Exact solution of the two-mode quantum Rabi model

Qiong-Tao Xie

Communications in Theoretical Physics ›› 2020, Vol. 72 ›› Issue (6) : 65105.

PDF(355 KB)
Welcome to visit Communications in Theoretical Physics, May 8, 2025
PDF(355 KB)
Communications in Theoretical Physics ›› 2020, Vol. 72 ›› Issue (6) : 65105. DOI: 10.1088/1572-9494/ab8a1f
Quantum Physics and Quantum Information

Exact solution of the two-mode quantum Rabi model

Author information +
History +

Abstract

An analytical method is developed to study the two-mode quantum Rabi model. For certain specific parameter conditions, especially for the resonant conditions, we obtain an infinite number of the exact solutions of the eigenfunctions and associated energies. It is shown that there exist new types of the exact energies which do not correspond to the level-crossings. Our analytical method may find applications in some related models.

Key words

exact solutions / two-mode quantum Rabi model / level-crossings

Cite this article

Download Citations
Qiong-Tao Xie. Exact solution of the two-mode quantum Rabi model[J]. Communications in Theoretical Physics, 2020, 72(6): 65105 https://doi.org/10.1088/1572-9494/ab8a1f

1. Introduction

Since the exact solutions to the quantum Rabi model (QRM) have been presented [14], the search of the analytical solutions has attracted renewed interest of research in some related physical models, such as the anisotropic QRM [5, 6], the two-photon QRM [711], the two-mode QRM [1216], and several related models [1721]. These models can provide a good theoretical description of the interaction between matter and light in different situations. Parts of these models have been realized in a variety of physical systems [22, 23].
Recent studies have shown that these models have two interesting features. Firstly, for some specially chosen parameter conditions, there are an infinite number of the exact energies and associated eigenfunctions. The exact eigenfunctions are usually given in terms of elementary functions [13, 5]. In the two-photon QRM, certain exact eigenfunctions are even expressed by some special functions [11]. For general parameter conditions, the energies and associated eigenfunctions cannot be obtained algebraically and explicitly. Secondly, it was shown that if the underlying parity symmetries in these models are not broken by some additional terms, all the known exact energies are related to the level-crossings.
The two-mode QRM is an important extension of the QRM. In Bargmann Hilbert spaces of entire functions, the Bethe ansatz method and the algebraic method have been applied to obtain the exact parts of the energy spectrum associated with polynomial eigenfunctions [12, 15]. Later, these exact energies have been obtained by using extended squeezed states. The resulting eigenfunctions are given by a finite expansion of the extended squeezed states [13, 16].
In the present work, we develop a different analytical method to study the solutions of the two-mode QRM. The eigenfunctions are expanded as a series of the confluent hypergeometric functions (CHFs). It is shown that when the parameters satisfy certain specific conditions, the series expansion become finite, thereby resulting in an infinite number of the exact solutions of the energies. In particular, it is found that at the resonant conditions, there are also an infinite number of the exact energies. In a comparison with the previously existing exact energies, it is further found that the obtained exact results contain the previous results as well as the new results. These new exact energies are associated the non-polynomial eigenfunctions, and do not correspond to the level-crossings. Therefore, our analytical results show that there are two types of exact eigenfunctions, polynomial and non-polynomial eigenfunctions. The polynomial eigenfunctions are associated with the level-crossings, and the non-polynomial eigenfunctions are not. In addition, it is shown that the analytical method may be applied to present the exact energies and eigenfunctions of some related physical models.

