The mixed quantum Rabi model

The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.

where Δ is the tunneling amplitude of the qubit, ω is the frequency of cavity, σ x,z are Pauli matrices describing the two-level system, a ( † a ) is the annihilation (creation) bosonic operator of the cavity mode, and g 1 (g 2 ) is the linear (nonlinear) qubit-cavity coupling constant.
The nonlinear coupling appears naturally as an effective model for a three-level system when the third (off-resonant) state can be eliminated. The two-photon model has been proposed to apply to certain Rydberg atoms in superconducting microwave cavities 7,8 . Recently, a realistic implementation of the two-photon QRM using trapped ions has been proposed 9 . In the trapped ions, the atom-cavity coupling could be tuned to the collapse regime.
The mixed QRM described by Eq. (1) can also be implemented in the proposal of the circuit QED 10 if non-zero DC current biases are applied. Using alternative methods, both linear and nonlinear interaction terms can be present in different circuit QED setup by Bertet et al. 11,12 . Besides the one-photon process, the two-photon process was also detected in the superconducting qubit and oscillator coupling system 13 . It was shown recently that a general Hamiltonian realized in the microwave driven ions can be used to simulate the QRM with nonlinear coupling 14 by chosing properly the time dependent phase and in a suitable interaction picture. The combined linear and non-linear couplings can also be attained. More recently, Pedernales et al. proposed that a background of a (1 + 1)-dimensional black hole requires a QRM with both one-and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime 15 . So the QRM with both one-and two-photon couplings is not only a generic model in the circuit QED and trapped ions, but also has applications in other realm of physics.
The unmixed QRMs, where either linear or nonlinear coupling is present, have been extensively studied for a few decades (for a review, please refer to refs. [16][17][18] ). The solution based on the well-defined G-function with pole structures was only found for one-photon model by Braak 19 in the Bargmann representation and two-photon model by Chen et al. 20 using Bogoliubov transformations. These solutions have stimulated extensive

Solutions within the Bogoliubov operators Approach
In the basis of spin-up and spin-down states, the Hamiltonian (1) can be transformed to the following matrix form in units of First, we perform Bogoliubov transformation to generate a new bosonic operator, where S(r) is the squeezing operator and D(w) is the displaced operator 1 with β = − g 1 4 2 2 , we have a simple quadratic form of one diagonal Hamiltonian matrix element The eigenstates of H 11 are the number states |n〉 A which can be written in terms of the Fock states |n〉 in original bosonic operator a as A Similarly, we can introduce another operator with which yields a simple quadratic form of the other diagonal Hamiltonian matrix element Note that if g 2 = 0, we have w = g 1 and w′ = −g 1 , which are exactly the same as those in the one-photon QRM 20 . Similarly, the eigenstates of H 22 are the number states |n〉 B which can be written in terms of the Fock states |n〉 in original bosonic operator a as In terms of the Bogoliubov operator A, the Hamiltonian can be written as where In principle, the eigenfunctions of the Hamiltonian can be expanded in terms of the number states of operator A  This is actually a five terms recurrence relation for f n . All coefficients e n and f n for n > 1 are determined in terms of f 0 and f 1 linearly.
In terms of the Bogoliubov operator B, the Hamiltonian can be written as where We can express the eigenfunctions as Similarly, we can get and the similar five terms recurrence relation for ′ f n . Analogously, all coefficients ′ e n and ′ f n for n > 1 are determined through ′ f 0 and ′ f 1 linearly. Except for the crossing points in the energy spectra, the eigenstates are nondegenerate. Two wavefunctions in terms of operators A and B correspond to the same eigenstate. Therefore, they should be proportional with each other by a constant r,   In the unmixed QRM, the well-defined G-functions can be derived by using the lowest number state |0〉 in the original Fock basis for the one-photon model 19,20 , and two lowest number states |0〉 and |1〉 for the two-photon model 20,30 . Here, we also project Eqs. (18) and (19) onto two original number states |0〉 and |1〉, and then obtain the following 4 equations where elements G i,j s are just coefficients before f 0 , f 1 , ′ f 0 , and ′ f 1 in Eqs. (20)(21)(22)(23). Equation (24) is just the G-function of the present mixed QRM. Its zeros thus give all regular eigenvalues of the mixed QRM, which in turn give the eigenstates using Eq. (12) or Eq. (15). Note from the coefficients in Eqs. (13) and (14) that this G-function is a well-defined transcendental function. Thus analytical exact solutions have been formally found. In the next section, we will employ it to analyze the characteristics of the spectra.
