Generating high-order quantum exceptional points in synthetic dimensions
Ievgen I. Arkhipov, Fabrizio Minganti, Adam Miranowicz, Franco Nori
GGenerating high-order quantum exceptional points
Ievgen I. Arkhipov, ∗ Fabrizio Minganti, † Adam Miranowicz,
2, 3, ‡ and Franco Nori
2, 4, § Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS,Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Institute of Spintronics and Quantum Information, Faculty of Physics,Adam Mickiewicz University, 61-614 Poznań, Poland Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Dated: March 1, 2021)Recently, there has been intense research in proposing and developing various methods for con-structing high-order exceptional points (EPs) in dissipative systems. These EPs can possess anumber of intriguing properties related to, e.g., chiral transport and enhanced sensitivity. Proposalsto realize high-order EPs have been based on the use of non-Hermitian Hamiltonians (NHHs) ofcomposite systems, i.e., the operators describing the evolution of coupled post-selected systems orcoupled intense light fields. In both cases, quantum jumps play no role. Here, by considering thefull quantum dynamics of a quadratic Liouvillian superoperator, we introduce a simple and effectivemethod for engineering NHHs with high-order quantum EPs, derived from evolution matrices ofsystem operators moments. That is, by quantizing higher-order moments of system operators, e.g.,of a quadratic two-mode system, the resulting evolution matrices can be interpreted as the newNHHs describing, e.g., networks of coupled resonators. Notably, such a mapping allows to correctlyreproduce the results of the Liouvillian dynamics, including quantum jumps. By applying thismapping, we demonstrate that quantum EPs of any order can be engineered in dissipative systemsand can, thus, be probed by the coherence and spectral functions. As an example, we consider a U (1) -symmetric quadratic Liouvillian describing an optical cavity with incoherent mode coupling,which can also possess anti- PT -symmetry. Compared to their PT -symmetric counterparts, suchanti- PT -symmetric systems could be easier to scale and, thus, can serve as a promising platformfor engineering quantum systems with high-order EPs. I. INTRODUCTION
Recently, the field of open quantum systems has at-tracted much interest. While dissipation is often seen asdetrimental, there exist a whole class of processes whichcan never take place for Hermitian (i.e., non-dissipative)systems. In these systems, the existence of exotic spec-tral degenaracies called exceptional points (EPs) has at-tracted much attention [1–3]. At an EP, two or moreeigenvalues, along with their eigenstates, coalesce. Sincethe eigenstates of a Hermitian operator are always or-thogonal, EPs require non-Hermitian operators. Histori-cally, EPs where first investigated in the context of non-Hermitian Hamiltonians (NHHs), primarily in the frame-work of parity-time ( PT )-symmetric systems, i.e., thosefor which a NHH commutes with the PT operator [4].Note that non-Hermitian Hamiltonians do not lead to theviolation of no-go theorems as explicitly demonstrated inRef. [5]. The existence of EPs has been further general-ized to any NHH exhibiting pseudo-Hermiticity [6], forwhich the PT -symmetry is a particular case.Beyond linear optical systems (see Refs. [1, 2] and ref-erences therein), EPs have been realized in various ex- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] perimental platforms, e.g., in nonlinear optics [7–9], elec-tronics [10], optomechanics [11–14], acoustics [15, 16],plasmonics [17], metamaterials [18], and ion trapped sys-tems [19].Many interesting and nontrivial effects are associatedwith the presence of EPs [20–34]. One of these is en-hanced system sensitivity to external perturbations inthe vicinity of EPs [35–44]. If n eigenstates coalesce (sothe order of an EP is n ), the response of a system to aperturbation of intensity (cid:15) scales as n √ (cid:15) . Although somerecent studies (both theoretical and experimental) havequestioned the presence of enhanced sensing at EPs [45–50], Refs. [48, 51, 52] have argued that EPs lead to en-hanced sensitivity.The interesting properties of high-order EPs ignitedthe search for methods which enable one to constructhigher-order EPs [53–56]. For NHHs, the proposed tech-niques require to realize complex networks of coherentlycoupled resonators. A major drawback is that one hasto finely tune the system parameters, due to the incom-mensurate mode couplings arising from the form of themode coupling in NHHs [53–55]. In a recent work [56],the authors proposed a novel approach for constructingtight-binding networks with higher-order EPs, based onchiral-mode coupling instead.Non-Hermiticity naturally emerges in the context ofopen quantum systems. Indeed, the Lindblad masterequation of a Markovian open system, although Hermitic-ity preserving, has a well-defined arrow of time. There-fore, the Liouvillian superoperator associated with the a r X i v : . [ qu a n t - ph ] F e b master equation is non-Hermitian. With respect to anNHH, the Liouvillian also accounts for the presence ofquantum jumps. The extension of EPs of NHHs to thosebased on Liouvillians [57] has shown that quantum jumpscan play a crucial role in the properties of EPs [44, 58–63]. Furthermore, the evolution of a density matrix ofan open quantum system is described by a completely-positive and trace-preserving (CPTP) linear map. Assuch, a NHH cannot describe the evolution of an arbi-trary quantum system.Despite the fact that an NHH may not describe thetime evolution of a Lindblad master equation (i.e., theeigenstates of a NHH do not reproduce those of the Li-ouvillian), the dynamics of some operators can be derivedin terms of the action of an NHH [63]. This apparent con-tradiction results from the fact that the operators in theHeisenberg picture do not evolve under a CPTP map.In other words, the NHH can describe the evolution ofoperators even in the quantum limit.In this article, we propose a method to properly de-fine a new class of effective NHHs for quadratic Li-ouvillian systems. These NHHs are derived from evo-lution matrices governing the moments of system op-erators and as such are called moments-based NHHs .Most importantly, these moments-based NHHs can re-veal higher-order quantum EPs, residing in the Liouvil-lian eigenspace [63]. That is, by quantizing system oper-ators moments, i.e., by mapping the corresponding evolu-tion matrices to moments-based NHHs, one can engineerquantum systems with high-order EPs.Such constructed moments-based NHHs with high-order EPs substantially differ from the standard
NHHs.Whereas the latter are constructed by expanding the
