Generalized Rotating-Wave Approximation for the Quantum Rabi Model with Optomechanical Interaction

The spectrum of energy and eigenstates of an hybrid cavity optomechanical system, where a cavity ﬁeld mode interacts with a mechanical mode of a vibrating end mirror via radiation pressure and with a two level atom via electric dipole interaction are investigated. In the spirit of approximations developed for the quantum Rabi model beyond rotating-wave approximation (RWA), the so-called generalized RWA (GRWA) to diagonalize the tripartite Hamiltonian for arbitrary large couplings is implemented. Notably, the GRWA approach still allows to rewrite the hybrid Hamiltonian in a bipartite form, like a Rabi model with dressed atom-ﬁeld states (polaritons) coupled to mechanical modes through reparametrized coupling strength and Rabi frequency. A more accurate energy spectrum for a wide range of values of the atom-photon and photon–phonon couplings, when compared to the RWA results is found. The ﬁdelity between the numerical eigenstates and its approximated counterparts is also calculated. The degree of polariton-phonon entanglement of the eigenstates presents a non-monotonic behavior as the atom-photon coupling varies, in contrast to the characteristic monotonic increase in the RWA treatment.


I. INTRODUCTION
Cavity quantum electrodynamics (cQED) and cavity optomechanics are two paradigmatic fields to study lightmatter interaction and the coupling between confined optical field and mechanical degrees of freedom, at a full quantum level of description.The simplest model in cQED is the celebrated quantum Rabi model (QRM), which describes the interaction between a two-level quantum system (the "atom") and a single mode of a bosonic field [1][2][3].It has been extensively applied in a wide variety of fields, ranging from solid state [4,5] and quantum optics [6][7][8][9] to molecular [10][11][12] and trapped ions physics [13][14][15][16] and recently, it has been proposed as basis for the development of quantum gates [17,18], information protocols [19] and nondemolition measurement [20,21].The simplest counterpart in optomechanics is the model consisting of an optical cavity with photons exerting radiation pressure through reflection on a movable end mirror.This model and related variants have played a role in the goal of reaching quantum control of mechanical motion and detect small forces or displacements beyond the standard quantum limit [22], and also in a number of applications like gravitational wave astronomy [23,24], cold atom experiments and creation of nonclassical macroscopic states [25].
Recently these models have been combined in an analytically solvable atom-photon-oscillator model [26].Its cQED part was described by the Jaynes-Cummings model (JCM), while the optomechanical part by a quantum harmonic oscillator whose canonical position operator is coupled to the photon number operator.The model predicts cooling of mechanical motion at the single-polariton level, antibunched states of mechanical motion, and quantum interference and correlation effects due to the interplay of all the coupling mechanisms [27].
As is well known, the JCM is derived from the QRM by using the rotating-wave approximation (RWA) [28,29], which works well under quasi-resonance condition and for small atom-field coupling strength compared to the energy of the photon field.This approximation neglects the anti-resonant terms and restricts the Hilbert space to an infinity set of 2D subspaces, each one characterized by the conservation of the number of excitations [28,30,31].Over the past decade, the so-called ultrastrong and deep strong coupling regimes of the light-matter interaction have been reached in experiments with superconducting circuits, semiconductor quantum wells, optomechanical systems, and other hybrid platforms [32,33].On the theoretical side, this motivated the development of several approximation methods for the QRM beyond the RWA [34][35][36][37][38].The exact solution of the QRM, for arbitrary magnitude of the parameters, is already know since more than a decade [3,39]; it is given in terms of the poles of transcendental functions, ant it lacks of analytical expressions of the spectrum of energy and eigenstates.Thus, it is still convenient to appeal to approximated models that lead to intuitive understanding of the physics, like in the RWA approach [40].
Here, motivated by the work of Restrepo et al. [26], we calculate the spectrum of energy, the eigenfunctions and its entanglement properties, of the same atom-photon-oscillator model, but with the cQED part described by the QRM and for arbitrary values of the atom-photon and photon-phonon coupling strengths.To this end, we adapted the generalized version of the RWA (GRWA), developed by K. Irish [35] for the QRM, to the full hybrid Hamiltonian.As a result, we found that the approach leads to accurate expressions of the levels and states, in the whole range of couplings and for large detuning, and to a degree of entanglement which differs strongly from the RWA results.
The paper is organized as follows.In Section II we introduced the hybrid Hamiltonian, and developed the GRWA strategy to obtain an approximated Hamiltonian and its corresponding spectrum of energy and states.In Section III a discussion of the entanglement properties of the eigenstates is presented, based on the comparison between the RWA and GRWA results for both, the QRM and the hybrid system.Section IV is devoted to summarize our findings.Appendix A enumerates some basis states associated to the GRWA for the QRM and needed for the application of the GRWA approach to the hybrid model under consideration.

