Double degeneracy associated with hidden symmetries in the asymmetric two-photon Rabi model

In this paper, we uncover the elusive level crossings in a subspace of the asymmetric two-photon quantum Rabi model (tpQRM) when the bias parameter of qubit is an even multiple of the renormalized cavity frequency. Due to the absence of any explicit symmetry in the subspace, this double degeneracy implies the existence of the hidden symmetry. The non-degenerate exceptional points are also given completely. It is found that the number of the doubly degenerate crossing points in the asymmetric tpQRM is comparable to that in asymmetric one-photon QRM in terms of the same order of the constrained conditions. The bias parameter required for occurrence of level crossings in the asymmetric tpQRM is characteristically different from that at a multiple of the cavity frequency in the asymmetric one-photon QRM, suggesting the different hidden symmetries in the two asymmetric QRMs.


INTRODUCTION
The simplest interaction between a two-level system (qubit) and a single mode bosonic cavity (oscillator) was described by the quantum Rabi model (QRM) [1,2], which is thus a fundamental textbook model in quantum optics [3]. It has been demonstrated in many advanced solid 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, even deep strong coupling between the artificial atom and resonators [8][9][10].
In contrast to the conventional cavity QED system, the artificial qubit appears in modern solid devices usually contains both the splitting ∆ and the bias ǫ between the two qubit states, thus the so-called asymmetric QRM is ubiquitous. Driven by the proposals and experimental realizations of the various QRMs model, the asymmetric two-photon QRM (tpQRM) are also realized or stimulated to explore new quantum effects [11][12][13]. The typical two asymmetric QRMs can be generally written in a unified way as where the first two terms fully describe a qubit with the energy splitting ∆ and the bias ǫ, σ x,z are the Pauli matrices, a † and a are the creation and annihilation operators with the cavity frequency ω, and g is the qubit-cavity coupling strength. p = 1, 2 denote the one-photon and two-photon QRMs, respectively. In the superconducting flux qubit [8,9], ∆ is the tunnel coupling between the two persistent current states, ǫ = 2I p (Φ − Φ 0 /2) with I p the persistent current in the qubit loop, Φ the an externally applied magnetic flux, and Φ 0 the flux quantum. The flux qubit is usually manipulated by the external magnetic flux and the persistent currents. For the symmetric case (ǫ = 0), the one-photon QRM possesses Z 2 -symmetry (parity), i.e. [H 0 1 ,P 1 ] = 0 with the parity operatorP 1 = σ z exp(iπa † a), whose eigenvalues are ±1, while the tpQRM has Z 4 -symmetry, i.e. [H 0 2 ,P 2 ] = 0 with the parity operatorP 2 = σ z exp(iπa † a/2), whose eigenvalues are the quartic roots of unity ±1, ±i [14,15]. Hence the whole Hilbert space separates therefore into two and four infinite-dimensional subspaces in one-photon QRM and the tpQRM, respectively.
An analytical exact solution of the one-photon QRM has been found by Braak in the Bargmann space representation [16]. It was quickly reproduced in the more familiar Hilbert space using the Bogoliubov operator approach (BOA) by Chen et al. [17]. Moreover, the BOA can be easily extended to the tpQRM, and solutions in terms of a G-function, which shares the common pole structure with Braak's G-function for the one-photon QRM, are also found. It was soon realized that the Gfunction can be constructed in terms of the mathematically well-defined Heun confluent function [18]. These studies have stimulated extensive interests in various QRMs [19][20][21][22][23][24][25][26][27]. For more theoretical details in this field, one may refer to recent review articles [28][29][30].
The presence of the qubit bias term ǫ 2 σ x breaks Z 2symmetry of the QRM, so no any obvious symmetry remains in the asymmetric QRM [18,24,31], while in the tpQRM, it reduces the original Z 4 -symmetry to the Z 2symmetry. In the asymmetric tpQRM, the Z 2 -symmetry corresponding to the parity operatorP 2p = exp(iπa † a) only acts in the bosonic Hilbert spaces, the whole Hilbert space then only divides into two invariant subspaces: even and odd number Fock states, which can be still labeled by the Bargmann index q = 1/4 and 3/4 [14].