2. Exact solutions of the two-mode QRM

We consider the two-mode QRM described by the following Hamiltonian (ℏ = 1)
H=ω(a1a1+a2a2)+Δσz+gσx(a1a2+a1a2),
(1)
where a1,2 and a1,2 are the creation and annihilation operators for two bosonic modes with the same frequency ω, σx,z are the usual Pauli matrices, 2Δ is the energy separation between the two levels, and g denotes the interaction strength. In the two-mode QRM, the eigenstate |ψ is expanded as
|ψ=ψ1(a1,a2)|0|+ψ2(a1,a2)|0|.
(2)
Here as in the QRM, ψ1,2 must be analytical entire functions of the creation operators a1,2. |0|01|02 is the vacuum state for the two bosonic modes. | and | are the eigenstates of σz with eigenvalues 1 and −1. Substituting this expansion into the Schrödinger equation H|ψ=E|ψ, we get the eigenvalue equation
([H,ψ1]+ψ1HEψ1)|0|+([H,ψ2]+ψ2HEψ2)|0|=0.
(3)
By applying the well-known relations, a1,2|0=0, [a1,ψ1,2]=0, [a2,ψ1,2]=0, [a1,ψ1,2]=ψ1,2/a1, and [a2,ψ1,2]=ψ1,2/a2, together with the linear combinations, φ1=(ψ1+ψ2)/2 and φ2=(ψ1ψ2)/2, we have two coupled operator-type differential equations
z1φ1z1+z2φ1z2+g2φ1z1z2+(gz1z2E)φ1+Δφ2=0,
(4)
z1φ2z1+z2φ2z2g2φ2z1z2(gz1z2+E)φ2+Δφ1=0,
(5)
where z1=a1, z2=a2, and we have set ω = 1 for brevity. It is evident that all the terms in this equation are mutually commutable. Aa a result, one can formally regard them as c-number differential equations [24, 25]. In addition, the two-mode QRM has a conserved operator
C=14(n1n2+1)(n1n21),
(6)
where n1=a1a1 and a2a2 are the photon-number operators of mode 1 and mode 2, respectively. This conserved operator allows us to decouple the Fock–Hilbert space into the direct sum of infinite number of subspaces labeled by κ=(n1n2+1)/2 [12, 15]. Here κ is usually called as the Bargmann index, and can take any positive integers or half-integers, κ=1/2,1,3/2,. Due to the presence of the conserved operator C, φ1,2 can take the form
φ1,2=z12κ1f1,2(z),
(7)
with z=z1z2. After a simple calculation, we have the coupled equations for f1,2(z)
gzd2f1dz2+(2κg+2z)df1dz+(2κ1E+gz)f1+Δf2=0,
(8)
gzd2f2dz2+(2κg2z)df2dz+(12κ+E+gz)f2Δf1=0.
(9)
It is easily found that if (f1(z), f2(z)) is a solution, (f2(z),f1(z)) is also a solution. This property can simplify our discussions. If we further take the following transformations
f1,2(z)=eηzϕ1,2(y),y=ξz,
(10)
we get
yd2ϕ1dy2+(2κ+λ1+y)dϕ1dy+(λ2++λ3+y)ϕ1+Δgξϕ2=0,
(11)
yd2ϕ2dy2+(2κ+λ1y)dϕ2dy+(λ2+λ3y)ϕ2Δgξϕ1=0,
(12)
where
λ1±=2ηg±2gξ,
(13)
λ2±=2ηgκ±(2κ1E)gξ,
(14)
λ3±=gη22η+ggξ2.
(15)
Here the two parameters η and ξ are constants to be determined.
The power series expansion is a conventional method to solve equations (11) and (12) by expanding φ1,2 by power series of y with unknown coefficients. This method has been used for the QRM and its related models. For a general parameter condition, one has a higher-order recurrence relation for the coefficients. If the model parameters satisfy certain special condition, the recurrence relation can be terminated. Such termination leads to the exact energies in many Rabi-related models. In the two-mode QRM, by using the power series expansion, we get the exact energies [12, 15]
EN,κIω=1+2(N+κ)1g2ω2.
(16)
Here N ≥ 1 , η±=±(ωω2g2)/g, and the parameter ξ can be chosen as an arbitrary non-zero constant. This exact energies were first found in [12, 15] via different methods, and correspond to the level-crossings in the energy spectrum [13]. The exact energies exist for certain specific parameter relations. For example, in the cases of N = 1, 2, the parameter relations are given as
Δ2ω2+4(2κ+1)g2ω24=0,
(17)
Δ4ω4+(4(7+6κ)g2ω220)Δ2ω2+128(1+κ)g2ω21)2+64((1+κ)g4ω41)=0.
(18)
It is evident that if we plot the energies as a function of g/ω with the fixed values of Δ/ω and κ, the exact energies (16) appear in some isolated values of g/ω [13].
In this work, our aim is to find new expansion functions which allow us to analytically obtain more exact energies. It is observed that under the condition of λ3±=0 and Δ = 0, equations (11) and (12) decouple, and have a similar form with the confluent hypergeometric equation [26]. This motivates us to search for solutions with a series expansion in terms of the CHFs. The series expansion in terms of the CHFs has been used in different systems [2730]. We shall show that under certain special parameter conditions, the series expansion can be terminated. This gives the exact analytical results of the energies. In the following, we discuss the series expansion of the solutions of equations (11) and (12) in terms of the CHFs.