In various unmixed QRMs, such as one-photon 19,20 , two-photon 30 or two-mode 53 QRM, the coefficients of the eigenstates satisfy a three-term recurrence relation which can be achieved by performing the Bogoliubov transformation. All these G-functions with explicit pole structures have been summarized in Eq. (27) of ref. 53 . But in the mixed model, the Hilbert space cannot be separated into invariant subspaces due to the lack of parity symmetry, so the recurrence relation of the coefficients {f n } is of higher order, as seen in Eq. (14). A similar behavior also happens in the Dicke model 54,55 . The possible reason is that the symmetry does not suffice to label each state uniquely, indicating that mixed QRM is non-integrable according to Braak's criterion for quantum intergrability 19 . www.nature.com/scientificreports www.nature.com/scientificreports/ exact Spectra Regular spectra. To show the validity of the G-function (24), we first check with independent numerics. The G-curves as a function of E for Δ = 0.5, g 1 = 0.1, g 2 = 0.2 and 0.47 are depicted in Fig. 1. We find that the zeros of G-function indeed yield the true eigenvalues by comparing with the numerical diagonalization in truncated Hilbert spaces with sufficiently high dimension.
Then we plot the energy spectra calculated by the G-function (24) in Fig. 2. Checking with numerics, our G-function reproduces completely all eigenvalues of the present mixed model. When g 2 is close to 1/2, the energy spectra collapse to negative infinity. The parity symmetry in this mixed QRM is lacking, so in principle, the energy degeneracy should be relieved, and level crossings should be absent. However, as shown in Fig. 2(a-c), it seems that some crossings still occur for the small g 1 . It will be shown later that these "crossing" can be actually discerned as avoided crossings. www.nature.com/scientificreports www.nature.com/scientificreports/ pole structure and collapse. From Eqs. (13) and (16), we can find two kinds of poles associated with the A and B operators respectively, which will lead to the divergency of the recurrence relations, With the same n, the difference of two poles is independent of n.
In the limit of diverge to −∞. It seems that there are two kinds of collapse energies. But actually, all energy levels tend to B-poles only, namely −∞, if g 2 → 1/2, as shown in Fig. 2. This spectral characteristics is quite different from the two-photon QRM where the energy spectra collapse to the finite value 9,30,56 . For the mixed QRM, the divergence of the eigenenergies to negative infinity for g 2 → 1/2 suggests some underlying unphysicality, which deserves further studies. The energies of the high excited states cross the pole A curves and then asymptotically converge to the pole B, which lead to exceptional solutions.
exceptional solutions. As shown in Fig. 3(a), most energy level curves pass through the pole curves on the way to g 2 = 1/2, which results in so-called exceptional solutions. They can be located in the following way. At the intersecting point of the energy levels and the m-th pole line associated with the A-operator (26), the coefficient f m must vanish so that the pole is lift. Otherwise, the coefficient e m would diverge due to zero denominator in Eq. (13). In the unmixed QRM, f m = 0 can uniquely yield the necessary and sufficient condition for the occurrence of the exceptional solution. But here it is not that case, because f m depends on two initial variable f 0 , f 1 , and cannot be determined uniquely. The corresponding coefficient e m should be finite and can be regarded as an www.nature.com/scientificreports www.nature.com/scientificreports/ unknown variable. In all summations in Eqs. (20)-(23), the m-th terms should be treated specially, i.e. let f m be 0 and e m be a new variable. By the recurrence relation (14), we can add a new equation for this case So for the exceptional solution, we have a set of linear homogeneous equations (Eqs. (20)(21)(22)(23) and (28) We call this function as exceptional G-function. Here the energy is not an explicit variable, but determined by Eq. (26). The m-th exceptional solution associated with the B operator can be detected in the same way by zero of an exceptional G-function With the help of the exceptional G-function, we can determine the intersecting points of the energy levels and the pole curves as shown in Fig. 3(a) for Δ = 0.5, g 1 = 0.1. The G exc curves associated with A-pole as a function of g 2 for m = 0 and 1 are shown in Fig. 3(c,d) respectively, and the G exc curve associated with B-pole for m = 1 is shown in Fig. 3(b). The detected exceptional solutions in the G exc curves are marked by the same symbols as those in the enlarged spectra graph. One can find many zeros for G exc curve associated with A-pole in Fig. 3(c,d), which are corresponding to intersecting points of energy levels and the m-th A-pole curves as displayed in Fig. 3(a). For G exc curve associated with B-pole, there is only one zero for m = 1 as exhibited in Fig. 3(b), also consistent with the single intersecting point shown in Fig. 3(a). No exceptional solution exists even for B-pole with m = 0. All g 2 obtained from G exc = 0 can find their corresponding intersecting points in the energy spectra exactly. Now we can judge whether it is the true level crossing or avoided crossing in the Fig. 2(a-c). Around this regime, we have not found any exceptional solutions, indicating that the energy level cannot intersect with the pole curves. So although two energy levels are very close but blocked off by two pole curves with difference ΔE (p) ∝ g 1 2 , they neither collide nor cross with each other. It is actually avoided crossing. For small g 1 , ΔE (p) is very small, so it looks like a "level crossing" as depicted in Fig. 2(a-c). For large g 1 , the avoided crossing is quite clear, as shown in Fig. 2(d-f). Actually, these avoided crossings are just remnants of the traces of the doubly degeneracy in the unmixed model, which is relieved in the mixed model.

