in non-Hermitian physics. Ultimately, the 43-fold symmetry classification is reduced to a 38-fold classification by addressing issues of overcounting and recognizing non-Hermitian symmetry classes.

Topological matter is notable for its remarkable properties and applications, primarily due to the existence of topologically robust edge states. The central principle that relates the bulk topological invariant to these robust edge states is known as the bulk-boundary correspondence (BBC)^[60,61]. It says that the number of edge states observed under open boundary conditions (OBCs) corresponds to the bulk topological number defined under the periodic boundary conditions (PBCs). However, a wide range of non-Hermitian systems show markedly different spectra under OBCs and PBCs, suggesting a breakdown of the BBC^[62–65]. In addition, in contrast to the Bloch bulk eigenstates in Hermitian systems with both PBC and OBC, eigenstates in non-Hermitian Hamiltonians are highly sensitive to boundary conditions. Although the eigenstates under the PBC remain extended, those under the OBC can all become localized at the boundary, a phenomenon known as the non-Hermitian skin effect^[66–68]. To correctly describe this unique phenomenon, the non-Bloch band theory is proposed to rebuild the generalized BBC for non-Hermitian systems^[69,70]. The presence of the non-Hermitian skin effect is linked to the existence of a closed loop or a finite region in the periodic-boundary spectrum of one-dimensional or higher-dimensional systems on the complex plane^[71–73]. Therefore, the topological origin of the non-Hermitian skin effect is referred to as intrinsic non-Hermitian topology, which can be characterized by spectral windings.

In addition to theoretical advancements, non-Hermitian physics offers fundamental insights for exploring novel phenomena and achieving practical applications across various physical platforms. Non-Hermiticity naturally occurs in a diverse range of systems, including mechanics^[51,74,75], acoustics^[26,50,76], magnetism^[77,78], electrical circuits^[79,80], atomic systems^[81–83], superconducting qubits^[84,85], and photonic systems^[14,86,87] (to name a few). In this review, we primarily focus on photonic systems, where non-Hermitian physics has been extensively explored. Without the need for stringent experimental conditions to implement non-Hermiticity, the flexibility of controlling photonic systems makes it an ideal platform for constructing and manipulating non-Hermitian systems. Non-Hermitian physics also introduces a wealth of new functionalities in photonics by expanding the tunable parameter space into the complex domain^[88]. Specifically, various photonic platforms, such as the optical waveguides^[89], optical cavities^[90], and fiber optics^[91], are readily available for studying non-Hermitian systems. Integrating non-Hermitian physics into photonics paves the way for new applications, including sensing^[37–39], quantum engines^[92,93], chiral state transfer^[45,46], and state engineering^[94], which extend beyond the capabilities of Hermitian photonic systems.

Building on all of these theoretical advancements and optical progress in non-Hermitian physics, we provide this review to give a comprehensive overview of recent developments. The review starts by introducing the basic concepts from a methodological perspective, discussing the relationships between the master equations and non-Hermitian Hamiltonians, $P T$ -symmetric systems, and conserved quantities. We then review the notable properties of EPs, which are the paramount characteristics unique to non-Hermitian Hamiltonians, to give an intuitive understanding of their generation and application. The concept of biorthogonal eigenstates for the non-Hermitian Hamiltonian will also be addressed in this part. Primarily in this review, we will discuss methods for constructing non-Hermitian systems using different degrees of freedom of photons, such as polarization, spatial modes, time bins, orbital angular momentum, and frequency. Focusing on these advanced methods, we will briefly introduce the implementation of non-Hermiticity in photonic platforms of bulk optics, optical waveguides, optical cavities, fiber optics, synthetic dimension in photonics, and optical metamaterials. Our review further covers the progress on the applications of non-Bloch band theory in demonstrating the unique phenomena of skin effects and edge bursts in non-Hermitian systems. In addition, we will outline the key concepts of non-Hermitian linear response theory, non-Bloch EPs, and the reconstruction of the generalized BBC with the generalized Brillouin zone to illustrate the power of the non-Bloch band theory. We will also review emerging non-Hermitian topological matters and their simulations in photonics. The novel functional applications induced by non-Hermitian physics in photonics will also be presented in this review. Finally, the review will conclude with a summary and outlook on the future of non-Hermitian physics in photonic systems.

2 Fundamental Theory and Background of Non-Hermitian Physics

2.1 Master Equations

In the framework of classical quantum mechanics, the temporal evolution of an isolated system is fundamentally described by the Schrödinger equation $i \frac{d}{d t} | ψ (t) ⟩ = H | ψ (t) ⟩,$ (1)where $| ψ (t) ⟩$ represents the state of the system at time $t$ , and $H$ denotes the Hamiltonian of the complete system, which acts on the corresponding Hilbert space.

A fundamental postulate of quantum theory states that the Hamiltonian $H$ of an isolated system must be Hermitian. This Hermiticity guarantees real, single-valued energy eigenvalues and ensures a unitary time evolution of the system. However, the real physical system is called an open system since it is interacting with an environment so that the full dynamics of the system and environment are unitary^[95,96]. Thus, for an open system interacting with an environment, the description of physical states through wave functions as in Eq. (1) is no longer complete when dealing with composite systems^[97]. Here, we use the density operators, which are in particular an appropriate description of subsystems to describe physical states instead of wave functions in general^[98]. The Schrödinger equation of an open system turns into $\dot{ρ} (t) = - i [H, ρ (t)],$ (2)where $ρ (t) = | ψ (t) ⟩ ⟨ ψ (t) |$ is the corresponding density operator of composite systems that contain a real physical system $S$ and the environment $E$ . The Hamiltonian $H = H_{S} + H_{E} + H_{S E}$ , with $H_{S E}$ representing the Hamiltonian of the interaction between two subsystems^[99]. Assuming there is no interaction between $S$ and $E$ at $t = 0$ , the density matrix $ρ (t)$ can be decomposed into $ρ (0) = ρ_{S} (0) \otimes ρ_{E} (0)$ ^[100]. The density operators $ρ_{S (E)} (t)$ representing the state of the subsystems $S$ ( $E$ ) are obtained by tracing out $E$ ( $S$ ), respectively. Therefore, the reduced dynamical equation for the physical system can be written as ${\dot{ρ}}_{S} (t) = - i {tr}_{E} [H, ρ (t)] .$ (3)

