Double degeneracy associated with hidden symmetries in the asymmetric two-photon Rabi model
aa r X i v : . [ qu a n t - ph ] M a r Double degeneracy associated with hidden symmetries in the asymmetric two-photonRabi model
You-Fei Xie and Qing-Hu Chen ∗ Zhejiang Province Key Laboratory of Quantum Technology and Device,Department of Physics, Zhejiang University, Hangzhou 310027, China (Dated: March 3, 2021)In this paper, we uncover the elusive level crossings in a subspace of the asymmetric two-photonquantum Rabi model (tpQRM) when the bias parameter of qubit is an even multiple of the renor-malized cavity frequency. Due to the absence of any explicit symmetry in the subspace, this doubledegeneracy implies the existence of the hidden symmetry. The non-degenerate exceptional pointsare also given completely. It is found that the number of the doubly degenerate crossing points inthe asymmetric tpQRM is comparable to that in asymmetric one-photon QRM in terms of the sameorder of the constrained conditions. The bias parameter required for occurrence of level crossingsin the asymmetric tpQRM is characteristically different from that at a multiple of the cavity fre-quency in the asymmetric one-photon QRM, suggesting the different hidden symmetries in the twoasymmetric QRMs.
PACS numbers: 03.65.Yz, 03.65.Ud, 71.27.+a, 71.38.k
I. INTRODUCTION
The simplest interaction between a two-level system(qubit) and a single mode bosonic cavity (oscillator) wasdescribed by the quantum Rabi model (QRM) [1, 2],which is thus a fundamental textbook model in quantumoptics [3]. It has been demonstrated in many advancedsolid devices, such as circuit quantum electrodynamics(QED) system [4, 5], trapped ions [6], and quantum dots[7] from weak coupling to the ultra-strong coupling, evendeep strong coupling between the artificial atom and res-onators [8–10].In contrast to the conventional cavity QED system,the artificial qubit appears in modern solid devices usu-ally contains both the splitting ∆ and the bias ǫ betweenthe two qubit states, thus the so-called asymmetric QRMis ubiquitous. Driven by the proposals and experimentalrealizations of the various QRMs model, the asymmetrictwo-photon QRM (tpQRM) are also realized or stimu-lated to explore new quantum effects [11–13]. The typi-cal two asymmetric QRMs can be generally written in aunified way as H ǫp = ∆2 σ z + ǫ σ x + ωa † a + g h(cid:0) a † (cid:1) p + a p i σ x , (1)where the first two terms fully describe a qubit with theenergy splitting ∆ and the bias ǫ , σ x,z are the Pauli ma-trices, a † and a are the creation and annihilation opera-tors with the cavity frequency ω , and g is the qubit-cavitycoupling strength. p = 1 , ǫ = 2 I p (Φ − Φ /
2) with I p the persistent current in the qubit loop, Φ the an exter-nally applied magnetic flux, and Φ the flux quantum.The flux qubit is usually manipulated by the externalmagnetic flux and the persistent currents.For the symmetric case ( ǫ = 0), the one-photon QRM possesses Z -symmetry (parity), i.e. [ H , ˆ P ] = 0 withthe parity operator ˆ P = σ z exp( iπa † a ) , whose eigen-values are ±
1, while the tpQRM has Z -symmetry,i.e. [ H , ˆ P ] = 0 with the parity operator ˆ P = σ z exp( iπa † a/ ± , ± i [14, 15]. Hence the whole Hilbert spaceseparates therefore into two and four infinite-dimensionalsubspaces in one-photon QRM and the tpQRM, respec-tively.An analytical exact solution of the one-photon QRMhas been found by Braak in the Bargmann space rep-resentation [16]. It was quickly reproduced in the morefamiliar Hilbert space using the Bogoliubov operator ap-proach (BOA) by Chen et al. [17]. Moreover, the BOAcan be easily extended to the tpQRM, and solutions interms of a G-function, which shares the common polestructure with Braak’s G-function for the one-photonQRM, are also found. It was soon realized that the G-function can be constructed in terms of the mathemat-ically well-defined Heun confluent function [18]. Thesestudies have stimulated extensive interests in variousQRMs [19–27]. For more theoretical details in this field,one may refer to recent review articles [28–30].The presence of the qubit bias term ǫ σ x breaks Z -symmetry of the QRM, so no any obvious symmetry re-mains in the asymmetric QRM [18, 24, 31], while in thetpQRM, it reduces the original Z -symmetry to the Z -symmetry. In the asymmetric tpQRM, the Z -symmetrycorresponding to the parity operator ˆ P p = exp( iπa † a )only acts in the bosonic Hilbert spaces, the whole Hilbertspace then only divides into two invariant subspaces:even and odd number Fock states, which can be still la-beled by the Bargmann index q = 1 / / G -function inboth the one-photon [16] and the two-photon [17] QRMs.Surprisingly, the level crossing even exists without Z -symmetry in the asymmetric one-photon QRM, when ǫ is a multiple of the cavity frequency ω [18]. In thesespecial cases, the hidden symmetry beyond any knownsymmetry is recently discussed based on the numericalcalculation on the energy eigenstates [34] and conservedoperators [35, 36].For the symmetric tpQRM, the standard Juddian solu-tions are level crossings within the same q subspace. Thesecond type of level crossings of the eigenstates in differ-ent q subspaces [32] was also found recently [37, 38]. Inthe asymmetric tpQRM, since Z -symmetry reduces to Z -symmetry, the level crossings within the same q sub-space would generally disappear, while the second typeof the level crossings in the different q subspaces remainsrobust due to the remaining Z -symmetry. Contrary tothe one-photon asymmetric QRM [18], the level crossingwithin the same q subspace in the asymmetric tpQRM iselusive, and has not been observed to date. In this work,we will uncover such a kind of level crossings irrelevantto any explicit symmetry.The paper is structured as follows: In Sec. II, webriefly review the solutions to the asymmetric one-photonQRM in the framework of BOA approach, and corrobo-rate the previous observed doubly degenerate states inBOA frame. We extend the BOA to study the asymmet-ric tpQRM, and derive the analytical exact solutions inSec. III. In Sec. IV, we discuss the non-degenerate excep-tional solutions for the asymmetric tpQRM. We demon-strate the level crossings within the same q subspace ofthe asymmetric tpQRM in Sec. V. The characteristics oflevel crossings in the two asymmetric QRMs is discussedin Sec. VI. The last section contains some concludingremarks. Appendix A confirms the conjecture that thetwo vanishing coefficients give the same solutions in bothasymmetric QRMs both analytically in the low order ofand numerically in the large order of the constrained con-ditions for the level crossings. II. ASYMMETRIC QUANTUM RABI MODELIN BOA
