Technique for studying the coalescence of eigenstates and eigenvalues in non-Hermitian systems

Abstract

In our study, we explore high-order exceptional points (EPs), which are crucial for enhancing the sensitivity of open physical systems to external changes. We utilize the Hilbert−Schmidt speed (HSS), a measure of quantum statistical speed, to accurately identify EPs in non-Hermitian systems. These points are characterized by the simultaneous coalescence of eigenvalues and their associated eigenstates. One of the main benefits of using HSS is that it eliminates the need to diagonalize the evolved density matrix, simplifying the identification process. Our method is shown to be effective even in complex, multi-dimensional and interacting Hamiltonian systems. In certain cases, a generalized evolved state may be employed over the conventional normalized state. This necessitates the use of a metric operator to define the inner product between states, thereby introducing additional complexity. Our research confirms that HSS is a reliable and practical tool for detecting EPs, even in these demanding situations.

Graphical abstract

Keywords

non-Hermitian physics / exceptional points / Hilbert−Schmidt speed / quantum statistical speed

1 Introduction

In quantum mechanics, systems are associated with specific Hilbert spaces, in which their state descriptions reside. The evolution of these quantum systems is governed by Schrödinger’s equation, which characterizes the dynamic behavior of the system over time in the presence of the system’s Hamiltonian operator. Conventionally, it has been assumed that Hamiltonians possess the Hermiticity property, ensuring that the eigenvalues are real [1]. The reality of the spectrum is a characteristic feature of Hermitian matrices. Additionally, it is widely acknowledged that a Hermitian Hamiltonian results in a unitary time evolution, thereby guaranteeing the preservation of probability. However, in nature, decoherence effects are pervasive since physical systems inevitably interact with their environment. Under such conditions, the system dynamics are more appropriately described by a Lindbladian generalized master equation [2].

In quantum physics, the non-Hermitian Hamiltonian (NHH) [1, 3–36] indicates the reality is that the system considered is actually open: it is incorporated into the continuum of scattering wavefunctions which all the time exists, and its features are affected by the coupling to the environment [37]. Furthermore, these Hamiltonians often have complex eigenvalue spectra and do not conserve probabilities [38]. This research field seeks to extend the conventional framework of quantum mechanics to encompass non-standard quantum theories and establish connections with open- and lossy-quantum systems [39–43]. Notably, four distinct classes of non-Hermitian quantum systems have garnered significant attention: P-pseudo-Hermitian systems [44–49], \mathcal{P}\mathcal{T}-symmetric systems [1, 9, 11–13, 15, 17, 19, 23, 28, 30–33, 50–72], and their anti-symmetric counterparts [73–85].

It is noteworthy that the requirement of Hermiticity is not rigorously essential for the real-valued nature of the eigenvalues. This observation was initially highlighted in 1959 within the framework of the hard-core Bose gas, using Fermi’s pseudo-potential [86]. Further investigations revealed that the spectrum of a non-Hermitian version of the Toda lattice was likewise found to be completely real [87], demonstrating that real eigenvalues can still emerge in non-Hermitian systems. The concept of pseudo-Hermiticity plays a crucial role in describing a category of non-Hermitian matrices possessing entirely real spectra [88]. Its historical roots can be traced back to the work of Dirac and Pauli, who introduced the notion of pseudo-Hermiticity in the context of quantum field theory with an indefinite metric, accounting for negative norms [89]. Subsequently, the interest in pseudo-Hermitian matrices was further motivated by investigations of the Yang−Lee edge singularity, within the framework of nonunitary quantum field theory [90, 91]. Bessis and Zinn−Justin formulated a conjecture positing that the single-particle Hamiltonian, featuring a cubic imaginary potential and rooted in this nonunitary quantum field theory, is expected to display a completely real spectrum [92]. The conjecture put forth by Bessis and Zinn−Justin regarding the entirely real spectrum of a single-particle Hamiltonian with a cubic imaginary potential, based on nonunitary quantum field theory, has been subject to rigorous examination.

In 1998, Bender et al. [3, 93, 94] found that a NHH, possessing both parity and time-reversal (\mathcal{P}\mathcal{T}) symmetry, can also have a real spectrum. This observation raises intriguing questions about the physical significance of a \mathcal{P}\mathcal{T}-symmetric NHH and its potential utilization as an alternative to Hermiticity in quantum mechanics [7, 95, 96]. Building upon this idea, Mostafazadeh et al. [44, 88, 97–99] further developed the concept of pseudo-Hermiticity, enabling the identification of general conditions under which a spectrum can be entirely real. Furthermore, after growing interest in \mathcal{P}\mathcal{T}-symmetry, Ge and Türeci [73] introduced anti-\mathcal{P}\mathcal{T} (\mathcal{A}\mathcal{P}\mathcal{T}) symmetry in optics and they demonstrated that \mathcal{A}\mathcal{P}\mathcal{T} systems can exhibit intriguing properties.

Non-Hermitian \mathcal{P}\mathcal{T}-symmetric Hamiltonians have found extensive applications in optics [100], mechanical oscillators [101], acoustics [102], LRC circuits [103], atomic physics [76, 104, 105], single-spin systems [106], microwave cavities [52], superconducting wire [107], optical scattering systems [108], photonic graphene [109], and nuclear magnetic resonance (NMR) [110]. In \mathcal{A}\mathcal{P}\mathcal{T}-symmetric systems, fascinating physical phenomena have been reported, including constant refraction in coupled optical systems [80] and materials with balanced positive and negative indices [73]. Moreover, relevant experimental investigations have been conducted in atomic systems [79, 83, 111], diffusive systems [84], molecular systems [74], and cold atoms [112].

Recently, non-Hermitian systems have garnered significant attention in research, particularly in studies reported in Refs. [7, 113–117]. This surge of interest is due to a multitude of rich physical phenomena inherent to non-Hermitian systems, including exceptional points (EPs), non-Bloch band theory, the non-Hermitian skin effect, exotic topological phases, and extended symmetry classes. Remarkably, these phenomena find no counterpart in the contemporary Hermitian realm. Among the most captivating features of non-Hermitian systems in quantum mechanics is the presence of EPs [113, 118–121], where eigenvalues and eigenvectors coalesce [122]. It should be emphasized that the Hamiltonian matrix at EPs, gets to be imperfect [123]. Furthermore, EPs may occur in various types of non-Hermitian systems.

An EP of NHH is a spectral degeneracy, which signifies that two or more eigenvalues and eigenvectors of the NHH become identical at a certain parameter value [34]. For simplicity, we will refer to an EP of an NHH as a Hamiltonian EP (HEP). A NHH is a mathematical operator that describes the energy and dynamics of a quantum system that is not isolated from its environment, such as an open system that exchanges energy or information with its surroundings. At an HEP, the NHH becomes non-diagonalizable and loses its spectral resolution, leading to a breakdown of the adiabatic theorem. The adiabatic theorem, which states that a quantum system can follow its instantaneous eigenstate if the Hamiltonian changes slowly enough, is violated at EPs [25]. This violation results in the quantum system jumping to another eigenstate even if the Hamiltonian changes infinitesimally. This phenomenon is referred to as the EP-induced breakdown of adiabaticity [34].

As an instance, non-Hermitian systems with \mathcal{P}\mathcal{T} or \mathcal{A}\mathcal{P}\mathcal{T} symmetry manifest two phases: the unbroken symmetry phase, wherein the energy spectrum consists entirely of real eigenvalues, and the broken symmetry phase, characterized by the emergence of complex conjugate pairs for some or all of the eigenvalues [3]. The number of eigenvectors n that coalesce at the non-Hermitian degeneracy determines the order of the exceptional point, denoted as EPn. The most usual case is an EP2, where a pair of eigenvalues become degenerate, and exceptional points of order greater than two are conventionally termed high-order EPs [121, 122, 124]. The realization of high-order EPs has greatly contributed to the development of this research field, most of them aimed at enhancing the response of open physical systems. For example, when the EP order increases then the sensitivity to perturbations increases [122]. Recently, there has been significant attention directed towards EPs, leading to the discovery of a wide range of intriguing phenomena in optical microcavities [125], band merging [126, 127], enhanced sensitivity of frequency detection [128], thresholdless phonon lasers [11, 18], coupled ridge lasers [129], topological chirality [130, 131], atom-cavity quantum composite [132], unidirectional invisibility [9, 133], exceptional photon blockade [70], and loss-induced transparency [134].