In the interaction picture, the evolution equation of the composite system is $\dot{\tilde{ρ}} (t) = - i [\tilde{H_{S E}}, \tilde{ρ} (t)]$ , with $\tilde{ρ} (t) = e^{i (H_{S} + H_{E}) t} ρ (t) e^{- i (H_{S} + H_{E}) t}$ and $\tilde{H_{S E}} = e^{i (H_{S} + H_{E}) t} H_{S E} e^{- i (H_{S} + H_{E}) t}$ . Iteratively substituting the system’s density operator into the evolution equation and taking the partial trace over the environment on both sides of the equation, the following formula can be derived: $\frac{d}{d t} \tilde{ρ} (t) = - \int_{0}^{t} d τ {tr}_{E} {{\tilde{H_{S E}} (t), [\tilde{H_{S E}} (τ), \tilde{ρ} (τ)]}} .$ (4)

The expected value of the coupling observable ${tr}_{E} {\tilde{H_{S E}}, ρ_{E} (0)}$ is zero under the initial state $ρ_{E} (0)$ since there is no interaction between $S$ and $E$ at $t = 0$ , which ensures that ${tr}_{E} [\tilde{H_{S E}}, ρ (0)] = 0$ . As time goes on, the interaction between the physical system and the environment involves a connection. Assuming the interaction is weak, that is, under the weak-coupling approximation or the so-called Born approximation^[95], the environment is virtually unaffected by the physical system, which gives rise to the fact that the environment remains in its initial state at any time. Then, the density operator of composite systems in Eq. (4) can be approximated as $\tilde{ρ} (τ) = {\tilde{ρ}}_{S} (τ) \otimes ρ_{E} (0)$ .

The Hamiltonian of the interaction between two subsystems $\tilde{H_{S E}}$ can be analyzed using a low-order expansion over time $t$ . However, the specific form of the density operator remains highly complex, even though the evolution of the physical system is non-Markovian indeed. Fortunately, if the rate of environmental re-equilibration is very fast, the time scale over which the state of the system varies appreciably is large compared to the time over which the environment correlation functions decay. In this case, the environment is approximately equivalent to erasing the memory of the system, which leads to the results that the state of the physical system can be regarded as determined by the current time, that is, ${\tilde{ρ}}_{S} (t) = {\tilde{ρ}}_{S} (τ)$ . This method is called Markov approximation^[95]. Last, assuming the environment is composed of a series of resonators and $H_{E} = Σ_{k} Ω_{k} Γ_{k}^{†} Γ_{k}$ , with $Γ_{k}$ representing the boson operator, the interaction Hamiltonian in the interaction picture can be written as $\tilde{H_{S E}} = Σ_{i} {\tilde{s}}_{i} (t) {\tilde{Γ}}_{i} (t)$ , where $s_{i}$ is the creation operator. By introducing the rotating-wave approximation for the interaction and switching back to the Schrödinger picture, the master equation for the physical system can be written as^[95] ${\dot{ρ}}_{S} = L [ρ_{S}] = - i [H_{S}, ρ_{S}] + \sum_{k} γ_{k} (L_{k} ρ_{S} L_{k}^{†} - \frac{1}{2} {L_{k}^{†} L_{k}, ρ_{S}}),$ (5)where $γ_{k}$ represents the transition, dissipation, or decoherence rate of the corresponding interaction channel. $L_{k} = \sqrt{γ_{k}} s_{k}$ is called jump operators, which represent the coupling between the physical system and the environment.

Equation (5) is called the Lindblad master equation and it describes the dynamics of an open quantum system interacting with an environment^[4,101]. $L$ is the generalized Liouvillian operator, which can be regarded as the generalization of the Hamiltonian in open systems. The first term in Eq. (5) represents the unitary evolution, and the other terms describe the changes that occur in the physical system as a result of the interaction between the physical system and the environment.

2.2 Non-Hermitian Hamiltonian

From the Lindblad master equation, it is clear that the dynamics of the quantum states are not only governed by the Hamiltonian of the physical system, but by additional interaction terms posed by the environment. Although a large variety of physical systems are well described by this equation, such as the atomic systems and quantum optical systems, the solutions to the master equation are often complicated. Here, we introduce a unique way to address the problems with various kinds of open systems. By constructing a complex quantum potential, the Lindblad master equation in Eq. (5) can now be rewritten as ${\dot{ρ}}_{S} = - i (H_{eff} ρ_{S} - ρ_{S} H_{eff}^{†}) + \sum_{k} γ_{k} L_{k} ρ_{S} L_{k}^{†} .$ (6)

The second term in Eq. (6) represents the quantum jumps, which can be considered as decoherence arising from the interaction between the physical system and the environment. The remaining gain-loss terms are incorporated into the system’s evolution, making it non-Hermitian.

The occurrence of quantum jumps requires a time accumulation process; thus, their effects on the system may not be significant within extremely short time windows. Additionally, if the system resides in a low excited state or if the dissipation rate is much smaller than the system’s coherent evolution rate, neglecting quantum jumps becomes reasonable. For example, in cavity quantum electrodynamics systems, when the cavity’s loss rate is low and the measurement duration is extremely short, a non-Hermitian Hamiltonian can be used to approximate the cavity mode decay^[102]. In this case, photon escape is treated as continuous loss, and ignoring quantum jumps may lead to theoretical predictions aligning well with experimental observations on short timescales.

However, over longer time periods or when the system is in a highly excited state, the cumulative effects of quantum jumps become significant, and their occurrence frequency increases. Ignoring jumps in these scenarios can result in inaccurate predictions. Furthermore, in photonic experiments, non-Hermitian Hamiltonians induce effective losses or gains, which may alter the system’s spectrum. Ignoring quantum jumps may affect photon pair generation efficiency or introduce discrepancies in photon statistics measurements. For instance, in photon pair correlation measurements, non-Hermitian models may predict different second-order coherence functions, but actual quantum jumps introduce additional noise or decoherence effects^[103]. While the evolution of the average photon number may align with the predictions of a non-Hermitian Hamiltonian, higher-order correlation functions may deviate significantly. Moreover, in entanglement measurements, neglecting quantum jumps may overestimate the entanglement lifetime, as actual dissipation destroys entanglement, whereas the non-Hermitian model may only account for amplitude decay, ignoring the collapse of the state due to quantum jumps.

