Quantumness and its hierarchies in PT-symmetric down-conversion models

PHYSICAL REVIEW A 112, 043545 (2025) Quantumness and its hierarchies in PT -symmetric down-conversion models Jan Peˇ rina, Jr. * Joint Laboratory of Optics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic and Institute of Physics of the Academy of Sciences of the Czech Republic, Joint Laboratory of Optics of Palacký University and Institute of Physics AS CR, 17. listopadu 50a, Olomouc 772 07, Czech Republic Karol Bartkiewicz , Grzegorz Chimczak , Anna Kowalewska-Kudlaszyk , and Adam Miranowicz Institute of Spintronics and Quantum Information, Faculty of Physics and Astronomy, Adam Mickiewicz University, 61-614 Pozna´n, Poland Joanna K. Kalaga and Wiesław Leo´ nski Quantum Optics and Engineering Division, Faculty of Physics and Astronomy, University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland (Received 14 May 2025; accepted 23 September 2025; published 29 October 2025) We investigate the hierarchy of quantum correlations in a quadratic bosonic parity-time-symmetric system (PTSS) featuring distinct dissipation and amplification channels. The hierarchy includes global nonclassicality, entanglement, asymmetric quantum steering, and Bell nonlocality. We elucidate the interplay between the system physical nonlinearity (which serves as a source of quantumness) and the specific dynamics of bosonic PTSSs, which are qualitatively influenced by their damping and amplification characteristics. Using a set of quantifiers (including local and global nonclassicality depths, negativity, steering parameters, and the Bell parameter) we demonstrate that the standard PTSS typically exhibits weaker quantumness than its counterparts affected solely by damping or solely by amplification. Both the maximum values attained by these quantifiers and the speed and duration of their generation are generally lower in the standard PTSS. A comparative analysis of three two-mode PTSSs (standard, passive, and active) with identical eigenvectors and real parts of eigenfrequencies, but differing in their damping and amplification strengths, reveals the crucial role of quantum fluctuations associated with gain and loss. Among them, the passive PTSS yields the most strongly nonclassical states. Nevertheless, under suitable conditions, the standard PTSS can also generate highly nonclassical states. The supremacy of the passive PTSS is further supported by its fundamental advantages in practical realizations. DOI: 10.1103/9vty-ctf7 I. INTRODUCTION Since the seminal works on parity-time-symmetric systems (PTSSs) by Bender et al. [1,2], non-Hermitian Hamiltonians with real eigenvalues have attracted considerable attention in the physics community [3]. This interest stems from the fact that PTSSs possess unique structures in their Hilbert (or Liouville) spaces, which exhibit degeneracies, both in eigenvalues and eigenvectors, at specific parameter values known as exceptional points (EPs). These degeneracies give rise to a variety of intriguing physical phenomena (for details, see Ref. [4,5]). Systems operating at or near EPs can be harnessed for a range of applications, including enhanced sensing [6–8], enhanced nonlinear interactions [9–12], unidirectional light propagation [13,14], and even invisibility cloaking [15,16]. *Contact author: [email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Interesting behaviors of nonlinear optical systems have also been studied under PTSS conditions, i.e., when damping and amplification are balanced. It has been shown that highly nonclassical states can be generated in such systems under suitable conditions [9–12], despite the unavoidable noise present in quantum systems involving damping and/or amplification [17]. This raises an important question: to what extent does the specific dynamics in the Hilbert space of bosonic PTSSs influence the ability of system physical nonlinearities (though analyzed in many cases in their linearized versions) to generate quantumness [18–21]? The system’s dynamics affect the rate at which different forms of quantumness emerge, the maximal values attained by various quantifiers, and their asymptotic limits. A fundamental problem thus arises: Can the specific dynamics in the Hilbert space of a PTSS enhance the ability of physical nonlinearities to generate diverse forms of nonclassical states? In this paper, we address in detail this complex issue by the extended numerical analysis of different versions of two-mode bosonic PT -symmetric system with parametric down-conversion that covers the entire system’s parameter space and includes all common forms of quantumness (system nonclassicality, entanglement, steering, and the Bell 2469-9926/2025/112(4)/043545(17) 043545-1 Published by the American Physical Society JAN PE ˇ RINA JR. et al. PHYSICAL REVIEW A 112, 043545 (2025) FIG. 1. Schematics of the two-mode bosonic system analyzed under different conditions: (a) standard PT -symmetric system, where mode 1 is damped (γ1>0) and mode 2 is amplified (γ2<0); (b) system with only mode 1 damped (γ1>0, γ2=0); (c) system with only mode 2 amplified (γ2<0, γ1=0); (d) passive PT -symmetric system with mode 1 doubly damped (2γ1>0, γ2=0); (e) active PT -symmetric system with mode 2 doubly amplified (2γ2<0, γ1=0). Various parameters calculated for these systems are consistently distinguished in the following figures using the superscripts: (a) ad,(b)d,(c)a,(d)dd,and(e)aa. nonlocality). The emergence