Quantum statistical speeds (QSSs) [135], such as quantum Fisher information (QFI) [136–144] and Hilbert−Schmidt speed (HSS) [145–155], have been effectively utilized in Ref. [38] to detect EPs of NHH in \mathcal{P}\mathcal{T} and \mathcal{A}\mathcal{P}\mathcal{T} symmetric systems. These findings have significantly advanced our understanding of these specific systems. However, the realm of non-Hermitian systems extends beyond \mathcal{P}\mathcal{T} and \mathcal{A}\mathcal{P}\mathcal{T} symmetric systems. A wide variety of other non-Hermitian systems, especially those of high dimensions, remain unexplored in this context. Therefore, the aim of our current research is to extend the application of QSS quantifiers for detecting EPs to these other non-Hermitian systems. In addition to extending the application of QSS quantifiers, our research also addresses the challenge of identifying EPs within generalized, as opposed to normalized, non-Hermitian systems. This is particularly important as sometimes, instead of using the normalized evolved state, we have to use the generalized evolved state. In general, the inner products in these generalized systems differ from conventional ones. For a Hermitian Hamiltonian in quantum mechanics, the norm squared of the corresponding vector in a Hilbert space is preserved over time, given that H(t)= H^{†}(t). This allows for the easy determination of the states’ inner product. However, this is not readily achievable in NHHs. To structure the formulation, we employ a metric operator to establish connections between states. It is worth noting that in Hermitian systems, the metrics can always be chosen to be the identity, aligning the inner products with conventional ones [1]. Furthermore, when the Hamiltonian’s eigenstates form a complete set of bases and a specific metric is chosen, biorthogonal quantum mechanics can be recovered [156]. This approach provides a promising avenue for exploring EPs in high-dimensional non-Hermitian systems. An important point to note is that the calculation of HSS does not require the diagonalization of the normalized evolved density matrix. This simplifies the exploration of EPs in high-dimensional NHHs [157]. By doing so, we hope to broaden the scope of our knowledge and understanding of EPs in non-Hermitian systems, thereby contributing to the ongoing advancements in this field.

This paper is structured as follows: Section 2 defines the preliminaries. Section 3 presents the main idea of this paper. Section 4 explores the problem of identifying EPs using HSS in various models of non-Hermitian systems governed by normalized evolved states. We also check the validity of the results obtained from the normalized evolved state when the evolved state is generalized. Section 5 discusses physical concepts and measurement of the quantum statistical speed from experimental data. Finally, Section 6 summarizes and discusses our key findings.

2 Preliminaries

2.1 Hilbert−Schmidt speed

Since the HSS is a particular case of quantum statistical speed, we begin by recalling the general framework leading to its definition. Hence, we consider the family of distance measures as [135]

(1)[d_{\alpha }(p,q ){]}^{\alpha }={\displaystyle \frac{1}{2}}{\displaystyle \sum _x|(p_x{)}^{\frac{1}{\alpha }}-(q_x{)}^{\frac{1}{\alpha }}{|}^{\alpha },}

where p=\{ p_x{\}}_x and q=\{q_x{\}}_x are probability distributions and \alpha \ge 1. To calculate the statistical speed from any statistical distance, one can assess the distance between extremely close distributions taken from a one-parameter family p_x({\varphi }_0) with parameter \varphi. Hence, the classical statistical speed is obtained by

(2)s_{\alpha }[p({\varphi }_0)]= {\displaystyle \frac{\mathrm{d}}{\mathrm{d}\varphi }}{d}_{\alpha }(p({\varphi }_0+\varphi ),p( {\varphi }_0)).

Now, we consider a given pair of quantum states \rho and \sigma, one can extend these classical notions to the quantum case by taking as the measurement p_x=\text{Tr} [E_x\rho ] and q_x=\text{Tr}[E_x\sigma ] probabilities associated with the positive-operator-valued measure (POVM) given by the set of \{E_x\ge 0\} satisfying {\sum }_n{p}_x=\mathbb{I}, where \mathbb{I} is the identity operator. The corresponding quantum distance, with maximizing the classical distance over all possible choices of POVMs, can be obtained as

(3)D_{\alpha }(\rho ,\sigma )=\underset{E_x}{max}{d}_{\alpha }(\rho ,\sigma ).

Extending Eq. (2) to the quantum case and maximizing the classical distance of Eq. (1) over all possible choices of POVMs, we obtain the quantum statistical speed called the Hilbert–Schmidt distance as [135]

(4)S_{\alpha }[\rho (\varphi )]=\underset{E_x}{max}{s}_{\alpha }[p( \varphi )]=({\displaystyle \frac{1}{2}}\text{Tr}[{\displaystyle \frac{\mathrm{d}\rho (\varphi )}{\mathrm{d}\varphi }}{]}^{\alpha }{)}^{1/\alpha }.

In the special case when \alpha =2, the quantum statistical speed is given by the HSS

(5)\text{HSS}({\rho }_{\varphi })=({\displaystyle \frac{1}{2}}\text{Tr}[{\displaystyle \frac{\mathrm{d}\rho (\varphi )}{\mathrm{d}\varphi }}{]}^2{)}^{1/2},

which does not require the diagonalization of \mathrm{d}{\rho }_{\varphi }/\mathrm{d}\varphi. Noted that, the explicit expression of HSS for various models in this paper has a cumbersome form and is not reported here.

2.2 Time evolution of the system governed by an NHH

For a quantum system with an n-dimensional Hilbert space \mathcal{H}, let us consider an initial state {\rho }_0= | {\psi }_0⟩⟨{\psi }_0| as we have

(6)| {\psi }_0⟩={\displaystyle \frac{1}{\sqrt{n}}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+\cdots +|{\psi }_n⟩),

where \varphi is an unknown phase shift and \{|{\psi }_i⟩,i=1,\cdots ,n\} represents the computational orthonormal basis. Furthermore, in general, the time evolution operator of a non-Hermitian system with time-independent Hamiltonian is defined as [72, 158]

(7)U_{\mathrm{N}\mathrm{H}}( t)={\text{e}}^{-\mathrm{i}{H}_{\mathrm{N}\mathrm{H}}t}.

2.2.1 Normalized evolved state of the system

We directly apply conventional quantum mechanics on non-Hermitian systems to obtain the evolved state. Accordingly, the dynamics governed by a non-Hermitian system with NHH H_{\mathrm{N}\mathrm{H}} is given by [158, 159]

(8){\rho }_N(t)={\displaystyle \frac{\rho (t)}{\text{Tr}[\rho (t)]}} ={\displaystyle \frac{U(t){\rho }_0{U}^{†}(t)}{\text{Tr}[U(t){\rho }_0{U}^{†}(t)]}},

where {\rho }_N(t) called normalized density matrix [160]. In this relation, the time-evolution operator U(t) can be associated with time-independent Hamiltonian [U_{\mathrm{N}\mathrm{H}}( t) in Eq. (7)].

2.2.2 Generalized evolved state of the system and biorthogonal quantum mechanics

The notion of probability in finite-dimensional quantum mechanics (QM) is closely related to the corresponding inner product. It is well-established that the total probability of a system must sum to unity. Any alterations to the total probability are physically nonsensical. In finite-dimensional non-Hermitian quantum mechanics (NHQM), the generalized or metricized inner product maintains the principle of probability conservation by drawing inspiration from concepts found in fiber bundles. In this section, we consider the representation of quantum states as vectors, as is customary. However, dual states (covectors), on the other hand, undergo an additional correction beyond Hermitian conjugation: they are modified by a “metric operator”. This metric operator serves as an effective mathematical tool for elegantly formulating the theory of quantum systems. Importantly, it should be emphasized that the conventional metric, as discussed in Ref. [38], yields results entirely consistent with experimental observations when correctly applied. There are some corrections to the inner products suggested in the papers which either do not consider the relation between quantum mechanics and Hilbert space [161–163] or were bound to some particular cases of non-Hermitian systems [94, 156, 164–166]. To discover a general inner product that conserves the notion of Hilbert space, we define a compatible metric presented in Ref. [1].