2.1. Series expansion in terms of CHFs

The series expansion of the solutions to equations (11) and (12) in terms of the CHFs is given as
ϕ1(y)=n=0an F(αn,γ,y),  ϕ2(y)=n=0bnF(αn,γ,y),
(19)
where F(αn, γ, y) is the CHF with αn=α0+n. Here α0 and γ are constants to be determined. After substituting these expansions into equations (11) and (12), and collecting the like terms, we obtain the recurrence relation for the coefficients an and bn (see details in appendix A)
An+1Wn+1+BnWn+Cn1Wn1=0,
(20)
with
Wn=(anbn).
(21)
It is a hard task to obtain an analytical solution for this higher-order recurrence relation. However under the condition
Wn=0  if  n>N
(22)
one may cut off the recurrence expressions, and solve them step by step. Such a termination condition may allow us to find the exact analytical results of the energies.

2.2. Exact energies

From the terminated condition (22), we have the explicit expressions of the exact energies (see details in appendix B)
EN,κIIω=1+N1g2ω2.
(23)
Here N ≥ 1, η±=ξ±/2=±(ωω2g2)/g, α0=κN/2, and γ = 2κ. The exact energies exist for certain specific parameter relations. For example, in the cases of N = 1, 2, 3, we have
(Δω2κ+1)(Δω+2κ1)=0,
(24)
Δ4ω4+4(g2ω21+2(1κ)κ)Δ2ω2+16(1κ)2κ2=0,
(25)
Δ3ω3±(12κ)Δ2ω2+(8g2ω24(12κ)2)Δω±(4κ21)(2κ3)=0.
(26)
Now we discuss the relation between the two exact energies EN,κI and EN,κII. Due to 2(N+κ)3 in EN,κI, it follows that the exact energies EN,κII with N = 1,2 are beyond the exact energies EN,κI. In addition, for certain given values of N and κ, the resulting CHFs become polynomial. In such situations, we get
EN,κI=E2(N+κ),κII.
(27)
This result indicates that all the exact energies EN,κI are contained in the exact energies EN,κII. For example, from the two parameter relations (17) and (26), we have E1,1/2I=E3,1/2II. But the exact energies E3,κII with κ>1/2 are not included in the exact energies EN,κI. These obtained new exact energies are not found previously. In the following, we shall show that they do not correspond to the level-crossings.
We first study the interesting exact energies EN,κII with N = 1, since the parameter g/ω does not appear in the parameter relation (24). Without loss of generality, we take Δ > 0. From the parameter relation (24), we have the parameter relation Δ/ω = (2κ − 1) corresponding to the resonant conditions. This means that at the resonant conditions, there exist the exact energies E1,κII/ω=1+1g2/ω2. The corresponding solutions are not expressed in terms of elementary functions (see details in appendix C). We note that in the two-photon QRM, there also exists the exact energy at the resonant condition (see details in appendix D). The reason behind this deserves to be studied in future. In figure 1, we show the numerical and exact analytical results of the energy spectrum of the lowest energy levels as a function of g/ω with ω = 1 and Δ/ω = 1. This choice of Δ/ω = 1 corresponds to the resonant condition of Δ/ω = (2κ − 1) with κ = 1. Here we discuss four sets of the exact energies, marked by square, triangles, circles, and dots. The square is for the exact energy E2,1II. The triangles denote the exact energies E3,κII with κ=1/2,1,3/2. The circles denote the exact energies E1,1II. The dots denote the exact energies E1,κI with κ = 1/2, 1. It is seen from the analytical and numerical results that the exact energies E2,1II and E3,3/2II do not correspond to the level-crossings. Furthermore, E1,1II shows a continuous dependence on the parameter g/ω up to g/ω = 1 where it meets E3,1II. In addition, as shown figure 2, we observe that depending on the chosen values of Δ/ω and κ, the position of the exact energies E1,κII changes. Therefore, these numerical and analytical results demonstrate that the two-mode QRM supports new exact energies which are not associated with the level-crossings.
Figure 1. Energy spectrum of the two-mode QRM as a function of g/ω with ω = 1 and Δ/ω = 1. The square denotes the exact energy E2,1II. The triangles denote the exact energies E3,κII with κ=1/2,1,3/2. The circles denote the exact energies E1,1II. The dots denote the exact energies E1,κI with κ=1/2,1.