of quantumness is discussed as competition between the system’s coherent dynamics and detrimental influence of the reservoir fluctuating forces. The obtained complete numerical analysis allows us to draw even several general conclusions. Whereas the coherent dynamics does not allow to fully compensate for the influence of the reservoir forces for the most of the parameters in the standard PTSS, it leads to the generation of the states with high levels of quantumness in the passive PTSS. The paper is structured as follows. In Sec. II we discuss the general behavior of bosonic PT -symmetric systems whose properties emerge in the competition between their coherent evolution and the influence of the noise inevitably accompanying damping and amplification present in the system. In Sec. III, the considered PTSS is introduced, its solution is found, and statistical properties of its modes are described considering Gaussian fields. Nonclassicality depths, negativity, steering parameter and the Bell parameter are introduced and determined for the Gaussian fields in Sec. IV.Therole of standard PT symmetry in nonclassical-state generation is elucidated in Sec. Vusing the comparison with the systems influenced only either by damping or amplification. In Sec. VI, relying on similarity of coherent dynamics in the standard PTSS, passive PTSS with doubled damping, and active PTSS system with doubled amplification [see the scheme in Figs. 1(a),1(d), and 1(e)], the role of reservoir fluctuations accompanying damping and amplification in nonclassicalstate generation is elucidated. Section VII is devoted to the comparison of the PTSSs ability to generate different forms of quantumness and quantum correlations. Time and speed aspects of the nonclassical-state generation are addressed in Sec. VIII. Conclusions are drawn in Sec. IX. II. COMPETING EFFECTS OF COHERENT DYNAMICS AND RESERVOIR NOISE IN QUANTUM PT -SYMMETRIC SYSTEMS When we look back at the history, PTSSs were extensively studied within the framework of classical physics (see Refs. [22,23] and Refs. [4,24–27]), particularly in optics, where their specific coherent dynamics proved especially beneficial. A hallmark of PTSSs is the simplification of system dynamics at EPs, often accompanied by the enhancement of certain system properties. The extension of these concepts to quantum optical bosonic systems, via the use of Glauber coherent states and the Glauber-Sudarshan Prepresentation of the statistical operator [28,29], appeared straightforward. The idea of employing PTSSs endowed with some form of physical nonlinearity to generate various nonclassical and entangled states promised significant and attractive outcomes. Indeed, a range of nonlinear PTSSs have been used to produce nonclassical states with unusual properties [9–12]. However, a critical problem was identified: the presence of chaotic fluctuating forces, which, according to both the fluctuation-dissipation and analogous amplificationfluctuation theorems, inevitably accompany damping and amplification. Due to their chaotic nature, these forces tend to degrade all forms of system quantumness [17]. As a result, two competing effects come into play in PTSSs with respect to the generation of quantumness: while the coherent PT -symmetric dynamics tends to support and enhance quantumness, the accompanying chaotic fluctuations tend to suppress it [30]. This raises a fundamental question: Can the coherent PT -symmetric dynamics fully compensate for the detrimental effects of the fluctuating forces, or even enhance the system’s quantumness despite their presence? While it is difficult, if not impossible, to answer this question in full generality, valuable physical insight can be gained by analyzing specific, well-defined models. One of the simplest models that satisfies the necessary criteria involves two interacting bosonic modes, mutually coupled through both linear and (physically) nonlinear interactions and subject to damping and amplification. An important technical advantage of this model lies in the fact that, under the physically relevant conditions, the model can be analyzed in its linearized version in which its dynamical operator equations remain linear, enabling a fully analytical treatment. By focusing on Gaussian states, we are able to analytically determine all relevant parameters characterizing the system, including the effects of averaging over the chaotic fluctuating forces. This analytical approach makes it possible to systematically explore the entire parameter space of the model, as well as to examine its temporal evolution. The study of this evolution is supported by both numerical simulations and asymptotic (long-time) analytical formulas. Regarding the effect of noise in our system, it originates from quantum sources, as it arises from interactions with quantum reservoirs composed of two-level atoms. These 043545-2 QUANTUMNESS AND ITS HIERARCHIES IN … PHYSICAL REVIEW A 112, 043545 (2025) atoms are either in the ground state (in the case of damping) or in the excited state (for amplification). The quantum state of the reservoir atoms fundamentally alters the character of the reservoir-induced noise: excited atoms can induce both spontaneous and stimulated emission in the system, whereas ground-state atoms only allow for stimulated absorption. This distinction leads to the fact that amplification-related noise is more detrimental than damping-related noise: it is simply stronger [17]. However, as already mentioned above, the generation of quantum features in the system is governed by more than just the noise level; it also strongly depends on the system’s physical nonlinearity, which is determined by the product of the nonlinear coupling constant and the mode amplitudes. The coherent part of the dynamics typically leads to larger mode amplitudes, especially in the presence of amplification, which can in turn enhance the generation of quantumness. This stands in contrast to the detrimental effects introduced by noise. As a result, the system’s behavior emerges from the competition between these two opposing factors: chaotic noise versus coherent physically nonlinear dynamics. This interplay lies at the heart of our investigation and constitutes the central motivation behind it. While noise often dominates and at least partially suppresses quantum effects, there exist regions in the parameter space where coherent dynamics prevail, enabling the emergence of nonclassical behavior despite the presence of noise. This delicate balance is what makes the results fundamentally interesting and potentially attractive. The selection of an appropriate reference system is an important issue. The hallmark features of PTSSs arise from a balance between gain and loss in the modes, and these features should be absent in any suitable reference system. Such reference configurations include systems in which only one mode is subject to damping while the other evolves freely, or vice versa, one mode is amplified while the other remains unaffected. However, detailed analysis across the full parameter space reveals that, in most cases, the PT -symmetric dynamics does not offer a significant advantage in generating different forms of quantumness. This suggests that the influence of chaotic fluctuating forces is typically too strong to be compensated for by coherent PT -symmetric evolution. This naturally leads us to consider more general PT - symmetric-like systems, namely, their passive and active variants [31]. In passive (active) PT -symmetric systems, the modes experience unequal levels of damping (amplification). However, the eigenvalues of their corresponding dynamical matrices share a common damping (amplification) component, while the remaining parts (along with the associated eigenvectors) retain the essential characteristics of standard PTSSs. As such, aside from the global damping (or amplification) factor, the coherent dynamics remain effectively identical to those of the standard PTSS counterpart. This structural similarity is promising for the generation of quantumness, provided that the impact of chaotic fluctuations is sufficiently reduced. Indeed, the contributions of chaotic fluctuating forces are not symmetric: those associated with damping are generally weaker than those arising from amplification. This is because amplification typically involves coupling to inverted two-level atoms, which are susceptible to spontaneous photon emission, a process that significantly enhances the destructive influence of noise. Based on this observation, we analyze a passive PTSS in which one mode is subject to double damping, while the other evolves freely. As demonstrated below, this configuration turns out to be optimal for the generation of quantumness. Nevertheless, active PTSSs, where one mode is doubly amplified and the other evolves without gain or loss, should not be dismissed apriori. Despite the stronger fluctuating forces inherent to amplification, the shared amplification factor can significantly enhance the system’s physical nonlinearity through its influence on the coherent PT -symmetric-like dynamics. While under typical conditions the detrimental effects of noise dominate, our results reveal specific parameter regimes in which amplified coherent dynamics prevail, leading to enhanced nonclassicality, as discussed in detail below. In the paper, we analyze a two-mode bosonic system governed by a quadratic Hamiltonian [11,12,30], the original nonlinear Hamiltonian belonging to the three-mode optical nonlinear interaction is linearized by assuming a strong undepleted pump mode (parametric approximation) [18,19]. This linearized Hamiltonian includes both linear mode coupling and nonlinear interaction arising from parametric downconversion. Though the nonlinear interaction is effectively described by its linearized form it enables the generation of quantum states. The quadratic form of the Hamiltonian yields linear Heisenberg equations of motion, which can be solved analytically [21]. These solutions incorporate fluctuating Langevin noise operators associated with damping and amplification, providing a rigorous framework for the system’s dynamical analysis. We note that we refer to the investigated model as (physically) nonlinear because the Hamiltonian in Eq. (1) below contains the terms a1a2+a† 1a† 2, which correspond to the two-mode squeezing interaction that belongs to the group of three-mode optical nonlinear interactions described by second-order susceptibilities χ(2). These interactions are capable of generating or enhancing nonclassicality and increasing the total number of excitations during evolution, in contrast to linear optical systems characterized solely by firstorder susceptibilities χ(1). This fundamental distinction then underpins the definition of certain nonclassicality measures, such as potentials of quantum entanglement, steering, and Bell nonlocality [32]. The approach based on linearizing the nonlinear-system dynamics and subsequent analytical treatment allow for a detailed comparison between the standard PTSS, affected by both damping and amplification, and two related configurations involving only damping or only amplification [see Figs. 1(a) to 1(c)]. By evaluating various quantifiers of quantumness, characterizing both nonclassicality and quantum correlations, we investigate how the system’s specific structure impacts the formation of nonclassical states. The analyzed quantifiers include local and global nonclassicality depths [33], negativity [34,35] as a measure of entanglement, the steering parameter [36] for asymmetric quantum steering, and the Bell (nonlocality) parameter [37] for identifying the strongest type of quantum correlations among those considered. 