Analytical Solution for the Anisotropic Rabi Model: Effects of Counter-Rotating Terms

The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.

cant key for us to understand the characters of the counter-rotating terms. The independence of two coupling constants allows us to explore the effects of the counter-rotating terms, while in the original Rabi model it is hard to separate the influences of two types of interaction terms. Although this model can be solved exactly 19 by the method originally developed by Braak 12 , the results are strongly dependent of the composite transcendental function defining through its power series in the interacting strength and the frequency, and are difficult to extract the fundamental physics of the model.

Elimination of the counter-rotating terms
We now begin to eliminate the counter-rotating terms. The Hamiltonian of the anisotropic Rabi model has the equivalent form H~va z azVs z zg 1 s x aza z ð Þzig 2 s y a z {a ð Þ, ð3Þ where g 1 5 (g 1 g9)/2 and g 2 5 (g9 2 g)/2. We note that the anisotropic Rabi model also possesses Z 2 -symmetry. If we define a parity operator P~s z e ipa z a , it is obvious that [H, P] 5 0. Thus the state space can be decomposed into two subspaces H + , where the parity operator P has eigenvalues 61 respectively. Now we apply the unitary transformation U 5 exp[ls x (a 1 2 a)]. Here l is the dimensionless parameter determined by the later calculations. By performing this transformation, the Hamiltonian becomes H9 5 UHU 1 The operator cosh[2l(a 1 2 a)] can be expanded as: The matrix elements in Eq. (8) can be calculated directly by using the formula From Eq. (9) we know that the remote matrix elements in Eq. (8) are the high-order terms of l. When l is much smaller than 1, these terms can be discarded and only the diagonal elements are retained. Then we have: This approximation can be interpreted as neglecting the multiphoton process in the effective Hamiltonian H9. When l=1, the multi-photon process is relatively weak and can be ignored. Following the same approximation procedure: As a result, the effective Hamiltonian reduces to the form where The expressions of G n and R n can be derived straightforwardly by using the Eq. (9) Here L n (y) is the Laguerre polynomial and L i n y ð Þ is the associated Laguerre polynomial.
Since s x 5 s 1 1 s 2 , 2is y 5 2s 1 1 s 2 , the effective Hamiltonian becomes Now we have transformed the original Hamiltonian into a form in which the counter-rotating terms can be adjusted by tuning the dimensionless parameter l.

Energy Spectrums and Wavefunctions
We now begin to derive the energy spectrums of the anisotropic Rabi model. In Eq. (19), the counter-rotating terms s 2 jn 2 1aeAEnj 1 s 1 jnaeAEn 21jcan be eliminated if the dimensionless parameter l is chosen as When n 5 1, the elimination of the term s 2 j0aeAE1j 1 s 1 j1aeAE0j makes {j2z, 0ae} becomes an invariant subspace of H9. Then the ground-state energy can be written as The ground-state wavefunction can be carry out immediately as Since l n11 ? l n21 , it seems unable to eliminate the terms s2 jn 2 2aeAEn 2 1j 1 s 1 jn 2 1aeAEn 2 2j and s 2 jnaeAEn 1 1j 1 s 1 jn 1 1aeAEnj simultaneously. But the numerical results show that jl n11 2 l n j and jl n21 2 l n j is much smaller than l n , so we can neglect the difference between l n61 and l n , and choose l 5 l n , then {j1z, n21ae, j2z, nae} becomes an invariant subspace of H9. When written in the basis of the states j1z, n 2 1ae, j2z, nae, the Hamiltonian becomes And the excited-state energy can be given by The eigenvectors for H n are given by where Then we obtain the excited-state wavefunctions: Since the anisotropic Rabi model also possesses Z 2 -symmetry, the inexistence of level crossings within the subspaces H + allows us to label each eigenstate with two quantum numbers jn 0 , n 1 ae, the same as the case of the Rabi model in the article 12 . The parity quantum number n 0 takes the values 61 which corresponds to the subspaces H + . Within each subspace the states are labeled with the quantum number n 1 5 0, 1, 2, …. Using this notation, our analytical state can be labelled as Fig. 1 shows the lowest part of the energy spectrum from our analytical results for g9 5 2g and g9 5 g/2 respectively. For comparison purposes, the energy spectrum obtained from the numerical calculations is also shown. In this figure we find that the analytical energy spectrum agrees well with the numerical results both for g9 . g and g9 , g in the regime g # 0.5. In the case of g 5 g9, the model returns to the original Rabi model and has been discuss by the article 11 .
It is necessary to discuss the valid parameter regime of our approximation. Our approximation procedures in Eqs. (10)(11)(12)(13) require the dimensionless parameter l to be less than 1. When l approaches 1, this approximation is no longer valid and the analytical results start to fail considerably. In Fig. 2, the dimensionless parameter l 5 l 1 as a function of the interaction strength g and g9 are plotted. In the regime of l # 0.5 our approximation results have a good agreement with the numerical results.

Limit of the weak counter-rotating coupling
We now discuss a particular interesting and significant situation of g 0 =v, which corresponds to the weak counter-rotating coupling limit. When the anisotropic Rabi model return to the JC model (g9 5 0), it is obvious that l 5 0. This indicates that we have l=1 when g 0 =v. This can also be confirmed in Fig. 2.
In the previous discussion, we know that l satisfies the equation Since L n m x ð Þ< mzn m when x=1, by neglecting the high-order terms of l, the Eq. (31) reduces to the form g 1 2 lv 2 2Vl 1 g 2 5 0. It leads to the solution: Now we have obtained the analytical expression of l when l=1. Therefore, in the weak counter-rotating coupling limit, the dimensionless parameter l is proportional to g9, this reveals the physical meaning of l. In the anisotropic Rabi model, g9 describes the 'absolute deviation' from the JC model, which is the coupling strength of the counter-rotating wave interaction. Then we may regard l as the 'relative deviation' from the JC model.
When l=1, we have where Dv~4g 2 l, DV~2g 2 l, Here we have obtained a modified JC model with an additional term s z a 1 a. The counter-rotating terms have been considered as shifts to the parameters instead of being abandoned directly in the RWA. This is an improved approximation compares to the RWA.
This modification of JC model allows us to discuss the validness of RWA. In the regime of near resonance 2V < v, we have Dg < 0. Hence it is reasonable to ignore the shift to the coupling strength, so RWA captures the changes of atom-field interactions very good in this regime. When g 0 =v, such as the optical cavity with strong coupling, the shifts to the other parameters is relatively small, then the RWA becomes a successful approximation. When the system is under ultrastrong coupling (g , 0.1v), the shifts to the other parameters become nonnegligible. Therefore, the effects of counter-rotating terms begin to appear and RWA fails to capture it, such as the Bloch-Siegert shift. The effects of counter-rotating terms on physics quantities Based on the energy spectrums and the wavefunctions, we are able to derive the corresponding physics quantities, and discuss the effects of counter-rotating terms on them. Firstly we calculate the Bloch-Siegert shift, which is the energy shift of the level transition due to the counter-rotating terms 21 . The Bloch-Siegert shift with the transition E 12 R E G can be calculated immediately as It is obvious that d 5 0 when the anisotropic Rabi model returns to the JC model (l 5 0). In Fig. 3 we show the absolute value of the Bloch-Siegert shift with the ransition E 12 R E G as a function of g and g9. In this figure, the analytical results agree perfectly with the numerical calculation even when l,0.6.
The mean photon number AEa 1 aae and AEs z ae can also be evaluated. For the ground state, they can be given by