Hilbert space of system operators [53–55], the former areconstructed by expanding rather the system operators moments space . Indeed, if one uses the eigenstates ofthe standard NHH for low-order operators moments toexpress higher-order ones, the result would be unphysi-cal and give different results when compared to the fullLiouvillian dynamics [63]. The moments-based NHHs,instead, are derived via the quantization of the systemoperators moments obtained from the Liouvillian, and,thus, correctly captures the dynamics of high-order sys-tem operators without approximations (i.e., including theeffects of quantum jumps [57, 59]). The dynamics of suchmoments-based NHH can describe, e.g., a network of cou-pled resonators. In other words, by starting from a quad-tratic Liouvillian, describing, e.g., a two-mode system,the resulting evolution matrices, governing the higher-order field moments, can be cast to the new NHHs, whichcan describe networks of coupled cavities instead.These newly obtained moments-based NHHs can re-veal the presence of arbitrarily-high order quantum EPsin the Liouvillian dynamics. Physically speaking, onecan witness the presence of high-order quantum EPsby means of the coherence and spectral functions [63]or by properly initializing the system. To put it an-other way, instead of considering networks of n resonators (where high-order EPs can be engineered), by consider-ing higher-order moments of, e.g., two coupled resonators,one can obtain the same spectral degeneracies. Apartfrom a theoretical interest in defining the NHHs in thequantum limit, the advantage in the use of moments-based NHHs with respect to standard NHHs lies in itsmanifesting simplicity and in the possibility to preservethe commensurate character of modes coupling.As an example, we implement our method for the U (1) -symmetric quadratic Liouvillian that describes atwo-mode optical cavity with incoherent mode cou-pling. This model is also characterized by the anti- PT -symmetry [63–68], as defined in Eq. (20). We showhow the Liouvillian eigenspace of such a two-mode sys-tem, expressed via field moments, can be mapped to theeigenspace of an effective NHH of a multimode system.The benefit of considering anti- PT -symmetric systems,compared to their PT -symmetric counterparts with ex-clusively coherent mode coupling, is the absence of anyactive elements, and their scalability, which is seeminglyeasier to realize. That is, one does not need to build upcomplex networks to achieve higher-order EPs, but only excite additional modes in the anti- PT -symmetric cavity;at the same time ensuring the incoherent character of thecoupling between the newly excited modes. Recent stud-ies show that such systems, with incoherent mode inter-actions, can be realized via incoherent mode backscatter-ing in waveguide networks [69] and cavity-based photonicdevices [70–73]. Moreover, the dissipative couplings canplay a prominent role in the experimental realizationsof photonic and quantum computing in time-multiplexedoptical systems [74, 75].As a byproduct of our method, we also highlight therich structure of quadratic Liouvillians. Indeed, thecorrespondence between multimode systems and higher-order moments is a peculiarity of the Liouvillian spacestructure [44]. We argue that, although the corre-spondence between the Liouvillian evolution of higher-order correlation functions and lower-order correlationfunctions of more complex multimode systems is exactonly for quadratic Liouvillians (i.e., describing Gaussianstates), such a procedure should be valid also in the pres-ence of weak nonlinearity, where a Gaussian state approx-imation can still be valid.We note that our method can be implemented irre-spective of the knowledge of the specific details of agiven quadratic Liouvillian. That is, in order to real-ize a physical system with high-order EPs, initially onehas only to have some physically realizable NHH (whichof course can again be related to some quadratic Liouvil-lian), whose matrix mode representation reveals an EPat least of order two. Then, by increasing the order of theEP, according to the method described here, one obtainsa non-Hermitian model, which can be a guide to realizeextended lattice systems whose NHH has a higher-orderEP.The paper is organized as follows. In Sec. II, we in-troduce a general model of quadratic Liouvillians. InSec. III, we analyze the dynamics of higher-order mo-ments of system operators in the model, expressed via thecorresponding evolution matrices. Then, by performinga second quantization of the operator moments, we intro-duce a map between moments evolution matrices and thenew class of NHHs, called moments-based NHHs, whichcan genuinely capture the quantum effects in a system.In Sec. IV, we describe a method to engineer higher-orderEPs determined by the moments-based NHHs. In Sec. V,we implement the proposed scheme on the example ofthe U (1) and anti- PT -symmetric cavity with incoherentmode coupling. The discussion of the proposed methodand its comparison to existing methods along with con-clusions are given in Sec. VI. II. GENERAL MODEL OF A SYSTEMDESCRIBED BY A QUADRATIC LIOUVILLIAN
The evolution of a density matrix ˆ ρ ( t ) is described bythe master equation dd t ˆ ρ ( t ) = L ˆ ρ ( t ) , (1)which for a quadratic Liouvillian superoperator L inthe Gorini-Kossakowski-Sudarshan-Lindblad form reads( (cid:126) = 1 ) L ˆ ρ ( t ) = − i (cid:16) ˆ H eff ˆ ρ ( t ) − ˆ ρ ( t ) ˆ H † eff (cid:17) +2 (cid:88) jkl Γ ljk ˆ s ( l ) j ˆ ρ ( t ) (cid:16) ˆ s ( l ) k (cid:17) † , (2)where ˆ H eff is the effective NHH given by: ˆ H eff = (cid:88) κ lmjk ˆ s lj ˆ s mk − i (cid:88) Γ ljk ˆ s ( l ) j (cid:16) ˆ s ( l ) k (cid:17) † . (3)In Eqs. (2) and (3), the indices { j, k } indicate the sites(modes) of a system, and { l, m } = { , } are such that ˆ s (1 , q = { ˆ a q , ˆ a † q } , where ˆ a q ( ˆ a † q ) is the annihilation (cre-ation) operator of a particle at the q th site (e.g., a pho-ton). The coefficients κ lmjk , Γ lmjk ∈ R describe the coher-ent and incoherent parts of the system evolution, re-spectively. Such a quadratic Liouvillian describes thedissipative dynamics of Gaussian states that, in optics,describes, e.g., linearly coupled waveguides (cavities) ornonlinear parametric processes [76].From now on, we will describe ˆ H eff in Eq. (3) as the effective NHH to distinguish it from the moments-basedNHH , which is associated with system operators mo-ments of higher order and which we will introduce inSec. III C.According to the quantum trajectory theory, the Liou-villian in Eq. (2) can be divided in two parts; namely, ina continuous nonunitary evolution described by the ef-fective NHH ˆ H eff , and in the action of discrete randomchanges expressed by quantum jumps [77, 78]. The effec-tive NHH is especially useful in the semiclassical limit,where the jump action can be neglected. It can also be used to describe systems where it is possible to deter-mine if a quantum jump took place, e.g., cavities with asmall photon number and with a very high finesse [79] orpostselected systems [80]. In other cases, the last termin Eq. (2), describing the quantum jumps effects (in thiscase, a sudden creation or annihilation of a particle) isessential to faithfully capture the system dynamics at thequantum level. Indeed, for the case studied here of open linear systems, the effective NHH ˆ H eff does not reflectthe non-conservative character of dissipation [60]. III. DYNAMICS OF THE MOMENTS OFSYSTEM OPERATORS AND THENON-HERMITIAN HAMILTONIAN
In the following, we assume that the dissipators inEq. (2) induce no incoherent amplification. Otherwise,the dynamics of some moments is affected by an addi-tional noise vector [81, 82], which is of no relevance here.
