Abstract
The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one and twophoton terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These socalled Gfunctions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of Gfunctions 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 twophoton 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 twophoton coupling regime, the mixed QRM is equivalent to an onephoton QRM with an effective positive bias, suppressed photon frequency and enhanced onephoton coupling, which may pave a highly efficient and economic way to access the deepstrong onephoton coupling regime.
Introduction
The quantum Rabi model (QRM) describes the simplest and at the same time most important interaction between a twolevel system (or qubit) and a singlemode bosonic cavity which is linear in the quadrature operators^{1}. This model is a paradigmatic one in quantum optics for a long time. It has been reactivated in the past decade, due to the progress in many solidstate devices, such as the circuit quantum electrodynamics (QED)^{2,3}, trapped ions^{4,5}, and quantum dots^{6}, where the strong coupling even ultrastrong coupling has been realized. Here we study a natural generalization of the QRM which exhibits both linear and nonlinear couplings between the qubit and the cavity, i.e. the mixed QRM having both one and twophoton terms, with Hamiltonian
where Δ is the tunneling amplitude of the qubit, ω is the frequency of cavity, σ_{x,z} are Pauli matrices describing the twolevel system, a (\({a}^{\dagger }\)) is the annihilation (creation) bosonic operator of the cavity mode, and g_{1} (g_{2}) is the linear (nonlinear) qubitcavity coupling constant.
The nonlinear coupling appears naturally as an effective model for a threelevel system when the third (offresonant) state can be eliminated. The twophoton model has been proposed to apply to certain Rydberg atoms in superconducting microwave cavities^{7,8}. Recently, a realistic implementation of the twophoton QRM using trapped ions has been proposed^{9}. In the trapped ions, the atomcavity 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 nonzero 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 onephoton process, the twophoton 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 nonlinear 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 twophoton 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 twophoton 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 welldefined Gfunction with pole structures was only found for onephoton model by Braak^{19} in the Bargmann representation and twophoton model by Chen et al.^{20} using Bogoliubov transformations. These solutions have stimulated extensive research interests in the exact solutions to the unmixed QRMs and their variants with either onephoton^{21,22,23,24,25,26} or twophoton term^{27,28,29,30,31,32,33,34}. In the literature, many analytical approximate but still very accurate results have also been given^{35,36,37,38,39,40,41,42,43,44,45,46,47}. In some limits of model parameters, the dynamics and quantum criticality have been studied exactly as well^{48,49,50}. In the unmixed QRMs, the parity symmetry is very crucial to get the analytical solution in the closed system. Recently, the role of the parity symmetry has been characterized in the excitationrelaxation dynamics of the system as a function of lightmatter coupling in the open systems^{51}.
In the mixed QRM with both linear and nonlinear couplings, the parity symmetry is however broken naturally, and the analytical solution thus becomes more difficult^{52}, compared to the unmixed models. In this paper, we propose an analytical exact solutions to this mixed QRM. We derive a Gfunction by the Bogoliubov transformations, which can be reduced to the previous Gfunctions for both one and twophoton QRMs if either of the couplings appears. We demonstrate that the derived Gfunction can really yield the regular spectra by checking with the numerics. The exceptional eigenvalues are also given with the help of the nondegeneracy property in this mixed model due to the absence of any symmetry. Two kinds of formulae for the collapse points are derived. The avoided crossings are confirmed. The level collapse in the strong twophoton coupling regime is also discussed. Finally, we study the influence of mixed coupling by constructing an equivalent onephoton QRM with an effective positive bias where the photon frequency is suppressed and the onephoton coupling is enhanced.