043545-3 JAN PE ˇ RINA JR. et al. PHYSICAL REVIEW A 112, 043545 (2025) We note that, in our analysis, we consider the so-called quantum exceptional points that occur in the dynamics of open quantum systems characterized by their Liouvillians. Contrary to the usual Hamiltonian EPs, they are fully compatible with the general dynamics of open quantum systems [38–41]. We note that, rather than analyzing the system’s evolution via the master equation and its associated Liouvillian, we employ a more tractable approach based on the analytical solution of the corresponding Langevin-Heisenberg equations, including their Langevin noise terms [20,42]. This method not only reveals the system’s eigenfrequencies (associated with the Liouvillian spectrum, cf. [39]), but also facilitates the derivation of Gaussian-state parameters through averaging over the chaotic noise forces. III. TWO-MODE BOSONIC SYSTEM WITH DAMPING AND AMPLIFICATION The modes are assumed to mutually interact via the linear exchange of energy (described by linear coupling constant ) and the physically nonlinear addition or subtraction of photon pairs into both modes that originates in parametric downconversion [19] (nonlinear coupling constant κ). Damping and amplification of modes, that causes the presence of additional Langevin fluctuating operator forces representing the backaction of the reservoir, describe loss and addition of energy into the modes (for the sketch of the system analyzed under different conditions, see Fig. 1). Introducing the photon annihilation (ˆaj) and creation (ˆa† j) operators of the modes denoted as 1 and 2 together with the corresponding Langevin operator forces ˆ ljand ˆ l† jand system-reservoir coupling constants rj,j=1,2, we express the appropriate system interaction Hamiltonian ˆ Has follows [20,30,39]: ˆ H=[ˆa† 1ˆa2+κˆa1ˆa2+H.c.]+[r1ˆa1ˆ l† 1+r2ˆa2ˆ l† 2+H.c.], (1) where the symbol H.c. replaces the Hermitian conjugated terms. To allow consistent description of both damped and amplified modes, we chose the Langevin operator forces ˆ lj and ˆ l† j,j=1,2, as the Pauli spin-flip operators that describe the reservoir two-level atoms [43]. Invoking the second-order perturbation theory in the system-reservoir coupling constants r1and r2and eliminating the reservoir operators by replacing them by their reservoir mean values [with the two-level reservoir atoms in the ground (excited) state for damping (amplification)] we reveal the corresponding damping and amplification constants as well as the appropriate mean values of the Langevin operator forces, see Eqs. (4) and (2) below as well as detailed derivation in Refs. [20,39,42]. We note that the above approach based on the HeisenbergLangevin operator equations with the Langevin fluctuating operator forces [20] represents an alternative to the commonly used approach based on the (generalized) master equation for a statistical operator [42,44]. The convenience of the application of these approaches differs according to the situation that includes both the structure of the model and the type of the states investigated. In our case in which we pay attention to Gaussian fields [20,45], the use of the Heisenberg-Langevin operator equations is more convenient owing to the linearity that arises in the parametric approximation applied to the nonlinear interaction [21]. In contrast, the solution of the corresponding master equation transformed into the form of the Fokker-Planck equation [46] would involve the temporal solution for the mean-field-operator amplitudes and also statistical coefficients of the Gaussian states [see Eq. (20) below]. In the applied approach, these coefficients are derived directly from the operator solution of the Heisenberg-Langevin equations. This considerably simplifies the calculations. In the model, we assume that mode 1 is damped (damping constant γ1) whereas mode 2 is amplified (amplification constant −γ2). The reservoirs responsible for damping and amplification are assumed to be described by independent quantum random Gaussian and Markovian processes with the following characteristics [43,47,48], j=1,2: ˆ lj(t)=ˆ l† j(t)=0, ˆ l† j(t)ˆ lj(t)=˜ ljδ(t−t), ˆ lj(t)ˆ l† j(t)=ljδ(t−t).