A. Field moments evolution for a quadratic U (1) -symmetric two-mode Liouvillian We begin by considering the simplest case, i.e., thatof a quadratic and U (1) symmetric systems describingtwo cavities (its generalization is provided in Sec. III B).We refer to the corresponding L as a linear Liouvillianbecause it emerges in the context of dissipative coupledlinear systems, e.g., coupled waveguides or cavities.Any Liouvillian for coupled bosonic systems is said tobe U (1) -symmetric if it commutes with a phase rotationoperator U , defined as U ˆ ρ = exp iφ (cid:88) j ˆ a † j ˆ a j ˆ ρ exp − iφ (cid:88) j ˆ a † j ˆ a j . (4)In other words, the master equation must be invariantunder a simultaneous arbitrary phase shift ˆ a j → ˆ a j e iφ .For two coupled cavities, i.e., j, k = 1 , in Eq. (2), U constraints the rate equations for the field moments (cid:104) ˆ a † m ˆ a † n ˆ a p ˆ a q (cid:105) , for m, n, p, q = 0 , , . . . . Normally, the dynamics of the field moments relates allthe possible combinations of { m, n, p, q } . However, in thepresence of the U symmetry only moments that have thesame order ( m + n − p − q ) can be coupled. Since the con-sidered Liouvillians here are also quadratic, one obtains p + q = m + n . Thus, the dynamics of the moments iscaptured by closed sets of coupled equations . For exam-ple, the first-order moment (cid:104) ˆ a (cid:105) ( (cid:104) ˆ a † (cid:105) ) would be coupledonly to the moment (cid:104) ˆ a (cid:105) ( (cid:104) ˆ a † (cid:105) ).Given the closed structure of the rate equations for thefirst-order moments we have: dd t (cid:104) (cid:126)A (cid:105) = M A (cid:104) (cid:126)A (cid:105) , (5)where (cid:126)A = [ˆ a , ˆ a ] T , and M A is a × evolution matrix.The matrix M A is the building block to obtain theevolution matrix for higher-order field moments by con-structing various Kronecker products of the vectors (cid:126)A and (cid:126)A † ≡ [ˆ a † , ˆ a † ] . For instance, given the second-ordermoments (cid:104) (cid:126)B (cid:105) = (cid:104) (cid:126)A ⊗ (cid:126)A (cid:105) = (cid:104) [ˆ a , ˆ a ˆ a , ˆ a ˆ a , ˆ a ] T (cid:105) , (6)the evolution matrix M B , such that ∂ t (cid:104) (cid:126)B (cid:105) = M B (cid:104) (cid:126)B (cid:105) , isthe Kronecker sum of the same two matrices M A , M B = M A ⊕ M A = M A ⊗ I + I ⊗ M A . (7)Here, the symbol ⊕ denotes a Kronecker sum, ⊗ is theKronecker product, and I is the × identity matrix.Note that we keep the order between the products of theoperators ˆ a and ˆ a in Eq. (6). Although, in this case,it is not relevant because [ˆ a , ˆ a ] = 0 , but in generalthe order should be preserved. We remark that a Kro-necker sum naturally appears in problems when solvingLyapunov and/or Sylvester equations [83].The same procedure can recursively be applied to ob-tain the evolution matrices for higher-order moments.The generalization of Eq. (7) is given by: (cid:104) (cid:126)γ (cid:105) = (cid:104) (cid:126)α ⊗ (cid:126)β (cid:105) , M γ = M α ⊕ M β = M α ⊗ I β + I α ⊗ M β , (8)where the vectors of operators, (cid:126)α and (cid:126)β , are the Kro-necker products of the initial vectors (cid:126)A and/or (cid:126)A † , while M α,β are the corresponding evolution matrices. The re-sulting evolution matrix M γ of higher-order field mo-ments is the recursive Kronecker sum of the matrices M A . The dimension N γ of the matrix M γ is the prod-uct of the dimensions of the matrices M α and M β , i.e., N γ = N α N β . Moreover, due to the standard propertiesof the Kronecker sum, the symmetry of the matrix M A is retained by all the matrices M γ . B. Evolution of field moments for a genericquadratic Liouvillian
The previous derivation for a U (1) quadratic two-modesystem can be extended to generic quadratic Liouvillians,even if the form of M γ is slightly more involved.For an arbitrary quadratic Liouvillian (i.e., not neces-sarily U (1) -symmetric), describing an n -mode system, allfield moments are generated by the tensor product of the n -dimensional vector (cid:126)A = [ˆ a , ˆ a † , . . . , ˆ a n , ˆ a † n ] T . (9)The time evolution of (cid:104) (cid:126)A (cid:105) is given by ∂ t (cid:104) (cid:126)A (cid:105) = M A (cid:104) (cid:126)A (cid:105) ,which is the same as in Eq. (5).Given the building block M A , the previously detailedprocedure, to obtain the evolution matrices for higher-order moments, remains valid. Similarly to the U (1) case, the evolution matrices M γ , which determine thedynamics of various higher-order moments, are obtainedby taking a recursive Kronecker sum of the correspond-ing n × n matrix M A . That is, Eq. (8) is valid for anyquadratic Liouvillian. Also, as it was stressed earlier, theoperators order should be kept when constructing Kro-necker tensors out of the vector of operators (cid:126)A in Eq. (9).In other words, no permutations are allowed in the ob-tained products of the operators. C. Second quantization of field moments and adefinition of the moments-based non-HermitianHamiltonian