Level crossing is very helpful to identify the symmetry in quantum systems. The quasi-exact energies in the symmetric QRMs, also called Juddian solutions [33], have been found 20 years ago [32]. The Juddian solutions are corresponding to the doubly degenerate states, and can be constructed with the terminated polynomials. These quasi-exact energies now can also be easily derived with the help of the pole structure of the G-function in both the one-photon [16] and the two-photon [17] QRMs. Surprisingly, the level crossing even exists without Z 2symmetry in the asymmetric one-photon QRM, when ǫ is a multiple of the cavity frequency ω [18]. In these special cases, the hidden symmetry beyond any known symmetry is recently discussed based on the numerical calculation on the energy eigenstates [34] and conserved operators [35,36].
For the symmetric tpQRM, the standard Juddian solutions are level crossings within the same q subspace. The second type of level crossings of the eigenstates in different q subspaces [32] was also found recently [37,38]. In the asymmetric tpQRM, since Z 4 -symmetry reduces to Z 2 -symmetry, the level crossings within the same q subspace would generally disappear, while the second type of the level crossings in the different q subspaces remains robust due to the remaining Z 2 -symmetry. Contrary to the one-photon asymmetric QRM [18], the level crossing within the same q subspace in the asymmetric tpQRM is elusive, and has not been observed to date. In this work, we will uncover such a kind of level crossings irrelevant to any explicit symmetry.
The paper is structured as follows: In Sec. II, we briefly review the solutions to the asymmetric one-photon QRM in the framework of BOA approach, and corroborate the previous observed doubly degenerate states in BOA frame. We extend the BOA to study the asymmetric tpQRM, and derive the analytical exact solutions in Sec. III. In Sec. IV, we discuss the non-degenerate exceptional solutions for the asymmetric tpQRM. We demonstrate the level crossings within the same q subspace of the asymmetric tpQRM in Sec. V. The characteristics of level crossings in the two asymmetric QRMs is discussed in Sec. VI. The last section contains some concluding remarks. Appendix A confirms the conjecture that the two vanishing coefficients give the same solutions in both asymmetric QRMs both analytically in the low order of and numerically in the large order of the constrained conditions for the level crossings.

II. ASYMMETRIC QUANTUM RABI MODEL IN BOA
For the asymmetric one-photon QRM, when the bias parameter ǫ is a multiple of the cavity frequency, the level crossings appear again in the spectra even without any explicit known symmetry in the system [18,24]. It should be noted that here ǫ in accord with the standard qubit Hamiltonian [8,9,34,39] is twice of that used in [16,18,24].
In this section, we revisit the asymmetric one-photon QRM by BOA. We first briefly review the solutions in the BOA framework [17], then we can describe the level crossings in the BOA alternatively, which is essentially equivalent to the Bargmann space approach. Furthermore, by BOA, we can obtain all the non-degenerate exceptional points in a more concise and complete way. Most importantly, this scheme can be easily extended to the asymmetric tpQRM in the next sections.

By two Bogoliubov transformations
the wavefunction can be expressed as the series expansions in terms of A operator where e n and f n are the expansion coefficients, and also in terms of B operator with two coefficients c n and d n . |n A and |n B are called extended coherent states [40]. By the Schr ..
odinger equation, we get the linear relation for two coefficients e m and f m with the same index m as [17] and the coefficient f m can be defined recursively, with f 0 = 1. Similarly, the two coefficients c m and d m satisfy and the recursive relation is given by with c 0 = 1. If both wavefunctions (3) and (4) are the true eigenfunction for a non-degenerate eigenstate with eigenvalue E, they should be in principle only different by a complex constant z, i.e. |A = z |B . Projecting both sides onto the original vacuum state |0 , using √ n! 0|n A = (−1) n √ n! 0|n B = e −( g ω ) 2 /2 g ω n and eliminating the ratio constant z gives with the help of Eqs. (5) and (7), one arrives at onephoton G-function This G-function was first derived by Braak [16] using Bargmann space approach, and later reproduced by Chen et al. [17]. We then discuss the level crossing of this asymmetric QRM in terms of BOA framework described above.