Here, we analyze the cases where the effects of quantum jumps can be neglected. By ignoring the quantum jumps of the composite systems, the density matrix of the physical system can be considered to approximate a non-Hermitian Hamiltonian^[95]: $H_{eff} = H - \frac{i}{2} \sum_{k} γ_{k} L_{k}^{†} L_{k} .$ (7)

One typical feature of the non-Hermitian system is that it cannot have both real energy spectra and mutually orthogonal eigenvectors, that is, the energy of the physical system can be complex. The complex energy corresponds to the time evolution $e^{- i (E + i Γ) t} = e^{- i E t} e^{Γ t}$ . It can be clearly seen that the imaginary part of the energy corresponds to a gain or loss. Thus, the non-Hermitian system can be realized by applying controlled gain-loss to the Hermitian system in general^[104,105].

2.3 $P T$ Symmetry

While generic non-Hermitian systems are characterized by complex energy spectra and the non-orthogonality of eigenstates, further constraints on the Hamiltonian structure can lead to remarkable spectral properties. Among them, $P T$ symmetry—where the combined action of parity ( $P$ ) and time-reversal ( $T$ ) commutes with the Hamiltonian—has emerged as a particularly fruitful framework. Although there have been some studies on real energy spectra of non-Hermitian Hamiltonians in different fields at an early stage^[106–112], a complete theory has not yet been formed. Until 1998, Bender and Boettcher proposed a special series of $P T$ -symmetric Hamiltonians and elaborated the relationship between $P T$ symmetry and real energy spectrum^[10]. As a result, non-Hermitian quantum mechanics began to gain widespread attention. In 2002, Mostafazadeh constructed the $η$ -pseudo-Hermitian symmetric quantum system based on a biorthogonal basis mathematically, and explained the conditions required for the real energy spectrum of non-Hermitian Hamiltonians^[113–115]. Since then, generally speaking, non-Hermitian quantum mechanics has been largely discussed along the two branches of $P T$ -symmetric and $η$ -pseudo-Hermitian quantum systems, as we discuss in the following.

The discussion of $P T$ -symmetric systems was initiated by a deformed harmonic-oscillator Hamiltonian^[10,116,117]: $H = p^{2} + x^{2} + i x^{3},$ (8)where $p$ is the momentum operator and $x$ is the position operator. This Hamiltonian is a complex non-Hermitian but has a real energy spectrum while most people used to think that only Hermitian operators had this property previously. It is further found that $H$ in Eq. (8) is $P T$ symmetric, that is, satisfies the following equation: $H = (P T)^{- 1} H (P T),$ (9)where $P$ and $T$ are the parity operator and time reversal operator, respectively, and $[P, T] = 0$ . The parity operator is a linear Hermitian operator that performs a space reflection and $P^{- 1} = P$ , $P^{2} = I$ , so under parity reflection both $x$ and $p$ change sign: $P p P^{- 1} = - p$ , $P x P^{- 1} = - x$ . The time-reversal operator $T$ is an anti-linear reflection operator and for bosonic systems $T^{- 1} = T$ , $T^{2} = I$ . Under time-reversal reflection, it can be shown that $T p T^{- 1} = - p$ , $T x T^{- 1} = x$ .

In order to reveal whether the real energy spectrum of the non-Hermitian Hamiltonian is related to $P T$ symmetry, Bender applied a special perturbation method to expand the $P T$ -symmetric Hamiltonian in terms of $δ$ , and replaced with $H = p^{2} + x^{2} + x^{2} {(i x)}^{δ}$ ^[10,116]. Here, $δ$ is a small real number and does not change the $P T$ symmetry of the non-Hermitian Hamiltonian above. Therefore, focusing on the formula of the replaced Hamiltonian, the range of $δ$ is further extended and processed by numerical calculation. Finally, according to the different values of $δ$ , he obtained the solution of the system with real and conjugate complex eigenvalues.

The publication of Ref. [10] began to arouse people’s attention to non-Hermitian quantum theory, and studies based on the real energy spectra of $P T$ -symmetric Hamiltonians are appearing with increasing frequency^{[116,118–125]}. In addition, the $P T$ symmetry is studied theoretically from the mathematical point of view^[126–133]. Among the many novel characteristics of non-Hermitian $P T$ -symmetric systems, the most noteworthy is the case that the $P T$ -symmetric Hamiltonian has real energy spectra. Since in traditional quantum mechanics, a Hermitian Hamiltonian with positive real energy spectra corresponds to an observable, a non-Hermitian Hamiltonian with real energy spectra can also theoretically be regarded as an observable. In order to give a self-consistent interpretation to such a non-Hermitian system in physics, several conditions must be satisfied: the positive definiteness of the inner product of the system, the completeness of the system, and the unitarity of the time evolution.

The eigenstates of $P T$ -symmetric non-Hermitian Hamiltonians do not satisfy the standard completeness relations^[134–137]. We can define $| υ_{i} ⟩$ as energy eigenstates of a particle on the real line subjected to a $P T$ -symmetric potential. Unlike traditional quantum mechanics where we know what the inner product is even before the given Hamiltonian, for a $P T$ -symmetric Hamiltonian, a plausible guess for what the inner product might be should be considered first. Suppose the energy eigenstates associated with $P T$ -symmetric systems are also orthogonal; numerical evidence implies that the inner product has been proposed^[138,139]: ${⟨ υ_{i} υ_{j} ⟩}^{P T} \int d x υ_{i}^{P T} (x) υ_{j} (x) {(- 1)}^{n} δ_{i j} .$ (10)Here we replace the symbol $†$ with $P T$ in order to distinguish the Hermitian Hamiltonian. The coefficient ${(- 1)}^{n}$ originates from the nonstandard completeness relations of the $P T$ -symmetric non-Hermitian Hamiltonian $\sum_{n} {(- 1)}^{n} υ_{n} (x) υ_{n} (y) = δ (x - y)$ , which is mathematically proven in Ref. [138]. The function $υ_{n} (x) = ⟨ x | E_{n} ⟩$ is the energy eigenstate of a particle on the real line subjected to a $P T$ -symmetric potential and $E_{n}$ is the eigenvalue of the $n$ th energy level. The negative norm makes the metric of the inner product of $P T$ -symmetric systems uncertain, and it is difficult to maintain the familiar probabilistic interpretation of quantum theory. However, the inner product has the same number of positive and negative terms that are alternating; thus, the extra ${(- 1)}^{n}$ implies that there should be some unknown hidden symmetry within the system.