(2) In Eq. (2), the real constants ljand ˜ lj,j=1,2, have to be chosen such that the commutation relations for the photon creation and annihilation operators are fulfilled. These are [ˆaj,ˆa† j]=1forj=1,2 and the remaining commutation relations among the operators ˆajand ˆa† jare zero. The symbol δ stands for the Dirac function. In the case of damping in mode 1 and the reservoir two-level atoms in the ground state, we have l1=2γ1and ˜ l1=0. However, the amplification in mode 2 and the reservoir two-level atoms in the excited state gives l2=0 and ˜ l2=2|γ2|. We note that, for standard PTSSs, amplification just compensates for damping, i.e., γ1=−γ2≡γ. However, we assume γ1≡2γand γ2≡0 for the analyzed passive PTSS. Similarly, we have γ1≡0 and γ2≡−2γfor the analyzed active PTSS. We also note that a more general PTSS containing also the nonlinear Kerr and cross-Kerr terms was analyzed in Refs. [11,12] using the method of small operator corrections to mean values. We note that, for the reservoir dynamics, we assume Markovian processes. However, the analysis can also be extended to non-Markovian dynamics. For example, a system coupled to a non-Markovian bath can be modeled by introducing an ancilla that is linearly coupled to the system and interacts with a standard Markovian reservoir. Such an approach enables the analysis of non-Markovian quantum exceptional points [49]. The Langevin-Heisenberg equations derived from the Hamiltonian ˆ Hin Eq. (1), with the help of the theory describing the system interaction with the reservoir, can conveniently be written in the following matrix form using the vectors ˆ AT=(ˆa1,ˆa† 1,ˆa2,ˆa† 2) and ˆ LT=(ˆ l1,ˆ l† 1,ˆ l2,ˆ l† 2): dˆ A(t) dt =−iMˆ A(t)+ˆ L(t),(3) M=⎡ ⎢ ⎢ ⎣ −iγ10κ 0−iγ1−κ− κ−iγ20 −κ−0−iγ2 ⎤ ⎥ ⎥ ⎦ .(4) In Eq. (3), we assume for simplicity real and κ. 043545-4 QUANTUMNESS AND ITS HIERARCHIES IN … PHYSICAL REVIEW A 112, 043545 (2025) We note that the model can readily be reformulated in terms of a master equation in the Lindblad form. Since it is naturally expressed in the coherent-state basis, this leads to a Fokker-Planck equation for the corresponding quasiprobability distribution. When assuming Gaussian states, the problem reduces to solving a set of ordinary differential equations for the first-order moments (mean amplitudes) and the secondorder moments [the statistical coefficients defined later in Eq. (21)]. Introducing the evolution matrix P(t,t)[21], P(t,t)=exp[−iM(t−t)],(5) the solution of Eq. (3) is obtained as ˆ A(t)=P(t,0) ˆ A(0) +ˆ F(t),(6) ˆ F(t)=t 0 dtP(t,t)ˆ L(t),(7) and ˆ FT≡(ˆ f1,ˆ f† 1,ˆ f2,ˆ f† 2). The properties of the Langevin fluctuating operator forces in Eq. (2) result in the nonzero second-order correlation functions of the forces ˆ Fgiven as [21] ˆ F(t)ˆ F†T(t)=t 0 d˜ tt 0 d˜ tP(t,˜ t)ˆ L(˜ t)ˆ L†T(˜ t)P†T(t,˜ t), (8) where the symbol Tstands for the transposed matrix. According to Eq. (5), the eigenvectors of the evolution matrix Pcoincide with those of the dynamical matrix M and the corresponding eigenvalues P(t,t)aregivenas exp[−iM(t−t)], where Mcontains the eigenvalues of matrix M. Diagonalization of the dynamical matrix Mleaves us with the following eigenvectors and eigenvalues: M=TMT−1;(9) M=−iγ+diag(1,1,1,1) +μdiag(1,1,−1,−1),(10) T=(T1,T2,T3,T4), TT 1,2=1 2√(ζ±,−ζ∓,±ζ±ψ+,∓ζ∓ψ+), TT 3,4=1 2√(ζ±,−ζ∓,∓ζ±ψ−,±ζ∓ψ−), T−1=(T−1 1,T−1 2,T−1 3,T−1 4), T−1T 1,2=√ 2√μ(ζ±ψ−,−ζ∓ψ−,ζ±ψ+,−ζ∓ψ+), T−1T 3,4=√ 2√μ(ζ±,ζ∓,−ζ±,−ζ∓),(11) and γ+=(γ1+γ2)/2, γ−=(γ1−γ2)/2, ξ=√2−κ2, ζ±=√±ξ,μ=2−κ2−γ2 −, and ψ±=(μ±iγ−)/ξ. We note that the structure of the eigenvectors and eigenvalues of the matrix Mclosely resembles that obtained for the special case γ1=−γ2(PT -symmetric case) analyzed in detail in Ref. [30] when studying the problem of nonclassicality and entanglement losses in the long-time limit caused by fluctuating forces and related to the properties of the fluctuating forces. According to Eq. (10) there exist two doubly degenerated eigenvalues ν1,2=−i(γ1+γ2)/2±2−κ2−(γ1−γ2)2/4.(12) We have the real eigenvalues νad 1,2for the standard PTSS: νad 1,2=2−κ2−γ2.(13) However, the eigenvalues νdd j(νaa j) for the passive (active) PTSS additionally contain a common damping (amplification) factor γ(−γ): νdd 1,2=−iγ±2−κ2−γ2,(14) νaa 1,2=iγ±2−κ2−γ2.(15) Importantly, the eigenvectors of the matrix Mcorresponding to the eigenvalues ν1,2as given in Eq. (11) are identical for the standard, passive, and active PTSSs. This means that, when we express the system evolution in the basis of these eigenvectors, the dynamics in the three discussed PTSSs differ just by common multiplicative functions describing exponential damping in the passive PTSS and exponential amplification in the active PTSS. This property provides a strong foundation for a meaningful comparison of the statistical characteristics of the three systems discussed below. It also accounts for the fact that all three systems exhibit EPs at identical locations in the parameter space. The existence and degeneracies of these EPs, interpreted as Liouvillian EPs, were examined in detail in Ref. [39]. The solution of the Heisenberg equations (3) can then be written in the following form that explicitly expresses the symmetry contained in the above four-dimensional matrix formulation: ˆ a(t)=U(t)ˆ a(0) +V(t)ˆ a†(0) +ˆ f(t).(16) In Eq. (16), we introduce ˆ aT≡(ˆa1,ˆa2), Uj,k(t)= P2j−1,2k−1(t,0), Vjk(t)=P2j−1,2k(t,0), and ˆ fj(t)=ˆ F2j−1(t), j,k=1,2. The matrices Uand Vattain the form U(t)=1 μμc(t)−γ−s(t)−is(t) −is(t)μc(t)+γ−s(t)exp(−γ+t), V(t)=−iκs(t) μ01 10 exp(−γ+t),(17) where s(t)≡sin(μt) and c(t)≡cos(μt). Incorporation of the solution into Eq. (8) for the matrix ˆ F(t)ˆ F†T(t)of correlation functions of the fluctuating forces results in the following formulas for its elements: ˆ f2 1(t)=(l2+˜ l2)κh(t)θ, ˆ f1(t)ˆ f† 1(t)=[l1h−(t)−(2l2+κ2˜ l2)h(t)]θ, ˆ f† 1(t)ˆ f1(t)=[˜ l1h−(t)−(κ2l2+2˜ l2)h(t)]θ, ˆ f2 2(t)=(l1+˜ l1)κh(t)θ, ˆ f2(t)ˆ f† 2(t)=[l2h+(t)−(2l1+κ2˜ l1)h(t)]θ, ˆ f† 2(t)ˆ f2(t)=[˜ l2h−(t)−(κ2l1+2˜ l1)h(t)]θ, 043545-5 JAN PE ˇ RINA JR. et al. PHYSICAL REVIEW A 112, 043545 (2025) ˆ f1(t)ˆ f2(t)=[l1iκd−(t)+˜ l2iκd+(t)]θ, ˆ f2(t)ˆ f1(t)=[˜ l1iκd−(t)+l2iκd+(t)]θ, ˆ f† 1(t)ˆ f2(t)=[˜ l1id−(t)−˜ l2id+(t)]θ, ˆ f2(t)ˆ f† 1(t)=[l1id−(t)−l2id+(t)]θ, (18) where θ=1/(2μ2). We introduce the following functions in Eq. (18): f(t)=1−exp(−2γ+t)exp(−2iμt) 2(γ++iμ), g(t)=[1 −exp(−2γ+t)]/(2γ+), h(t)=Re{f(t)}−g(t), h±(t)=Re{(μ±iγ−)2f(t)}+ξ2g(t), d±(t)=Im{(μ±iγ−)f(t)}∓γ−g(t).