The state probability in quantum mechanics, is the norm squared of the corresponding vector in a Hilbert space [1]. The dynamics of the vector |\psi (t) ⟩ has to satisfy the Schrödinger equation,

(9)\mathrm{i}\hslash {\mathrm{\partial }}_t|\psi (t) ⟩=H (t)|\psi (t )⟩,

where H refers to Hamiltonian, but it should be considered as a generalized Hamiltonian-type operator. In general, the generalized inner products are different from the conventional ones. If the Hamiltonian H(t) is Hermitian, then the norm squared is preserved in time, because we have H(t)= H^{†}(t), so that the inner product can be easily obtained. On the other hand, for a Hamiltonian that does not have such a clear symmetry, the inner product that conserves the norm squared can not be easy to find. Then, Eq. (9) can be written as

(10){\mathrm{\nabla }}_t|\psi (t)⟩ =({\mathrm{\partial }}_t+{\displaystyle \frac{\mathrm{i}}{\hslash }}H(t))|\psi ( t)⟩=0,

where {\mathrm{\nabla }}_t refers to a connection role in a vector bundle that connects the geometries of nearby points and |\psi ( t)⟩ is therefore parallel transported along the time direction. This analogy hints that there is a “connection-compatible metric” such that the inner products between the parallel transported vectors are time-independent. Hence, the norm squared that is given by the vector inner product and itself, is also time independent [1]. To recognize the changed inner product from the conventional one, the changed inner product is denoted as [[{\psi }_1(t) | {\psi }_2(t)]]. The dual vectors are exposed to a linear map as follows:

(11)[[\psi (t)|=⟨\psi (t)|\eta ,

where ⟨\psi (t) | is a conventional Hermitian conjugate of |\psi (t) ⟩ and \eta will be called metric operator. It should be noted that there is no difference between the generalized state |\psi (t)]] and the conventional one |\psi (t)⟩, because both of which evolve according to the Schrödinger equation.

The generalized evolved state governed by NHH H_{\mathrm{N}\mathrm{H}} is defined by [160, 167]

(12){\rho }_G(t)=\rho (t) \eta .

Moreover, \eta is the metric operator and one can introduce as bellows [1, 160, 168–170]:

(13)\eta =({\displaystyle \sum _i|E_i⟩⟨E_i|{)}^{-1}={\displaystyle \sum _i|E_i⟩⟩⟨⟨E_i|,}}

where |E_i⟩ and |E_i⟩⟩ are the normalized right and left eigenvectors of NHH H_{\text{NH}}, respectively. For more details, we consider NHH operator {\hat{H}}_{\text{NH}} and its conjugate {\hat{H}}_{\text{NH}}^{†}, so that they have complete sets of eigenstates [171]:

(14){\hat{H}}_{\text{NH}}|e_i⟩= E_i|e_i⟩,{\hat{H}}_{\text{NH}}^{†}|e_i⟩⟩=E^i|e_i⟩⟩, i=1,2,\cdots ,

where E_i=(E^i{)}^{\ast }. Moreover, |e_i⟩ and |e_i⟩⟩ represent respectively right and left eigenvectors of H_{\text{NH}} which should be normalized. We can then define normalized right eigenvectors |E_i⟩ and normalized left eigenvectors |E_i⟩⟩ for the respective eigenvalues E_i of the Hamiltonian operator H_{\text{NH}} by performing the following definitions [169, 170]:

(15)| E_i⟩\equiv {\displaystyle \frac{|e_i⟩}{\sqrt{⟨⟨e_i|e_i⟩}}}, | E_i⟩⟩\equiv {\displaystyle \frac{|e_i⟩⟩}{\sqrt{⟨e_i|e_i⟩⟩}}},

and

(16)⟨E_i | \equiv {\displaystyle \frac{⟨e_i|}{\sqrt{⟨e_i|e_i⟩⟩}}},⟨ ⟨E_i | \equiv {\displaystyle \frac{⟨⟨e_i|}{\sqrt{⟨⟨e_i|e_i⟩}}}.

So that, they satisfy orthonormality relations:

(17)⟨E_n | E_m⟩⟩=⟨⟨ E_m | E_n⟩={\delta }_{n,m},n ,m=1, 2,\cdots .

The eigenstates constitute biorthonormal bases in Hilbert space \mathcal{H} with two resolutions of unity and satisfy the following completeness relations:

(18){\displaystyle \sum _i|E_i⟩⟨⟨E_i|={\displaystyle \sum _i|E_i⟩⟩⟨E_i|=\hat{\mathbb{I}}.}}

The metric \eta in Eq. (13) is Hermitian, positive-definite and satisfying the equation of motion \mathrm{\partial }\eta /\mathrm{\partial }t=(\mathrm{i}/\hslash )[\eta (t )H(t)- H^{†}(t) \eta (t)] [1, 160]. In many cases, the metric can be independent of time; for example, in Hermitian [172] and pseudo-Hermitian [164] systems. Therefore, the metric is not necessarily time-dependent [1].

3 Probing EPs via HSS tool in non-Hermitian systems

In Ref. [38], a significant rule for detecting EPs in (anti-)\mathcal{P}\mathcal{T}-symmetric systems is introduced. This rule is based on a quantum system with an n-dimensional Hilbert space \mathcal{H} and an initial state defined as

(19)| {\psi }_0⟩={\displaystyle \frac{1}{\sqrt{n}}}\left({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+\cdots +|{\psi }_n⟩\right) ,

where \phi is an unknown phase shift and \{|{\psi }_i⟩,i=1,\cdots ,n\} forms the computational orthonormal basis. The authors discovered that the dynamics of the HSS, calculated with respect to the initial phase \varphi, can indicate the phase of the (anti-)\mathcal{P}\mathcal{T}-symmetric system. Specifically, an oscillating pattern signifies an (anti-)\mathcal{P}\mathcal{T}-unbroken (broken) phase, while the absence of oscillation corresponds to EPs or a broken (unbroken) phase.

The most crucial aspect of this work is the potential for generalization. The EPs detection, as facilitated by the aforementioned rule, is not limited to the system under study. Instead, it opens up new possibilities for exploring other non-Hermitian systems.We find that the dynamics of the Hilbert−Schmidt speed transition from an oscillatory to a non-oscillatory behavior pattern, or vice versa, upon crossing an exceptional point. This general applicability significantly broadens the scope of this rule, making it a powerful and efficient tool in the study of non-Hermitian quantum mechanics.

4 Main results and discussion

4.1 Hybrid \mathcal{P}\mathcal{T}−\mathcal{A}\mathcal{P}\mathcal{T}-symmetric system

As the first example, we consider the model of the \mathcal{P}\mathcal{T}−\mathcal{A}\mathcal{P}\mathcal{T}-symmetric system that is an implementation of possible nanophotonic in the form of four nanoring resonators coupled to a bus waveguide [173]. It is formed by two pairs of \mathcal{P}\mathcal{T}-symmetric resonators with resonances \pm {\omega }_0, gain/loss rates \pm {\gamma }_0, and coupling coefficient \kappa. Therefore, the Hamiltonian of this system is given by

(20)H_{\mathrm{P}\mathrm{T}-\mathrm{A}\mathrm{P}\mathrm{T}}=\left(\begin{array}{cccc}{\omega }_0+\mathrm{i}{\gamma }_0& \kappa & 0& \mathrm{i}\kappa \\ \kappa & {\omega }_0-\mathrm{i}{\gamma }_0& -\mathrm{i}\kappa & 0\\ 0& -\mathrm{i}\kappa & -{\omega }_0-\mathrm{i}{\gamma }_0& \kappa \\ \mathrm{i}\kappa & 0& \kappa & -{\omega }_0+\mathrm{i}{\gamma }_0\end{array}\right).