B. Doubly degenerate states
The two types of pole energies appear in the onephoton G-function (10) as They are labeled with the type-A and type-B pole energy, respectively. If these two pole energies are the same 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 the coefficient e N (d M ) would diverge at the same pole energy (14). It does not make sense if some coefficients in the series expansion of a wavefunction really become infinity. A normalizable wavefunction should consist of the global property, i.e. the finite inner product, so the series 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), the numerator of right-hand-side of Eq. (5) [(7)] should also vanish, so that e N (d M ) remains finite, which result in Note that f N and c M can be obtained using the following three-terms recurrence relation from (6) and (8) with energy (14), respectively If ∆, M, N are given, two equations in (15) would provide the coupling strength g in the energy spectra where the energy levels intersect with the same pole line described by Eq. (14). A mathematical proof to the conjecture that f N (M, g) = 0 and c M (N, g) = 0 could give the same real and positive solutions for the coupling strength g was given in [31]. Li and Batchelor [24] have analyzed the relation between the number of the exceptional points and the model parameters (∆ and ǫ), and numerically found that the number of positive roots from these two equations are the same for integer ǫ/ω. But we confine us 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 condition will be derived and we will also do the similar things because a similar conjecture will be proposed but cannot be proven at the present stage.
To this end, we present our discussions only in terms of the fixed integers N and M . In this case, ǫ is a multiple of the cavity frequency is known immediately, and the remaining task is to show the same crossing points by two equations in Eq. (15), which is illustrated in Appendix A1.
In the left panels of Figs. 1 and 2, we present the energy spectrum E + g 2 /ω + ǫ 2 for ǫ = 1, 2 at ∆ = 1.5 and ∆ = 3 with ω = 1, respectively. The dotted and dashed horizontal lines denote different types of pole lines. Obviously, if the two types of pole lines cannot coincide, the level crossings cannot happen. f N (M, g) and c M (N, g) curves are plotted in the right panels. By Eq. (A1), one finds g = 0.5995 for 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 be found in this case, so level crossings cannot occur in the overlapped line with N = 1 in Fig. 2

(a).
At ∆ = 1.5, we can obtain two solutions for g as 0.4804, 1.0287 for N = 2 and M = 3 by Eq. (A4), agreeing well with two crossing points in the second type-A pole line with N = 2 and M = 3 shown in Fig. 1 (a). For ∆ = 3, we only find one real positive g = 0.8356, consistent with the spectra in Fig. 2

(a).
Associated with the overlapped N = 1 type-A and M = 3 pole lines, one can obtain g = 0.7806 for ∆ = 1.5, and g = 0.4330 for ∆ = 3 by Eq. (A6), consistent with the crossing points in the calculated spectrum in Fig. 1 (a) and Fig. 2 (a).
For large value of M and N , as shown in the right curves provide the same zeros for all cases. Now we will further demonstrate explicitly that any crossing point found above is corresponding to a doubly degenerate state in the BOA framework. At the crossing point, looking at (5), since both the numerator f N (g) and denominator vanish, e N would be arbitrary. If we set from Eq. (6) we know f N +1 = 0, further e N +1 = 0, and all coefficients f k and e k for k > N + 1 vanish. So the infinite series expansion in the wavefunction (4) terminates with finite N as Similarly, the infinite series expansion in the wavefunction (4) terminates with finite M as where Interestingly, both wavefunction terminates at finite terms. Because these two wavefuntions are not obtained from the G-function based on the proportionality (9), so they are different |A N = |B M , leading to doubly degenerate states. Since the degenerate eigenfunctions, |A N and |B M , are given as finite polynomials in the extended coherent state basis {|n A } and {|n B } (see also [41]). These states are the quasi-exact solutions of the asymmetric QRM.
At this stage, we can simply discuss the number of the doubly degenerate crossing points associated with the given N type-A pole line. f N (M > N, x = 4g 2 ) derived by Eq. (16) is a polynomial with N terms. Its zero would generally give around N roots, indicating that there are around 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 have just N roots.

C. Non-degenerate exceptional points
The non-degenerate exceptional points can be generated if only one energy level intersects with the energy pole line alone. In principle, all non-degenerate states including non-degenerate exceptional ones can be obtained by the G-function (10) because it is built based on the proportionality (9), only excluding the degenerate states. These states have been first analyzed for the symmetric QRM with the Bargmann space technique in [19] and later in [41][42][43]. We believe that the BOA has advantages with regard to the non-degenerate exceptional solutions, which cannot be found with any ansatz.