(19) We note that we also have ˆ F(t)=ˆ F†(t)=0. IV. NONCLASSICALITY, ENTANGLEMENT, STEERING, AND BELL NONLOCALITY IN TWO-MODE BOSONIC SYSTEMS In the analysis of nonclassicality and quantum correlations, we consider only the Gaussian states [20,45] that, however, represent the most useful states both in the analysis of fundamental physical experiments and applications. Moreover and most importantly, the linear Heisenberg-Langevin equations in Eq. (3) describe the state evolution inside this group of states. They are conveniently described by their normal characteristic function CNwritten in the general form as [20,45] CN(μ1,μ 2,t)=exp j=1,2(α∗ j(t)μj−c.c.) −Bj(t)|μj|2+Cj(t)μ2∗ j+c.c. 2 +[D(t)μ∗ 1μ∗ 2+¯ D(t)μ1μ∗ 2+c.c.],(20) and c.c. replaces the complex conjugated term. The parameters Bj,Cj,D, and ¯ Dthat, together with the mode complex amplitudes, identify the state are obtained according to the formulas valid for the initial coherent states with amplitudes α1(0) and α2(0): Bj(t)≡δˆa† j(t)δˆaj(t)= l=1,2 [|Vjl (t)|2+ˆ f† j(t)ˆ fj(t)], Cj(t)≡[δˆa2 j(t)]2= l=1,2 [Ujl (t)Vjl (t)+ˆ f2 j(t)], D(t)≡δˆa1(t)δˆa2(t) = l=1,2 [U1l(t)V2l(t)+ˆ f1(t)ˆ f2(t)], ¯ D(t)≡−δˆa† 1(t)δˆa2(t) =− l=1,2 [V∗ 1l(t)V2l(t)+ˆ f† 1(t)ˆ f2(t)],(21) where δˆaj=ˆaj−ˆajfor j=1,2. Substituting Eq. (17)into Eq. (21), we arrive at the formulas appropriate for our model: Bj(t)=(κ/μ)2˜s(t)+f† j(t)fj(t), Cj(t)=−(κ/μ2)˜s(t)+f2 j(t),j=1,2, D(t)=−i(κ/μ)˜c(t)+i(κγ+/μ2)˜s(t)+f1(t)f2(t), ¯ D(t)=−f† 1(t)f2(t),(22) where ˜s(t)=sin2(μt)exp(−2γ+t), ˜c(t)=sin(μt) cos(μt) exp(−2γ+t), and the correlation functions of the fluctuating forces are given in Eq. (18). At an EP, we have μ=0 and the formulas (22)forthe statistical parameters considerably simplify (μ→0): BEP j(t)=κ2t2exp(−2γ+t)+˜ ljg(t)/2−(κ2l3−j +2˜ l3−j)˜ h0(t)/2, CEP j(t)=−κt2exp(−2γ+t)+(l3−j+˜ l3−j)κ˜ h0(t)/2, j=1,2, DEP(t)=−iκ(t−γ+t2)exp(−2γ+t)+(l1+˜ l2)iκ ×[texp(−2γ+t)−g(t)]/(2γ+)−(l1−˜ l2) ×iκγ−˜ h0(t)/2, ¯ DEP(t)=(˜ l2−˜ l1)i[texp(−2γ+t)−g(t)]/(2γ+) +(˜ l1+˜ l2)iγ−˜ h0(t)/2,(23) where ˜ h0(t)≡[(t+γ+t2)exp(−2γ+t)−g(t)]/γ 2 +. We can see that the original oscillatory behavior of the standard PTSS is replaced by the polynomial one at the EP. For the passive (active) PTSS additional exponential damping (amplification) occurs. A. Nonclassicality Coefficients of the quadratic terms in the argument of exponential function in Eq. (20) can be arranged into the matrix KCsusing the vector (μ1,μ ∗ 1,μ 2,μ ∗ 2). The matrix KCs describes the characteristic function Cswritten in the general sordering of the field operators [20] KCs(s)=1 2⎡ ⎢ ⎢ ⎢ ⎣ −B1,s(s)C∗ 1¯ D∗D C1−B1,s(s)D∗¯ D ¯ DD−B2,s(s)C∗ 2 D∗¯ D∗C2−B2,s(s) ⎤ ⎥ ⎥ ⎥ ⎦ , (24) and Bj,s(s)=(1 −s)/2+Bjfor j=1,2. Eigenvalues of the matrix KCs(s), that depend on the ordering parameter s, bear the information about the state nonclassicality [19,28]. Detailed analysis reveals that the Lee nonclassicality depth τ [30,33] of the state is equal to the greatest positive eigenvalue of the matrix KCs(s=1) written for the normal field-operator ordering. Applying this procedure to the individual modes, we immediately arrive at the formula for the local nonclassicality depths τjfor j=1,2: τj=max{0,|Cj|−Bj}.(25) 043545-6 QUANTUMNESS AND ITS HIERARCHIES IN … PHYSICAL REVIEW A 112, 043545 (2025) B. Steering Quantum correlations of the Gaussian states are described by their coherence matrix σdefined for the vector (ˆq1,ˆp1,ˆq2,ˆp2)[45]: σ=σ1σ12 [σ12]Tσ2,(26) σj=1+2Bj+2Re{Cj}2Im{Cj} 2Im{Cj}1+2Bj−2Re{Cj}, σ12 =2Re{D−¯ D}Im{D−¯ D} Im{D+¯ D}−Re{D+¯ D},(27) where ˆqj=(ˆaj+ˆa† j)/2, ˆpj=(ˆaj−ˆa† j)/(2i), j=1,2. The steering of mode (3 −j) by mode jfor j=1,2 is then expressed using the formula [36] Sj→3−j=max{0,det{σj}/det{σ}}/2.(28) C. Entanglement The (logarithmic) negativity EN[34,35] as a commonly accepted measure of entanglement is inferred from the coherence matrix σPT defined for the vector ( ˆq1,ˆp1,ˆq2,−ˆp2), i.e., for the partially transposed state of mode 2 [50–52] σPT =σ1σPT 12 σPT 12 TσPT 2,(29) σPT 2=1+2B2+2Re{C2}−2Im{C2} −2Im{C2}1+2B2−2Re{C2}, σPT 12 =2Re{D−¯ D}Im{−D+¯ D} Im{D+¯ D}Re{D+¯ D}.(30) The symplectic eigenvalue ν−determined with the help of the invariants and δ[45] ν−=δ 2−δ2 4−, =det{σPT }, δ=det{σ1}+detσPT 2+2detσPT 12 ,(31) then gives the negativity EN=max{0,−ln(ν−)}.(32) D. Bell nonlocality The strongest quantum correlations, that imply the Bell nonlocality, are quantified by the Bell parameter BBell [37] that is in our case a specific linear combination of the mean values of suitably displaced parity operators ˆ (β1,β 2). Introducing two sets of displacements (β1,β 2) and (β 1,β 2) the Bell parameter BBell is determined along the formula BBell(β1,β 2;β 1,β 2)=ˆ (β1,β 2)+ˆ (β 1,β 2) +ˆ (β1,β 2)−ˆ (β 1,β 2).(33) If |BBell|>2 for any suitable choice of the displacements (β1,β 2) and (β 1,β 2), the state exhibits the Bell nonlocality manifested by the violation of the Bell inequalities [37]. According to Refs. [53,54], the mean value of a displaced parity operator is directly obtained from the Wigner function s=0 using the formula ˆ (β1,β 2)=π2 4s=0(β1,β 2).