Full size|PPT slide

Figure 2. Energy spectrum of the two-mode QRM as a function of g/ω for two different values of (a) Δ/ω = 2 and (b) Δ/ω = 3 with ω = 1. In (a), the circles denote the exact energies E1,3/2II at the resonant condition of Δ/ω=2κ1=2, and in (b), the circles denote the exact energies E1,2II at the resonant condition of Δ/ω=2κ1=3.

Full size|PPT slide

3. Conclusion

In conclusion, the series expansion of the eigenfunctions in terms of the CHFs has been introduced to study the two-mode QRM. The conditions for the finite series expansion have been applied to find an infinite number of the exact energies. Depending on whether they are associated with the level-crossings, two different types of the exact energies are identified. They correspond to the polynomial and non-polynomial eigenfunctions, respectively. In particular, it has been found that there exist the exact energies at the resonant conditions.
This analytical method has been applied to present the exact energies for the two-photon QRM. It is believed that our analytical method could be used to find new exact energies associated with the non-polynomial eigenfunctions in some related physical models, such as the anisotropic two-photon QRM, and the anisotropic two-mode QRM [16]. In addition, one could use the new exact energies to discuss the spectral collapse phenomenon [13]. It is an interesting problem to explore whether there is a hidden Lie algebraic structure which explains the existence of the non-polynomial eigenfunctions in the two-mode QRM [15].

Acknowledgments

Supported by the National Natural Science Foundation of China under Grant No. 11965011.

Appendix A. Derivation of the recurrence relation (20)

In this section, we present a detailed derivation of the recurrence relation (20). Substituting the series expansion ϕ1(y)=n=0anun(y) and ϕ2(y)=n=0bnun(y) with un(y)=F(αn,γ,y) into equations (11) and (12) gives
n=0an(yd2undy2+(2κ+λ1+y)dundy+(λ2++λ3+y)un)+Δgξn=0bnun=0,
(A1)
n=0bn(yd2undy2+(2κ+λ1y)dundy+(λ2+λ3y)un)Δgξn=0anun=0.
(A2)
We use the following relations [26]
yd2undy2=(yγ)dundy+αnun,
(A3)
ydundy=αn(un+1un),
(A4)
yun=(αnγ)un1+(γ2αn)un+αnun+1.
(A5)
To further simplify the above equations
n=0an(2κγ)dundy+Qn+un+1+Pn+un+Rn+un1)+Δgξn=0bnun=0,
(A6)
n=0bn(2κγ)dundy+Qnun+1+Pnun+Rnun1)Δgξn=0anun=0,
(A7)
with
Rn±=λ3±(αnγ),Pn±=λ3±(γ2αn)+λ2±λ1±αn,Qn±=(1+λ1±+λ3±)αn.
Since dun/dy cannot be expressed as a linear combination of functions un, we demand γ=2κ. After equating coefficient of un(y) = F(αn, γ, y), we have
Rn+1+an+1+Pn+an+Δgξbn+Qn1+an1=0,
(A8)
Rn+1bn+1+PnbnΔgξan+Qn1bn1=0.
(A9)
If we let
Wn=(anbn),
(A10)
we have the recurrence relation (20)
An+1Wn+1+BnWn+Cn1Wn1=0,
(A11)
where An, Bn and Cn are given as
An=(Rn+00Rn),Bn=(Pn+ΔgξΔgξPn),Cn=(Qn+00Qn).