A. The model
We consider a hybrid system that combines cQED and cavity optomechanics, with Hamiltonian (Fig. 1) The first three terms corresponds to the well known quantum Rabi model[1-3] ĤR , where a two-level atom (level spacing ω a ) is coupled (parameter g ac ) to a single-mode quantized electromagnetic field of an optical cavity (frequency ω c ).The fourth and fifth terms describe the standard optomechanical model, which assumes that one of the cavity mirrors oscillates harmonically (frequency ω m ) due to radiation pressure caused by the interchange of momentum of bouncing photons [41] (coupling g om ).The boson operators â, â † ( b, b † ) are the annihilation and creation operators for photons (phonons), and σi are the spin-1/2 Pauli matrices; we use = 1.For the sake of simplicity, the atom-oscillator coupling is not included in model ( 1), however it can be added without major changes in the derivation that follows.
FIG. 1. Hybrid system combining the atom-photon interaction and quantum optomechanics.A two-level system (the "atom") lies within a optical cavity (photons) one of whose mirrors oscillates harmonically (phonons).

B. GRWA Hamiltonian
The use of the JCM in the cQED part of the Hamiltonian (1) allows to break the Hilbert space into a set of invariant subspaces characterized by the conservation of the atom-cavity excitation number NJC = â † â + (σ z + 1)/2 [26,27].When NJC = 0, further order-of-magnitude simplifications lead to a mathematical structure in which each subspace writes as a Rabi Hamiltonian.At this point, an additional RWA to neglect counter-rotating terms in the effective bipartite system formed by the atom-cavity dressed states (polaritons) and the mechanical vibrations, implies further reduction to 2×2 blocks of the JC type [26,27].This allows in turn to obtain analytical expressions for the non-trivial part of the energy spectrum [26,27].The case for NJC = 0 corresponds to a subspace of a simple displaced harmonic oscillator in the phonon operators.
In what follows, we adopt a similar procedure to derive an approximate Hamiltonian to model (1), but introducing a generalized version of the RWA (the GRWA) instead to proceed within the weak coupling and quasi-resonant restrictions.The GRWA is an approach to the QRM developed to go beyond the JCM conditions, allowing to explore a wider range of frequency detuning values and larger magnitudes of the atom-cavity coupling [35].As we will see, within the context of the hybrid model (1), this approximation allows also a better description of the atom-cavity polaritons, the generalized Rabi frequency, energy spectrum and eigenstates, than its RWA counterpart.
As will be discussed below, the conserved quantity within the GRWA is the excitation number involving adiabatic states and displaced photons.Correspondingly, the atom-cavity polaritonic states becomes dressed differently with respect to the RWA.On the other hand, the optomechanical interaction mixes the GRWA polaritonic states of different subspaces.In order to obtain disconnected invariant subspaces, the GRWA should be applied to the complete Hamiltonian (1).In our derivation we will use several quantities associated to the GRWA of the QRM, like the displaced oscillator basis or the adiabatic basis states, the corresponding energies, and frequency parameters.These are presented in an appendix, to simplify the application of the GRWA to the full Hamiltonian (1).
We start by writing an approximate (GRWA) Hamiltonian for the Rabi part of the tripartite model (1), through the spectral decomposition with the last term containing the non-Rabi contribution, namely the optomechanical part Ĥom = ω m b †b −g om â † â( b † + b).The substitution Ĥgrwa R → ĤJC should produce the results reported in references 26 and 27.As is well known, the GRWA breaks the Rabi Hamiltonian ĤR into 2 × 2 invariant subspaces (see Appendix A), and we note that each one is characterized by a conserved number given by where D(±ν) = e ±ν(â † −â) is the displacement operator and Î(2) is the 2 × 2 identity matrix.We recall that the starting point of the GRWA is to express the Rabi Hamiltonian ĤR in the adiabatic basis [34] {|ψ ad ±,N , N = 0, 1, . ..