(34) To reveal the Wigner function s=0, we first have to rewrite the matrix KCs(s=0) into that written for the vector μreal,T≡(Re{μ1},Im{μ1},Re{μ2},Im{μ2}): Kreal Cs=⎡ ⎢ ⎢ ⎣ −B1,s(0) +Re{C1}Im{C1} Im{C1}−B1,s(0) −Re{C1} Re{D+¯ D}Im{D−¯ D} Im{D+¯ D}Re{−D+¯ D} Re{D+¯ D}Im{D+¯ D} Im{D−¯ D}Re{−D+¯ D} −B2,s(0) +Re{C2}Im{C2} Im{C2}−B2,s(0) −Re{C2} ⎤ ⎥ ⎥ ⎦ .(35) Forming the vector αreal,T≡(Im{α1},Re{α1},Im{α2},Re{α2}) from the arguments α1and α2of the Wigner function s=0(α1,α 2) the four-dimensional Fourier transform of the characteristic function Cs=0(αreal,T) related to the symmetric ordering of field operators leaves the Wigner function in the form s=0(α1,α 2)=exp αreal,TKreal,−1 Csαreal π2det{Kreal Cs} .(36) In Eq. (36), we use the inverse to the matrix Kreal Csand its determinant. The Bell parameter BBell depends on two sets of displacements that have to be suitably chosen to reveal the violation of the Bell inequalities. In the numerical analysis, inspired by Refs. [53,55,56], we set (β1,β 2)=(0,0) and systematically scan the remaining complex displacements (β 1,β 2) (expressed in radial coordinates) such that |βj|⩽2Bj,s(0) for j=1,2. V. RO L E O F PT -SYMMETRY IN NONCLASSICAL-STATE GENERATION Before diving into a detailed discussion of the system’s behavior, it is important to note that its dynamics, particularly regarding damping and amplification, are shaped by two competing effects. The first involves the influence of damping or amplification on the coherent component of the system’s evolution. In this context, amplification is generally considered beneficial compared to damping, as it increases mode amplitudes. This increase, in turn, enhances the system’s effective physical nonlinearity, defined as the product of the nonlinear coupling constant and the mode amplitudes. The second effect stems from random fluctuating forces (i.e., quantum noise) that tend to degrade nonclassical features and quantum correlations by disrupting phase coherence within the system. Notably, the noise accompanying amplification is generally stronger than that associated with damping, due in part to spontaneous photon emission from reservoir atoms in excited states. In detail, mutual balance between damping and amplification in standard PTSSs gives their specific dynamical 043545-7 JAN PE ˇ RINA JR. et al. PHYSICAL REVIEW A 112, 043545 (2025) behavior. In our case, PT symmetry is reached assuming γ1=γ,γ2=−γ,l1=2γ,˜ l2=2γ,˜ l1=l2=0. In specific cases, at EPs, spectral degeneracies of the dynamical matrix Moccur being accompanied by the corresponding eigenvector degeneracies. This means that the system evolution considerably simplifies and only a single eigenfrequency is sufficient to describe the evolution. Following Eq. (10), or Eq. (13), such situation occurs provided that μ≡2−κ2−γ2=0. For the Hamiltonian ˆ Hin Eq. (1) and assuming γ1=−γ2=γ, EPs occur for κ2 2+γ2 2=1.(37) In the analyzed system, this specific dynamics influences the ability to generate nonclassical states of different kinds. The nonclassicality of a generated state reflects either local nonclassicalities of the constituting modes 1 and 2 or quantum correlations between these modes. Whereas we quantify below the nonclassicalities by the corresponding Lee nonclassicality depths τ,τ1, and τ2, quantum correlations with the increasing quantumness are in turn quantified by the negativity EN(entanglement), steering parameters S1→2and S2→1 (steering), and the Bell parameter (Bell nonlocality). We examine the system behavior by assuming the modes initially in their vacuum states (arbitrary initial coherent states in both modes can be considered as well) and follow their temporal evolution. To assess the behavior of the investigated quantities, we determine their maximal values along the time taxis τ=max t{τ(t)},EN=max t{EN(t)}, τj=max t{τj(t)},Sj→3−j=max t{Sj→3−j(t)}, BBell =max t{BBell(t)},(38) where j=1,2 and compare these maximal values for the whole parameter space of the investigated system. We note that, due to linearity of the corresponding Heisenberg equations, the parameter space is effectively two-dimensional and is spanned by the variables κ/ and γ/. To reveal the role of balance between damping and amplification in the standard PTSS, we compare its behavior with two specific cases (systems) in which only damping (γ1=γ,γ2=0, l1=2γ,˜ l1=l2=˜ l2=0) and only amplification (γ1=0, γ2=−γ,˜ l2=2γ,l1=˜ l1=l2=0) influence the system dynamics. Both systems for comparison, owing to their physical nonlinearities, preserve the ability to generate the nonclassical states. Mutual comparison of the maximal values of the quantities characterising both nonclassicality and different types of quantum correlations then sheds light on how beneficial the balance between damping and amplification in standard PTSSs is when generating the nonclassical states. We note that in the systems for comparison different conditions for EPs occur: κ2 2+γ2 42=1.(39) The change of the system dynamics caused by the absence of amplification even results in the observation of nonclassical properties of the modes in the asymptotic limit t→∞[t→∞]. In this case, only the following coefficients from Eq. (22) attain asymptotically nonzero values B2(∞)=κ2 2−κ2,C2(∞)=− κ 2−κ2.(40) They imply the nonclassicality in mode 2 quantified by the nonclassicality depth τ2: τ2(∞)=κ +κ.