For the asymmetric one-photon QRM, when the biasparameter ǫ is a multiple of the cavity frequency, the level crossings appear again in the spectra even withoutany explicit known symmetry in the system [18, 24]. Itshould be noted that here ǫ in accord with the standardqubit Hamiltonian [8, 9, 34, 39] is twice of that used in[16, 18, 24].In this section, we revisit the asymmetric one-photonQRM by BOA. We first briefly review the solutions inthe BOA framework [17], then we can describe the levelcrossings in the BOA alternatively, which is essentiallyequivalent to the Bargmann space approach. Further-more, by BOA, we can obtain all the non-degenerateexceptional points in a more concise and complete way.Most importantly, this scheme can be easily extended tothe asymmetric tpQRM in the next sections. A. Solutions in BOA
By two Bogoliubov transformations A = a + g/ω, B = a − g/ω, (2)the wavefunction can be expressed as the series expan-sions in terms of A operator | A i = (cid:18) P ∞ n =0 √ n ! e n | n i A P ∞ n =0 √ n ! f n | n i A (cid:19) , (3)where e n and f n are the expansion coefficients, and alsoin terms of B operator | B i = (cid:18) P ∞ n =0 ( − n √ n ! c n | n i B P ∞ n =0 ( − n √ n ! d n | n i B (cid:19) , (4)with two coefficients c n and d n . | n i A and | n i B are calledextended coherent states [40].By the Schr .. o dinger equation, we get the linear relationfor two coefficients e m and f m with the same index m as [17] e m = ∆2 (cid:0) mω − g /ω + ǫ − E (cid:1) f m , (5)and the coefficient f m can be defined recursively,( m + 1) f m +1 = 12 g mω + 3 g /ω − ǫ − E − ∆ (cid:0) mω − g /ω + ǫ − E (cid:1) ! f m − f m − , (6)with f = 1. Similarly, the two coefficients c m and d m satisfy d m = ∆2 (cid:0) mω − g /ω − ǫ − E (cid:1) c m , (7)and the recursive relation is given by( m + 1) c m +1 = 12 g mω + 3 g /ω + ǫ − E − ∆ (cid:0) mω − g /ω − ǫ − E (cid:1) ! c m − c m − , (8)with c = 1 . If both wavefunctions (3) and (4) are the true eigen-function for a non-degenerate eigenstate with eigenvalue E , they should be in principle only different by a com-plex constant z , i.e. | A i = z | B i . Projecting both sidesonto the original vacuum state | i , using √ n ! h | n i A =( − n √ n ! h | n i B = e − ( gω ) / (cid:0) gω (cid:1) n and eliminating theratio constant z gives ∞ X n =0 e n (cid:16) gω (cid:17) n ∞ X n =0 d n (cid:16) gω (cid:17) n = ∞ X n =0 f n (cid:16) gω (cid:17) n ∞ X n =0 c n (cid:16) gω (cid:17) n , (9)with the help of Eqs. (5) and (7), one arrives at one-photon G-function G p = (cid:18) ∆2 (cid:19) " ∞ X n =0 f n nω − g /ω + ǫ − E (cid:16) gω (cid:17) n × " ∞ X n =0 c n nω − g /ω − ǫ − E (cid:16) gω (cid:17) n − ∞ X n =0 f n (cid:16) gω (cid:17) n ∞ X n =0 c n (cid:16) gω (cid:17) n . (10)This G-function was first derived by Braak [16] usingBargmann space approach, and later reproduced by Chenet al. [17]. We then discuss the level crossing of thisasymmetric QRM in terms of BOA framework describedabove. B. Doubly degenerate states
The two types of pole energies appear in the one-photon G-function (10) as E AN = N ω − g /ω + ǫ , N = 0 , , , ... (11) E BM = M ω − g /ω − ǫ , M = 0 , , , ... (12)They are labeled with the type-A and type-B pole energy,respectively. If ǫ = ( M − N ) ω, (13)these two pole energies are the same E AN ( g ) = E BM ( g ) = 12 ( M + N ) ω − g /ω. (14) Note that ǫ should be a multiple of the cavity frequency ω under the condition (13). In this paper, we only consider M > N , so that ǫ is positive. For the case of M < N ,the extension is achieved straightforwardly by changing ǫ into − ǫ and interchanging M and N .From Eqs. (5) [(7)], one immediately notes that thecoefficient e N ( d M ) would diverge at the same pole en-ergy (14). It does not make sense if some coefficientsin the series expansion of a wavefunction really becomeinfinity. A normalizable wavefunction should consist ofthe global property, i.e. the finite inner product, so theseries expansion coefficients in the wavefunction (3) and(4) should be analytic and vanish as or before n → ∞ .To achieve a physics state, at the pole energy (14), thenumerator of right-hand-side of Eq. (5) [(7)] should alsovanish, so that e N ( d M ) remains finite, which result in f N ( M, g ) = 0; c M ( N, g ) = 0 . (15)Note that f N and c M can be obtained using the follow-ing three-terms recurrence relation from (6) and (8) withenergy (14), respectively( n + 1) f n +1 = 12 g (cid:20) g /ω + ( n − M ) ω − ∆ ω ( n − N ) (cid:21) f n − f n − , (16)( n + 1) c n +1 = 12 g (cid:20) g /ω + ( n − N ) ω − ∆ ω ( n − M ) (cid:21) c n − c n − . (17)If ∆ , M, N are given, two equations in (15) would providethe coupling strength g in the energy spectra where theenergy levels intersect with the same pole line describedby Eq. (14).A mathematical proof to the conjecture that f N ( M, g ) = 0 and c M ( N, g ) = 0 could give the same realand positive solutions for the coupling strength g wasgiven in [31]. Li and Batchelor [24] have analyzed therelation between the number of the exceptional pointsand the model parameters (∆ and ǫ ), and numericallyfound that the number of positive roots from these twoequations are the same for integer ǫ/ω . But we confineus here to a closed-form proof for small values of N and M , and numerically confirmation for large N and M .In the asymmetric tpQRM, similar constrained condi-tion will be derived and we will also do the similar things f N ( c M ) N = 1 ,M = 2 f N ( c M ) N = 2 ,M = 3 f N ( c M ) N = 3 ,M = 4 f N ( c M ) N = 4 ,M = 5 E + g + ǫ / ( a ) ∆ = 1 . , ǫ = 1 g -1012345 E + g + ǫ / ( b ) ∆ = 1 . , ǫ = 2 f N ( c M ) N = 3 ,M = 5 f N ( c M ) N = 2 ,M = 4 g -505 f N ( c M ) N = 1 ,M = 3 FIG. 1: (Color online) Energy spectrum E + g + ǫ/ ω = 1 , ∆ = 1 . , ǫ = 1 (a) and ǫ = 2 (b) in left panels. Thehorizontal blue dotted lines correspond to the pole energy onesfor E AN and the red dashed lines to E BM . Only the overlappedpole lines with N > f N ( M, g ) (blue) and c M ( N, g ) (red) curves are shown inthe right panels. The zeros are exactly corresponding to thetriangles in the left spectrum. because a similar conjecture will be proposed but cannotbe proven at the present stage.To this end, we present our discussions only in terms ofthe fixed integers N and M . In this case, ǫ is a multipleof the cavity frequency is known immediately, and theremaining task is to show the same crossing points by twoequations in Eq. (15), which is illustrated in AppendixA1.In the left panels of Figs. 1 and 2, we present theenergy spectrum E + g /ω + ǫ for ǫ = 1, 2 at ∆ =1 . ω = 1, respectively. The dottedand dashed horizontal lines denote different types of polelines. Obviously, if the two types of pole lines cannotcoincide, the level crossings cannot happen. f N ( M, g )and c M ( N, g ) curves are plotted in the right panels. ByEq. (A1), one finds g = 0 . N = 1 , M = 2 at∆ = 1 .