} (see Appendix A), instead of using the eigenbasis of the noninteracting Hamiltonian ωa 2 σz + ω c â † â.As a consequence, the number (4) involves σx and the adiabatic states |ψ ad +,N .Note however that [ Ĥom , N grwa R ] = 0, and then the optomechanical part, after introducing the projectors used in (2), becomes These terms connect projectors with different N index, having terms like g om ( b † + b) P−,N (â † â) P−,N+2 = 0 or g om ( b † + b) P−,N (â † â) P+,N+2 = 0 , among others.Proceeding in the same spirit of the GRWA [35], these crossed terms involving different labels N will be neglected.As we will show below, this help us to find and approximated Hamiltonian which commutes with the operator N grwa R (4), in complete analogy to the hybrid RWA Hamiltonian of Ref.26 which commutes with the conserved number NJC of the JCM.Such a simplification of (5) allows to breaks the Hilbert space into a set of disconnected invariant subspaces, each with = N + 1 atom-cavity polaritons.Therefore, the Hamiltonian Ĥgrwa hyb will have projectors involving only the same index N , with the state |ψ grwa ±,N containing exactly N grwa R polaritons, while the ground state |ψ grwa G contains zero.Droping the mixing terms is equivalent to perform the GRWA to Ĥom .This contrasts to the RWA approach [26,27], where such a mixing is absent.
The photon number operator in the optomechanical term become where Ω N = 2g ac √ N + 1 is the characteristic Rabi frequency for N photons of the JCM [29].The angle α N is defined by tan , where N − |N + is the overlap between oppositely displaced Fock states (see (A3) and states (A2) in Appendix A).To write these expressions, we defined Pauli matrices σ(N) x,y,z from the doublet |ψ grwa ±,N , and the identity Î(N) = P+,N + P−,N .In order to rationalize what is neglected it is convenient to move to the interaction picture through the unitary transformation Û (t) = exp (i Ĥgrwa R t).After this, each term of the effective Hamiltonian Û (t) Ĥom Û † (t) rotates at a speed determined by the energy difference of the GRWA states that it connects.For a given N , the diagonal terms become time independent and off-diagonal terms oscillating with the generalized GRWA Rabi frequency )).In addition, there also appear non diagonal terms connecting states with different quantum numbers N = N , revolving at higher frequencies ω c (N − N ) + (T N ± T N )/2.These terms are non-energyconserving and will be neglected in the same way as the counter-rotating terms arising in the GRWA for the Rabi Hamiltonian ĤR .
As a consequence, the hybrid system can be viewed as two coupled subsystems defined by the atom-cavity polaritons (a dressed two-level system for a given N in the GRWA basis) and by displaced phonons.The Hilbert space reads as a direct sum of invariant subspaces . Correspondingly, the total Hamiltonian, as obtained from ( 2) and (3) without mixing terms, reads as where is associated to the polaritonic ground state, and to an effective spin-boson Hamiltonian.The additive constant k N and the static shift of the mechanical resonator q N , for the subspace N , are given by Note that the operator (4) remains a conserved quantity for the hybrid GRWA Hamiltonian ( 7), [ Ĥgrwa hyb , N grwa R ] = 0.The first term on the RHS of ( 7) is just a displaced harmonic oscillator tensor product with the polaritonic ground state projector.Although the second term looks more complicated, note however that for each N we have an interacting spin-boson like Hamiltonian that can be independently treated.To this end, we simplify the Hamiltonian (9) to a Rabi model-like form.Note at this point that when g ac /ω c 1, all the quantities in our Hamiltonian (9) reduces to that reported in reference 26.
In contrast to the RWA version of the Eq. ( 9), where the quasi-resonance condition |ω a −ω c | Ω N allows to simplify the angle α N to π/2 for all values of g ac , the GRWA Hamiltonian involves an angle which depends on ∆ N and Ω N,N +1 , and therefore such a simplification does not occur.Further reduction is achieved noting that, in the range of values of couplings and quantum number N considered in this study, the coupling factor g  energy shift (see Fig. 2(a)).Note however that the effective coupling g ωc cos α N should be kept for the whole coupling range.A convenient change to a displaced frame leads to the Rabi-like Hamiltonian where bN = D(q N /ω m .Following reference 27, we have ignored in (12) an off-diagonal term (2q N /ω m )g x , which produces a Stark shift of the energies, in order to maintain analytical solvability.
Equations ( 8) and ( 12) constitute the GRWA version of the hybrid Hamiltonian (1).In the following we use them to calculate the energy spectrum and eigenstates for arbitrary large values of the couplings.