Note from G-function (10) that, at the pole energy either (11) or (12), the denominator of the associated term become zero, so this term would diverge and should be treated specially. For a physics state, to avoid the divergence, 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) could vanish in two different ways. First, f n (c m ) can be obtained by using the three-terms recurrence relation (6) [ (8) (6)[(8)]. This is to say, we have two ways to overcome the divergence. In the infinite summation where the diverging term is present, we may cut off all the terms either after or before this diverging one. E.g. for the N th type-A pole line, we may terminate the infinite summation at the diverging term following the same idea outlined in the last section for the degenerate states. So the first non-degenerate exceptional G-function can be written as where e N is given by Eq. (18). Note that the remaining terms vanish because all coefficients become zero. We can also remove all terms before the diverging term in the summation, and give the second non-degenerate exceptional G-function as with the initial condition e N = 1. The non-degenerate exceptional G-functions G non,1B 1p and G non,2B 1p associated with the type-B pole line can be obtained similarly by modifying the other infinite summation, which are not shown here.
Two non-degenerate exceptional G-functions (21) and (22)  Interestingly, for integer ǫ, two types of pole line may merge together. At the same pole energy (14), the second non-degenerate exceptional G-function Eq. (22) would be further modified as where e n<N = 0, e N = 1, and d n<M = 0, d M = 1, the other coefficients can still be obtained from the threeterms recurrence relations (6) and (8).
In the left panels of Figs In the end of this section, we would like to point out that the previous main results in the asymmetric QRM based on the Bargmann space approach, see Ref [24] and reference therein, can be well described in the BOA framework in a self-contained way. The asymmetric tpQRM has not been studied in the literature, much less the level crossings irrelevant to the explicit symmetry, to our knowledge. Note that the G-function by the direct application of the Bargmann space approach to the tpQRM [44] has no pole structure, and thus could not give qualitative insight into the behavior of the spectral collapse [15] and the level crossing. As far as we know, the G-function with its pole structure for the tpQRM has only been found using the BOA [14,17,45] and, in particular, has so far not been derived using the Bargmann space method in the literature. Therefore, it is perhaps irreplaceable, at the moment, to employ the BOA to study the asymmetric tpQRM, which is the main topic of this paper.

III. ASYMMETRIC TWO-PHOTON RABI MODEL AND SOLUTIONS USING BOA
For convenience, we rewrite the Hamiltonian H ǫ 2 on the σ z basis by rotating it around the y-axis with an angle π/2. The transformed Hamiltonian is given by the following matrix form (24) The Hamiltonian above is connected with su(1, 1) Lie algebra which obey spin-like commutation relations [K 0 , K ± ] = ±K ± , [K + , K − ] = −2K 0 . The quadratic invariant Casimir operator is given by Then we apply a squeezing operator S 1 = e r 2 (a 2 −a †2 ) to diagonalize the bosonic part of the above Hamiltonian and the parameter r is to be fixed later. In terms of the K 0 , K ± , the transformed Hamiltonian is derived as where β = ω 1 − 4 g ω 2 < ω can be termed as the renormalized cavity frequency owing to the fact that it is just a g-dependent pre-factor of the free photon number operators 2K 0 , and β = ω if g = 0. It will be shown later that β plays a key role in two-photon QRM. The second diagonal element is , and the squeezing parameter It is obvious that the coupling strength g < ω/2 leads to a real squeezing parameter. Based on the squeezing transformation, we propose the corresponding wavefunction as m are to be determined in the following.
In the case of the Lie algebra considered here, K 0 |q, 0 A = q |q, 0 A where q = 1 4 and 3 4 divide the whole Hilbert space H into even and odd sectors and label them, respectively. For the even subspace, H 1/4 = a †n |0 , n = 0, 2, 4, ... , and for the odd subspace, H 3/4 = a †n |0 , n = 1, 3, 5, ... , corresponding to even or odd Fock number basis. The Casimir element C 2 = 3 16 in both cases. The Bargmann index q allows us to deal with both cases independently.