Appendix B. Derivation of the exact energies in equation (23)

In this section, a detailed derivation of the exact energies (23) is given. By setting n = N + 1 in the recurrence relation (20), we have CNWN=0. For non-zero solution of aN and bN, the determinant of the matrix CN must vanish, thereby resulting in the condition, (1+λ1++λ3+)(1+λ1+λ3)αN2=0. If we choose αN=0. we get α0=N corresponding to the power series expansion. We do not consider this situation. By setting n = − 1 in the recurrence relation (20), we have A0W0=0, and thus obtain the condition, λ3+λ3(α0γ)2=0. After a detailed calculation, it is found that for a physically acceptable solutions, we take two sets of conditions, (1) λ3=0, 1+λ1++λ3+=0, a0=0, and bN = 0, and (2) λ3+=0, 1+λ1+λ3=0, aN = 0, and b0=0. For the first set of conditions, by setting n = 0 and n = N in the recurrence relation (20), we further get λ2λ1α0=0 and λ3+(γ2αN)+αN+λ2+(1+λ1+)αN=0. For the second set of conditions, by setting n = 0 and n = N in the recurrence relation (20), we have λ2+λ1+α0=0 and λ3+(γ2αN)+λ2+λ1+αN=0. The parameter η is determined by the conditions, λ3±=0. After substituting η into the conditions, 1+λ1+λ3=0, the parameters ξ are given. Once the parameters η and ξ are determined, the parameters α0 and the energies E are given by the conditions, λ2λ1α0=0 and λ3±(γ2αN)+λ2±λ1±αN=0.
In the first situation with λ3=0, we have the parameters and the energies
η=ξ2=11g2g,α0,=κN2,E,N=1+N1g2.
(B1)
The expansion coefficients a=[a1,a2,,aN]T and b=[b0,b1,,bN1]T are obtained from the recurrence relation
Rn+1+an+1+Pn+an+Δgξbn=0,
(B2)
PnbnΔgξan+Qn1bn1=0,
(B3)
with a0=0. For brevity, we present the matrix form
M(ab)=0,
(B4)
with
M=(M11M12M21M22).
(B5)
Here the elements of the matrix M are given as
M11=(R1+P1+R2+P2+PN1+RN+),M12=(ΔgξΔgξΔgξ),
(B6)
M21=(ΔgξΔgξΔgξ),M22=(Q0P1Q1P2PN1QN1).
(B7)
For non-zero values for the expansion coefficients a and b, we further require that the determinant of the matrix M must vanish, thereby leading to the parameter relations for the exact results. For example, in the simplest case of N=1,2,3, we get the parameter relations
N=1,(Δ2κ+1)(Δ+2κ1)=0,
(B8)
N=2,Δ4+4(g21+2(1κ)κ)Δ2+16(1κ)2κ2=0,
(B9)
N=3,Δ3±(12κ)Δ2+(8g24(12κ)2)Δ±(4κ21)(2κ3)=0.
(B10)
Under the general parameter conditions, the expansion is infinite. One can still choose the following parameters η=(11g2)/g and ξ=2(11g2)/g, so that λ3=0 and 1+λ1++λ3+=0. If we further take the initial conditions a0 = 0 and b0 = 1, we require α=κ+(E1)/21g2. Therefore, we have the three-term recurrence relations for an and bn
Rn+1+Pnan+1+(Rn+Qn1+Pn+Pn+Δ2g2ξ2)an+Pn1+Qn1an1=0,
(B11)
Rn+1+Pn+1bn+1+(Rn+1+Qn+Pn+Pn+Δ2g2ξ2)bn+Pn+Qn1bn1=0.
(B12)
In the second situation with λ3+=0, we have the parameters and the energies
η+=ξ+2=11g2g,α0,+=κN2,E+,N=1+N1g2.
(B13)
The expansion coefficients a=[a0,a2,,aN1]T and b=[b1,b2,,bN]T satisfy the following equation
M(ab)=0,
(B14)
with
M=(M11M12M21M22).
(B15)
Here the elements of the matrix M are given as
M11=(ΔgξΔgξΔgξ),M12=(R1P1R2P2PN1RN),
(B16)
M21=(Q0+P1+Q1+P2+PN1+QN1+),M22=(ΔgξΔgξΔgξ).
(B17)
In this situation with λ3+=0, the requirement for the non-zero values for the expansion coefficients a and b leads to the same parameter relations with the situation with λ3=0.