C. Eigenstates and energy spectrum
We first consider the spectrum associated to the contribution ĤG (8), which involves the polaritonic ground state term PG and a displaced quantum harmonic oscillator.Thus, the corresponding energy eigenvalue is given by E grwa G (see Eq.(A8)) plus the quantized energy spectrum of such an oscillator.Explicitly, The associated eigenstates are where |ψ grwa G is the ground state of the Rabi Hamiltonian under the GRWA (2), and |M (q G ) = D (q G /ω m ) |M is a phonon number state displaced by the quantity q G /ω m , where q G = g om g 2 ac /ω 2 c .We note that the state |ψ grwa G coincide with the ground state of the Rabi Hamiltonian under the adiabatic approximation (see Eq. (A11)) and contains exactly zero atom-cavity polaritons (N grwa R = 0).Figure 3(a) shows the GRWA energy (13) for several values of the phonon quantum number M as a function of the atom-cavity coupling, the corresponding RWA spectra [27], and the values obtained from a numerical evaluation of the spectrum of Hamiltonian (1), at resonance ω c = ω a = 5 ω m with g om = 0.1ω m .For small g ac , the RWA and GRWA energies coincide and both fit well the numerical results.At intermediate values of g ac the RWA lines remain constant while the GRWA curves follow the exact (numerical) results, with the agreement improving for larger couplings and for ω c = ω a .
The remaining part of the spectrum corresponds to the states of the Rabi-like polariton-phonon Hamiltonian Ĥ(N) (12).For each N , there will be an isolated state and a set of doublets with dressed polariton-phonon states, in analogy to the well known Jayne-Cummings or the GRWA spectrum [35].The former is given by where are the eigenkets of σx We added a prime label to emphasize the phononic nature of these displaced states in comparison with the displaced photonic states |N ± (see Eq.(A2)).The displacement of |M by q N arises from the introduction of the displaced phonon operators b N , while the shift by ±g ef f /ω m is a consequence of a further GRWA treatment of the Rabi Hamiltonian (12) ("GRWA-GRWA" approach).The isolated state (15) contains N grwa R = N +1 atom-cavity polaritons and, unlike those of a RWA-RWA treatment [27], is not separable.The corresponding energy is Note that the last term introduces a dependence on the coupling parameters g ac and g om , in remarkably contrast to the RWA approach.G (g ac ) for N = 0, 1, ..., 5 at resonance ω a = ω c .After an initial agreement between all the calculations for small coupling, only the GRWA curve (18) accurately follows the exact results for increasing coupling strength.The RWA-RWA presents a linearly decreasing behavior without any curvature, which it is apparent in the numerical calculation.It is clear how the GRWA method describes very precisely the energies even in the deep strong coupling.We also calculated some off-resonant ω c > ω a cases (not shown) and found that the deviation of the RWA-RWA results start for even smaller values of the coupling.
Finally, the spectrum of dressed polariton-phonon states, arising from the doublets structure in H (N ) (12), are given by Motivated by the GRWA of the QRM, we wrote these energies in terms of a polariton-phonon Rabi frequency and a polariton-phonon detuning, defined respectively as The eigenstates read as which are written in terms of the hybrid adiabatic basis The angle φ   ±,2 and their numerical counterpart, under resonant and off-resonant conditions respectively, in comparison with the RWA-RWA result.Within the GRWA, the fidelity lies above 80%, obtaining the highest value for the state |Ψ (0) −,2 , while the RWA fidelity shows a departure from the optimal coincidence for increasing values of coupling g ac .
The behavior of the spectrum E (N ) ±,M as a function of the optomechanical coupling g om , at resonance ω c = ω a , is shown in Fig. 4. At small g ac = 0.5 ω m (Fig. 4(a)), both approximations display a good agreement with the exact results.At higher values, g ac = 2.5 ω m , the RWA-RWA curves clearly fail, while the GRWA-GRWA energies with S = +1 follow closely the exact results (Fig. 4(b)).Note however that the branches with S = −1 show a moderate discrepancy.On the other hand, under the off-resonance condition ω c > ω a , all the spectra (not shown) and E (N ) G , approximate the numerical solution much better.In Ref. 35 the limited accuracy of the GRWA of the QRM for large positive detuning ω a − ω c (∼ ω c ) was discussed.By analogy, from (12) one could expect a limited precision for large T N − ω m .The increase of the g ac coupling implies the increase of the generalized GRWA Rabi frequency T N (see Eq.(A10)) in the range considered in Fig. 3, and the fairly good agreement observed in Fig. 3(c) means that the GRWA-GRWA approach works for a wider range of positive detuning T N − ω m .