(41) A. Nonclassicality As documented in Figs. 2(a) to 2(d), (first row), the analyzed standard PTSS allows for the generation of highly nonclassical states with τ,τ1,τ 2→1/2 for small γ/ and κ/ close to 1, i.e., when the system damping and amplification are small. We note that 1/2 gives the greatest value of nonclassicality depth attained by a Gaussian state.As the nonlinear coupling constant κrepresents the source of nonclassicality, the greater the value of κis the better the ability of the system to generate nonclassical states is. Despite the balance between damping and amplification, the greater the damping and amplification are the worse the system ability to provide nonclassical states is. This is because stronger damping and amplification are accompanied by more intense fluctuating forces. This relationship is quantified by the fluctuation-dissipation and fluctuation-amplification theorems [43]. These fluctuating forces then weaken the system ability to generate nonclassical states. We can see in Fig. 2(b) that we also reach greatest field intensities nad in the area of parameters optimal for nonclassical-state generation. The comparison of maximal values of the nonclassicality depths τand τ1with those characterizing the systems with only damping [see Figs. 2(a) and 2(c), (second row)] and only amplification [see Figs. 2(a) and 2(c), (third row)] leads us to the conclusion that both systems for comparison provide greater maximal values than those of the corresponding standard PTSS for most system parameters. The only exception is the narrow region in the graph of the ratio τad 1/τd 1of nonclassicality depths [see Fig. 2(c), second row] for values of κ/ close to 1. In this region, a large nonlinear coupling constant κ, combined with the high amplitudes of the amplified mode 2 in the standard PTSS, leads to enhanced effective physical nonlinearity. This, in turn, results in stronger nonclassicality compared to the case without the amplified mode 2. Notably, this region naturally occurs near the curve of EPs and within the domain of exponential mode amplitude growth, in contrast to the periodic amplitude behavior observed in the PT -symmetric region. However, for fixed model parameters, the nonclassicality depth τ2of mode 2 is highest for the system with only damping [see Fig. 2(d), (second row)] and lowest for the system with only amplification [see Fig. 2(d), (third row)]. Moreover, in the area of parameters optimal for nonclassical-state generation, both systems for comparison give more intense (and more nonclassical) nonclassical states. B. Entanglement and steering The analysis of entanglement quantified by the negativity ENand steering described by the steering parameters 043545-8 QUANTUMNESS AND ITS HIERARCHIES IN … PHYSICAL REVIEW A 112, 043545 (2025) (a) (b) (c) (d) FIG. 2. (a) Nonclassicality depth τad and (b) the corresponding mean photon number nad τ, (c), [(d)] local nonclassicality depth τad 1[τad 2] of mode 1 [2] of standard PTSS as they depend on model parameters γ/ and κ/. The values of the drawn parameters are compared with those originating in the model with considered only damping (superscript d) and only amplification (superscript a). In white areas, τad 2=0. Solid (dashed) black curves identify positions of EPs in PTSS (systems with only damping and only amplification). The superscript notation is explained in Fig. 1. S1→2and S2→1provides us the graphs in Figs. 3(a) to 3(d), (first row) for the standard PTSS and the graphs in Figs. 3(a) to 3(d), (second row) when the system with only damping is considered and the graphs in Figs. 3(a) to 3(d), (third row) when the system with only amplification is addressed. The conclusions drawn from these graphs are similar to the above ones for the nonclassicality depth τ: Both systems for comparison allow for greater values of the negativity ENand the steering parameters S1→2and S2→1than the standard PTSS. We note that, whereas greater nonlinearity constant κ/ and small damping and amplification constants are required to allow steering of the amplified mode 2 by the damped mode 1, the amplified mode 2 steers the damped mode 1 for any value of the system parameters. C. Bell nonlocality The advantage of the system with only damping in nonclassicality-state generation over the other two investigated systems manifests dramatically when generating the states that exhibit the Bell nonlocality. Only this system allows to violate the Bell inequalities in the wide area of the system parameters: Only the small nonlinearity constant κ/ and greater damping and amplification constants γ/prevent from the violation of the Bell inequalities [see Fig. 4(a)]. Contrary to this, only very small values of damping and amplification constants γ/ are compatible with the states violating the Bell inequalities in the standard PTSS and its variant with only amplification [see Figs. 4(b) and 4(c),γ/ <≈0.03]. Even under these conditions the attained values of the Bell parameters Bad Bell and Ba Bell are smaller than the parameters Bd Bell belonging to the system with only damping. In summary, the coexistence of damping and amplification under balanced conditions in the standard PTSS offers a clear advantage only when the nonlinear coupling constant κis large. In this case, the increased amplitudes of the amplified mode 2 enhance the system’s effective physical nonlinearity, leading to higher nonclassicality depths τ1for the damped mode 1. 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