In an attempt to construct an inner product endowed with a positive-definite norm and establish that the $P T$ -symmetric Hamiltonian defines a unitary quantum mechanical theory, a remedy against the indefinite metric in Hilbert space has been proposed in the form of a linear charge operator $C$ , whose properties closely resemble those of the charge conjugation operator $C$ in particle physics^[140], known as the $C P T$ theorem.

In coordinate space, the linear operator $C$ , which represents the additional symmetry of $P T$ -symmetric Hamiltonian, is a sum over the $P T$ -normalized eigenfunctions and satisfies^[141] $C (x, y) = \sum_{n = 0}^{\infty} υ_{n} (x) υ_{n} (y) .$ (11)

It is easy to verify that $C^{2} = I$ and $[C, P T] = [C, H] = 0$ . Therefore, the operator $C$ is a measure of the inner product for the $P T$ -symmetric Hamiltonian, and satisfies $C | υ_{n} (x) ⟩ = (- {1)}^{n} | υ_{n} (x) ⟩$ . Thus, by introducing $C$ into the inner product, the negative signs therein can be cancelled out. The advantage of the $C P T$ inner product is that, unlike the $P T$ inner product, the $C P T$ inner product is positive definite, and the completeness condition of the $P T$ -symmetric system in coordinate space is rewritten as^[138] $\sum_{n = 0}^{\infty} υ_{n} (x) [C P T υ_{n} (y)] = δ (x - y) .$ (12)

Last, the coherent, non-unitary time evolution operator for the system based on the Schrödinger equation is given by $U t e^{- i H_{P T} t} .$ (13)

The time evolution operator is the same as the Hermitian quantum system, except that the Hermitian Hamiltonian is replaced by the $P T$ -symmetric Hamiltonian. Consider $| ψ (0) ⟩$ as a given initial state; the final state at $t$ is defined as $| ψ (t) ⟩ = e^{- i H_{P T} t} | ψ (0) ⟩$ . It is straightforward to calculate that the norm of the final state ${⟨ ψ (t) | ψ (t) ⟩}^{C P T} = [C P T e^{- i H_{P T} t} | ψ (0) ⟩]^{T} e^{- i H_{P T} t} | ψ (0) ⟩ = {⟨ ψ (0) | ψ (0) ⟩}^{C P T} .$ (14)Here, $T$ is the matrix transpose and $H_{P T} = H_{P T}^{T}$ . It can be seen that the unitarity of the time evolution is also guaranteed in $P T$ -symmetric systems.

In traditional Hermitian quantum mechanics, a linear operator $A$ must be Hermitian to be an observable, that is, $A = A^{†}$ . This condition is obtained by complex conjugation in positive definite inner products of Hermitian operators, which ensures that the expectation value of $A$ in a state is real. Similar to the Hermitian operators, in $P T$ -symmetric quantum mechanics the condition for an operator $A$ to be an observable is^[141] $A = (C P T)^{- 1} A^{T} C P T .$ (15)

This condition also guarantees that the expectation value of $A$ in any state is real since ${⟨ ψ | A | ψ ⟩}^{C P T} = {⟨ A ψ ⟩}^{C P T}$ . It is easy to prove that the operator $C$ and the $P T$ -symmetric Hamiltonian themselves satisfy Eq. (15), so they are all observables.

Besides, the operator $A$ satisfies $A = A^{T}$ , which is an extremely strong constraint. Otherwise, the norm of the system would not be guaranteed to be conserved under a set of orthogonal positive definite $C P T$ inner products. However, there exists $A \neq A^{T}$ in other non-Hermitian systems. For this case, the pseudo-Hermitian inner product or biorthogonal basis inner product needs to be introduced. For example, in Ref. [142], Jones obtained the pseudo-Hermitian inner product of the system by transforming the wave function of the harmonic oscillator when he analyzed the Swanson Hamiltonian with $P T$ symmetry but transposed asymmetrically. We will discuss the biorthogonal quantum mechanics and pseudo-Hermitian systems in the next subsection in detail.

2.4 Biorthogonal Quantum Mechanics and Pseudo-Hermitian Systems

$P T$ -symmetric systems show that non-Hermitian Hamiltonians can still have real spectra under certain conditions. However, due to the non-Hermiticity, their eigenstates are generally not orthogonal, and the usual rules of quantum mechanics no longer apply directly. To handle these issues, a more general framework is needed. This leads to biorthogonal quantum mechanics and pseudo-Hermitian systems, which provide consistent ways to describe measurements, inner products, and time evolution in non-Hermitian settings.

Compared to the Hermitian systems with real eigenvalues and orthogonal eigenstates, the eigenvalues and eigenstates in a non-Hermitian Hamiltonian are not necessarily real and orthogonal. In biorthogonal quantum mechanics^{[131,143–146]}, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. Consider a non-Hermitian Hamiltonian $H$ ; the eigenvalue equations of $H$ and $H^{†}$ with the $n$ th energy level are given by $H | φ_{n} ⟩ = E_{n} | φ_{n} ⟩, H^{†} | ϕ_{n} ⟩ = E_{n}^{'} | ϕ_{n} ⟩,$ (16)where $E_{n}$ and $E_{n}^{'}$ are the $n$ th eigenvalues, and $| φ_{n} ⟩$ and $| ϕ_{n} ⟩$ are the right eigenstates of $H$ and $H^{†}$ , respectively. Using the transpose complex conjugation on both sides of Eq. (16), it can be obtained that $⟨ φ_{n} | H^{†} = E_{n}^{*} ⟨ φ_{n} |, ⟨ φ_{n} | H = E_{n}^{' *} ⟨ ϕ_{n} | .$ (17)