III. BIPARTITE ENTANGLEMENT
In this section we address the entanglement properties of the GRWA states (22) and (23).The reduction of the hybrid Hamiltonian (1) to an effective spin-boson Hamiltonian (12), which describes a two polariton level system interacting with a displaced phononic field, allows to study the entanglement as that of a bipartite system.Thus, the Schmidt number criterion can be applied to reveal the non-separability of the states [43].To this end, we calculate the participation ratio ξ = 1/Tr(ρ 2 ) in order to quantify the degree of entanglement of the dressed GRWA states, where ρ is a reduced density matrix obtained by a partial trace on any of the subsystems.The tripartite entanglement properties of a JCM with optomechanical interaction was recently studied, although in the resonant weak coupling regime only [44].

A. Entanglement of QRM states
The density matrices ρgrwa where f + (x) = sin x and f − (x) = cos x, such that f 2 ± (α n /2) = (T N ± ∆ N )/2T N , and |ψ ad ±,N are the adiabatic states (A4).Taking the trace over the photonic subsystem, we obtain the reduced density matrices where The corresponding participation ratio is This expression resembles the participation ratio of the Jaynes-Cummings states which reads as where tan β N = −2g ac √ N + 1/(ω a − ω c ). Figure 5 shows a comparison between the results (28), (30), and the numerical, for a particular state.The participation ratio of the JC states predicts maximum entanglement for any coupling value, under resonance conditions ω c = ω a (Fig. 5(a)).In contrast, the GRWA states present an oscillatory behavior following that of the numerical solution.For the off-resonant case ω c = 2ω a (Fig. 5(b)), now ξ rwa N increase monotonically, suggesting increase of the entanglement.However, ξ grwa ±,N still behaves non-monotonically, fitting the numerical result closely.The observed oscillations of the GRWA result arise from the overlap between displaced photonic states (A3).