5, consistent with the spectra in Fig. 1 (a). For∆ = 3, see Eq. (A2), no real positive solution can befound in this case, so level crossings cannot occur in theoverlapped line with N = 1 in Fig. 2 (a).At ∆ = 1 .
5, we can obtain two solutions for g as 0 . . N = 2 and M = 3 by Eq. (A4), agreeing wellwith two crossing points in the second type-A pole linewith N = 2 and M = 3 shown in Fig. 1 (a). For ∆ = 3 , we only find one real positive g = 0 . N = 1 type-A and M = 3 pole lines, one can obtain g = 0 . . g = 0 . M and N , as shown in the right g -1012345 E + g + ǫ / ( b ) ∆ = 3 , ǫ = 2 g -505 f N ( c M ) N = 1 ,M = 3 f N ( c M ) N = 2 ,M = 4 f N ( c M ) N = 3 ,M = 5 E + g + ǫ / ( a ) ∆ = 3 , ǫ = 1 f N ( c M ) N = 4 ,M = 5 f N ( c M ) N = 3 ,M = 4 f N ( c M ) N = 2 ,M = 3 FIG. 2: (Color online) Notations are the same as those inFig. 1 except for ∆ = 3. Note that the doubly degeneratecrossing point is absent in the N = 1 , M = 2 overlapped linein (a) due to ∆ > √ panels of Figs. 1 and 2, both f N ( M, g ) and c M ( N, g )curves provide the same zeros for all cases.Now we will further demonstrate explicitly that anycrossing point found above is corresponding to a doublydegenerate state in the BOA framework. At the crossingpoint, looking at (5), since both the numerator f N ( g ) anddenominator vanish, e N would be arbitrary. If we set e N = − g ∆ f N − , (18)from Eq. (6) we know f N +1 = 0, further e N +1 = 0, andall coefficients f k and e k for k > N + 1 vanish. So the in-finite series expansion in the wavefunction (4) terminateswith finite N as | A i N = (cid:18) P Nn =0 √ n ! e n | n i A P N − n =0 √ n ! f n | n i A (cid:19) . (19)Similarly, the infinite series expansion in the wavefunc-tion (4) terminates with finite M as | B i M = (cid:18) P M − n =0 ( − n √ n ! c n | n i B P Mn =0 ( − n √ n ! d n | n i B (cid:19) , (20)where d M = − g ∆ c M − . Interestingly, both wavefunction terminates at finiteterms. Because these two wavefuntions are not obtainedfrom the G-function based on the proportionality (9), sothey are different | A i N = | B i M , leading to doubly degen-erate states. Since the degenerate eigenfunctions, | A i N and | B i M , are given as finite polynomials in the extendedcoherent state basis {| n i A } and {| n i B } (see also [41]).These states are the quasi-exact solutions of the asym-metric QRM.At this stage, we can simply discuss the number ofthe doubly degenerate crossing points associated with thegiven N type-A pole line. f N ( M > N, x = 4 g ) derivedby Eq. (16) is a polynomial with N terms. Its zero wouldgenerally give around N roots, indicating that there arearound N doubly degenerate crossing points along the N type-A pole line in the energy spectra. Note that for large∆, the number of the roots could be slightly less than N ,as shown in Fig. 2. For small ∆, we can actually havejust N roots. C. Non-degenerate exceptional points
The non-degenerate exceptional points can be gener-ated if only one energy level intersects with the energypole line alone. In principle, all non-degenerate states in-cluding non-degenerate exceptional ones can be obtainedby the G-function (10) because it is built based on theproportionality (9), only excluding the degenerate states.These states have been first analyzed for the symmetricQRM with the Bargmann space technique in [19] andlater in [41–43]. We believe that the BOA has advantageswith regard to the non-degenerate exceptional solutions,which cannot be found with any ansatz.Note from G-function (10) that, at the pole energy ei-ther (11) or (12), the denominator of the associated termbecome zero, so this term would diverge and should betreated specially. For a physics state, to avoid the diver-gence, the numerator f N ( g ) or c M ( g ) should also vanish.It is very important to see that f N ( g ) or c M ( g ) couldvanish in two different ways. First, f n ( c m ) can be ob-tained by using the three-terms recurrence relation (6)[(8)] from f = 1 ( c = 1), and f N = 0 ( c M = 0) un-til n = N ( m = M ). Second, one can set f n N = 0( c m M = 0) and e n = N = 1 ( d m = M = 1) at the begin-ning directly and obtain all remaining coefficients by therecurrence relation (6)[(8)]. This is to say, we have twoways to overcome the divergence. In the infinite summa-tion where the diverging term is present, we may cut offall the terms either after or before this diverging one. E.g.for the N th type-A pole line, we may terminate the infi-nite summation at the diverging term following the sameidea outlined in the last section for the degenerate states.So the first non-degenerate exceptional G-function can bewritten as G non, A p = " N − X n =0 ∆ f n ω ( n − N ) (cid:16) gω (cid:17) n + e N (cid:16) gω (cid:17) N × " ∞ X n =0 ∆ c n nω − g /ω − ǫ ) (cid:16) gω (cid:17) n − N − X n =0 f n (cid:16) gω (cid:17) n ∞ X n =0 c n (cid:16) gω (cid:17) n = 0 , (21)where e N is given by Eq. (18). Note that the remainingterms vanish because all coefficients become zero. We can also remove all terms before the diverging term inthe summation, and give the second non-degenerate ex-ceptional G-function as G non, A p = "(cid:16) gω (cid:17) N + ∞ X n = N +1 ∆ f n ω ( n − N ) (cid:16) gω (cid:17) n × " ∞ X n =0 ∆ c n nω − g /ω − ǫ ) (cid:16) gω (cid:17) n − ∞ X n = N +1 f n (cid:16) gω (cid:17) n ∞ X n =0 c n (cid:16) gω (cid:17) n = 0 , (22)with the initial condition e N = 1. The non-degenerateexceptional G-functions G non, B p and G non, B p associatedwith the type-B pole line can be obtained similarly bymodifying the other infinite summation, which are notshown here.Two non-degenerate exceptional G-functions (21) and(22) provide different exceptional solutions, which com-prise the full non-degenerate exceptional points associ-ated with the Type-A pole lines. Particularly, f N = 0or c M = 0 is implied Eq. (21) or G non, B p = 0, thus canbe also used to give the same non-degenerate exceptionalpoints in a simpler way. Just as pointed out in Ref [24],for noninteger ǫ , a subset of the non-degenerate excep-tional points associated with the pole lines can be givenby the vanishing coefficients f m or c m , equivalently, us-ing Eq. (21) or G non, B p = 0 here. However Eq. (21) and G non, B p = 0 fail at integer ǫ including ǫ = 0, because f N = 0 or c M = 0 actually results in the doubly degen-erate states, which results in nonzero G-function in thiscase.Interestingly, for integer ǫ , two types of pole line maymerge together. At the same pole energy (14), the secondnon-degenerate exceptional G-function Eq. (22) wouldbe further modified as G non p = "(cid:16) gω (cid:17) N + ∞ X n = N +1 ∆ f n ω ( n − N ) (cid:16) gω (cid:17) n × "(cid:16) gω (cid:17) M + ∞ X n = M +1 ∆ c n ω ( n − M ) (cid:16) gω (cid:17) n − ∞ X n = N +1 f n (cid:16) gω (cid:17) n ∞ X n = M +1 c n (cid:16) gω (cid:17) n = 0 , (23)where e n For convenience, we rewrite the Hamiltonian H ǫ onthe σ z basis by rotating it around the y -axis with anangle π/ 2. The transformed Hamiltonian is given by thefollowing matrix form H ǫ ,r = (cid:18) ωa † a + g ( a † + a ) + ǫ − ∆2 − ∆2 ωa † a − g ( a † + a ) − ǫ (cid:19) . (24)The Hamiltonian above is connected with su (1 , 1) Liealgebra K = 12 ( a † a + 12 ) , K + = 12 a † , K − = 12 a , (25)which obey spin-like commutation relations [ K , K ± ] = ± K ± , [ K + , K − ] = − K . The quadratic invariantCasimir operator is given by C = K + K − + K (1 − K ) . Then we apply a squeezing operator S = e r ( a − a † ) todiagonalize the bosonic part of the above Hamiltonianand the parameter r is to be fixed later. In terms of the K , K ± , the transformed Hamiltonian is derived as H ǫ ,r = (cid:18) β (2 K ) + ǫ − ω − ∆2 − ∆2 H (cid:19) , (26)where β = ω q − (cid:0) gω (cid:1) < ω can be termed as therenormalized cavity frequency owing to the fact that it is just a g-dependent pre-factor of the free photon numberoperators 2 K , and β = ω if g = 0. It will be shown laterthat β plays a key role in two-photon QRM. The seconddiagonal element is H = (2 ω cosh 2 r − g sinh 2 r ) K +( ω sinh 2 r − g cosh 2 r )( K + + K − ) − ǫ + ω , and the squeezing parameter r = 14 ln (cid:18) − g/ω g/ω (cid:19) . (27)It is obvious that the coupling strength g < ω/ | Ψ A i q = (cid:18)P ∞ m =0 q [2( m + q − )]! e ( q ) m | q, m i A P ∞ m =0 q [2( m + q − )]! f ( q ) m | q, m i A (cid:19) , (28)where the new basis | q, m i A = S A | q, m i with | q, m i isthe Fock state. The coefficients e ( q ) m and f ( q ) m are to bedetermined in the following.In the case of the Lie algebra considered here, K | q, i A = q | q, i A where q = and dividethe whole Hilbert space H into even and odd sectorsand label them, respectively. For the even subspace, H / = (cid:8) a † n | i , n = 0 , , , ... (cid:9) , and for the odd sub-space, H / = (cid:8) a † n | i , n = 1 , , , ... (cid:9) , corresponding toeven or odd Fock number basis. The Casimir element C = in both cases. The Bargmann index q allows usto deal with both cases independently.The su (1 , 1) Lie algebra operators satisfy K | q, n i A = ( n + q ) | q, n i A ,K + | q, n i A = r ( n + q + 34 )( n + q + 14 ) | q, n + 1 i A ,K − | q, n i A = r ( n + q − 34 )( n + q − 14 ) | q, n − i A . Projecting both sides of the Schr¨ o dinger equation onto | q, n i A gives a linear relation between coefficients e ( q ) n and f ( q ) n , e ( q ) n = ∆ / β ( n + q ) − E + ǫ − ω f ( q ) n , (29)and a three-term linear recurrence relation is given by f ( q ) n +1 = 2(2 ω − β )( n + q ) − β (cid:0) E + ǫ + ω (cid:1) − ∆ β/ β ( n + q ) − E + ǫ − ω gω ( n + q + )( n + q + ) f ( q ) n − n + q + )( n + q + ) f ( q ) n − . (30)All coefficients f ( q ) n and e ( q ) n can be calculated with initialconditions f ( q ) − = 0 and f ( q )0 = 1 . We then apply the second squeezing operator S B = e − r ( a − a † ) to the Hamiltonian (24) and suggest thewavefunction as | Ψ B i q = (cid:18)P ∞ m =0 ( − m q [2( m + q − )]! c ( q ) m | q, m i B P ∞ m =0 ( − m q [2( m + q − )]! d ( q ) m | q, m i B (cid:19) , (31) where | q, m i B = S B | q, m i . Similarly, we can obtain alinear relation between the coefficients c ( q ) n and d ( q ) n d ( q ) n = ∆ / β ( n + q ) − E − ǫ + ω c ( q ) n , (32)and the three-term linear recurrence relation is c ( q ) n +1 = 2(2 ω − β )( n + q ) − β ( E − ǫ − ω ) − ∆ β/ β ( n + q ) − E − ǫ + ω gω ( n + q + )( n + q + ) c ( q ) n − n + q + )( n + q + ) c ( q ) n − . (33)Left-multiplying the vacuum state h q, | to the extendedsqueezed state | q, m i A and | q, m i B , we can obtain theinner product h q, | | q, m i A = ( − tanh r ) m √ cosh r q [2( m + q − )]!2 m m ! , h q, | | q, m i B = (tanh r ) m √ cosh r q [2( m + q − )]!2 m m ! . (34)If both wavefunction | Ψ A i q and | Ψ B i q for the same q are the true eigenfunction for a non-degenerate eigen-state with eigenvalue E , they should be proportional witheach other, i.e. | Ψ A i q = z | Ψ B i q , where z is a complexconstant. Projecting both sides of this identity onto theoriginal vacuum state | q, i , we obtain a transcendentalfunction below defined as G-function G ( q ) = (cid:18) ∆2 (cid:19) " ∞ X m =0 f ( q ) m Ω ( q ) m β ( m + q ) + ǫ − ω − E × " ∞ X m =0 c ( q ) m Ω ( q ) m β ( m + q ) − ǫ + ω − E − ∞ X m =0 f ( q ) m Ω ( q ) m ∞ X m =0 c ( q ) m Ω ( q ) m , (35)with Ω ( q ) m = ( − tanh r ) m √ cosh r [2( m + q − )]!2 m m ! . If set ǫ = 0, the G-function for the symmetric tpQRM[17] is recovered. The zeros of the G ( q ) -function give theregular spectrum in the q subspace of the asymmetrictpQRM.From Eqs. (29) and (32), we find the G-function di-verges when its denominators vanishes, the condition ofthe denominators being zero can be obtained as E Am = 2 β ( m + q ) + ǫ − ω , (36)and E Bm = 2 β ( m + q ) − ǫ + ω , (37)with m = 0 , , , ... . They are also labeled as two types(A and B) pole energies, similar to the asymmetric QRM.G-curves at q = 1 / / ǫ = 0 . , g = 0 . , and∆ = 3 with ω = 1 are plotted in Fig. 3. The zeros areeasily detected. As usual, one