Solutions within the Bogoliubov Operators Approach
In the basis of spinup and spindown states, the Hamiltonian (1) can be transformed to the following matrix form in units of ω = 1
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
with \(r=arc\,\cosh \,u\). If set
with \(\beta =\sqrt{14{g}_{2}^{2}}\), 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
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 onephoton 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
with
In principle, the eigenfunctions of the Hamiltonian can be expanded in terms of the number states of operator A
Projecting the Schrödinger equation onto n〉_{A} gives
with
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
with
We can express the eigenfunctions as
Similarly, we can get
and the similar five terms recurrence relation for \({f^{\prime} }_{n}\). Analogously, all coefficients \({e^{\prime} }_{n}\) and \({f^{\prime} }_{n}\) for n > 1 are determined through \({f^{\prime} }_{0}\) and \({f^{\prime} }_{1}\) 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,
We will set r = 1, because only ratios among f_{0}, f_{1}, \(r{f^{\prime} }_{0}\) and \(r{f^{\prime} }_{1}\) are relevant. In this case we can absorb r into new \({f^{\prime} }_{0}\) and \({f^{\prime} }_{1}\). Then we have
In the unmixed QRM, the welldefined Gfunctions can be derived by using the lowest number state 0〉 in the original Fock basis for the onephoton model^{19,20}, and two lowest number states 0〉 and 1〉 for the twophoton 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
They form 4 sets of linear homogeneous equations with 4 unknown variables f_{0}, f_{1}, \({f^{\prime} }_{0}\), and \({f^{\prime} }_{1}\). Nonzero solutions require the vanishing of the following 4 × 4 determinant
where elements G_{i,j}s are just coefficients before f_{0}, f_{1}, \({f^{\prime} }_{0}\), and \({f^{\prime} }_{1}\) in Eqs. (20–23).
Equation (24) is just the Gfunction 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 Gfunction is a welldefined 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.
Note that for the onephoton QRM (g_{2} = 0), the parity symmetry leads to
then Eq. (20) becomes
which is just the Gfunction of onephoton QRM (g_{2} = 0)^{19}. Similarly, Eq. (20) (Eq. (21)) can be reduced to the previous ones^{20,30} of the twophoton QRM (g_{1} = 0) in the subspace with even (odd) bosonic number.
In various unmixed QRMs, such as onephoton^{19,20}, twophoton^{30} or twomode^{53} QRM, the coefficients of the eigenstates satisfy a threeterm recurrence relation which can be achieved by performing the Bogoliubov transformation. All these Gfunctions 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 nonintegrable according to Braak’s criterion for quantum intergrability^{19}.
Exact Spectra
Regular spectra
To show the validity of the Gfunction (24), we first check with independent numerics. The Gcurves 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 Gfunction 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 Gfunction (24) in Fig. 2. Checking with numerics, our Gfunction 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.
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 g_{2} → 1/2, β → 0, all E_{n}^{(pole_A)} are squeezed into a single finite value \(\frac{1}{2}(1+{g}_{1}^{2})\), while all E_{n}^{(pole_B)} diverge to −∞. It seems that there are two kinds of collapse energies. But actually, all energy levels tend to Bpoles only, namely −∞, if g_{2} → 1/2, as shown in Fig. 2. This spectral characteristics is quite different from the twophoton 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 socalled exceptional solutions. They can be located in the following way.
At the intersecting point of the energy levels and the mth pole line associated with the Aoperator (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 unknown variable. In all summations in Eqs. (20)–(23), the mth 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. (20–23) and (28)) with 5 unknown variables f_{0}, f_{1}, \({f^{\prime} }_{0}\), \({f^{\prime} }_{1}\) and e_{m} for m ≥ 2. While for 0 ≤ m < 2, f_{m} = 0, we have only 4 unknown variables f_{1−m}, \({f^{\prime} }_{0}\), \({f^{\prime} }_{1}\) and e_{m}, which can be determined by solving another set of linear homogeneous equations (Eqs. (20–23)). Nonzero solution requires the vanishing of the 5 × 5 (4 × 4) determinant whose elements are just coefficients before f_{0}, f_{1}, \({f^{\prime} }_{0}\), \({f^{\prime} }_{1}\) and e_{m} in Eqs. (20–23) and (28) (f_{1−m}, \({f^{\prime} }_{0}\), \({f^{\prime} }_{1}\) and e_{m} in Eqs. (20–23)) for m ≥ 2 (0 ≤ m < 2),
We call this function as exceptional Gfunction. Here the energy is not an explicit variable, but determined by Eq. (26). The mth exceptional solution associated with the B operator can be detected in the same way by zero of an exceptional Gfunction