B. Entanglement of hybrid states
We consider now the entanglement between the atom-cavity polaritons and phonons.The density matrix for the dressed states ( 22) and ( 23) is Taking the trace over the phononic subsystem, we obtain the reduced density operator ±2f ± (φ , where the angle φ M is defined below the states ( 22) and ( 23), and the states |M in (17).This leads to the following participation ratio, The same procedure for the hybrid eigenstates within the RWA-RWA approach gives with the angle θ N is the well known generalized Rabi frequency of the Jaynes-Cummings model.
Figure 6(a) shows a color map of ξ ±,M (g ac , g om ) (34) for the eigenstate |Ψ +,2 at resonance ω a = ω c = ω m .For small values of g om , the polariton-phonon entanglement occurs maximally for a set of narrow ranges of g ac .For increasing magnitude of the optomechanical coupling the structure widens, displaying a larger region of values (g ac , g om ) with maximal entanglement ξ (2) +,2 ≈ 2. In order to explain this structure, we show in Fig. 6(b) a cut of the map with the line g om = 0.05ω m .The participation ratio ξ (2) ±,2 presents three peaks in the whole range of g ac , from the weak coupling to the deep strong coupling regime.This behavior can be understood by noting that when g ≈ 1 (see Eq.( 17)), and therefore Λ M ] (dashed line in Fig. 6(b)).It can be verified that the term cos φ (2) 2 develops a structure with three minima, arising from a combination of the Laguerre polynomials L 0 2 (x), L 0 3 (x), L 1 2 (x) with x = 2g (2) ef f /ω m , where g (2) ef f involves in turn the coupling g ac and the photonic overlaps 2 − |2 + , 3 − |3 + , and 2 − |3 + (see (A3)), through the frequency Ω N and the angle α 2 given by tan α 2 = −Ω 2,3 /∆ 2 .For increasing g (2) ef f (or g om ), the phononic overlaps in Λ (2) +,2 (34) start to deviate from trivial values (0 or 1), leading to the more structured map observed in Fig. 6(a), where ξ +,2 (g ac , g om ) ≈ 2. We can explain the behavior of the map shown in Fig. 6(a) as arising from the photonic overlaps and phononic overlaps.
In contrast, the corresponding RWA-RWA participation ratio ξ rwa N,M (g ac , g om ) (35) develops only one triangular structure with maximal entanglement, as is illustrated in Fig. 6(c).For a fixed value of g om , only one hump appears as a function of g ac .This is illustrated in Fig. 6(b) for small optomechanical coupling.In this case, the peak arises from the single zero of cos 2 θ match for weak coupling, as expected, but for strong coupling strengths only the GRWA-GRWA approach predicts maximum entanglement.The slight discrepancy between the first peaks predicted by GRWA and RWA calculations becomes more apparent when the frequency T N differs appreciably from R N .
In Figure 6(d) a cut of the map with the lines g ac = 0.3ω m and g ac = 1.5ω m are shown.For small g ac the RWA result grows monotonically and saturates close to ξ rwa 2,2 ≈ 2, while ξ +,2 starts to develop a hump.For the larger value of g ac , within the deep-strong-coupling regime, now ξ   A remaining part of the spectrum is the isolated state |Ψ (N ) G (15), for which For small (large) effective coupling g ef f , the overlap between oppositely displaced phonon states is close to one (zero) and thus ξ begins to be non-separable (separable).The corresponding RWA-RWA state exhibits an absence of entanglement for any value of g ac and g om .
It is clear that the zero atom-cavity polaritons states |Ψ M (14) are separable and thus there will be no polaritonphonon entanglement in both approaches, ξ M = 1.

IV. SUMMARY
Following the strategy of an approximation developed to be valid in the strong coupling regime of the quantum Rabi model, we apply a generalized rotating wave approximation (GRWA) to an atom-photon-oscillator system.This hybrid model was introduced by Restrepo et al. [26], with the atom-photon interaction given by the Jaynes-Cummings model, which assumes weak coupling and quasi-resonance (RWA).
We found that GRWA approach allows to break the full hybrid Hamiltonian into an infinite set of 2 × 2 blocks, each one described by a spin-boson model and characterized by a composed conserved quantity.Now the role of the twolevel atom is played by the dressesed atom-photon states (polaritons) and the bosonic part by the quantized mechanical modes.This effective JCM is treated in turn by another GRWA, in order to go beyond the RWA conditions.Through this double approximation, we obtained analytical expressions for the energies which show a very good agreement with the numerical exact solution in a wide range of atom-photon couplings, from the weak to the deep-strong-coupling regime, and for large detunings.Importantly, the use of the adiabatic basis states involve the interference between oppositely displaced photonic states and between displaced phononic states.As a consequence, the effective couplings, the generalized Rabi frequencies and detunings introduce oscillations in the energies, which follow closely those of the numerical solution.These oscillations are absent in the RWA-RWA treatment of the hybrid model [26].On the other hand, as a function of the photon-phonon coupling, the agreement is moderate for large values, but still much better than the RWA-RWA result.Similarly, the calculated eigenstates also demonstrate the good quality of the GRWA-GRWA approach, showing good fielity when overlaped with the numerical states.Because of the analytical character of the approach, we were able to obtain also a closed form of the participation ratio of the GRWA-GRWA states of the bipartite Hamiltonian.This quantity displays a structure with maxima and minima, in strong contrast to the monotonic behavior of the RWA-RWA results.The participation ratio predicts polariton-phonon entanglement in coupling regions where the solution based on the RWA does not.
A number of experiments on a variety of settings have demonstrated the emergence of novel optical phenomena in the ultrastrong and deep-strong coupling regime of light-matter interaction, as described by the quantum Rabi model [32].Thus, in a quantum platform combining cavity QED and cavity optomechanics like that studied here, new effects can be expected in the large coupling regime of the atom-photon-phonon interaction.Our work might be useful about this.Geometric phases, quantum correlations, quasiparticles statistics, polariton-assisted cooling of the mechanical motion, among other properties of the quantum dynamic of the strongly coupled hybrid system, are some of the phenomena that could be investigated, taking our work as a basis.We hope it encourages further studies.