can check it easily withnumerics, an excellent agreement can be achieved. Thepoles given in Eqs. (36) and (37) are marked with verticallines. The G-curves indeed show diverging behavior whenapproaching the poles.In the limit of g → ω/ 2, the type-A pole energies aresqueezed into a single finite value E Am ( g = ω/ 2) = ǫ − ω ,and type-B pole energies into E Bm ( g = ω/ 2) = − ǫ + ω . Itseems that there are two kinds of collapse energies − ω ± ǫ .But actually, at g = ω/ 2, our obtained energy levels tendto the smaller one − ω −| ǫ | , except some low lying stateswhich split off from the continuum. The ground-state -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 E -10010 G ( / ) ( b ) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-303 G ( / ) ∆ = 3 , ǫ = 0 . , g = 0 . -2-1.5-1-0.500.511.522.533.54-505 G ( / ) E Am E Bm ED ( a ) FIG. 3: (Color online) G -curves for ω = 1 , ∆ = 3 , ǫ =0 . , g = 0 . q = 1 / q = 3 / E Am in Eq. (36) and red dashed lines are E Bm in Eq. (37), m = 0 , , ... . The data by numerical diagonal-izations are indicated by black dots, which agree excellentlywith the zeros of the G-functions (35). is always separated from the continuum by a finite ex-citation gap. In the spectrum shown in the next sec-tions, we will indeed observe that the plotted energy lev-els always collapse to the smaller one at g = ω/ 2, whichgenerate more non-degenerate exceptional points when g −→ ω/ 2. However, we cannot rule out the possibilitythat some energy levels would stay between two limit en-ergies − ω −| ǫ | and − ω + | ǫ | , and thus these levels could notcollapse. Since the analytical solution at g = ω/ IV. NON-DEGENERATE EXCEPTIONALSOLUTIONS IN THE ASYMMETRIC TPQRM As outlined in the Sec. II (c) for the asymmetricone-photon QRM, we can easily find the non-degenerateexceptional solutions in the spectra for the asymmetrictpQRM by the pole structures of the G-function. Whenthe energy levels cross the pole lines, the coefficientsin the G-function would diverge, and therefore shouldbe treated specially. For any real physical systems, thewavefunction should be analytic, so the numerators f ( q ) m in Eq. (29) or c ( q ) m Eq. (32) should also vanish, which fur-ther gives the condition for the model parameters g, ∆ , ǫ ,for fixed value of m associated with one pole line.In parallel to the asymmetric one-photon QRM, thefirst non-degenerate exceptional G-function associatedwith the N-th type-A pole lines (36) for the asymmet- ric tpQRM is easily given by G ( q ) non, A = " N − X n =0 ∆ f ( q ) n Ω ( q ) n / β ( n − N ) + e N Ω ( q ) N × " ∞ X n =0 ∆ c ( q ) n Ω ( q ) n / β ( n − N ) − ǫ − N − X n =0 f ( q ) n Ω ( q ) n ∞ X n =0 c ( q ) n Ω ( q ) n = 0 , (38)where e ( q ) N = − gω ∆ β f ( q ) N − , (39)and that associated with the M-th type-B pole lines (37)reads G ( q ) non, B = " ∞ X n =0 ∆ f ( q ) n Ω ( q ) n / β ( n − M ) + ǫ × " M − X n =0 ∆ c ( q ) n Ω ( q ) n / β ( n − M ) + d M Ω ( q ) M − ∞ X n =0 f ( q ) n Ω ( q ) n M − X n =0 c ( q ) n Ω ( q ) n = 0 , (40)where d ( q ) M = − gω ∆ β c ( q ) M − . (41)Note that by Eq. (39) [Eq. (41)], all the remaining co-efficients for n > N [ n > M ] vanish. Zeros of the firstnon-degenerate exceptional G-functions are equivalent to f ( q ) N = 0 or c ( q ) M = 0. Obviously, the later ones are obvi-ously simpler in practical calculations, while the formerones are more conceptually interesting, both can give thesame solutions.Similarly, the second non-degenerate exceptional G-function associated with the type-A pole lines (36) is G ( q ) non, A = " Ω ( q ) N + ∞ X n = N +1 ∆ f ( q ) n Ω ( q ) n / β ( n − N ) × " ∞ X n =0 ∆ c ( q ) n Ω ( q ) n / β ( n − N ) − ǫ − ∞ X n = N +1 f ( q ) n Ω ( q ) n ∞ X n =0 c ( q ) n Ω ( q ) n = 0 , (42)where we have set e ( q ) N = 1, and the coefficients e ( q ) n 4. The bluedashed lines are E Am =0 , by Eq. (36) and red dashed lines are E Bm =0 , , by Eq. (37). The inset on an enlarged scale showsthe avoided crossing instead of the true level crossing. f (1 / N =1 , c (1 / M =1 , curves are exhibited in (b), whose zeros are indicatedby open triangles, agreeing with the same symbols in the spec-tra (a). The non-degenerate exceptional G-function G (1 / non, B in Eq. (43) for M = 2, and G (1 / non, A in Eq. (42) for N = 1 , the type-B pole lines (37) can be obtained in a straight-forward way as G ( q ) non, B = " ∞ X n =0 ∆ f ( q ) n Ω ( q ) n / β ( n − M ) + ǫ × " Ω ( q ) M + ∞ X n = M +1 ∆ c ( q ) n Ω ( q ) n / β ( n − M ) − ∞ X n =0 f ( q ) n Ω ( q ) n ∞ X n = M +1 c ( q ) n Ω ( q ) n = 0 , (43)where d ( q ) M = 1, and the coefficients d ( q ) n In the asymmetric tpQRM, can we also find level cross-ings in the same q subspace? According to the pole en-ergies (36) and (37), if E AM = E BN , then ǫ = 2 β ( M − N ) , (44)the same pole energy takes E = ( M + N + 2 q ) β − ω. (45)Interestingly, Eq. (44) entails ǫ to be an even multipleof the renormalized cavity frequency β , in contrast tothe asymmetric one-photon QRM where ǫ should be anmultiple of the cavity frequency ω under the condition(13) for level crossings. It makes sense that only thetwo-photon process is involved in the two-photon model,while the single photon process in the one-photon model.Without loss of generality, we also only consider M >N here. From Eq. (30) [(33)], one immediately note thatthe coefficient e ( q ) N in (29) ( d ( q ) M in (32)) would diverge atthe same pole energy (45). Similar to the asymmetricQRM case, the series expansion coefficients in the wave-function (28) and (31) should be analytic and vanish asor before n → ∞ .Regarding states with the energy (45), the numeratorof right-hand-side of (30) [(32)] should also vanish, sothat e ( q ) N ( d ( q ) M ) remains finite, which requires f ( q ) N ( M, g ) = 0; c ( q ) M ( N, g ) = 0 . (46)Note that f ( q ) N and c ( q ) M can be respectively obtained fromthe recurrence relations (30) and (33) by using the samepole energy (45) f ( q ) n +1 = 2 ω ( n + q ) − β ( n + M + 2 q ) + ∆ N − n ) gω ( n + q + )( n + q + ) f ( q ) n − n + q + )( n + q + ) f ( q ) n − , (47)0 c ( q ) n +1 = 2 ω ( n + q ) − β ( n + N + 2 q ) + ∆ M − n ) gω ( n + q + )( n + q + ) c ( q ) n − n + q + )( n + q + ) c ( q ) n − . (48) f q N ( c q M ) ( a ) q = 1 / N = 1 , M = 2 ( b ) q = 1 / g -20246 E ǫ = 1 . g -20246 ǫ = 1 . ( c ) q = 3 / g -20246 ǫ = 2 . N = 1 , M = 3 N = 1 , M = 3 FIG. 5: (Color online) Energy spectrum are list in low pan-els. The black lines are energy levels, the blue dashed linesare E Am and the red dashed lines are E Bm . Open triangles indi-cate the doubly degenerate level crossings. f ( q ) N =1 ( M, g ) (blue)and c ( q ) M ( N = 1 , g ) (red) curves are displayed in the upperpanels. ( q, ǫ ) = (1 / , . / , . / , . ω = 1. Similar to the asymmetric one-photon QRM, we con-jecture that both f ( q ) N ( M, g ) = 0 and c ( q ) M ( N, g ) = 0 couldgive the same positive real g and ǫ under the constrainedcondition (44), leading to levels crossing at the same poleenergy. While it would be interesting to rigorously provethe conjecture in the two-photon case mathematically,we also confine us here to an analytical closed-form proofonly for small values of N and M , and numerically con-firmation for large N and M , in searching for physicallyreasonable coupling strength g . Similar to the asymmet-ric QRM, we also present our discussions only in termsof fixed values of N and M, but here ǫ cannot be deter-mined independently, and would be determined togetherwith g by Eqs. (44) and (46).In Appendix A3, we analytically prove that, for somesmall values of N and M , both f ( q ) N ( M, g ) = 0 and c ( q ) M ( N, g ) = 0 in (46) give the same values for ǫ and g .Two energy levels cross the corresponding pole lines atthe same values of ǫ and g , where the two pole lines alsocross. Thus true level crossings also happen in the asym-metric tpQRM. Compare to the one-photon QRM where ǫ can be determined independently, in the asymmetrictpQRM, we need to solve two equations simultaneouslyto determine ǫ and g .We show the energy spectrum of the asymmetrictpQRM at N = 1, ∆ = 2 with ω = 1, for q = 1 / , M = 2(left), q = 1 / , M = 3 (middle), and q = 3 / , M = 3(right) in the low panels of Fig. 5. The correspondingvalues of ǫ are just those determined by Eq. (A9), whichin turn are ǫ = 1 . , . . g -0.500.5 f q N ( c q M ) ( a ) q = 1 / g -202468 ( c ) ǫ = 0 . g -202468 E ( b ) ǫ = 1 . f / N =2 c / M =3 FIG. 6: (Color online) f (1 / N =2 and c (1 / M =3 curves are exhibitedin (a). Zeros are the same for both curves. Energy spectrumare presented in (b) for ǫ = 1 . ǫ = 0 . E Am , and the red dashed lines are E Bm . Open triangles indicate thedoubly degenerate level crossings. ∆ = 2 and ω = 1. right. Interestingly, one level crossing point indicated bythe open triangle really appears in each spectra, confirm-ing the analytical prediction.The upper panels in Fig. 5 present the curves for f ( q )1 ( M, g ) and c ( q ) M (1 , g ). It is clear that the zeros ofboth functions are the same, and are consistent with thecoupling strength at the level crossing points. For exam-ple, for q = 1 / 4, ∆ = 2, two energy levels cross exactlyat g = 0 . E AN =1 and the type-B pole lines E BM also crossat the degenerate points in the low panels of Fig. 5.For N = 1, no matter what is the value of M > N ,from Eq. (A7), we can at most find one solution for g, which is M dependent. For N > f ( q ) N ( M, g ) = 0 isa polynomial equation with N terms, which would givemore than one solutions for g , and further correspondingsolutions for ǫ in terms of Eq. (44).As shown in left panel of Fig. 6 for ∆ = 2 , ω =1 , q = 1 / , N = 2 and M = 3 , both f (1 / (3 , g ) = 0and c (1 / (2 , g ) = 0 yield the same solutions for g , =0 . , . ǫ , =1 . , . ǫ in themiddle and right panels of Fig. 6 for ∆ = 2. The levelcrossings are clearly shown at the analytical predictedcoupling strength. Note that the N = 2 type-A pole lineand the M = 3 type-B pole line indeed cross at the samedoubly degenerate points.Finally, the doubly degenerate states at the true levelcrossing points can be expressed explicitly in terms of the1BOA as | Ψ A i q = (cid:18)P Nm =0 q [2( m + q − )]! e ( q ) m | q, m i A P N − m =0 q [2( m + q − )]! f ( q ) m | q, m i A (cid:19) , and | Ψ B i q = (cid:18)P M − m =0 ( − m q [2( m + q − )]! c ( q ) m | q, m i B P Mm =0 ( − m q [2( m + q − )]! d ( q ) m | q, m i B (cid:19) , respectively, where e ( q ) N and d ( q ) M are given by Eqs (39) and(41). Because these two wavefuntions are not obtainedfrom the G-function based on the proportionality, so theyare different, leading to doubly degenerate states. Bothwavefunction terminates at finite terms, so they are thequasi-exact solutions of the asymmetric tpQRM. VI. DISCUSSIONS From the spectrum in Figs. 5 and 6, one might specu-late that level crossings seldom happen in the asymmetrictpQRM. Actually it is not that case. If we incorporateEq. (44) required by the level crossings, we may plotthe similar spectra graph as Figs. 1 and 2 in one-photoncase. In doing so, we calculate the energy as a functionof g , and at the same time ǫ also changes as Eq. (44). Todisplay the level crossings in asymmetric tpQRM moreclearly, we can make the pole lines horizontal, thus weplot the normalized energy E ′ = E + ω/ β − q + ǫ β as afunction of g and simultaneously varying ǫ = kβ in Fig.7 for k = 0 , , , ω = 1 , q = 1 / ǫ is an even multiple of the normalized cavityfrequency β entailed in Eq. (44), i.e. k is an even integerincluding the symmetric case k = 0, we find that the twoequations in Eq. (46) result in the same positive solu-tions for the coupling strength, as indicated with opentriangles (a), (c) and (d). One can note that the levelcrossings happen regularly. The crossing points at the N = 1 type-A pole line in (c) and (d) are just corre-sponding to those in Figs. 5 (a) and (b), while the twocrossing points at the N = 2 type-A pole line in (c) tothose