1. Moments-based non-Hermitian Hamiltonian of U (1) quadratic Liouvillians In the case of U (1) quadratic Liouvillians, the evo-lution matrix M A for the first-order field moments be-comes equivalent (up to the imaginary factor) to the ma-trix form of the corresponding NHH, i.e.,: M A = − i H eff , (10)where the matrix form of the NHH is defined as follows ˆ H eff ≡ (cid:16) (cid:126)A (cid:17) † H eff (cid:126)A, (11)and vector of operators (cid:126)A , for a two-mode case, is given inEq. (5). In other words, there is a one-to-one correspon-dence between the evolution of the operator (cid:126)A and itsfirst-order moments (cid:104) (cid:126)A (cid:105) for the U (1) systems. This cor-respondence is quite intriguing, since it provides a clearphysical meaning to the effective NHH of a two coupledbosonic systems via the introduction of its first-ordermoments, and which has been intensively exploited ina number of previous works on quantum EPs [59, 84, 85].The described above correspondence however cannotbe simply extended to higher-order field moments, sincethe evolution matrices governing the dynamics of the op-erators and their moments would be in general different.This stems from the fact that the dynamics of the fieldmoments of any order is determined by the Liouvillianwhich include quantum jump effects, whereas the effec-tive NHH applied to the same moments (in Heisenbergpicture) in general fails to incorporate them [63].Nevertheless, one can assign to any evolution matrix M γ a new NHH in analogy to Eq. (10), which we thus calla moments-based NHH, by quantizing the correspondinghigher-order field moments (cid:104) (cid:126)γ (cid:105) (see also Fig. 1). Thedetermination of such moments-based NHHs however re-quires an intermediate passage. Namely, in general, (cid:126)γ contains terms which are identical [e.g., ˆ a ˆ a = ˆ a ˆ a inEq. (6)], and it cannot be straightforwardly quantized.The degeneracy of M γ can be eliminated by introduc-ing the reduced matrix M red γ . For instance, M γ of anynon-Hermitian moment of the vector (cid:126)A , i.e., (cid:126)γ = m (cid:78) i =1 (cid:126)A ,has an “initial” dimension N γ = 2 m . By collecting iden-tical terms, the resulting matrix M red γ has the dimension N eff γ = m + 1 . The price to pay for such reduction is,in general, the loss of some initial symmetry of the ma-trix M γ . For instance, if M γ = M Tγ then, in general, M red γ (cid:54) = (cid:16) M red γ (cid:17) T . That is, if the mode coupling in thematrix M γ is symmetric, then that coupling in the ef-fective evolution matrix M eff γ is, in general, asymmetric.Having eliminated the redundant variables, any N di-mensional vector ˆ γ in Eq. (8) can be quantized as (cid:126)γ → (cid:126)γ (cid:48) ,where (cid:126)γ (cid:48) = [ˆ b , . . . , ˆ b N ] is the vector of the boson annihi-lation operators ˆ b j , which describe new fields, as shownin Fig. 1. The emerging physics is that a new lineardissipative system is constructed. This new system isdescribed by N coupled fields ˆ b j and it evolves under amoments-based NHH ˆ H mb γ given by [c.f. Eq. (10)]: H mb γ = i M red γ , (12)where H mb γ is, as before, a matrix form of the moments-based NHH ˆ H mb γ = ( (cid:126)γ (cid:48) ) † H mb γ (cid:126)γ (cid:48) .This procedure is easily and recursively extended toany set of higher-order moments. Thus, moments-basedNHHs representing large systems can be constructed byconsidering an initial U (1) two-mode system with anevolution matrix M A for the first-order field moments.Vice versa, one can obtain higher-order EPs by using themoments-based NHHs as a guideline to realize dissipa-tive lattices of coupled resonator, whose effective NHHwill have higher-order EPs (see Fig. 1). Notably, the ef-fective NHH of the lattice could be characterized by asmaller decay rate of the observables, resulting in a bet-ter visibility of the EP.
