Introduction

The Rabi model1, one of the simplest model which deals with the matter-light interaction, describes the interaction of a two-level atom with a single bosonic mode. This ubiquitous model is applied to a great variety of physical systems, such as microwave and optical cavity quantum electrodynamics (QED)2, ion traps3, quantum dots and circuit QED4,5,6,7. The Rabi model Hamiltonian reads

where a+ and a are creation and annihilation operators of the mode with frequency ω, σx,y,z are the Pauli spin operators associated to a two-level system with transition frequency 2Ω and g is the interaction strength. For simplicity, we already take the unit of .

Although much attention has been paid over the last decades, until now the exact analytical solution of Rabi model is still lacking. To overcome this problem, the most used analytical method is the rotating-wave approximation (RWA), where the counter-rotating terms a+σ+ + aσ− is neglected. This approximation leads to the Jaynes-Cummings (JC) model8, which can be solved exactly. In the regime of near resonance and relatively weak coupling, the JC model captures the quantum dynamics of Rabi model successfully. However, solid-state systems such as superconductor systems have allowed the coupling strength to reach the ultrastrong coupling regime (g/ω~0.1)6. In this regime, the counter-rotating terms are no longer negligible. This leads to the break-down of RWA and the system can only be described by the Rabi model. Consequently, a variety of method has been developed to go beyond the RWA9,10,11,12,13,14,15,16,17,18.

Recently, a generalization of the Rabi model called the anisotropic Rabi model was proposed19,20. The Hamiltonian of anisotropic Rabi model can be written as:

Here σ± = (σx ± iσy)/2, g and g′ denote the coupling strength of the rotating terms σ+a + σ−a+ and counter-rotating terms σ−a + σ+a+ respectively. In this model, the coupling strength of the rotating wave interaction is different from that of the counter-rotating wave interaction. The anisotropic Rabi model includes the JC model (g′ = 0) and the original Rabi model (g′ = g) and has the applications to different physical fields, such as quantum optics, solid state physics and mesoscopic physics19. Besides these practical applications, this model is a significant 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 exactly19 by the method originally developed by Braak12, 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

where g1 = (g + g′)/2 and g2 = (g′ − g)/2. We note that the anisotropic Rabi model also possesses -symmetry. If we define a parity operator , it is obvious that [H, P] = 0. Thus the state space can be decomposed into two subspaces , where the parity operator P has eigenvalues ±1 respectively.

Now we apply the unitary transformation U = exp[λσx(a+ − a)]. Here λ is the dimensionless parameter determined by the later calculations. By performing this transformation, the Hamiltonian becomes H′ = UHU+ + H1 + H2 + H3 + H4, where

The operator cosh[2λ(a+ − 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 λ. When λ 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 multi-photon process in the effective Hamiltonian H′. When , 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 Gn and Rn can be derived straightforwardly by using the Eq. (9)

Here Ln(y) is the Laguerre polynomial and is the associated Laguerre polynomial.

Since σx = σ+ + σ−, −iσy = −σ+ + σ−, 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 λ.

Energy Spectrums and Wavefunctions

We now begin to derive the energy spectrums of the anisotropic Rabi model. In Eq. (19), the counter-rotating terms σ−|n − 1〉〈n| + σ+|n〉〈n −1|can be eliminated if the dimensionless parameter λ is chosen as

When n = 1, the elimination of the term σ−|0〉〈1| + σ+|1〉〈0| makes {|−z, 0〉} becomes an invariant subspace of H′. Then the ground-state energy can be written as

The ground-state wavefunction can be carry out immediately as

Since λn+1 ≠ λn−1, it seems unable to eliminate the terms σ−|n − 2〉〈n − 1| + σ+|n − 1〉〈n − 2| and σ−|n〉〈n + 1| + σ+|n + 1〉〈n| simultaneously. But the numerical results show that |λn+1 − λn| and |λn−1 − λn| is much smaller than λn, so we can neglect the difference between λn±1 and λn and choose λ = λn, then {|+z, n−1〉, |−z, n〉} becomes an invariant subspace of H′. When written in the basis of the states |+z, n − 1〉, |−z, n〉, the Hamiltonian becomes

And the excited-state energy can be given by

The eigenvectors for Hn are given by

where

Then we obtain the excited-state wavefunctions:

Since the anisotropic Rabi model also possesses -symmetry, the inexistence of level crossings within the subspaces allows us to label each eigenstate with two quantum numbers |n0, n1〉, the same as the case of the Rabi model in the article12. The parity quantum number n0 takes the values ±1 which corresponds to the subspaces . Within each subspace the states are labeled with the quantum number n1 = 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 g′ = 2g and g′ = 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 g′ > g and g′ < g in the regime g ≤ 0.5. In the case of g = g′, the model returns to the original Rabi model and has been discuss by the article11.