The su(1, 1) Lie algebra operators satisfy K 0 |q, n A = (n + q) |q, n A , K + |q, n A = (n + q + 3 4 )(n + q + 1 4 ) |q, n + 1 A , Projecting both sides of the Schrödinger equation onto |q, n A gives a linear relation between coefficients e (q) and a three-term linear recurrence relation is given by All coefficients f We then apply the second squeezing operator S B = e − r 2 (a 2 −a †2 ) to the Hamiltonian (24) and suggest the wavefunction as where |q, m B = S B |q, m . Similarly, we can obtain a linear relation between the coefficients c (q) and the three-term linear recurrence relation is Left-multiplying the vacuum state q, 0| to the extended squeezed state |q, m A and |q, m B , we can obtain the inner product If both wavefunction |Ψ A q and |Ψ B q for the same q are the true eigenfunction for a non-degenerate eigenstate with eigenvalue E, they should be proportional with each other, i.e. |Ψ A q = z |Ψ B q , where z is a complex constant. Projecting both sides of this identity onto the original vacuum state |q, 0 , we obtain a transcendental function below defined as G-function with If set ǫ = 0, the G-function for the symmetric tpQRM [17] is recovered. The zeros of the G (q) -function give the regular spectrum in the q subspace of the asymmetric tpQRM. From Eqs. (29) and (32), we find the G-function diverges when its denominators vanishes, the condition of the denominators being zero can be obtained as and with m = 0, 1, 2, .... They are also labeled as two types (A and B) pole energies, similar to the asymmetric QRM. G-curves at q = 1/4 and 3/4 for ǫ = 0.4, g = 0.35, and ∆ = 3 with ω = 1 are plotted in Fig. 3. The zeros are easily detected. As usual, one can check it easily with numerics, an excellent agreement can be achieved. The poles given in Eqs. (36) and (37) are marked with vertical lines. The G-curves indeed show diverging behavior when approaching the poles.
In the limit of g → ω/2, the type-A pole energies are squeezed into a single finite value E A m (g = ω/2) = ǫ−ω 2 , and type-B pole energies into E B m (g = ω/2) = − ǫ+ω 2 . It seems that there are two kinds of collapse energies −ω±ǫ 2 . But actually, at g = ω/2, our obtained energy levels tend to the smaller one −ω−|ǫ| 2 , except some low lying states which split off from the continuum. The ground-state is always separated from the continuum by a finite excitation gap. In the spectrum shown in the next sections, we will indeed observe that the plotted energy levels always collapse to the smaller one at g = ω/2, which generate more non-degenerate exceptional points when g −→ ω/2. However, we cannot rule out the possibility that some energy levels would stay between two limit energies −ω−|ǫ| 2 and −ω+|ǫ| 2 , and thus these levels could not collapse. Since the analytical solution at g = ω/2 in the asymmetric tpQRM is lacking, the collapse issue in this model is somehow challenging.

IV. NON-DEGENERATE EXCEPTIONAL SOLUTIONS IN THE ASYMMETRIC TPQRM
As outlined in the Sec. II (c) for the asymmetric one-photon QRM, we can easily find the non-degenerate exceptional solutions in the spectra for the asymmetric tpQRM by the pole structures of the G-function. When the energy levels cross the pole lines, the coefficients in the G-function would diverge, and therefore should be treated specially. For any real physical systems, the wavefunction should be analytic, so the numerators f In parallel to the asymmetric one-photon QRM, the first non-degenerate exceptional G-function associated with the N-th type-A pole lines (36) for the asymmet-ric tpQRM is easily given by (38) where e (q) and that associated with the M-th type-B pole lines (37) reads Note that by Eq. Similarly, the second non-degenerate exceptional Gfunction associated with the type-A pole lines (36) is where we have set e the type-B pole lines (37) can be obtained in a straightforward way as n Ω (q) n = 0, (43) where d  Fig. 4 (b), and denoted with the same symbols in Fig. 4 (a). 7 zeros of the non-degenerate exceptional G-functions (42) and (43) corresponding to 7 open circles in Fig. 4 (c-e) are indicated by the 7 same symbols in Fig. 4 (a).
As revealed on an enlarged scale in the inset of Fig. 4  (a) and (b) that two open triangles do not coincide, indicating an avoided crossing at this bias parameter ǫ = 1. We will show in the left panels of Fig. 5 at ǫ = 1.0954 in the next section that, the two open triangles also obtained from f In the asymmetric tpQRM, can we also find level crossings in the same q subspace? According to the pole energies (36) and (37) the same pole energy takes Interestingly, Eq. (44) entails ǫ to be an even multiple of the renormalized cavity frequency β, in contrast to the asymmetric one-photon QRM where ǫ should be an multiple of the cavity frequency ω under the condition (13) for level crossings. It makes sense that only the two-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 that the coefficient e (32)) would diverge at the same pole energy (45). Similar to the asymmetric QRM case, the series expansion coefficients in the wavefunction (28) and (31) should be analytic and vanish as or before n → ∞.