Appendix C. Solutions at the resonant conditions

In this section, we present the solutions at the resonant conditions of Δ=(2κ1). Following the results in appendix B, in the case of λ3=0, we have the exact solutions
f1(z)=a1e(11g2)z/g×F(12+κ,2κ,221g2gz),f2(z)=b0e(11g2)z/g×F(12+κ,2κ,221g2gz).
(C1)
Here a1=b0. In the case of λ3+=0, we have the exact solutions
f1+(z)=a0+e(11g2)z/g×F(12+κ,2κ,221g2gz),f2+(z)=b1+e(11g2)z/g×F(12+κ,2κ,221g2gz).
(C2)
Here b1+=a0+. From the relations for the CHF, F(α,γ,z)=ezF(γα,γ,z), it follows that the two eigenfunctions (f1(z),f2(z)) and (f1+(z),f2+(z)) are linearly dependent, and represent the same states. This exact energies are non-degenerate.

Appendix D. Application in the two-photon QRM

In this section, we show that this method can be applied to the two-photon QRM described by the following Hamiltonian (=1)
H=ωaa+Δσz+ϵσx+gσx(a2+a2).
(D1)
Here we introduce an additional coupling term ϵσx. In the Fock–Bargmann representation, the bosonic creation and annihilation operators have the form
addz,az.
(D2)
The eigenstate |ψ can be expressed as
|ψ=(ψ1(z)ψ2(z)).
(D3)
From the Schrödinger equation H|ψ=E|ψ, with the linear combinations, φ1=(ψ1+ψ2)/2 and φ2=(ψ1ψ2)/2, we have the two coupled equations for the two component wave functions φ1,2(z)
gd2φ1dz2+zdφ1dz+(gz2+ϵE)φ1+Δφ2=0,
(D4)
gd2φ2dz2zdφ2dz+(gz2+ϵ+E)φ2Δφ1=0.
(D5)
Here we have set ω=1. With the following transformations
ψ1,2(z)=eηz2ϕ1,2(y),y=ξz2,
(D6)
we obtain the similar equations with equations (11) and (12)
yd2ϕ1dy2+(12+λ1+y)dϕ1dy+(λ2++λ3+y)ϕ1+Δ4gξϕ2=0,
(D7)
yd2ϕ2dy2+(12+λ1y)dϕ2dy+(λ2+λ3y)ϕ2Δ4gξϕ1=0
(D8)
with
λ1±=4ηg±12gξ,
(D9)
λ2±=2ηg+ϵE4gξ,
(D10)
λ3±=4η2+g2η4gξ2.
(D11)
It is evident that our analytical method can be applied directly. Since φ1,2(z) have the parity symmetry, φ1,2(z)=±φ1,2(z), one may seek the solutions with the forms of the even-parity and odd-parity functions of z. In the series expansion with the CHFs, the even-parity and odd-parity solutions are given as
ϕ1(y)=n=0an F(αn,γ,y),  ϕ2(y)=n=0bnF(αn,γ,y),
(D12)
and
ϕ1(y)=n=0anzF(αn,γ,y),  ϕ2(y)=n=0bnzF(αn,γ,y),
(D13)
where αn=α0+n. By applying our method, for both the even-parity and the odd-parity eigenfunctions, we have η±=ξ±/2=±(ωω24g2)/4g. γ=1/2 and γ =3/2 correspond to the even-parity and odd-parity eigenfunctions, respectively. The different choice of α0 gives the different types of the exact energies. With the choice of α0=N, we have the exact energies
ENI,eω=12+(2N+12)14g2ω2±ϵω,
(D14)
for the even-parity eigenfunctions, and
ENI,oω=12+(2N+32)14g2ω2±ϵω,
(D15)
for the odd-parity eigenfunctions. These exact results indicate that due to the presence of the additional term εσx, these exact energies associated with polynomial eigenfunctions do not correspond to the level-crossings. If we take α0=(12N±2ϵ/ω24g2)/4 and α0=(32N±2ϵ/ω24g2)/4 for the even-parity and the odd-parity eigenfunctions, we have the exact energies
ENII,e,oω=12+N14g2ω2.
(D16)
Here we have N ≥ 1. In the simplest case of N = 1, the parameter relation is given as
Δ2ω214+414g2/ω2ϵ2ω2=0.
(D17)
This means that for the resonance condition of 2Δ = ω with ε = 0, we have the exact energy ENII,e,o/ω=1/2+14g2/ω2.