Note that, our system exhibits both \mathcal{P}\mathcal{T}-symmetry and \mathcal{A}\mathcal{P}\mathcal{T}-symmetry. In this model, previously realized in quantum [173], we identify two EPs, denoted by EP(1) and EP(2). They are given by {\kappa }_{\text{EP(1)}}={\omega }_0{\gamma }_0/\sqrt{{\omega }_0^2+{\gamma }_0^2} and {\kappa }_{\text{EP(2)}}=\sqrt{{\omega }_0^2+{\gamma }_0^2}/2, respectively.

In order to investigate the performance of witness, first we compute the normalized evolved state of the system when it is prepared in the initial state {\rho }_0= | {\psi }_0⟩⟨{\psi }_0|, where utilizing Eq. (6) for the four-level system we have |{\psi }_0⟩=\frac{1}{\sqrt{4}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+|{\psi }_2⟩+|{\psi }_3⟩+|{\psi }_4⟩). We can simply calculate the HSS methodically by placing the evolved state Eq. (8) into Eq. (5).

The qualitative dynamics of the HSS to probe the EP(1) is illustrated in Fig.1 for the unbroken phase () and broken phase ({\kappa }_{\text{EP(1)}}>{\omega }_0{\gamma }_0/\sqrt{{\omega }_0^2+{\gamma }_0^2}). In the unbroken phase, we see that the HSS oscillates and eventually returns to its initial value. However, in the broken phase, it ultimately decreases over time, with no oscillations observed.

Moreover, the time variations of the HSS to detect the EP(2) is shown in Fig.2 for the unbroken phase () and broken phase ({\kappa }_{\text{EP(2)}}>\sqrt{{\omega }_0^2+{\gamma }_0^2}/2). The same results in Fig.1 can be obtained here.

4.2 System without \mathcal{P}\mathcal{T}−\mathcal{A}\mathcal{P}\mathcal{T}-symmetries

In this section, we consider a non-Hermitian system without \mathcal{P}\mathcal{T} and \mathcal{A}\mathcal{P}\mathcal{T}-symmetries. First, we examine the two-dimensional system and then extend it to higher dimensions. Consider a two-level system containing two coupled cavities A and B having the same resonant frequency {\omega }_2 [174]. The Hamiltonian of this system is given by

(21)H=\left(\begin{array}{cc}{\omega }_2-\mathrm{i}{\mathrm{\Gamma }}_0& \kappa \\ \kappa & {\omega }_2-\mathrm{i}\mathrm{\Gamma }\end{array}\right),

where \kappa is the coupling strength, {\mathrm{\Gamma }}_0 represents the intrinsic loss of each cavity, and \mathrm{\Gamma }= {\mathrm{\Gamma }}_0+ \mathrm{\Delta }\mathrm{\Gamma }, with \mathrm{\Delta }\mathrm{\Gamma } denoting an additional tunable loss introduced at cavity B. In this system, the EP occurs at the \mathrm{\Delta }\mathrm{\Gamma }=2|\kappa |.

To study the efficiency of HSS in this system, we first calculate the normalized evolved state of the system when it is prepared in the initial state {\rho }_0= | {\psi }_0⟩⟨{\psi }_0|, where employing Eq. (6) for the two-level system we have |{\psi }_0⟩=\frac{1}{\sqrt{2}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+|{\psi }_2⟩). We can simply compute the HSS numerically by placing the evolved state Eq. (8) into Eq. (5).

The time evolution of the HSS to explore the EP is plotted in Fig.3 for () and (\mathrm{\Delta }\mathrm{\Gamma }>2 | \kappa |). When (), the HSS oscillates and finally comes back to its initial value. However, when (\mathrm{\Delta }\mathrm{\Gamma }>2 | \kappa |), it exhibits no oscillations and is uniformly suppressed over time.

Now, we consider the system includes two pairs of coupled cavities with the same values of \kappa, {\mathrm{\Gamma }}_0, and \mathrm{\Gamma } but different resonant frequencies. Cavities A and B form one pair with resonant frequency {\omega }_2, and cavities C and D form another pair with resonant frequency {\omega }_1. Coupling between these two pairs is introduced by coupling cavities A and D with a small tube and cavities B and C with another small tube. The four-level NHH of this system is obtained as [174]

(22)H=\left(\begin{array}{cccc}{\omega }_2-\mathrm{i}{\mathrm{\Gamma }}_0& \kappa & 0& \tau \\ \kappa & {\omega }_2-\mathrm{i}\mathrm{\Gamma }& \tau & 0\\ 0& \tau & {\omega }_1-\mathrm{i}{\mathrm{\Gamma }}_0& \kappa \\ \tau & 0& \kappa & {\omega }_1-\mathrm{i}\mathrm{\Gamma }\end{array}\right),

where \tau denotes the strength of interpair coupling and additional losses are introduced in cavities B and D. The coalescence of states (CS) can occur under three assumptions, which we mention as CS-1: {\mathrm{\Delta }}_1\pm 4\sqrt{{\mathrm{\Delta }}_2}=0 for {\mathrm{\Delta }}_1\ne 0, {\mathrm{\Delta }}_2\ne 0, CS-2: for {\mathrm{\Delta }}_2= 0, {\mathrm{\Delta }}_1\ne 0, and CS-3: for {\mathrm{\Delta }}_1= {\mathrm{\Delta }}_2 =0 where {\mathrm{\Delta }}_1=-(\mathrm{\Delta }\mathrm{\Gamma }{)}^2+4{\kappa }^2+4 {\tau }^2+(\mathrm{\Delta }\omega {)}^2, {\mathrm{\Delta }}_2=4{\kappa }^2{\tau }^2+ {\kappa }^2(\mathrm{\Delta }\omega {)}^2-(\mathrm{\Delta }\mathrm{\Gamma }{)}^2\frac{(\mathrm{\Delta }\omega {)}^2}{4} in which \mathrm{\Delta }\omega = {\omega }_1-{\omega }_2. CS-1 relates to a normal EP. At CS-2, two different EPs coincide. CS-3 relates to the coalescence of four states.

The qualitative behavior of the HSS to identify the EP for four-level NHH in Eq. (22) in terms of time is displayed in Fig.4-Fig.6. Here, we investigate all three conditions mentioned above to find EPs for Hamiltonian described by Eq. (22). In Fig.4(a) and (c), the oscillating behavior of HSS can be obtained for {\mathrm{\Delta }}_1>\pm 4\sqrt{{\mathrm{\Delta }}_2}. On the other hand, in Fig.4(b) and (d), the dynamics of HSS decreases for .

In addition, the qualitative behaviors of oscillation and decrease in the HSS dynamics are evident in Fig.5(a) and (b) for and {\mathrm{\Delta }}_2> 0, respectively. Furthermore, the same result can be achieved for and in Fig.6(a) and (b).

4.3 P-pseudo and anti-P-pseudo Hermitian two-level system

Next, we consider P-pseudo Hermitian two-level system and the anti-symmetric counterpart to check the EPs via HSS in them. First, we begin P-pseudo Hermitian two-level system. The Hamiltonian of this system is defined as [174]

(23)H_{\mathrm{P}\mathrm{P}\mathrm{H}}=\left(\begin{array}{cc}r{\mathrm{e}}^{\mathrm{i}\theta }& v\\ u& r{\mathrm{e}}^{-\mathrm{i}\theta }\end{array}\right),

where r, u, v and \theta are four independent real parameters. Furthermore, the Hamiltonian of this anti-P-pseudo Hermitian two-level system is given by

(24)H_{\mathrm{A}\mathrm{P}\mathrm{P}\mathrm{H}}=\mathrm{i}\left(\begin{array}{cc}r{\mathrm{e}}^{\mathrm{i}\theta }& v\\ u& r{\mathrm{e}}^{-\mathrm{i}\theta }\end{array}\right).

The eigenvalues of H_{\text{PPH}} and H_{\text{APPH}} are {ϵ}_{\pm }=r \cos \theta \pm \sqrt{uv-r^2{\sin }^2\theta } and \mathrm{i}{ϵ}_{\pm }, respectively. The EP for two cases of Hamiltonian occurs at the r=\sqrt{uv}/\sin \theta.