Regarding states with the energy (45), the numerator of right-hand-side of (30) [(32)] should also vanish, so that e Note that f 4gω(n + q + 1 4 )(n + q + 3 4 ) c (q) n+1 = 2ω 2 (n + q) − β 2 (n + N + 2q) +  Similar to the asymmetric one-photon QRM, we conjecture that both f  (44), leading to levels crossing at the same pole energy. While it would be interesting to rigorously prove the conjecture in the two-photon case mathematically, we also confine us here to an analytical closed-form proof only for small values of N and M , and numerically confirmation for large N and M , in searching for physically reasonable coupling strength g. Similar to the asymmetric QRM, we also present our discussions only in terms of fixed values of N and M, but here ǫ cannot be determined independently, and would be determined together with g by Eqs. (44) and (46).
In Appendix A3, we analytically prove that, for some small values of N and M , both f  (46) give the same values for ǫ and g. Two energy levels cross the corresponding pole lines at the same values of ǫ and g, where the two pole lines also cross. Thus true level crossings also happen in the asymmetric tpQRM. Compare to the one-photon QRM where ǫ can be determined independently, in the asymmetric tpQRM, we need to solve two equations simultaneously to determine ǫ and g.
We Finally, the doubly degenerate states at the true level crossing points can be expressed explicitly in terms of the BOA as M are given by Eqs (39) and (41). Because these two wavefuntions are not obtained from the G-function based on the proportionality, so they are different, leading to doubly degenerate states. Both wavefunction terminates at finite terms, so they are the quasi-exact solutions of the asymmetric tpQRM.

VI. DISCUSSIONS
From the spectrum in Figs. 5 and 6, one might speculate that level crossings seldom happen in the asymmetric tpQRM. Actually it is not that case. If we incorporate Eq. (44) required by the level crossings, we may plot the similar spectra graph as Figs. 1 and 2 in one-photon case. In doing so, we calculate the energy as a function of g, and at the same time ǫ also changes as Eq. (44). To display the level crossings in asymmetric tpQRM more clearly, we can make the pole lines horizontal, thus we plot the normalized energy E ′ = E+ω/2 2β − q + ǫ 4β as a function of g and simultaneously varying ǫ = kβ in Fig.  7 for k = 0, 1, 2, 4 at ∆ = 2 with ω = 1, q = 1/4.
When ǫ is an even multiple of the normalized cavity frequency β entailed in Eq. (44), i.e. k is an even integer including the symmetric case k = 0, we find that the two equations in Eq. (46) Fig. 6.
However, if k is not an even integer, no level crossings happen, a lot of non-degenerate exceptional points emerges instead. As exhibited in Fig. 7 (b) for k = 1, the open triangles correspond to the non-degenerate exceptional points by Eqs. (38) or (40), while the open circles to those by Eqs. (42) and (43).
We can also estimate the number of the doubly degenerate crossing points associated with the given N type-A pole line. For any M > N , generally there are around N crossing points due to the polynomial equation with N terms, f around N degenerate crossing points for both asymmetric one-photon and two-photon QRMs. In Appendix A3, we have employed the constrained conditions in the asymmetric both one-and two-photon QRMs, and numerically found that they have nearly the same numbers of level crossing points in the range of integers of N and M in each case. Therefore, we could reach a conclusion that the number of the doubly degenerate crossing points in asymmetric tpQRM would be twice of that in the asymmetric QRM due to two Bargmann indices in the former model.
Braak proposed a new criterion of integrability that if the eigenstates of a quantum system can be uniquely labeled by i = i 1 + i 2 quantum numbers, where i 1 and i 2 are the numbers of the discrete and continuous degree of freedom, then it is integrable [16]. Both symmetric QRM and tpQRM may be considered integrable in terms of this criterion. As the bias term of qubit sets in, the integrability will be violated in the asymmetric QRM. However, if ǫ matches the multiple of the cavity frequency, the in-tegrability can be recovered in the asymmetric QRM, by using the hidden symmetry instead of the parity number. As shown in Figs. 1 and 2, the regular level crossings reappear when ǫ/ω is an integer, similar to that in the symmetric QRM which is considered to be integrable [16]. However, the asymmetric tpQRM with fixed ǫ is always non-integrable because the energy levels cannot be uniquely labeled by the only continuous degree of freedom. As displayed in the spectrum in Figs. 5 and 6 with special ǫ's, there is no regular level crossings, in sharp contrast to the integrable symmetric tpQRM [14]. Of course, if ǫ changes as kβ with k an even integer, the regular level crossings reappear in the asymmetric tpQRM as shown in Fig. 7, and it can be reconsidered to be integrable.