2. Moments-based non-Hermitian Hamiltonian of genericquadratic Liouvillians
In a general case, the correspondence between M A andthe matrix form of the NHH ˆ H eff , drawn in Eq. (10), doesnot hold anymore. Instead, one has: H eff ≡ i η M A + i η M † A η , (13)where η = ∼ (cid:77) n (cid:18) (cid:19) , η = ∼ (cid:77) n (cid:18) (cid:19) , η = ∼ (cid:77) n (cid:18) − (cid:19) . (14)In Eq. (14), the symbol ∼ (cid:76) means a direct sum (not theKronecker one). In Eq. (13), without loss of generality,we have also dropped a constant term related to the fieldfrequencies.Equation (13) reads as follows. The first term in itsr.h.s. takes into account the dynamics of only the anni-hilation operators ˆ a k , i.e., the odd elements of the vector ˆ A in Eq. (9). This term is equivalent to the r.h.s. ofEq. (10), when the system is U (1) symmetric and linear.The second term in Eq. (13) accounts for the dynamics ofthe creation operators of the fields, i.e., the even elementsof the vector ˆ A .Contrary to the case of the linear systems, Eq. (13) im-plies that the evolution matrix M A and the NHH ˆ H eff are, in general, not simply related. Even though the crit-ical points of the NHH and Eq. (13) coincide, the spec-tral properties of these matrices might differ. And it isthe spectral degeneracies of the evolution matrices thatdetermine the properties of the coherence and spectralfunctions, which are experimentally accessible via photo-count or homodyne measurements.Despite the fact that now there is no one-to-one cor-respondence between the evolution matrix for the first-order field moments and effective NHH in the case ofgeneric nonlinear quadratic Liouvillians, according toEq. (13), one still can map the evolution matrix forhigher-order moments to a new moments-based NHH,i.e., M γ → H mb γ . (15)The mapping in Eq. (15), in general, might require ad-ditional operations over the matrix M γ , compared toEqs. (12) and (13), e.g., row and column permutations,which depends on how a corresponding vector of opera-tors (cid:126)γ is constructed from the initial vector (cid:126)A .As it has been shown in Ref. [86], even for a classof Hermitian quadratic Hamiltonians, i.e., without dissi-pation, and which describes optical nonlinear processes,the corresponding evolution matrix M A can reveal anEP. Moreover, one can easily map the matrix M A to anew NHH as in Eq. (10) [86]. Based on the latter, onethen can straightforwardly apply Eq. (12) for the con-struction of new moments-based NHHs exhibiting higher-order EPs.Within this description, we can properly define themoments-based NHH of composite systems of higher-order moments. The advantages of this procedure aremanifold: (i) The moments-based NHH now correctlytakes into account the effects of quantum jumps. (ii)These NHH can now be correctly quantized, and thecommutation rules emerging from the physical moments-based NHH can correspond to those of the original sys-tem. In other words, the computation of quantum-relevant variables is unaffected by the algebraic construc-tion. (iii) The procedure can be easily implemented nu-merically, even for large systems. IV. ENGINEERING HIGHER-ORDEREXCEPTIONAL POINTS FROM QUADRATICLIOUVILLIANS
In the previous discussion, we proved that it is pos-sible to define a physical NHH using the full-Liouvilliandescription if one focuses on the dynamics of moments. ∂ t h ˆ a ih ˆ a ˆ a ih ˆ a i = M red γ h ˆ a ih ˆ a ˆ a ih ˆ a i ∂ t h ˆ a ih ˆ a ˆ a ih ˆ a ˆ a ih ˆ a i = M red γ h ˆ a ih ˆ a ˆ a ih ˆ a ˆ a ih ˆ a i ∼ = ∼ = n ˆ b , ˆ b , ˆ b on ˆ b , ˆ b , ˆ b , ˆ b o FIG. 1. Schematic representation of the procedure to define a moments-based NHH and its relation to a lattice system ofresonators in the U (1) symmetric case. A quadratic Liouvillian system (e.g., the two cavities on the left) is characterizedby coherent interactions (the red double arrows), dissipation (green arrows pointing outwards), and amplification channels(green arrows pointing inwards) that compete in determining the photonic field inside the cavity (orange balls). The equationsof motion for the various moments (cid:104) ˆ a † m ˆ a † n ˆ a p ˆ a q (cid:105) of such a system (middle part) form a finite set, which can be describedby the reduced evolution matrices M red γ (see the main text in Sec. III C 1 for details). By quantizing the moments, suchmatrices M red γ can be interpreted as an effective NHH of a more complex lattice of resonators (right part of the figure),according to Eq. (12). Indeed, each moment can be mathematically treated as a separate bosonic field ( ˆ b i ), i.e., a drivendissipative quadratic bosonic system, as shown in the right panel with cavities. Each of this field interact with the othersvia dissipative or coherent interactions (represented by green and red dashed lines). This procedure is general, and exploresa direct correspondence between the evolution of higher-order moments of a Liouvillian system and lower-order moments of alarger system. Interestingly, this correspondence can be exploited to observe higher-order EPs in simple lattices by consideringhigher-order correlation functions and, vice versa, as a guideline to engineer lattices which display higher-order EPs. Despite the lack of the correspondence between theeigenvectors describing the state and operators evolu-tion (namely, the right- and left-hand-side eigenstates ofthe Liouvillian) there is a correspondence between theireigenvalues and their degeneracies. Indeed, if the stateshave an EP, so do the moments. Here, by considering thespectra of evolution matrices for higher-order field mo-ments, and exploiting the described above mapping, weshow how to engineer moments-based NHHs with higher-order EPs in any quadratic Liouvillian systems.