The qualitative time-behavior of the HSS is used to examine EPs in a P-pseudo Hermitian two-level system over time, as demonstrated in Fig.7. For , the HSS oscillates before returning to its initial value. Conversely, for r>\frac{\sqrt{uv}}{\sin \theta }, the HSS does not oscillate and decreases monotonically over time.

Additionally, Fig.8 illustrates the time evolution of the HSS for investigating EPs in an anti-P-pseudo Hermitian two-level system. It is evident that for r>\frac{\sqrt{uv}}{\sin \theta }, the HSS oscillates and reverts to its starting point. When , however, there are no oscillations, and the HSS consistently diminishes with time.

4.4 More high dimensional systems

4.4.1 Three sites

In this section, we extend our idea to high dimensional systems based on the paradigmatic Hatano−Nelson model [123, 175, 176]. One can define the three-site Hamiltonian under open boundary conditions as [123]

(25)H_{L3}=\left(\begin{array}{ccc}0& \tau -\gamma /2& 0\\ \tau +\gamma /2& 0& \tau -\gamma /2\\ 0& \tau +\gamma /2& 0\end{array}\right),

where \tau \pm \gamma /2 represent non-reciprocal hopping terms, i.e., the unequal left and right hopping strengths. In this model, higher order EPs occur at \tau =\pm \gamma /2, and the order depends on the number of sites L. In Eq. (25), \tau \pm \gamma are the parameter space.

The qualitative behavior of the HSS in identifying EPs in the three-site system is shown in Fig.9 for and \tau >\frac{\gamma }{2}. When \tau >\frac{\gamma }{2}, the HSS exhibits oscillations and eventually returns to its initial value. However, when , it displays no oscillations and monotonically decreases over time.

Additionally, the qualitative dynamics of the HSS in detecting another EP in the three-site system is illustrated in Fig.10 for and \tau > -\gamma /2. When \tau >-\frac{\gamma }{2}, the HSS is observed to oscillate and, over time, return to its initial value. In contrast, when , it shows no oscillations and ultimately decreases with time.

4.4.2 Four sites

The four-site Hamiltonian based on the paradigmatic Hatano−Nelson model [123] is given by

(26)H_{L4}=\left(\begin{array}{cccc}0& \tau -\gamma /2& 0& 0\\ \tau +\gamma /2& 0& \tau -\gamma /2& 0\\ 0& \tau +\gamma /2& 0& \tau -\gamma /2\\ 0& 0& \tau +\gamma /2& 0\end{array}\right).

Similar to the three-site model, the higher order EPs occur at \tau =\pm \gamma /2. The time variation of the HSS to probe the EP in the four-site system is illustrated in Fig.11 for and \tau >\gamma /2. A similar outcome to that shown in Fig.9 can be observed here.

In addition, the time evolution of the HSS to detect the other EP in the four-site system is plotted in Fig.12 for and \tau > -\gamma /2. The same results in Fig.10 can be obtained.

In this section, another case of the four-site system can be considered, which may host both second and fourth-order EPs. For example, the Hamiltonian of the four-site system reads as [123, 177]

(27)H_{L4}=\left(\begin{array}{cccc}\mathrm{i}\delta & p& 0& 0\\ p& \mathrm{i}\gamma & q& 0\\ 0& q& -\mathrm{i}\gamma & p\\ 0& 0& p& -\mathrm{i}\delta \end{array}\right) .

Here p represents the coupling between sites one and two, and between sites three and four, q denotes the coupling between sites two and three, \pm \mathrm{i}\delta and \pm \mathrm{i}\gamma are gain and loss terms. To demonstrate the efficiency of HSS in detecting EPs in this system, we only consider EP2. Setting \gamma =1, the EP2 occurs at q=\sqrt{(p^4+{\delta }^2+2\delta p^2)/{\delta }^2}.

The qualitative behavior of the HSS to find the EP in the four-site system in terms of time is demonstrated in Fig.13 for and q>\sqrt{(p^4+{\delta }^2+2\delta p^2)/{\delta }^2}. When , the HSS is observed to oscillate and eventually return to its initial value. Conversely, when q>\sqrt{(p^4+{\delta }^2+2\delta p^2)/{\delta }^2}, it shows no oscillations and eventually decreases over time.

4.4.3 Five sites

The next model is described by the five-site Hamiltonian. The Hamiltonian of this system based on the paradigmatic Hatano−Nelson model [123] is given by

(28)H_{L5}=\left(\begin{array}{ccccc}0& \tau -\gamma /2& 0& 0& 0\\ \tau +\gamma /2& 0& \tau -\gamma /2& 0& 0\\ 0& \tau +\gamma /2& 0& \tau -\gamma /2& 0\\ 0& 0& \tau +\gamma /2& 0& \tau -\gamma /2\\ 0& 0& 0& \tau +\gamma /2& 0\end{array}\right).

In this model, same as the three-site and four-site models, the higher order EPs occur at \tau =\pm \gamma /2. The dynamics of the HSS to obtain the EP in the five-site system is illustrated in Fig.14 for and \tau > \gamma /2. The similar findings in Fig.9 can be achieved.

Moreover, the time variations of the HSS to find the EP in the five-site system is shown in Fig.15 for and \tau > -\gamma /2. The similar outcomes in Fig.10 can be derived here.

4.4.4 Six sites

The next model is described by the six-site Hamiltonian. The Hamiltonian of this system based on the paradigmatic Hatano−Nelson model [123] is given by

(29)\begin{array}{rl}& H_{L6}=\\ & \left(\begin{array}{cccccc}0& \tau -\gamma /2& 0& 0& 0& 0\\ \tau +\gamma /2& 0& \tau -\gamma /2& 0& 0& 0\\ 0& \tau +\gamma /2& 0& \tau -\gamma /2& 0& 0\\ 0& 0& \tau +\gamma /2& 0& \tau -\gamma /2& 0\\ 0& 0& 0& \tau +\gamma /2& 0& \tau -\gamma /2\\ 0& 0& 0& 0& \tau +\gamma /2& 0\end{array}\right).\end{array}

In this final model, as with the three-site, four-site, and five-site models, the higher-order EPs occur at \tau = \pm \gamma /2. Fig.16 demonstrates the time variations of the HSS to detect the EP in the six-site system for and \tau >\gamma /2. The same results as those presented in Fig.9 can be inferred.

Furthermore, the dynamics of the HSS to identify the EP in the six-site system for is shown in Fig.17 for and \tau > -\gamma /2. Similarly, we obtain the findings in Fig.10.

4.5 Interacting models

4.5.1 Non-Hermitian strongly interacting Dirac fermions

Here, we investigate the non-Hermitian extension of an interacting model on the honeycomb lattice [178]. We specifically examine asymmetric hopping to build a non-Hermitian form of the Hubbard model [179]. Considering the low-energy effective theory, one can express the effective Hamiltonian describing the low-energy Dirac fermions in the non-Hermitian model on the honeycomb lattice as follows [180]:

(30)\begin{array}{rl}{H}_{\text{DF}}(\mathit{q})=& -v_F(q_x{s}_0\otimes {\sigma }_1+q_y{s}_0\otimes {\sigma }_2-\frac{\mathrm{i}{q}_y\delta }{\tau }{s}_3\otimes {\sigma }_1\\ & -\frac{\mathrm{i}{q}_x\delta }{\tau }{s}_3\otimes {\sigma }_2),\end{array}

where s and \sigma represent Pauli matrices acting on spin and sublattice spaces, respectively. Moreover, v_F=\frac{3\tau }{2} denotes the Fermi velocity in the absence of non-Hermitian hopping, namely \delta =0. We set energy unit \tau =1 throughout this subsection. Note that the EP occurs at \delta =1.

In this model, the qualitative behavior of the HSS over time for the phase broken (\delta >1) and phase unbroken () scenarios to probe the EP is depicted in Fig.18. The discernible results within this figure indicate a behavioral shift in the HSS dynamics upon crossing the EP, thereby confirming the HSS capability to efficiently detect EPs.