Since the energy spectra are unique, it is straightforward to obtain the biorthonormal relation^[146] $⟨ ϕ_{m} | φ_{n} ⟩ = δ_{m n} .$ (18)Here $| φ_{n} ⟩$ and $| ϕ_{m} ⟩$ are two different sets of eigenstates that correspond to $H$ and $H^{†}$ due to $H \neq H^{†}$ . Equation (18) extends the orthogonality in traditional quantum mechanics. Accordingly, the completeness relation of non-Hermitian systems should also be generalized. An arbitrary quantum state $| ψ ⟩$ can be expressed by the eigenstates as $| ψ ⟩ = \sum_{n} a_{n} | ϕ_{n} ⟩ .$ (19)

Thus, the probability amplitude is calculated as $a_{n} = ⟨ φ_{n} | ψ ⟩$ using the biorthonormal relation. Obviously, the state $| ψ ⟩$ can be rewritten as $| ψ ⟩ = \sum_{n} | ϕ_{n} ⟩ ⟨ φ_{n} | ψ ⟩ .$ (20)

Therefore, the right and left eigenstates satisfy the completeness relation^[146] $\sum_{n} | ϕ_{n} ⟩ ⟨ φ_{n} | = I .$ (21)

In summary, the aforementioned relations are two basic properties of biorthogonal quantum mechanics in non-Hermitian systems, and ${| ϕ_{n} ⟩, | φ_{n} ⟩}$ are the so-called biorthogonal eigenstates.

The eigenstates of a biorthogonal system are no longer normalized under the completeness relation, which leads to the fact that the traditional probabilistic interpretation used in Hermitian quantum mechanics has changed. Because it is straightforward to calculate that $⟨ ϕ_{n} | ϕ_{n} ⟩ \geq 1$ , by assuming that all eigenstates have the same Hermitian norm $⟨ ϕ_{n} | ϕ_{n} ⟩ = ⟨ φ_{n} | φ_{n} ⟩$ for all $m (n)$ . In this case, the inner product should be redefined by introducing the so-called associated state that defines duality relations between elements of the Hilbert space and its dual space.

For an arbitrary state $| ψ ⟩$ in Eq. (19), the associated state $| \tilde{ψ} ⟩$ is defined according to the following relation: $| \tilde{ψ} ⟩ = \sum_{n} a_{n} | φ_{n} ⟩,$ (22)where the state $| \tilde{ψ} ⟩$ associated with $| ψ ⟩$ is then given by the Hermitian conjugate of the bra vector $⟨ \tilde{ψ} |$ with $⟨ \tilde{ψ} | = \sum_{n} a_{n}^{*} ⟨ φ_{n} |$ . Thus, by expressing another state $| ξ ⟩$ with the same eigenvectors, that is, $| ξ ⟩ = \sum_{n} b_{n} | ϕ_{n} ⟩$ , the inner product for a biorthogonal system is defined as^[146] $⟨ ξ, ψ ⟩ \equiv ⟨ \tilde{ξ} | ψ ⟩ = \sum_{n, m} b_{n}^{*} a_{m} ⟨ φ_{n} | ϕ_{m} ⟩ = \sum_{n} b_{n}^{*} a_{n} .$ (23)

Next, we will discuss the pseudo-Hermitian Hamiltonian system based on the biorthogonal quantum mechanics^[113–115]. A Hamiltonian is pseudo-Hermitian with respect to $η$ if it satisfies $H^{†} = η H η^{- 1},$ (24)where $η$ is called an intertwining operator, which is generally a linear Hermitian invertible transformation^[147].

In order to observe the equivalence of the existence of a complete biorthonormal set of eigenstates of $H$ , we introduce the degeneracy label $a$ into the completeness relation and the biorthonormality relation. Naturally, the evolution of the state can be written as^[113–115] $H (η^{- 1} | ϕ_{n}, a ⟩) = η^{- 1} H^{†} | ϕ_{n}, a ⟩ = E_{n}^{*} η^{- 1} | ϕ_{n}, a ⟩ .$ (25)

Thus, it can be seen that $η^{- 1} | ϕ_{n}, a ⟩$ is the eigenstate of $H$ corresponding to the eigenvalue $E_{n}^{*}$ . Moreover, the operator $η^{- 1}$ maps the eigensubspace with eigenvalue $E_{n}$ to the subspace with eigenvalue $E_{n}^{*}$ . The eigenfunctions corresponding to these two eigenvalues have the same degeneracy.

Furthermore, using the completeness relation in Eq. (21), it is easy to obtain the transition relation between the two biorthonormal eigenstates $| ϕ_{n} ⟩$ and $| φ_{n} ⟩$ as follows: $| ϕ_{n}, a ⟩ = η | φ_{n}, a ⟩ .$ (26)

Therefore, the complete biorthonormal set of eigenstates of pseudo-Hermitian Hamiltonian $H$ based on the indefinite inner product is given by^[113] ${⟨ ϕ_{n}, a | ϕ_{m}, b ⟩}_{η} = δ_{n m} δ_{a b} .$ (27)

Besides, it can be easily calculated to find the form of the invertible linear operator^[113]: $η = \sum_{n} \sum_{a = 1}^{d_{n}} | ϕ_{n}, a ⟩ ⟨ ϕ_{n}, a |, η^{- 1} = \sum_{n} \sum_{a = 1}^{d_{n}} | φ_{n}, a ⟩ ⟨ φ_{n}, a | .$ (28)Here, $d_{n}$ is the degree of degeneracy of eigenvalue $E_{n}$ . Meanwhile, the pseudo-Hermitian Hamiltonian $H$ can be expressed by $H = \sum_{n} \sum_{a = 1}^{d_{n}} E_{n} | φ_{n}, a ⟩ ⟨ ϕ_{n}, a | .$ (29)

It is clearly seen that the expansion based on a biorthogonal basis is self-consistent with the pseudo-Hermitian theory. The above analysis leads to two properties: i) $H$ is a pseudo-Hermitian Hamiltonian if and only if the eigenvalues are real or come in complex conjugate pairs; ii) if $H$ is pseudo-Hermitian with respect to two linear Hermitian invertible transformations $η_{1}$ and $η_{2}$ , then the intertwining operators satisfy $[H, η_{1}^{- 1} η_{2}] = 0$ .