A. High-order exceptional points
Due to the structure of the equations of motion of themoments, the spectral degeneracies of an evolution ma-trix M γ can be directly obtained from those of M A .Given the Kronecker sum in Eq. (8), one has the follow-ing relation between eigenvalues of the matrices [83]: λ ij ( M γ ) = λ i ( M α ) + λ j ( M β ) , (16)with i = 1 , . . . , N α and j = 1 , . . . , N β . Therefore, witheach new term M A in Eq. (8), the order of an EP of M γ increases by one. Note that, although the algebraic degeneracy of the eigenvalues in Eq. (16) grows propor-tionally to N γ , its geometric multiplicity does not. In-deed, if M A has an EP of order two, the evolution matrix M γ = m (cid:76) i =1 M A has an EP of order ( m + 1) .This result is an analogous prove of that in Ref. [63],were it was demonstrated that once the evolution matrix M A has an EP of second order, it immediately impliesthe presence of an EP of any higher-order n ≥ in the Li-ouvillian eigenspace. This eigenspace determines the evo-lution matrices M γ of field moments. Consequently, thequantized moments and the corresponding NHHs givenin Eqs. (12) and (15) are characterized by the same de-generacy as that of M γ .The matrix M γ determines also the dynamics of high-order correlation functions, according to the quantum re-gression theorem [82]. The Wick’s theorem for Gaussianstates (i.e., quadratic systems) indicates that correlationfunctions of any order can be expressed as a sum of prod-ucts of correlation functions of lower orders. This impliesthat if the matrix M A , which determines the first-ordercoherence function, exhibits an EP, higher-order coher- FIG. 2. Schematic representation of the model described bythe Liouvillian superoperator L in Eq. (17). S and S aretwo system modes, which dissipate to the environment E withrates Γ and Γ , respectively. The modes are also dissipa-tively coupled to the environment with the incoherent cou-pling strength Γ = Γ . ence functions reveal a higher order EP [63]. The sameconclusion can alternatively be drawn using the proper-ties of the matrix exponential of a Kronecker sum [87].Nevertheless, when constructing the moments-basedNHH, the reduced matrix M red γ is used instead, accord-ing to Eqs. (12) and (15). As a result, the order of an EPof the reduced matrix would correspond to its dimension,e.g., for the U (1) case, N ( M red γ ) = m + 1 , which is thesame as the order of the EP. As a result, the correspond-ing moments-based NHH ˆ H mb γ , describing ( m + 1) modes(resonators) would have an EP of the order ( m + 1) .Moreover, from Eqs. (8) and (16) it is evident that theconstruction of the moments-based NHH ˆ H mb γ for higher-order EPs can be realized with arbitrary matrices M α and M β . We also note that although our method im-plicitly assumes that the evolution matrix M A for first-order field moments already has an EP, there are meth-ods which allow to construct a new matrix having an EPfrom a combination of generic complex matrices with noinitial degeneracies [88]. V. EXAMPLE OF A U (1) ANTI- PT -SYMMETRIC CAVITY In this section, we implement our scheme for the ex-ample of a U (1) -symmetric two-mode cavity with inco-herent mode coupling, which additionally possesses theanti- PT -symmetry. Namely, we show how a Liouvillianeigenspace of such a two-mode system, expressed via fieldmoments and their evolution matrices, can be mappedto a new moments-based NHH representing a multimodesystem.In other words, first we reveal EPs of any order arisingfrom the evolution matrices of the bimodal system un-der consideration. Then, for any evolution matrix whichexhibits an EP of order n > , we assign a new moments-based NHH, which can, thus, correspond to a new cou-pled n -mode system with higher-order EPs. A. Model of an anti- PT -symmetric bimodal cavity The model under consideration is the same as inRef. [63]. Namely, we consider the Lindblad master equa-tion in Eq. (1) with the following Liouvillian superoper-ator [89–91]: L ˆ ρ = − i (cid:16) ˆ H eff ˆ ρ − ˆ ρ ˆ H † eff (cid:17) + (cid:88) j,k =1 , Γ jk ˆ a j ˆ ρ ˆ a † k , (17)with the effective NHH: ˆ H eff = (cid:88) j =1 , ω j ˆ a † j ˆ a j − i (cid:88) j,k =1 , Γ jk ˆ a † j ˆ a k . (18)The Liouvillian in Eq. (17) describes a dissipative linearsystem of two incoherently coupled modes. A schematicdiagram of the model under study is shown in Fig. 2. InEq. (17), ˆ a j ( ˆ a † j ) is the annihilation (creation) operator ofmode j with a bare frequency ω j ; the diagonal dampingcoefficient Γ kk denotes the inner k th mode decay rate,while the off-diagonal coefficient Γ jk = Γ kj (for j (cid:54) = k )accounts for the incoherent coupling strength betweenmodes j and k , due to the interaction of both modeswith the environment [89]. That is, without loss of gen-erality, we focus on a symmetric form of the decoherencematrix in Eq. (17), although, in general, Γ jk (cid:54) = Γ kj . Thelatter case can also result into the chiral character of theinteraction between modes. B. Second-order EP
In our previous study [63], we analyzed EPs, up totheir third order, of such an anti- PT -symmetric bimodalcavity. The evolution matrix for the first-order field mo-ments (cid:104) ˆ A (cid:105) = [ (cid:104) ˆ a (cid:105) , (cid:104) ˆ a (cid:105) ] T in Eq. (5) takes the form [63]: M A = (cid:18) − i ∆ − Γ − Γ − Γ i ∆ − Γ (cid:19) , (19)where ∆ = ω − ω is the frequency difference between thetwo modes, Γ = Γ = Γ is an inner loss rate of eachmode, and Γ = Γ is an incoherent mode couplingstrength (for details, see Ref. [63]).According to Eq. (19), the corresponding NHH is anti- PT -symmetric, since it anticommutes with the parity-time PT operator, i.e., PT ˆ H eff PT = − ˆ H eff , (20)which implies the PT -symmetry of the evolution matrix M A . Moreover, by appropriately rotating the anti- PT -symmetric NHH ˆ H eff , one can switch it to a passive PT -symmetric system [63].The eigenvalues of the matrix M A in Eq. (19) read λ , = − Γ ± (cid:113) Γ − ∆ . (21)Thus, the EP of the system (which is of second order forthe evolution matrix M A ) is observed at the point Γ EP12 = | ∆ | . (22)As a consequence, at the EP, the matrix M A , andthus ˆ H eff , acquire a Jordan form, i.e., it become non-diagonalizable. C. Third-order EP