in Fig. 6.However, if k is not an even integer, no level cross-ings happen, a lot of non-degenerate exceptional pointsemerges instead. As exhibited in Fig. 7 (b) for k = 1, theopen triangles correspond to the non-degenerate excep-tional points by Eqs. (38) or (40), while the open circlesto those by Eqs. (42) and (43).We can also estimate the number of the doubly degen-erate crossing points associated with the given N type-Apole line. For any M > N , generally there are around N crossing points due to the polynomial equation with N terms, f ( q ) N ( M, g ) = 0 in Eq. (46), the detailed poly-nomial equations is derived from Eq. (47). This is tosay, associated with N type-A pole line, we generally have g -1012345 E ′ ( d ) ǫ = 4 β g -1012345 E ′ ( c ) ǫ = 2 β g -1012345 E ′ ( b ) ǫ = β g -1012345 E ′ ( a ) ǫ = 0 E Am =0 , , ... E Bm =0 , , ... FIG. 7: Energy spectrum E ′ = E +1 / β − q + ǫ β for ω =1 , ∆ = 2 , q = 1 / , ǫ = kβ . k = 0 (a), 1 (b), 2 (c), and 4 (d).Note particularly that ǫ changes with g along g -axis. Thehorizontal blue dotted lines correspond to the pole energy onesfor E AN and the red dashed lines to E BM . Only the overlappedpole lines with N > f ( q ) N = 0and c ( q ) M = 0 or equivalently from Eqs. ( 38) and (40), whileopen circles from Eqs. ( 42) and (43), all of them correspondto non-degenerate exceptional points. around N degenerate crossing points for both asymmetricone-photon and two-photon QRMs. In Appendix A3, wehave employed the constrained conditions in the asym-metric both one- and two-photon QRMs, and numericallyfound that they have nearly the same numbers of levelcrossing points in the range of integers of N and M ineach case. Therefore, we could reach a conclusion thatthe number of the doubly degenerate crossing points inasymmetric tpQRM would be twice of that in the asym-metric QRM due to two Bargmann indices in the formermodel.Braak proposed a new criterion of integrability that ifthe eigenstates of a quantum system can be uniquely la-beled by i = i + i quantum numbers, where i and i are the numbers of the discrete and continuous degree offreedom, then it is integrable [16]. Both symmetric QRMand tpQRM may be considered integrable in terms of thiscriterion. As the bias term of qubit sets in, the integra-bility will be violated in the asymmetric QRM. However,if ǫ matches the multiple of the cavity frequency, the in-2tegrability can be recovered in the asymmetric QRM, byusing the hidden symmetry instead of the parity num-ber. As shown in Figs. 1 and 2, the regular level cross-ings reappear when ǫ/ω is an integer, similar to that inthe symmetric QRM which is considered to be integrable[16]. However, the asymmetric tpQRM with fixed ǫ is al-ways non-integrable because the energy levels cannot beuniquely labeled by the only continuous degree of free-dom. As displayed in the spectrum in Figs. 5 and 6 withspecial ǫ ’s, there is no regular level crossings, in sharpcontrast to the integrable symmetric tpQRM [14]. Ofcourse, if ǫ changes as kβ with k an even integer, the reg-ular level crossings reappear in the asymmetric tpQRMas shown in Fig. 7, and it can be reconsidered to beintegrable.In the asymmetric QRM, the effort to look for the hid-den symmetry responsible for the level crossings in thesame ǫ , continues to be a great interest [31, 39, 46, 47].Since the doubly degenerate states within the same q subspace also exist in the asymmetric tpQRM, which isdefinitely not owing to an explicit symmetry. It shouldbe also interesting to rigorously find hidden symmetry inthe asymmetric tpQRM in the near future. VII. CONCLUSION In this paper, we have studied both the asymmetricQRM and the asymmetric tpQRM by the BOA in a uni-fied way. The previously observed level crossing when thebias parameter ǫ is a multiple of cavity frequency in theasymmetric QRM is illustrated by a closed-from prooffor low orders of the constrained polynomial equations ina transparent manner. For the asymmetric tpQRM, thebiased term breaks original Z symmetry to Z symme-try, so the Hilbert space only divides into even and oddbosonic number state subspaces. In each subspace, wederived the transcendental equation, called G-function,and obtain the regular spectrum exactly. The coefficientsat the pole energy vanish in two different ways, givingtwo kinds of non-degenerate exceptional G-functions, bywhich all non-degenerate exceptional points can be de-tected.Very interestingly, the true level crossings can also hap-pen in the same q subspace in the asymmetric tpQRM ifthe qubit bias parameter ǫ is an even multiple of the g -dependent renormalized cavity frequency, in contrast tothe asymmetric one-photon QRM where ǫ can be simplya multiple of the cavity frequency. We argue that theeven multiple is originated from the two-photon processinvolved in the two-photon model. The doubly degener-ate points can be also located analytically, similar to theasymmetric QRM. The number of the doubly degener-ate points within the same subspace in the asymmetrictpQRM should be comparable with that in asymmetricQRM. The subspace in the asymmetric tpQRM has noany explicit symmetry, the newly found double degener-acy thus also implies the hidden symmetry. The hidden symmetry in the asymmetric QRM could be identified atthe same integer ǫ/ω , while in the asymmetric tpQRMat the same integer ǫ/ (2 β ). The latter constraint on theparameter space for the occurrence of the double degen-eracy is illuminating in searching for a conserved operatorin two-photon case. The present results may shed somelights on the different nature of the hidden symmetriesin the two asymmetric QRMs. ACKNOWLEDGEMENTS This work is sup-ported by the National Science Foundation of China un-der No. 11834005, the National Key Research and Devel-opment Program of China under No. 2017YFA0303002. ∗ Email:[email protected] Appendix A: Demonstration for the same physicalsolutions of the two equations in the constrainedconditions in two asymmetric QRMs In this Appendix, we first present a closed-form prooffor the conjecture that f N ( M, g ) = 0 and c M ( N, g ) = 0in Eq. (15) could give the same real and positive solu-tions for the coupling strength g with small numbers of N and M in the asymmetric one-photon QRM. In par-allel, we then provide a closed-form proof for the conjec-ture that f ( q ) N ( M, g ) = 0 and c ( q ) M ( N, g ) = 0 in Eq. (46)could give the same real and positive solutions for thecoupling strength g with small numbers of N and M inthe asymmetric tpQRM. Finally, we provide numericalconfirmations on the conjecture with large range of inte-gers N and M in two asymmetric QRMs. We set ω = 1in both models for simplicity in the whole Appendix. 