Effect of the mixed couplings.
In the mixed QRM, if we combine Eqs. (5) and (8), the Hamiltonian (Eq. (2)) can be written as which can be reorganized and separated into three terms: where I is a unit matrix, and An effective bias ε (eff) appears naturally, as well as a total energy shift (1 − β)/2. Recalling the one-photon QRM 20 , www.nature.com/scientificreports www.nature.com/scientificreports/ this Hamiltonian can be expressed with a new set of bosonic operators = = + † P D g aD g a g ( ) ( ) 1 1 1 and = = − † Q D g aD g a g ( ) ( ) 1 1 Comparing Eqs. (33) and (36), we can introduce an effective photon frequency ω (eff) and an effective one-photon coupling strength g 1 and rewrite H 0 as ( ) Therefore, we construct an effective one-photon QRM to describe the mixed one, which provides a more intuitional description of the influence of the mixed coupling, Comparing P (Q) with A (B), the mainly difference is the lack of squeezing operator S(r). The two-photon interaction leads to the squeezed field state 10,47 , which can be well captured by the squeezing operator 20,30,34 . Therefore, the squeezing operator is introduced explicitly to deal with the mixed QRM, but it is not necessary for the one-photon model. The definitions of P (Q) in Eq. (33) is equivalent with that of A (B) in Eq. (36) only if g 2 = 0, so it is hard to use the effective Hamiltonian to deal with the strong two-photon coupling regime, especially the bosonic part due to the intense squeezing effect. We calculate the Wigner function W(α,α * ) of the ground state 57,58 in Fig. 4, which describes the probability distribution of the bosonic field in the phase space. When g 2 = 0.1, the differences of the Wigner functions calculated from H and H (eff) are negligible. However, It is shown in Fig. 4(c) that the squeezing effect becomes apparent for H when g 2 = 0.3. The effective Hamiltonian can hardly describe the squeezing effect as demonstrated in Fig. 4(d). Nevertheless, it shed light on the analysis of strong coupling case, especially the properties of the two-level system. The ground-state magnetization M = 〈ψ GS |σ z |ψ GS 〉 calculated from H (eff) is in good agreement with that calculated from H even in strong two-photon coupling regime, as shown in Fig. 5. When g 2 tends to 1/2, the effective bias ε (eff) will tend to infinity, and the two-level system will prefer to stay in the lower level as indicated by the ground-state magnetization M → −1 in Fig. 5. Therefore, the energy contributed by ε (eff) σ z /2 would be negative infinite, which is one of the reason for the negative divergence of the eigenenergies in the mixed QRM in the limit of g 2 → 1/2, as observed in Fig. 2. What is more, with the increase of g 2 , the effective photon frequency ω (eff) will decrease while the effective coupling g 1 (eff) will increase, which might provide a novel and economic way to reach deep-strong one-photon coupling regime.