According to Eqs. (6) and (7), the matrix M B = M A ⊕ M A written for the non-Hermitian second-ordermoments (cid:104) (cid:126)B (cid:105) [in the form given in Eq. (6)] reads as: M B = − i ∆ − − Γ − Γ − Γ −
2Γ 0 − Γ − Γ − − Γ − Γ − Γ i ∆ − , (23)would attain the same EP but of its third order. Indeed,the eigenvalues of this matrix are: λ , = − ± (cid:113) Γ − ∆ , λ , = − , (24)which at the EP in Eq. (22) become identical. The plotsfor these eigenvalues were presented in Ref. [63]. Al-though the algebraic multiplicity of the eigenvalues inEq. (24) at the EP equals four, the geometric multiplic-ity equals three; thus, indicating the coalescence of threemodes. The latter fact points that the EP is, indeed, ofthird order. Moreover, for the matrix M B in Eq. (23),as mentioned above, one can effectively reduce its dimen-sion to three, keeping the same order of the EP. That is,by reducing the vector of moments (cid:104) ˆ a (cid:105)(cid:104) ˆ a ˆ a (cid:105)(cid:104) ˆ a ˆ a (cid:105)(cid:104) ˆ a (cid:105) → (cid:104) ˆ a (cid:105)(cid:104) ˆ a ˆ a (cid:105)(cid:104) ˆ a (cid:105) , (25)one obtains the following effective matrix M B → M red B = − i ∆ − − − Γ − − Γ − i ∆ − . (26)By comparing Eqs. (23) and (26), one can see that thereduced matrix M red B has lost the symmetry of the initialmatrix M B , i.e., M red B (cid:54) = ( M red B ) T .One can directly obtain a NHH from M red B , accordingto Eq. (12). Namely, by additionally discarding the innerdecoherence terms in Eq. (26), the corresponding dis-sipative system for this model is a lattice of three bosonic FIG. 3. (a) Real and (b) imaginary parts of the eigenvalues λ ,according to Eq. (31), of the effective evolution matrix ˆ M eff C ,given in Eq. (30), for the third-order field moments. Its foureigenvalues coalesce at the EP in Eq. (22), thus indicatingthat the EP is of the fourth order. System parameters are setas Γ = 1 [arb. units] and ∆ = 1 [arb. units]. modes, interacting via the effective NHH ˆ H eff =2∆ (cid:16) ˆ b † ˆ b − ˆ b † ˆ b (cid:17) − i Γ (cid:16) ˆ b ˆ b † + 2ˆ b † ˆ b + ˆ b ˆ b † + 2ˆ b † ˆ b (cid:17) , (27)and whose explicit Liouvillian dynamics reads L ˆ ρ ( t ) = − i ( ˆ H eff ˆ ρ ( t ) − ˆ ρ ( t ) ˆ H eff ) + 2 (cid:88) i, j γ ij (cid:16) ˆ b i ˆ ρ ˆ b † j (cid:17) , (28)where the decoherence matrix in Eq. (28) has onlynonzero off-diagonal elements γ ji = 2 γ ij = 2Γ with i = 1 , , j = 2 . D. Fourth-order EP
In a similar way, one can generate an NHH with an EPof the fourth order, whose matrix form reads as follows H mb γ = i M C = i (cid:77) n =1 M A , (29)where (cid:126)γ = (cid:126)A ⊗ (cid:126)A ⊗ (cid:126)A , and M A is given in Eqs. (23)and (19), respectively. Note that the resulting matrix ˆ M C has dimension × . In other words, the numberof required modes increases to eight. Nonetheless, again,one can contract the resulting matrix to dimension four,but at the expense of losing the mode-coupling symme-try. Namely, the evolution matrix M C can be defined byeight moments of the form (cid:104) ˆ a i ˆ a j ˆ a k (cid:105) , where i, j, k = 1 , .As such, there are two sets of three equivalent moments (cid:104) ˆ a i ˆ a j (cid:105) = (cid:104) ˆ a i ˆ a j ˆ a i (cid:105) = (cid:104) ˆ a j ˆ a i (cid:105) for i, j = 1 , and i (cid:54) = j . Asa result, only four non-degenerate moments remain outof eight , which, thus, defines the × reduced matrix M red C , and which attains the following form M red C = − i ∆ − − − Γ − i ∆ − − − i ∆ − − Γ − i ∆ − , (30)and its eigenvalues read λ , = − ± (cid:113) Γ − ∆ ,λ , = − ± (cid:113) Γ − ∆ . (31)We plot these eigenvalues in Fig. 3. E. ( N + 1) th-order EP Clearly, one can obtain the effective evolution matricesfor any dimension ( N + 1) as M eff N +1 = ∆ − N Γ . . . . . . . . . . . . . . . − n Γ ∆ n − ( N − n )Γ . . . . . . . . . . . . . . . − N Γ ∆ N , (32)where ∆ n = i ( − N + 2 n )∆ − N Γ for n = 0 , . . . , N . Com-bining now Eqs. (16) and (21), the eigenvalues of thismatrix can be written as λ n = − N Γ ± ( N − n ) (cid:113) Γ − ∆ , n = 0 , . . . , N. (33)Hence, a dissipative system described by the followingeffective NHH (expressed via matrix form): H mb γ = i M red N +1 , (34)which is defined for ( N + 1) modes (cid:126)γ (cid:48) , exhibits the EP oforder ( N + 1) .Therefore, by recursively repeating the same proce-dure, one can create new effective NHHs with higher-order EPs out of various evolution matrices, generated bythe initial matrix M A corresponding to a bimodal anti- PT -symmetric cavity. Importantly, the same conclusionscan be drawn for any two-mode linear U (1) -symmetricopen systems exhibiting an EP, including those with co-herent mode coupling.We note, when constructing moments-based NHH, ac-cording to Eq. (34), one can always rescale the inner de-coherence rate Γ to ensure that the achieved resolution ofa given high-order EPs via coherence function and spec-tra is maximal. VI. DISCUSSION AND CONCLUSIONS