Now consider a $P T$ -symmetric Hamiltonian $H_{P T}$ that has the discrete energy spectrum with a set of biorthogonal bases. After using the eigenvalue equation $H_{P T} | ψ ⟩ = E | ψ ⟩$ , the commutation relation $[P T, H] = 0$ is used. The formula is changed into $P T H | ψ ⟩ = H P T | ψ ⟩ = E P T | ψ ⟩ .$ (30)

Obviously, $E$ is the eigenvalue of $H_{P T}$ corresponding to the eigenstate $P T | ψ ⟩$ . The energy spectra are real in the $P T$ -symmetric region and come in complex conjugate pairs in the $P T$ -symmetry broken region. Thus, a $P T$ -symmetric Hamiltonian system with discrete energy spectra and a set of biorthogonal bases belongs to the pseudo-Hermitian category.

The above pseudo-Hermitian theory is mainly based on the biorthogonal states with discrete energy spectra. However, people are more accustomed to discussing pseudo-Hermitian theory in the framework of a set of orthogonal bases. Consider ${| ν_{n} ⟩}$ to be a set of orthogonal bases in Hilbert space; the indefinite inner product of two arbitrary bases is defined as^[148] ${⟨ ν_{i} | ν_{j} ⟩}_{η} = ⟨ ν_{i} | η | ν_{j} ⟩ .$ (31)

The indefinite inner product is an extension of the ordinary Hermitian inner product. It degenerates into the Hermitian inner product when $η = 1$ ; meanwhile, the pseudo-Hermitian Hamiltonian changes into Hermitian.

The evolution of the pseudo-Hermitian system with initial state $| ν_{n} ⟩$ is governed by the Schrödinger equation $i \frac{d}{d t} | ν (t) ⟩ = H | ν (t) ⟩ .$ (32)

It is easy to check that the equation for the indefinite inner product is given by $i \frac{d}{d t} {⟨ ν_{i} | ν_{j} ⟩}_{η} = ⟨ ν_{i} | η H - H^{†} η | ν_{j} ⟩ = 0.$ (33)

Thus, it can be concluded that the probability of indefinite inner product space defined by a pseudo-Hermitian Hamiltonian is a conserved quantity.

Similar to the calculation above, based on the eigenvalue equation, the eigenvalues of the pseudo-Hermitian Hamiltonian $H$ satisfy $(E_{i}^{*} - E_{j}) {⟨ ν_{i} | ν_{j} ⟩}_{η} = 0.$ (34)

Therefore, the pseudo-Hermitian systems based on indefinite inner products can only have real energy spectra. Otherwise, when $i = j$ , the constraint condition of the above equation becomes ${⟨ ν_{j} | ν_{j} ⟩}_{η} = ⟨ ν_{j} | η | ν_{j} ⟩ = 0$ , which means that the inner product of the same state is zero under any circumstances, thereby violating the probability explanation in quantum theory obviously.

2.5 Exceptional Points

In non-Hermitian systems, the energy spectra can be complex with the corresponding eigenstates being non-orthogonal and the corresponding time evolution operators are non-unitary. When the skewed eigenstates completely coalesce at certain points in the parameter space, the eigenenergies also become degenerate^[149]. In this case, the non-Hermitian Hamiltonian cannot be diagonalized and the eigenspace collapses to a lower dimension and becomes incomplete, which is analogous to the reduction of the dimension of Hilbert space^[3,36]. The special points are referred to as the EPs and the number of degeneracies is the order of EPs.

It is often accompanied by novel phenomena such as ultrahigh sensitivity^[37,38] and phase transitions^[150,151] in the immediate vicinity of an EP. The special algebraic behavior allows a reduction of the full problem to the two-dimensional problem associated with the two coinciding levels^[152,153]. Here consider the two-dimensional Hamiltonian $H_{2} = (\begin{matrix} 0 & x \\ x & y \end{matrix}) .$ (35)

The Hamiltonian becomes non-Hermitian when either parameter $x$ or $y$ is subjected to analytic continuation. In an open system, for simplicity, we can set $x = 1$ and $y = t + i γ$ , where $t$ may be realized as detuned energy, and $i γ$ denotes the loss (gain) of the system. It is straightforward to calculate the eigenvalues and the eigenstates as $E_{\pm} = \frac{1}{2} (y \pm \sqrt{y^{2} + 4}), | ν_{EP} ⟩ = (\begin{matrix} \mp i \\ 1 \end{matrix}) .$ (36)

The eigenvalues and the eigenstates are identical at the EP, where $y = \pm 2 i$ , and the Hamiltonian becomes a defective matrix. Interestingly, the eigenstates at the EP can be mapped to the Jones vector describing circular polarization of light, which is referred to as chirality^[154].

Note that at the EP, the difference between degeneracy and coalescence is clearly manifested by the occurrence of only one eigenstate instead of the familiar two in the case of a genuine degeneracy. Using the biorthogonal system for a non-Hermitian Hamiltonian in the previous subsection, the inner product of the eigenstates at EP satisfies $⟨ ν_{EP} | ν_{EP} ⟩ = 0$ , which means that the eigenstates at EP are self-orthogonal^[155]. It is the vanishing of the inner product that enables the reduction of a high-dimensional problem to two dimensions in the close vicinity of an EP^[154].

In fact, even though the Hamiltonian at EPs is not diagonalizable, it can be brought to Jordan normal form via a similarity transformation $H = S H S^{- 1}$ , with $J$ being the block diagonal, consisting of Jordan blocks^[156,157]. Then, the existence of higher-order EPs depends on checking whether the eigenvalues and their corresponding eigenstates of the Jordan blocks coalesce. Note that EPs of different orders can coexist since the Hamiltonian may feature multiple such Jordan blocks of varying dimensions.