Adiabatic Approximation basis |ψ ad
±,N .The adiabatic approximation for the QRM assumes that the photonic frequency is much larger than the atomic frequency, ω a ω c .In the displaced basis |±x, N ± ≡ |s, N s (s = ±), the restoring of the energy separation ω a σz /2 in Ĥωa=0 The mixing occurs only between states displaced oppositely (s = s), that is involves only the overlaps M ± |N ∓ .
The adiabatic approximation consists of truncating the matrix to the block diagonal form, each block involving levels in opposite wells with the same energy, N ± |N ∓ .The assumption ω a ω c means that a transition in the two-level system can never excite the photonic field, and in turn means that the mixing occurring between displaced levels with different photon numbers (M = N ) can be ignored [34].The overlap of two oppositedisplaced Fock states is given by The Laguerre polynomials introduces a rippled structure on the smooth variation of the energy as a function of g ac /ω c .It is verified that N − |N + → 0 when g ac /ω c → ∞, which corresponds to two uncoupled identical harmonic oscillators and pairwise degenerate energy levels.The states (A4) appear in the GRWA conserved number N grwa R (4).The first step in the derivation of the GRWA is to write the Rabi Hamiltonian ĤR in the adiabatic basis (A4), instead of the eigenbasis |±z, N of the non-interacting Hamiltonian [35].In such a representation, ĤR becomes, in matrix form where Ω N,N = ω a N − |N + is the atomic frequency renormalized by the overlaps (A3).The order of columns and rows is |ψ ad −,0 , |ψ ad +,0 , |ψ ad −,1 , |ψ ad +,1 , . ... The following step is to proceed as the RWA approach for the QRM.This means the neglecting of the remote matrix elements (the "non-conserving energy", "counter-rotating", or "anti-resonant" terms), which involve two or more excitations, for example ψ ad −,0 | ĤR |ψ ad +,1 = −Ω 0,1 /2.In terms of the interaction picture, with respect to the non-interacting Hamiltonian, the discarded terms are those involving two or more net excitations, which oscillate very fast.This reduces the matrix to a 2 × 2 block-diagonal form (A7) Shif t ≡ gom2 Ω N ωc sin α N − cos α N is about an order of magnitude less than the generalized GRWA Rabi frequency T N , leaving the σ (N ) z term just as an

Figure 3 (
b) displays the function E (N )

M
is defined by tan φ (N )M = −Ω N M,M +1 /∆ N M .The energy levels E (N ) ±,M , at resonance ω a = ω c and out of resonance ω c > ω a , are shown in figures 3(c) and 3(d), respectively.The GRWA-GRWA energies(19) reproduce very well the exact (numerical) calculation in the whole range of couplings g ac .The discrepancy of the RWA-RWA curves is clearly visible.

R
lifts its degeneracy and introduces non-diagonal terms,s , M s | ĤR |s, N s = E N δ s s δ M N + (1 − δ s s ) M s |N s ω a /2.