1. Analytical proof for the small order of theconstrained conditions in asymmetric one-photonQRM Since f = 1, we begin with the N = 1 type-A poleenergy, Eq. (16) becomes f ( M, g ) = 12 g (cid:18) g + 14 ∆ − M (cid:19) , its zero is simply g = 12 s M − (cid:18) ∆2 (cid:19) , (A1)which is dependent on M . If ∆ > √ M , no real solutionexists, so the level crossing dose not occur along the N =1 pole line.If we set M = 2 i.e. ǫ = 1, we have g = 12 s − (cid:18) ∆2 (cid:19) . (A2)The second equation in (15) c (1 , g ) = 0 yields3 (cid:18) g + ∆ (cid:19) (cid:18) g + ∆ − (cid:19) − g = 0 , resulting in g = 12 s − (cid:18) ∆2 (cid:19) , which is exactly the same as Eq. (A2), the solution for f (2 , g ) = 0. It follows that two energy levels intersectwith the same pole line at the same coupling strength g in the spectra, indicating a true energy level crossing.For N = 2 type-A pole energy, the first equation in(15) f ( M, g ) = 0 becomes ( we set x = 4 g for simplic-ity) (cid:18) x + 14 ∆ − M + 1 (cid:19) (cid:18) x + 18 ∆ − M (cid:19) − x = 0 , (A3)yielding x = (cid:18) M − 316 ∆ (cid:19) ± s(cid:18) ∆ − (cid:19) + ( M − . If M = 3, i. e. ǫ is still 1, the solutions then read g = 18 r − − ± q (∆ − + 512 . (A4)On the other hand, the second equation in (15) c (2 , g )=0 is (cid:18) x + ∆ (cid:19) (cid:20)(cid:18) x + ∆ − (cid:19) (cid:18) x + ∆ − (cid:19) − x (cid:21) − x (cid:18) x + ∆ − (cid:19) = 0 , (A5)which interestingly gives the same solutions as in Eq.(A4), consistent with the conjecture. Here an unphysicalsolution x = − ∆ is omitted.Next, we set N = 1 , M = 3, thus ǫ = 2. f (3 , g ) = 0gives g = 14 p − ∆ . (A6)By c (1 , g ) = 0, we have (cid:18) x + 14 ∆ + 1 (cid:19) (cid:20)(cid:18) x + 18 ∆ (cid:19) (cid:18) x + 112 ∆ − (cid:19) − x (cid:21) − x (cid:18) x + 112 ∆ − (cid:19) = 0 . Its solutions are x = 3 − 14 ∆ ; x = − ∆ ± r − ! . Note that the second root is not a positive real value,and so omitted. The first root gives exactly the same g in Eq. (A6). 2. Analytical proof for the small order of theconstrained conditions in asymmetric tpQRM In this Appendix, we present a closed-form proof forthe conjecture that f ( q ) N ( M, g ) = 0 and c ( q ) M ( N, g ) = 0 inEq. (46) could give the same real and positive solutionsfor the coupling strength g with small numbers of N and M in the asymmetric tpQRM.For the most simply case, we set N = 1, then f ( q )1 ( M, g ) = 0 gives4 q − (2 M + 4 q ) β + ∆ , (A7)then the location of the degenerate point is obtained β = 2 q + ∆ / M + 2 q , (A8)which is dependent on M . Also note that the positivereal solution only exists for ∆ < √ M . Subject to theconstrained condition (44), we have ǫ = ( M − s q + ∆ / q + M . (A9)If set M = 2 , c ( q )2 (1 , g ) = 0 gives (cid:20) q + 1) (cid:0) − β (cid:1) + 18 ∆ (cid:21) × (cid:20) q + 1) (cid:0) − β (cid:1) + 116 ∆ − (cid:21) − q + 14 )( q + 34 ) (cid:0) − β (cid:1) = 0 , we then have β = 2 q + ∆ / 162 + 2 q , (A10)which is the same as that in Eq. (A8) for M = 2, consis-tent with our conjecture.Next, we set N = 2 , M = 3 . f ( q )2 (3 , g ) = 0 gives (cid:20) (cid:0) − β (cid:1) (2 + q ) + ∆ − (cid:21) × (cid:20) (cid:0) − β (cid:1) (3 + 2 q ) + ∆ − (cid:21) − (cid:0) − β (cid:1) ( q + 14 )( q + 34 ) = 0 . The solutions at q = are β = 13 + 23∆ ± r 256 + 212 ∆ + 1008 , and at q = are β = 511 + 29∆ ± r , c ( q )3 (2 , g ) = 0 results in { (cid:18) q + ∆ 32 (1 − β ) (cid:19) (cid:20) (cid:0) − β (cid:1) (3 + 2 q ) + ∆ − (cid:21) − ( q + 54 )( q + 74 ) } × (cid:20) (cid:0) − β (cid:1) (1 + q ) + ∆ − (cid:21) − (cid:20) (cid:0) − β (cid:1) (2 + q ) + ∆ (cid:21) ( q + 14 )( q + 34 ) = 0 . If q = , the solutions are β = 13 + 23∆ ± r 256 + 212 ∆ + 1008 ,β = 1 + ∆ . If q = , the solutions are β = 511 + 29∆ ± r ,β = 1 + ∆ . Omitting the unreasonable solutions β , we can find thatboth f ( q )2 (3 , g ) = 0 and c ( q )3 (2 , g ) give the same crossingcoupling strengths for q = 1 / / g (1 / , = 12 s − ± √ + 2688∆ + 2580482016 , (A11) g (3 / , = 12 s − ± √ + 1152∆ + 5529603168 . (A12)which also agree well with our conjecture. 3. Numerical confirmation for the two conjecturesin both asymmetric QRMs We extensively demonstrate that, for large N and M ( >N ), the two equations in either Eq. (15) or Eq. (46) givethe same physics solutions in both asymmetric QRMs.We sets N from 1 to 10 and M from 2 to 20 for bothone-photon QRM and tpQRM in the q = 1 / f N = 0 and c M = 0 are exactly the same in either case,confirming the conjectures numerically. Second, thereare 715 level crossings points for both cases, indicating N roots in the N order polynomial equations in bothmodels at ∆ = 2. Generally, the root number is equalto or slightly less than N for any ∆. This is to say, forany values of ∆, the numbers of the level crossings are generally nearly the same for the same ranges of N and M in asymmetric QRMs. FIG. 8: Three-dimensional view for the doubly degeneratelevel crossing points at ω = 1 , ∆ = 2 for the asymmetricone-photon QRM (a) and tpQRM in the q = 1 / N from 1 to 10 and M from 2 to 20 for bothcases, and the numbers of the true level crossing points arethe same. The data are drawn from part of the level crossingsin this range. 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