To further demonstrate the accuracy of the effective Hamiltonian, we calculate the dynamics of fidelity as shown in Fig. 6. The fidelity is defined as the overlap of the wavefunctions |ψ (eff) (t)〉 obtained from H (eff) (Eq. (37)) and |ψ(t)〉 obtained from H (Eq. (1)), namely F (eff) (t) = |〈ψ (eff) (t)|ψ(t)〉|, which can be used to judge how accurately the state of the effective Hamiltonian reproduces that of the original Hamiltonian. The initial states are |↑〉|0〉 for both of them. The fidelity of the unbiased one-photon QRM, namely H 1P (Eq. (35)), is also presented for comparison, i. e. F 1P (t) = |〈ψ 1P (t)|ψ(t)〉|. When g 2 is as small as 0.05, the corresponding effective bias reaches ε (eff) ≃ 0.202. This effective bias is large enough to play a significant role in the evolution of fidelity, as clearly seen in the Fig. 6(a). The fidelity of H (eff) tends to one, which is a strong evidence of the equivalence between |ψ (eff) (t)〉 and |ψ(t)〉. The fidelity of H 1P is much smaller, indicating that it deviates from the original Hamiltonian significantly. When we further increase g 2 , the effective Hamiltonian still gives considerably good results while the deviation of H 1P becomes more obvious. The fidelity of H (eff) drops slightly in the long time, which is mainly due to the error accumulation. The results of fidelity confirm the limitation of the effective Hamiltonian in dealing with the strong two-photon coupling and long-time limits.
The mixed QRM can be realized in the experiment by coupling the flux qubit to the plasma mode of its DC-SQUID detector 11 . We expect that the effective Hamiltonian can be employed to explain the experimental results. One of the most widely measured quantity in experiments is the transmission spectrum 3,59,60 . The transmission spectrum, i.e. δE n = E n − E 0 , from both original full model (1) and the effective model (37) are shown in Fig. 7. We introduce an additional bias εσ z /2 which is originated from the an externally applied magnetic flux in circuit QED system, and Eqs. (1) and (37)     Therefore, the total bias in the effective Hamiltonian is (ε (eff) + ε). It is obvious from Fig. 7(a) that the effective Hamiltonian can capture the effects of two-photon coupling in the weak coupling regime very well. Even for strong two-photon coupling g 2 = 0.45, the effective Hamiltonian (39) still provides quite accurate energy structure and almost captures all features of the mixed QRM, as shown in Fig. 7(b,c). With the decrease of g 1 , the deviation appears gradually because the two-photon interaction becomes dominated. One should also note that the energy differences of the effective Hamiltonian (Eq. (39)) is symmetry about ε = −ε (eff) , which is different from that of the original one (Eq. (38)), as shown in Fig. 7(c). For the effective Hamiltonian with an additional bias, we can easy confirm that only the absolute value of (ε + ε (eff) ) affects the eigenenergies, as well as the energy differences. For the mixed QRM, this symmetry is broken due to the two-photon interaction term. Therefore, the asymmetry in the transmission spectrum can be regarded as a signature of the mixed one-and two-photon couplings. Far from the symmetry point ε = −ε (eff) , the energy difference tends to be a multiple of the photon frequency. The energy difference decreases with the increase of g 2 , which can be explained by the suppressed effective photon frequency ω (eff) .  A negative M emerges in the mixed QRM due to the positive effective bias. Clearly, the photon number N ph increases with the increase of g 2 , which also indicates that the qubit-cavity interactions are enhanced and more photons are excited. For different two-photon coupling in the mixed QRM, the photons are considerably enhanced at almost the same scaled coupling around 1. Whether it is a signature of the quantum phase transitions in the mixed QRM as observed in ref. 50,61 . deserves further study.

Summary
In this paper, by using Bogoliubov operators, we exactly solve the mixed QRM with both one-and two-photon terms analytically. The G-functions with the pole structures are derived, which reproduce completely the regular spectra. They can also be reduced to the unmixed ones. It is found that there are two sets of poles associated with two Bogoliubov operators. Two types of exceptional eigenvalues are then derived, which cannot be obtained solely by requiring that the corresponding coefficients vanish like in the unmixed models. When the two-photon coupling strength g 2 is close to 1/2, two collapse energies are derived. One is finite, while the other diverges negatively. All energy levels collapse to the lower one, therefore diverge also, in sharp contrast to the unmixed two-photon model. The level degeneracy in the unmixed model is relieved due to the absence of parity symmetry. The avoided crossings are strictly discerned from the very close levels in the mixed model by the absence of exceptional eigenvalues around the "crossings".
We construct an effective one-photon Hamiltonian to describe the the mixed QRM, which is valid in weak two-photon coupling regime. The mixed QRM is equivalent to a single-photon one with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling. This feature in the mixed system is very helpful to the recent circuit QED experiments where the intense competition to increase one-photon coupling is performed in many groups 2,3,59,60 . We suggest that the simultaneous presence of both one-and two-photon couplings would cooperate to provide richer physics.