In this article, we have proposed a method to prop-erly define a new class of NHHs (so-called moments-based NHH) for quadratic Liouvillian systems, and whichcan exhibit higher-order quantum EPs. These moments-based NHHs are derived via a quantization of systemoperator moments obtained via a given quadratic Liou-villian. This approach correctly captures the dynam-ics of high-order system operators without semiclassi-cal approximations, i.e., including the effect of quantumjumps [57, 59]. In other words, we have proposed a simpleand effective method for engineering higher-order quan-tum EPs based on the moments-based NHH.The dynamics of the moments-based NHHs can be as-sociated with networks of dissipative coupled resonators.That is, by starting from a quadratic Liouvillian, describ-ing, e.g., a two-mode system, the resulting evolution ma-trices, governing the higher-order field moments, can becast to the new NHHs, which can, thus, describe net-works of coupled cavities instead. To put it another way,one can assign to moments-based NHHs a clear physi-cal meaning by mapping them onto the Liouvillian dy-namics of the high-order moments of coupled resonators.This can be prove also useful to design lattice of U (1) resonators with higher-order EPs, which could have ap-plications for transport properties [92].Compared to other existing methods for constructingNHHs with high-order EPs [53, 55, 56], the main ad-vantage of our approach lies in its simplicity and preser-vation of the commensurate character of mode couplingstrengths . Another outcome of the method developedhere is to reveal the rich structure of quadratic Liouvil-lians for the dynamics of higher-order moments of thesystem operators, which can, thus, be probed by coher-ence and spectral functions of any order.As an example, we have analyzed a U (1) -symmetriccavity with incoherent mode coupling, which can alsopossess the anti- PT -symmetry. The system studied inSec. V can serve as a promising platform for implement-ing structures with high-order EPs. Indeed, such sys-tems with incoherent mode coupling can combine fea-tures related to both PT and anti- PT -symmetries, ashas been demonstrated in Ref. [63]. The construction ofthe physical moments-based NHH allows to obtain suchEPs without building complex networks of coupled cav-ities, compared to the previous works [53–56]. Instead,one can just excite additional modes in a multimode cav-ity, ensuring that the system modes interact incoherently.Moreover, according to the previous section, the incoher-ent mode coupling strengths in such systems are commen-surate and do not require fine tuning as, e.g., in Ref. [53],where coupling constants have incommensurate irrationalprefactors, due to the expansion of the Hilbert space ofan effective NHH. This conclusion is also valid when theintermode interaction is coherent.The method proposed here allows to constructmoments-based NHHs which can be both symmetric0and asymmetric. Such asymmetrical mode coupling canbe engineered via backscattering processes, as has beenshown in Refs. [71, 72, 93]. Moreover, exploiting suchasymmetric incoherent mode interactions, one can alsoimplement a recently proposed scheme [56], which isbased on the chiral nature of coherently coupled cavi-ties, but rather using a much simpler single multimode cavity. Using our approach, one can also engineer blocktriangular moments-based NHHs highlighting the chiral-ity in the mode coupling [56]. The latter can be obtainedby combining Eq. (8) and Schur’s triangularization the-orem [94].Apart from the interest in exploiting higher-orderquantum Liouvillian EPs for constructing new moments-based NHHs, it also prompts the question of a furtherutility of such EPs, which can be revealed by coherenceand spectral functions [63]. After all, it is exactly theLiouvillian eigenspace which correctly captures the dy-namics of the fields, and therefore the moments-basedNHH derived from it can be seem more physically justi-fied.The Liouvillian eigenspace genuinely incorporates theeffect of quantum jumps, the averaged effect of whichis captured in the dynamics of the field moments ex-pressed via corresponding evolution matrices. As such,the mapping between the evolution matrices, governingthe moments of the system operators, and moments-based NHHs ensures that the latter has a clear physicalmeaning also in the quantum limit. Note that the pres-ence of quantum jumps can profoundly affect the system dynamical spectra, e.g., in some finite dimensional sys-tems, the jumps can even lead to a shift of an EP in theparameter space [60].The proposed method is universal in its nature. Ac-cording to Eqs. (8) and (13), it can be applied to anyphysically realizable matrices M α and M β , each exhibit- ing an EP of some order, and which can be related toarbitrary quadratic Liouvillians. For instance, matrices M α and M β can represent a Z -symmetric quadraticLiouvillian which describes a nonlinear parametric dissi-pative process.We note that with the help of the described here tech-nique, one can also analyze spectral degeneracies in sys-tems with a weak nonlinearity, e.g., with Kerr-like inter-actions between modes. In that case, one can still invokethe Gaussian approximation for the fields fluctuationsnear the steady state, where the fields intensities are as-sumed to be fixed. As such, one can analytically deriveand experimentally probe the system dynamical criticalpoints by means of higher-order coherence and spectralfunctions of the fields fluctuations even in the presenceof weak nonlinearities. ACKNOWLEDGMENTS
I.A. thanks the Grant Agency of the Czech Re-public (Project No. 18-08874S), and Project No.CZ.02.1.010.00.016_0190000754 of the Ministry of Ed-ucation, Youth and Sports of the Czech Republic.A.M. is supported by the Polish National ScienceCentre (NCN) under the Maestro Grant No. DEC-2019/34/A/ST2/00081. F.N. is supported in part by:NTT Research, Army Research Office (ARO) (GrantNo. W911NF-18-1-0358), Japan Science and Technol-ogy Agency (JST) (via the CREST Grant No. JP-MJCR1676), Japan Society for the Promotion of Sci-ence (JSPS) (via the KAKENHI Grant No. JP20H00134and the JSPS-RFBR Grant No. JPJSBP120194828), theAsian Office of Aerospace Research and Development(AOARD) (via Grant No. FA2386-20-1-4069), and theFoundational Questions Institute Fund (FQXi) via GrantNo. FQXi-IAF19-06. [1] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,”Nat. Mater. , 783 (2019).[2] M. Miri and A. 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