With the help of the exceptional Gfunction, 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 Apole 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 Bpole 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 Apole in Fig. 3(c,d), which are corresponding to intersecting points of energy levels and the mth Apole curves as displayed in Fig. 3(a). For G^{exc} curve associated with Bpole, 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 Bpole 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 onephoton QRM^{20},
this Hamiltonian can be expressed with a new set of bosonic operators \(P={D}^{\dagger }({g}_{1})aD({g}_{1})=a+{g}_{1}\) and \(Q=D({g}_{1})a{D}^{\dagger }({g}_{1})=a{g}_{1}\) as
Comparing Eqs. (33) and (36), we can introduce an effective photon frequency ω^{(eff)} and an effective onephoton coupling strength \({g}_{1}^{(\text{eff})}\),
and rewrite H_{0} as
Therefore, we construct an effective onephoton 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 twophoton 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 onephoton 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 twophoton 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 twolevel system. The groundstate magnetization M = 〈ψ_{GS}σ_{z}ψ_{GS}〉 calculated from H^{(eff)} is in good agreement with that calculated from H even in strong twophoton 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 twolevel system will prefer to stay in the lower level as indicated by the groundstate 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 \({g}_{1}^{(\text{eff})}\) will increase, which might provide a novel and economic way to reach deepstrong onephoton 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 onephoton 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 twophoton coupling and longtime limits.
The mixed QRM can be realized in the experiment by coupling the flux qubit to the plasma mode of its DCSQUID 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) become
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 twophoton coupling in the weak coupling regime very well. Even for strong twophoton 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 twophoton 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 twophoton interaction term. Therefore, the asymmetry in the transmission spectrum can be regarded as a signature of the mixed one and twophoton 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)}.
Recently, the quantum phase transition of onephoton QRM in Δ/ω → ∞ has drawn much attention^{48,49}. The magnetization M = 〈σ_{z}〉 serves as an order parameter which changes from zero in the normal phase to nonzero in the superradiant phase when g_{1} crosses the critical point \({g}_{1c}=\sqrt{\Delta \omega }/2\). Above g_{1c}, photons are extremely activated as well. The groundstate magnetization M and the photon number \({N}_{{\rm{ph}}}=\langle {a}^{\dagger }a\rangle \) calculated from H (Eq. (1)) as a function of the scaled coupling \({g}_{1}^{(\mathrm{eff})}/{g}_{1c}^{(\mathrm{eff})}\) for large values of Δ are shown in Fig. 8, where \({g}_{1c}^{({\rm{eff}})}=\sqrt{\Delta {\omega }^{({\rm{eff}})}}/2\). 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 qubitcavity interactions are enhanced and more photons are excited. For different twophoton 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 twophoton terms analytically. The Gfunctions 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 twophoton 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 twophoton 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 onephoton Hamiltonian to describe the the mixed QRM, which is valid in weak twophoton coupling regime. The mixed QRM is equivalent to a singlephoton one with an effective positive bias, suppressed photon frequency and enhanced onephoton coupling. This feature in the mixed system is very helpful to the recent circuit QED experiments where the intense competition to increase onephoton coupling is performed in many groups^{2,3,59,60}. We suggest that the simultaneous presence of both one and twophoton couplings would cooperate to provide richer physics.
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Acknowledgements
This work is supported by the National Key Research and Development Program of China (No. 2017YFA0303002), the National Science Foundation of China (Grants No. 11674285 and No. 11834005).
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Q.H. Chen conceived the idea and contributed to the theoretical analysis and interpretation of data, and wrote the manuscript. L. Duan performed the numerical calculations and contributed to the interpretation of the numerical results, and contributed to the writing of the manuscript. Y.F. Xie contributed to the interpretation of the numerical results. All authors reviewed the manuscript.
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Duan, L., Xie, YF. & Chen, QH. The mixed quantum Rabi model. Sci Rep 9, 18353 (2019). https://doi.org/10.1038/s41598019547560
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DOI: https://doi.org/10.1038/s41598019547560
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