Figure 1
figure 1

The lowest part of the energy spectrum of the anisotropic Rabi model as a function of the interaction strength g for ω = 1, Ω = 0.3 and (a) g′ = 2g, (b) g′ = g/2.

In all figures, the solid lines and dotted lines correspond to the analytical results and numerical results respectively.

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It is necessary to discuss the valid parameter regime of our approximation. Our approximation procedures in Eqs. (10–13) require the dimensionless parameter λ to be less than 1. When λ approaches 1, this approximation is no longer valid and the analytical results start to fail considerably. In Fig. 2, the dimensionless parameter λ = λ1 as a function of the interaction strength g and g′ are plotted. In the regime of λ ≤ 0.5 our approximation results have a good agreement with the numerical results.

Figure 2
figure 2

The dimensionless parameter λ = λ1 as a function of the interaction strengths g and g′ for ω = 1, Ω = 0.3.

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Limit of the weak counter-rotating coupling

We now discuss a particular interesting and significant situation of , which corresponds to the weak counter-rotating coupling limit. When the anisotropic Rabi model return to the JC model (g′ = 0), it is obvious that λ = 0. This indicates that we have when . This can also be confirmed in Fig. 2.

In the previous discussion, we know that λ satisfies the equation

Since when , by neglecting the high-order terms of λ, the Eq. (31) reduces to the form g1 − λω − 2Ωλ + g2 = 0. It leads to the solution:

Now we have obtained the analytical expression of λ when . Therefore, in the weak counter-rotating coupling limit, the dimensionless parameter λ is proportional to g′, this reveals the physical meaning of λ. In the anisotropic Rabi model, g′ describes the ‘absolute deviation’ from the JC model, which is the coupling strength of the counter-rotating wave interaction. Then we may regard λ as the ‘relative deviation’ from the JC model.

When , we have

On substituting Eq. (33) into the effective Hamiltonian Eq. (19), we obtain

which can be written as

where

Here we have obtained a modified JC model with an additional term σza+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 2Ω ≈ ω, we have Δg ≈ 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 , 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 ~ 0.1ω), 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 terms21. The Bloch-Siegert shift with the transition E1− → EG can be calculated immediately as

It is obvious that δ = 0 when the anisotropic Rabi model returns to the JC model (λ = 0). In Fig. 3 we show the absolute value of the Bloch-Siegert shift with the ransition E1− → EG as a function of g and g′. In this figure, the analytical results agree perfectly with the numerical calculation even when λ~0.6.

Figure 3
figure 3

The absolute value of the Bloch-Siegert shift with the transition E1− → EG as a function of g and g′ for ω = 1, Ω = 0.3.

The part (a) represents the analytical results, while the part (b) corresponds to the numerical results.

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The mean photon number 〈a+a〉 and 〈σz〉 can also be evaluated. For the ground state, they can be given by

For the excited state, we have

These results enable us to evaluate the number of polaritons . In the JC model, N is a conservation quantity and leads to the U(1)-symmetry. In the presence of the counter-rotating terms, the U(1)-symmetry is broken down to the -symmetry and N is no longer conserved. The mean number of polaritons of the ground state can be given by

The Eq. (42) shows that 〈N〉G is increased with λ. This is due to the transition from |−z, 0〉 to the upper level caused by the counter-rotating terms. In the JC model, this kind of transition is forbidden since no counter-rotating terms. This transition makes the ground state becomes the superposition state of |−z, 0〉 and the other states. Therefore, the mean number of polaritons of the ground state is increased with the coupling strength of the counter-rotating terms.

The uncertainty ΔN of the ground state can also be given as

From Eq. (43), We can see that as the counter-rotating terms arise, the uncertainty ΔN is no longer zero, which indicates the break-down of the U(1)-symmetry.

Conclusions

In this work, we have eliminated the counter-rotating terms approximately and obtained the analytical energy spectrum and wavefunctions. 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 JC model. This modification of JC model shows the validness of RWA on the assumption of near-resonance and relatively weak coupling. Finally, we derive the analytical expressions of the Bloch-Siegert shift and the uncertainties of the number of polaritons. The results show the break-down of the U(1)-symmetry and the deviation from the JC model.