4.5.2 Interacting non-Hermitian ultracold atoms

We now explore the dynamics of interacting ultracold atoms within a three-dimensional (3D) harmonic trap, subject to spin-selective dissipative processes. These phenomena are precisely characterized by non-Hermitian parity−time (\mathcal{P}\mathcal{T}) symmetric Hamiltonians. For three spin-1/2 bosons, the allowable total spins are S=\frac{3}{2} and S=\frac{1}{2}. The wave function for the ferromagnetic S=\frac{3}{2} state is completely symmetric, representing the ground state under s-wave interaction within this spin configuration. Given the anisotropy in the two-body interaction within the |\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩ channel, the corresponding Hamiltonian for the ferromagnetic spin basis — \{|S=\frac{3}{2},M=\frac{3}{2}⟩,\mid S= \frac{3}{2},M=\frac{1}{2}⟩,|S=\frac{3}{2},M=-\frac{1}{2}⟩,|S=\frac{3}{2},M=-\frac{3}{2}⟩\} — can be derived, as detailed in Ref. [119]:

(31)H_{\mathrm{U}\mathrm{A}}=\left(\begin{array}{cccc}{\displaystyle \frac{3}{2}}\mathrm{i}\mathrm{\Gamma }& {\displaystyle \frac{\sqrt{3}}{2}}\mathrm{\Omega }& 0& 0\\ {\displaystyle \frac{\sqrt{3}}{2}}\mathrm{\Omega }& {\displaystyle \frac{1}{2}}\mathrm{i}\mathrm{\Gamma }+2\delta & \mathrm{\Omega }& 0\\ 0& \mathrm{\Omega }& -{\displaystyle \frac{1}{2}}\mathrm{i}\mathrm{\Gamma }+2\delta & {\displaystyle \frac{\sqrt{3}}{2}}\mathrm{\Omega }\\ 0& 0& {\displaystyle \frac{\sqrt{3}}{2}}\mathrm{\Omega }& -{\displaystyle \frac{3}{2}}\mathrm{i}\mathrm{\Gamma }\end{array}\right),

where \mathrm{\Omega } represents the coupling strength of the two-spin states and \mathrm{\Gamma } denotes dissipation on the spin-down state. The EP is characterized by the condition \mathrm{\Gamma }= \mathrm{\Omega }.

In Fig.19, the temporal variations of the HSS are illustrated to detect the EP in the system of interacting non-Hermitian ultracold atoms confined within a harmonic trap, for both and \mathrm{\Gamma }>\mathrm{\Omega } scenarios. This indicates that the EP can be readily detected by monitoring the dynamics of the HSS.

In the next step, we analyze three spin-1 bosons with permissible total spin values of S=3,2,1,0. The Hamiltonian, expressed in the ferromagnetic basis \{|S= 3,M⟩\} (integer M\in [-3,3 ]), is formulated as detailed in Ref. [119]:

(32)\begin{array}{rl}& H_{\mathrm{U}\mathrm{A}}=\\ & \left(\begin{array}{ccccccc}3\mathrm{i}\mathrm{\Gamma }& {\displaystyle \frac{\sqrt{6}}{2}}\mathrm{\Omega }& 0& 0& 0& 0& 0\\ {\displaystyle \frac{\sqrt{6}}{2}}\mathrm{\Omega }& 2\mathrm{i}\mathrm{\Gamma }& {\displaystyle \frac{\sqrt{10}}{2}}\mathrm{\Omega }& 0& 0& 0& 0\\ 0& {\displaystyle \frac{\sqrt{10}}{2}}\mathrm{\Omega }& \mathrm{i}\mathrm{\Gamma }+{\displaystyle \frac{12\delta }{5}}& \sqrt{3}\mathrm{\Omega }& 0& 0& 0\\ 0& 0& \sqrt{3}\mathrm{\Omega }& {\displaystyle \frac{18\delta }{5}}& \sqrt{3}\mathrm{\Omega }& 0& 0\\ 0& 0& 0& \sqrt{3}\mathrm{\Omega }& -\mathrm{i}\mathrm{\Gamma }+{\displaystyle \frac{12\delta }{5}}& {\displaystyle \frac{\sqrt{10}}{2}}\mathrm{\Omega }& 0\\ 0& 0& 0& 0& {\displaystyle \frac{\sqrt{10}}{2}}\mathrm{\Omega }& -2\mathrm{i}\mathrm{\Gamma }& {\displaystyle \frac{\sqrt{6}}{2}}\mathrm{\Omega }\\ 0& 0& 0& 0& 0& {\displaystyle \frac{\sqrt{6}}{2}}\mathrm{\Omega }& -3\mathrm{i}\mathrm{\Gamma }\end{array}\right).\end{array}

As with the Hamiltonian in Eq. (31), the EP is determined to occur at \mathrm{\Gamma }=\mathrm{\Omega }. Fig.20 explores the dynamics of the HSS for detecting the EP in high-dimensional interacting non-Hermitian ultracold atoms within a harmonic trap, under conditions where and \mathrm{\Gamma }>\mathrm{\Omega }. This reinforces our main result, aligning with the findings presented in the analysis of Fig.19.

4.6 The quantum witness of the HSS with inserting metric

In the previous sections, we examined the dynamics governed by the normalized evolved state of Eq. (8). In the following, we investigate the dynamics of the system governed by the generalized evolved state of Eq. (12). Our goal here is to see if the results obtained in the previous sections still hold true when we insert the metric into the system state. Inserting the metric, we find that monitoring the HS{S}_{\varphi } (\eta {\rho }_G(t) ) dynamics can be used as an efficient tool to detect the EPs in non-Hermitian systems. In more detail, examining the dynamics, we observe that not only HS{S}_{\varphi }({\rho }_N(t) ) but also HS{S}_{\varphi } (\eta {\rho }_G(t) ) exhibit a qualitative transition at EPs of the non-Hermitian systems. This transition leads to a change in dynamics from monotonic to oscillatory behavior and vice versa.

This method is illustrated with a two-dimensional system with \mathcal{P}\mathcal{T}-symmetry and then it is generalized to higher dimensions and \mathcal{A}\mathcal{P}\mathcal{T}-symmetry systems.

We start with the two-dimensional \mathcal{P}\mathcal{T}-symmetric Hamiltonian given by [1, 94, 160, 168]

(33)H_{\mathrm{P}\mathrm{T}}=\left(\begin{array}{cc}r{\mathrm{e}}^{\mathrm{i}\theta }& s\\ s& r{\mathrm{e}}^{-\mathrm{i}\theta }\end{array}\right),

where \{r, \theta ,s\in \mathbb{R}\} and s\ne 0. The two eigenvalues are obtained by {ϵ}_{\pm }=r\cos \theta \pm \sqrt{s^2-r^2{\sin }^2\theta }. In this case, the EP occurs at s^2=r^2{\sin }^2\theta. Then, s^2>r^2{\sin }^2\theta and , correspond to the \mathcal{P}\mathcal{T}-unbroken and \mathcal{P}\mathcal{T}-broken regions, respectively [1]. Considering Eq. (15) and assuming \sin \alpha =(r/s)\sin \theta ,\cos \alpha =\sqrt{1-(r/s{)}^2{\sin }^2\theta }, one can construct two properly normalized right eigenvectors as

(34)| E_1⟩={\displaystyle \frac{1}{\sqrt{1+{\text{e}}^{-2\mathrm{i}\alpha }}}}\left(\begin{array}{c}-{\text{e}}^{-\mathrm{i}\alpha }\\ 1\end{array}\right),

and

(35)| E_2⟩={\displaystyle \frac{1}{\sqrt{1+{\text{e}}^{2\mathrm{i}\alpha }}}}\left(\begin{array}{c}{\text{e}}^{\mathrm{i}\alpha }\\ 1\end{array}\right).

Moreover, with the similar method one can construct two properly normalized left eigenvectors as

(36)| E_1⟩⟩={\displaystyle \frac{1}{\sqrt{1+{\text{e}}^{2\mathrm{i}\alpha }}}}\left(\begin{array}{c}-{\text{e}}^{\mathrm{i}\alpha }\\ 1\end{array}\right),

and we have

(37)| E_2⟩⟩={\displaystyle \frac{1}{\sqrt{1+{\text{e}}^{-2\mathrm{i}\alpha }}}}\left(\begin{array}{c}{\text{e}}^{-\mathrm{i}\alpha }\\ 1\end{array}\right).