In the asymmetric QRM, the effort to look for the hidden symmetry responsible for the level crossings in the same 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 is definitely not owing to an explicit symmetry. It should be also interesting to rigorously find hidden symmetry in the asymmetric tpQRM in the near future.

VII. CONCLUSION
In this paper, we have studied both the asymmetric QRM and the asymmetric tpQRM by the BOA in a unified way. The previously observed level crossing when the bias parameter ǫ is a multiple of cavity frequency in the asymmetric QRM is illustrated by a closed-from proof for low orders of the constrained polynomial equations in a transparent manner. For the asymmetric tpQRM, the biased term breaks original Z 4 symmetry to Z 2 symmetry, so the Hilbert space only divides into even and odd bosonic number state subspaces. In each subspace, we derived the transcendental equation, called G-function, and obtain the regular spectrum exactly. The coefficients at the pole energy vanish in two different ways, giving two kinds of non-degenerate exceptional G-functions, by which all non-degenerate exceptional points can be detected. Very interestingly, the true level crossings can also happen in the same q subspace in the asymmetric tpQRM if the qubit bias parameter ǫ is an even multiple of the gdependent renormalized cavity frequency, in contrast to the asymmetric one-photon QRM where ǫ can be simply a multiple of the cavity frequency. We argue that the even multiple is originated from the two-photon process involved in the two-photon model. The doubly degenerate points can be also located analytically, similar to the asymmetric QRM. The number of the doubly degenerate points within the same subspace in the asymmetric tpQRM should be comparable with that in asymmetric QRM. The subspace in the asymmetric tpQRM has no any explicit symmetry, the newly found double degeneracy thus also implies the hidden symmetry. The hidden symmetry in the asymmetric QRM could be identified at the same integer ǫ/ω, while in the asymmetric tpQRM at the same integer ǫ/(2β). The latter constraint on the parameter space for the occurrence of the double degeneracy is illuminating in searching for a conserved operator in two-photon case. The present results may shed some lights on the different nature of the hidden symmetries in the two asymmetric QRMs. Omitting the unreasonable solutions β 3 , we can find that both f (A12) which also agree well with our conjecture.
We extensively demonstrate that, for large N and M (> N ), the two equations in either Eq. (15) or Eq. (46) give the same physics solutions in both asymmetric QRMs. We sets N from 1 to 10 and M from 2 to 20 for both one-photon QRM and tpQRM in the q = 1/4 subspace at ∆ = 2. First, we find that physics solutions from f N = 0 and c M = 0 are exactly the same in either case, confirming the conjectures numerically. Second, there are 715 level crossings points for both cases, indicating N roots in the N order polynomial equations in both models at ∆ = 2. Generally, the root number is equal to or slightly less than N for any ∆. This is to say, for any values of ∆, the numbers of the level crossings are generally nearly the same for the same ranges of N and M in asymmetric QRMs. . We set N from 1 to 10 and M from 2 to 20 for both cases, and the numbers of the true level crossing points are the same. The data are drawn from part of the level crossings in this range.
In Fig.  8, the doubly degenerate level crossing points are visualized in a three-dimensional (3D) view in (ǫ, g, E)-space for the asymmetric one-photon QRM and in (ǫ/2β, g, E)-space for the asymmetric tpQRM at ∆ = 2. It is interesting to draw planes for level crossings in both cases, as ǫ is simply scaled by a gdependent factor, 2β, in the two-photon case. In the original 3D (ǫ, g, E)-space, all the degenerate crossing points in asymmetric QRM are confined in equally spaced integer ǫ/ω planes, while those in asymmetric tpQRM are actually locked in different cylindrical surfaces with integer ǫ/ (2β). Those different constrained surfaces in the model parameter spaces for the occurrence of the double degeneracy in two models should be considered in the definition of conserved operators and the detection of hidden symmetries.