In the non-Hermitian systems without symmetry, finding an $n$ th-order EP requires tuning $2 n - 2$ parameters, meaning that the appearance of higher-order EPs requires increasing the number of fine-tuned parameters, and they are thus assumed to play a much less prominent role^[158]. However, Mandal and Bergholtz proposed that third-order EPs generically require only two real tuning parameters in the presence of either a $P T$ symmetry or a generalized chiral symmetry^[54]. This shows that the requirement for tuning parameters to find an $n$ th-order EP can be reduced in non-Hermitian systems by introducing different symmetries. However, the constraints required for higher-order EPs under different symmetries remain an open problem.

Recent interest in higher-order EPs, where more than two eigenstates coalesce, has motivated significant efforts in their realization and application. While second-order EPs are relatively well-understood and widely demonstrated, constructing third- or higher-order EPs typically requires the fine-tuning of multiple parameters to enforce the degeneracy condition. A general $N$ th-order EP satisfies the condition ${(H - λ_{EP} I)}^{N} = 0$ , where all $N$ eigenvalues and eigenvectors coalesce. This gives rise to a characteristic eigenvalue sensitivity scaling as $Δ λ \sim ϵ^{1 / N}$ under perturbation strength $ϵ$ , offering potential advantages for sensing. In photonic systems, such EPs can be realized by coupling multiple microring resonators^[159,160], photonic crystal defect cavities^[161], or by exploiting synthetic dimensions, such as dynamically modulated or time-multiplexed structures^[162]. These approaches effectively construct higher-dimensional non-Hermitian Hamiltonians, enabling the coalescence of multiple modes.

Importantly, certain symmetries can significantly simplify the realization of higher-order EPs by reducing the number of tuning parameters. In particular, $P T$ symmetry allows real spectra to persist despite non-Hermiticity, and facilitates EP formation under constrained conditions. For example, in a three-mode $P T$ -symmetric system composed of gain, loss, and neutral components, only two independent parameters may suffice to reach a third-order EP^[163,164]. Likewise, chiral (or sublattice) symmetry enforces a symmetric spectrum about zero, leading to a block-off-diagonal Hamiltonian structure of the form $H = (\begin{matrix} 0 & A \\ B & 0 \end{matrix})$ . This naturally supports pairwise eigenvalue coalescence, aiding in the emergence of higher-order EPs with reduced tuning effort^[165]. These symmetry-induced constraints lower the codimension of the EP manifold, enhancing both the feasibility and stability of experimental realizations.

Despite these theoretical insights, observing the enhanced sensitivity associated with higher-order EPs remains experimentally challenging. While the $ϵ^{1 / N}$ scaling implies stronger response to target perturbations, it also amplifies the effects of unwanted noise, disorder, and fabrication imperfections^[166]. Moreover, the coalescence of eigenvectors at an EP leads to non-orthogonality, which can degrade the signal-to-noise ratio (SNR) and limit precision in practical measurements^[167]. Recent theoretical analyses further show that the quantum Fisher information (QFI), which bounds estimation precision, may saturate or even decline near EPs under realistic decoherence, thus challenging the assumption that EPs always enhance metrological performance^[168,169]. These issues highlight the need for noise-resilient architectures, dynamically tunable platforms, and hybrid schemes that can robustly harness the non-Hermitian features of higher-order EPs without sacrificing precision.

2.6 Conserved Quantities

The conserved quantities of a closed system are determined by its symmetries and offer global insights into its dynamical evolution, such as energy, the Runge–Lenz vector in Kepler dynamics^[170], or the electric charge^[171]. Constraints imposed by these time invariants have been instrumental in advances ranging from the prediction of neutrinos to the development of conserving approximations in many-body physics.

In closed systems, an observable $A$ is called conserved if it commutes with the Hermitian Hamiltonian of the system. Due to the equivalence between conservation and commutation, a unitary symmetry transformation on the quantum state space is generated by each conserved observable $A$ . One can construct simultaneous eigenstates of the Hamiltonian and the observable such that the Hamiltonian becomes block-diagonalized, thereby reducing the complexity of the eigenvalue problem. A complete set of conservation laws for a system is obtained by identifying all linearly independent observables that commute with the Hamiltonian.

However, for a non-Hermitian Hamiltonian $H$ , the norm of the wave function is not conserved. A complete characterization and observation of conserved quantities in open systems and their consequences is still an outstanding question. With an approach inspired by the pseudo-Hermiticity^[113–115], these questions can be addressed. If $η$ is an intertwining operator, i.e., it satisfies the relation $η H = H^{†} η$ , it is straightforward to show that $⟨ ψ (t) | η | ϕ (t) ⟩$ is constant for any arbitrary states $| ψ ⟩$ and $| ϕ ⟩$ at any arbitrary time $t$ . When the Hamiltonian is Hermitian, the intertwining relation has a trivial solution $η = I$ . This implies that the Dirac inner products, and in particular, Dirac norm, are time invariant. All other nontrivial solutions correspond to operators that commute with the Hamiltonian and thus correspond to conserved quantities in the traditional Hermitian quantum theory.

When $H \neq H^{†}$ , the non-unitary time evolution operator $G (t) = e^{- i H t}$ keeps the observable $η$ unchanged, that is, $G^{†} (t) η G (t) = η$ . Thus, an expectation value of $η$ in an arbitrary quantum state is a conserved quantity. The equivalence between conservation and commutation breaks down for a non-Hermitian Hamiltonian, meaning that the conserved observables $η$ and the Hamiltonian $H$ cannot be simultaneously diagonalized^[172].

The defining equation for the intertwining operator in open systems allows one to systematically solve for it. Note that since this process is linear, if $η_{1}$ is a solution and $η_{2}$ is another solution, then their linear combinations are also intertwining operators. Thus, one needs a suitable definition of $η$ to satisfy the conditions of orthogonality and linear independence.