Hence, one can obtained metric Eq. (12) as bellow:

(38)\eta ={\displaystyle \frac{1}{\cos \alpha }}\left(\begin{array}{cc}1& -\mathrm{i}\sin \alpha \\ \mathrm{i}\sin \alpha & 1\end{array}\right).

The metric \eta is positive-definite with determinant \text{det}[ \eta ]=1.

To examine the competence of our generalized HSS-based witness to detect EPs in this system, we first comfortably compute the generalized evolved state of the system Eq. (12), with inserting metric Eq. (38), when it is prepared in the initial state {\rho }_0= | {\psi }_0⟩⟨{\psi }_0| in which |{\psi }_0⟩=\frac{1}{\sqrt{2}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+|{\psi }_2⟩).

In order to obtain the EPs in the \mathcal{P}\mathcal{T}-symmetric system, the HSS dynamics of the generalized evolved state, i.e., {\text{HSS}}_{\varphi }(\eta {\rho }_G(t)), and that of the normalized evolved state, i.e., {\text{HSS}}_{\varphi }({\rho }_N(t) ) are illustrated in Fig.21 for the \mathcal{P}\mathcal{T}-unbroken phase region s^2>r^2{\sin }^2\theta and \mathcal{P}\mathcal{T}-broken phase region . In the \mathcal{P}\mathcal{T}-unbroken phase region, both HSS-based witnesses oscillate and eventually come back to their initial value. However, for the \mathcal{P}\mathcal{T}-broken phase region, they represent no oscillation and monotonously vary with time. These results indicate that the qualitative dynamics of the HSS-based witnesses, in the presence and absence of the metric, are completely similar. Furthermore, in the \mathcal{P}\mathcal{T}-unbroken phase, their corresponding maximum and minimum points coincide, and hence the periods of the oscillations do not change when inserting the metric.

Now we consider a high-dimensional system described by a 4×4 Hamiltonian [38, 181] H_{PT} =-J{S}_x+\mathrm{i}\gamma S_z in which S_x and S_z indicate the spin-3/2 representations of the SU(2) group. The Hamiltonian can be represented by

(39)H_{PT}={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}3\mathrm{i}\gamma & -\sqrt{3}J& 0& 0\\ -\sqrt{3}J& \mathrm{i}\gamma & -2J& 0\\ 0& -2J& -\mathrm{i}\gamma & -\sqrt{3}J\\ 0& 0& -\sqrt{3}J& -3\mathrm{i}\gamma \end{array}\right) .

The eigenvalues of H_{PT} are simply given by {ϵ}_k=\{-3/2,-1/2,1/2,3/2\}\sqrt{J^2-{\gamma }^2} (k=1,2,3,4), leading to an EP4 at the \mathcal{P}\mathcal{T}-breaking threshold \gamma =J [38]. Then, and \gamma >J, correspond to the \mathcal{P}\mathcal{T}-unbroken and \mathcal{P}\mathcal{T}-broken regions, respectively. The time evolution of the HSS respect to generalized evolved state {\text{HSS}}_{\varphi } (\eta {\rho }_G(t) ) and normalized evolved state {\text{HSS}}_{\varphi }({\rho }_N(t) ) in order to explore the EPs in the high dimensional \mathcal{P}\mathcal{T}-symmetric system are displayed on Fig.22 for the \mathcal{P}\mathcal{T}-unbroken phase region and \mathcal{P}\mathcal{T}-broken phase region \gamma > J. Again, we find that both {\text{HSS}}_{\varphi }(\eta {\rho }_G(t)) and {\text{HSS}}_{\varphi }({\rho }_N(t) ) dynamics exhibit a qualitative chabge at EPs where the transition from monotonic to oscillatory behavior or vice versa occurs. Here, we observe that the maximum or minimum points of {\text{HSS}}_{\varphi }(\eta {\rho }_G(t) ) and {\text{HSS}}_{\varphi }({\rho }_N(t) ) dynamics do not coincide with each other. However, their oscillation periods are exactly similar.

Now, we consider the two-dimensional \mathcal{A}\mathcal{P}\mathcal{T}-symmetric Hamiltonian as follows [38, 94]:

(40)H_{\mathrm{A}\mathrm{P}\mathrm{T}}=\left(\begin{array}{cc}\lambda n{\mathrm{e}}^{\mathrm{i}\vartheta }& \mathrm{i}n\\ \mathrm{i}n& -\lambda n{\mathrm{e}}^{-\mathrm{i}\vartheta }\end{array}\right),

where in this Hamiltonian, all of the parameters are real. Moreover, the eigenvalues of Hamiltonian H_{APT} are {ϵ}_{\pm }=\mathrm{i}\lambda n\sin \vartheta \pm \sqrt{{\lambda }^2{n}^2{\cos }^2\vartheta -n^2}, and the system is denoted in the regime of unbroken \mathcal{A}\mathcal{P}\mathcal{T}-symmetric phase if . For simplicity, we consider \vartheta =0 ,n\ge 0. Therefore, the EP2 is located at \lambda =1. Then, the \mathcal{A}\mathcal{P}\mathcal{T} unbroken and broken phases are equal and \lambda >1, respectively [38].

Similar to the method implemented for the \mathcal{P}\mathcal{T}-symmetric Hamiltonian [Eq. (33)], one can construct two properly normalized right eigenvectors:

(41)| E_1⟩={\displaystyle \frac{1}{\sqrt{1-(\lambda -\sqrt{{\lambda }^2-1}{)}^2}}}\left(\begin{array}{c}\mathrm{i}(-\lambda +\sqrt{{\lambda }^2-1})\\ 1\end{array}\right),

and

(42)| E_2⟩={\displaystyle \frac{1}{\sqrt{1-(\lambda +\sqrt{{\lambda }^2-1}{)}^2}}}\left(\begin{array}{c}-\mathrm{i}(\lambda +\sqrt{{\lambda }^2-1})\\ 1\end{array}\right).

Moreover, the two corresponding properly normalized left eigenvectors are given by

(43)|E_1⟩⟩={\displaystyle \frac{1}{\sqrt{1-(\lambda -\sqrt{{\lambda }^2-1}{)}^2}}}\left(\begin{array}{c}-\mathrm{i}(-\lambda +\sqrt{{\lambda }^2-1})\\ 1\end{array}\right),

and

(44)| E_2⟩⟩={\displaystyle \frac{1}{\sqrt{1-(\lambda +\sqrt{{\lambda }^2-1}{)}^2}}}\left(\begin{array}{c}\mathrm{i}(\lambda +\sqrt{{\lambda }^2-1})\\ 1\end{array}\right).

Hence, one can obtained metric Eq. (12) for two dimensional \mathcal{A}\mathcal{P}\mathcal{T}-symmetric system as following:

(45)\eta =\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right).

The metric \eta is positive-definite. To investigate the productivity of our HSS-based witnesses in this system, we compute numerically {\text{HSS}}_{\varphi }(\eta {\rho }_G(t) ) and {\text{HSS}}_{\varphi }({\rho }_N(t) ), starting from the initial state {\rho }_0= | {\psi }_0⟩⟨{\psi }_0|, where |{\psi }_0⟩=\frac{1}{\sqrt{2}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+|{\psi }_2⟩). Their qualitative behavior versus time to search the EPs in the \mathcal{A}\mathcal{P}\mathcal{T}-symmetric system are plotted in Fig.23 for the \mathcal{A}\mathcal{P}\mathcal{T}-unbroken phase region and \mathcal{A}\mathcal{P}\mathcal{T}-broken phase region \lambda > 1. In the \mathcal{A}\mathcal{P}\mathcal{T}-broken phase region, the HSS-based witnesses exhibit oscillatory behavior and eventually return to their initial value. Moreover, their oscillation periods are identical. In contrast, in the \mathcal{A}\mathcal{P}\mathcal{T}-unbroken phase region, they do not oscillate and change monotonically over time.