Consider a $P T$ -symmetric Hamiltonian for instance; the form of $H$ is given by $H = - J S_{x} + i γ S_{z},$ (37)where $S_{x}$ and $S_{z}$ are the spin operators that satisfy the canonical commutation relations $[S_{α}, S_{β}] = i ϵ_{α β γ} S_{γ}$ . The simplest representation for the $2^{2} - 1 = 3$ generators of SU(2) is the spin $S = \frac{1}{2}$ representation. This is given by $N = 2 S + 1 =$ two-dimensional matrices $S_{α} = σ_{α} / 2$ , where $σ_{α}$ are the three Pauli matrices. The analytical recursive construction is as follows^[172]. The transpose symmetry of the $P T$ -symmetric Hamiltonian implies $T H T = H^{†}$ , where the time-reversal operator $T$ is given by complex conjugation $T = *$ . It follows from the $P T$ symmetry of the Hamiltonian that the parity operator $P$ is a conserved observable, namely, $η_{1} = P$ . Therefore, a sequence of linearly independent and dimensionless observables can be constructed by $η_{i} = η_{i - 1} H / J$ . The intertwining nature of $η_{i - 1}$ implies that $η_{i}$ is also a conserved observable. This sequence terminates with $η_{d}$ since the characteristic equation for $H$ is a polynomial of dimensional order $d$ ^[147].

For $H_{2} = (- J σ_{x} + i γ σ_{z}) / 2$ , it is easy to check that $η_{1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), η_{2} = (\begin{matrix} J & i γ \\ - i γ & J \end{matrix})$ (38)are two linearly independent but not orthogonal intertwining operators. Thus, their expectation values are independent of time, irrespective of whether the system is in the $P T$ -symmetric region or the $P T$ -broken region. Furthermore, a coherent superposition of eigenstates of a conserved quantity cannot be generated from a single initial eigenstate in an isolated system. However, the eigenstates of a conserved observable exhibit nontrivial dynamics in $P T$ -symmetric systems^[172].

3 Experimental Implementation of Non-Hermitian Physics in Photonic Systems

Although non-Hermitian physics originates from quantum mechanics, optical systems have emerged as an ideal platform for its study. This is primarily due to the ability to control gain and loss effects in open systems by manipulating components within the optical system. The characteristics of non-unitary evolution are investigated by realizing non-Hermitian Hamiltonians. Non-Hermitian systems give rise to novel physical phenomena due to their diverse symmetries and exceptional points, offering new solutions for information processing, topological structures, precision measurement, and other areas of research.

Non-Hermitian physics has already been experimentally realized in various optical systems. It is helpful to briefly reflect on the distinctive advantages that non-Hermitian physics brings to photonic systems. Beyond theoretical interest, non-Hermitian effects offer new degrees of freedom—such as loss engineering, complex-valued coupling, and asymmetric mode interactions—that have led to a range of novel functionalities in photonics. For example, by leveraging asymmetric coupling or gain-loss control, non-Hermitian photonics has enabled the realization of mode-tunable lasers, where the lasing mode can be selectively addressed through parameter tuning near exceptional points^[173,174]. Similarly, sharp phase transitions and nonreciprocal dynamics in non-Hermitian systems have been harnessed for ultrafast and compact optical switches^[175]. Furthermore, by embedding non-Hermitian terms in topologically structured lattices, researchers have demonstrated reconfigurable topological photonic systems, allowing for flexible control of edge transport and robust light steering^[176,177].

These examples underscore the practical relevance of non-Hermitian models in photonics and motivate their continued experimental exploration. In this section, we examine the application potential of non-Hermitian physics in optical platforms by discussing its realization in different optical systems, including bulk systems, time-multiplexed systems, waveguides, optical lattices, optical cavities, optical metastructures, and optical combs.

3.1 Implementation of Non-Hermitian Physics in Bulk Optics

Non-Hermitian Hamiltonians with $P T$ symmetry exhibit real energy spectra, which have garnered significant attention in classical optical systems^[178–187] and have been employed in various applications^[188–193]. The currently reported realization of $P T$ -symmetric synthetic optical lattices with balanced gain and loss relies on classical light pulse propagation, analogous to quantum walks^[194].

Quantum walks^[195–199], as a versatile quantum simulation framework, describe the movement of particles with internal degrees of freedom on discrete lattices. However, implementing this approach in the quantum domain poses challenges, particularly due to the difficulty in amplifying single photons. In 2017, Xiao et al. experimentally realized passive $P T$ -symmetric single-photon quantum walks^[200], where photon losses were modulated intermittently in a quantum walk interferometer network. By alternately increasing and decreasing these losses, the study observed the dynamical behavior of the non-Hermitian system in regimes where $P T$ symmetry was both preserved and broken. They realize passive $P T$ -symmetric non-unitary quantum walks with single photons by employing a scheme that alternates photon losses^[201–204]. The dynamics of the quantum walk are dictated by a non-unitary time-evolution operator $U = L S C [θ_{2} (x)] L^{'} S C [θ_{1} (x)]$ , where $S$ is the conditional translation operator, and $C [θ (x)]$ is the position-dependent coin operator. The symbol $x$ is the position of walker and $| 0 ⟩, | 1 ⟩$ are the orthogonal coin states. $L$ is the loss operator.

As illustrated in Fig. 1, the spatial modes of photons encode the walker states, while their horizontal and vertical polarization states define the coin states. A position-dependent coin operator is achieved using individually placed unmounted half-wave plates (HWPs) along designated spatial paths. Conditional position shifts are introduced by birefringent beam displacers, which transmit vertically polarized photons straight through while displacing horizontally polarized photons laterally to an adjacent mode. The polarization-dependent loss operator $L$ is implemented using a single-input partially polarizing beam splitter (PPBS) with distinct transmission coefficients for horizontal and vertical polarizations. The raw probabilities are calculated by normalizing the coincidence counts against the total number of photon pairs generated prior to interacting with the experimental setup at each step.

Figure 1.Experimental setup for non-unitary quantum walks with alternating losses^[200]. A photon pair is generated in a Bell state through spontaneous parametric down-conversion. One photon serves as a trigger and can be prepared in any desired polarization state using a quarter-wave plate (QWP), a half-wave plate (HWP), and a polarizer for projection measurement. The other photon undergoes a series of processes, including controlled polarization rotations, polarization-dependent loss operations, and spatial translations. To explore the topological characteristics of the quantum walk, unmounted HWPs are strategically placed along specific paths to introduce spatially varying polarization rotations. The resulting probability distribution is recorded using avalanche photodiodes.

Table 1. Comparison of Key Metrics Across Different Photonic Platforms for Non-Hermitian Implementations.

Table 1. Comparison of Key Metrics Across Different Photonic Platforms for Non-Hermitian Implementations.