Upon reexamining the high-dimensional interacting models described by Eqs. (31) and (32), we consider the metric defined in Eq. (13) to assess the effectiveness of our generalized HSS-based witness in detecting EPs within these models. Initially, we compute the generalized evolved state of the system as indicated by Eq. (12), incorporating the metric (13) calculated with eigenstates of Hamiltonian (31). The system has been prepared in the initial state {\rho }_0= | {\psi }_0⟩⟨{\psi }_0|, where |{\psi }_0⟩=\frac{1}{\sqrt{4}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+|{\psi }_2⟩+\cdots +|{\psi }_4⟩). Fig.24 displays the temporal variations of the HSS for both the generalized evolved state {\text{HSS}}_{\varphi } (\eta {\rho }_G(t) ) and the normalized evolved state {\text{HSS}}_{\varphi }({\rho }_N(t) ), with the aim of locating EPs. The results demonstrate the HSS efficiency in detecting EPs in interacting models, even with the inclusion of metrics. A similar result is obtained when computing the metric with eigenstates of Hamiltonian (Eq. (32)) and starting from the initial state |{\psi }_0⟩=\frac{1}{\sqrt{7}}({\text{e}}^{\mathrm{i}\varphi }|{\psi }_1⟩+|{\psi }_2⟩+\cdots +|{\psi }_7⟩), as shown in Fig.25.

5 Measuring the quantum statistical speed from experimental data

To measure the HSS, quantum state tomography is often employed. This technique reconstructs a quantum system’s density matrix from a sequence of measurements. Once the density matrix \rho (\varphi ) is established for various values of the parameter \varphi, the derivative \frac{\mathrm{d}\rho (\varphi )}{\mathrm{d}\varphi } can be calculated. This is usually done numerically by observing the changes in \rho (\varphi ) as \varphi is altered. The Hilbert Schmidt norm of this derivative is then determined. Suppose we are designing an optical setup to measure the HSS for a qubit. This requires creating a system capable of performing quantum state tomography on the qubit state and calculating the changes in the state relative to a parameter. Initially, the qubit is represented by the polarization states of photons, where horizontal polarization |H⟩ and vertical polarization |V⟩ correspond to the qubit states |0⟩ and |1⟩, respectively. Photons are generated using a single-photon source. The initial state of the qubit is set using polarizers and wave plates to establish the desired polarization state, thereby encoding the parameter \varphi into the polarization. The photon then passes through an optical element that simulates the system evolution. A variable phase retarder, such as a liquid crystal display, is used to introduce a phase shift between the horizontal and vertical polarization components, effectively varying the parameter \varphi. A Mach−Zehnder interferometer, with the phase retarder in one arm, is then set up to measure the phase changes resulting from the variation in \varphi. This setup causes interference between the two polarization components of the photon, providing information about the state evolution. Polarizing beam splitters and single-photon detectors are placed at the output of the interferometer to measure the intensity of the different polarization components. This data is used to reconstruct the quantum state of the qubit. A series of measurements with various settings of the wave plates and polarizers are performed to fully reconstruct the density matrix of the qubit state. Finally, the reconstructed density matrix is used to numerically compute the derivative with respect to \varphi and to calculate the HSS using the formula: \text{HSS}( {\rho }_{\varphi })={\left(\frac{1}{2}\text{Tr}\left[{\left(\frac{\mathrm{d}\rho (\varphi )}{\mathrm{d}\varphi }\right)}^2\right]\right)}^{1/2}. This method describes a systematic way to determine the HSS of a qubit using optical techniques. It relies on computation based on measurements that reconstruct the density matrix, rather than direct measurement of HSS. The specific experimental methods for quantum state tomography can differ based on the system, whether photonic, spin, or atomic, and may require various detectors and measurement setups.

However, it is important to recognize that the practical application of this process may present challenges, including the need for precise measurements, the risk of quantum decoherence, and the computational demands of quantum state tomography. Moreover, the specific experimental methods for quantum state tomography can vary based on the system — such as photonic, spin, or atomic systems — and may require different detectors and setups. As a result, meticulous experimental design and thorough data analysis are essential for accurately measuring HSS.

In an alternative method [135] to experimentally determine quantum statistical speed from data, one starts by selecting a quantum system represented by \rho (\varphi ), which possesses a controllable and measurable observable. The first step is to measure the quantum state to establish the probability distribution p_x(\varphi ) for the outcomes of the observable. The system then undergoes a slight parameter change, \delta \varphi, leading to its evolution into a new state \rho (\varphi +\delta \varphi ). Subsequent measurements of this new state yield another probability distribution p_x(\varphi +\delta \varphi ). This process is repeated incrementally, generating a series of probability distributions p_x( \varphi ),p_x( \varphi +\delta \varphi ),p_x(\varphi +2\delta \varphi ),p_x(\varphi +3\delta \varphi ),\cdots, from which the classical distance can be inferred as shown in Eq. (1). By fitting the statistical distance as a function of \varphi to (2), one can derive the classical statistical speed. This speed serves as a lower bound to the quantum statistical speed. When the observable is optimally selected, the derived speed will match the actual quantum statistical speed. Further optimization of the observable can be performed to improve precision.

6 Conclusion

We have demonstrated the detection of exceptional points (EPs), where a simultaneous coalescence of eigenvalues and their corresponding eigenstates occurs, in various non-Hermitian Hamiltonians, using the simple and powerful tool of Hilbert−Schmidt speed (HSS), which is a special case of quantum statistical speed. Our principal finding is that the temporal behavior of the HSS exhibits a shift around the exceptional point, transitioning from an oscillatory to a non-oscillatory pattern, or the reverse. This phenomenon has been rigorously tested across a wide spectrum of non-Hermitian system models, encompassing two to six dimensions, both with and without \mathcal{P}\mathcal{T} or \mathcal{A}\mathcal{P}\mathcal{T} symmetry, including interacting models, and under normalized or generalized time evolution. In every instance, our proposed method has demonstrated high efficacy in detecting EPs.

Specifically, we have discussed the distinctions between Hermitian and non-Hermitian Hamiltonians concerning the preservation of the norm squared and the inner product of states. For a Hermitian Hamiltonian, H(t) =H^{†}(t), both the norm squared and the inner product are conserved over time, yielding a normalized evolved state. Conversely, for a non-Hermitian Hamiltonian, neither the norm squared nor the inner product is conserved, resulting in a generalized evolved state. In such instances, it is necessary to utilize a metric compatible with the inner product of the states to accurately describe their evolution. We have also tested the efficacy of our witness in these scenarios.

A relevant question that arises is how the HSS and EPs are interconnected. As the system approaches an EP, a subset of the Hamiltonian eigenvalues begin to converge. This convergence directly influences the time evolution of the system state, as represented by the density matrix \rho (t). Simultaneously, the associated eigenstates also start to merge. Since the eigenstates form the basis in which \rho ( t) is expressed, the coalescence of those eigenstates can lead to significant changes in the elements of \rho ( t). The HSS is acutely sensitive to these changes, particularly due to its reliance on the square of the density matrix’s derivative. Abrupt variations in the system state, especially those occur at an EP, are accentuated when squared, thereby enhancing the HSS ability to signal these critical transitions. It is precisely this mathematical characteristic that underpins the HSS ability to detect the critical juncture at an EP, where a convergence of certain eigenvalues and eigenvectors occurs. Such convergence causes the Hamiltonian to become non-diagonalizable, leading to a notable reconfiguration of the eigenstates involved. Such reconfiguration prompts the emergence of new dynamical patterns in the HSS, leading to a transition from oscillatory behavior to non-oscillatory one, or vice versa. It would be interesting to examine whether this transition in the HSS dynamics when crossing an EP is a response to possible topological changes in the eigenstates.

To conclude, we have introduced a new and reliable method for detecting EPs in non-Hermitian systems, using the HSS tool. We have demonstrated this method on different models of non-Hermitian systems and verified its efficiency and versatility. We expect that our method will facilitate further research on non-Hermitian physics and its applications.

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