References

1
Braak D 2011 Phys. Rev. Lett. 107 100401
2
Chen Q-H Wang C He S Liu T Wang K L 2012 Phys. Rev. A 86 023822
3
Zhong H Xie Q Batchelor M T Lee C 2013 J. Phys. A: Math. Theor. 46 415302
4
Maciejewski A J Przybylska M Stachowiak T 2014 Phys. Lett. A 378 16
5
Xie Q T Cui S Cao J P Amico L Fan H 2014 Phys. Rev. X 4 021046
6
Tomka M El Araby O Pletyukhov M Gritsev V 2014 Phys. Rev. A 90 063839
7
Travenec I 2012 Phys. Rev. A 85 043805
8
Maciejewski A J Przybylska M Stachowiak T 2015 Phys. Rev. A 91 037801
9
Duan L Xie Y-F Braak D Chen Q H 2016 J. Phys. A: Math. Theor. 49 464002
10
Maciejewski A J Stachowiak T 2017 J. Phys. A: Math. Theor. 50 244003
11
Maciejewski A J Stachowiak T 2019 J. Phys. A: Math. Theor. 52 485303
12
Zhang Y-Z 2013 J. Math. Phys. 54 102104
13
Duan L He S Braak D Chen Q-H 2015 Eur. Phys. Lett. 112 34003
14
Chilingaryan S A Rodriguez-Lara B M 2015 J. Phys. B: At. Mol. Opt. Phys. 48 245501
15
Zhang Y-Z 2016 Ann. Phys. 375 460
16
Cui S Cao J P Fan H Amico L 2017 J. Phys. A: Math. Theor. 50 204001
17
Zhong H Xie Q Guan X Batchelor M T Gao K Lee C 2014 J. Phys. A: Math. Theor. 47 045301
18
Peng J Ren Z Guo G Ju G Guo X 2013 Eur. Phys. J. D 67 162
19
Peng J Ren Z Braak D Guo G Ju G Zhang X Guo X 2014 J. Phys. A: Math. Theor. 47 265303
20
Wang H He S Duan L Zhao Y Chen Q H 2014 Eur. Phys. Lett. 106 54001
21
Duan L He S Chen Q H 2015 Ann. Phys. 355 121
22
Xie Q Zhong H Batchelor M T Lee C 2017 J. Phys. A: Math. Theor. 50 113001
23
Forn-Díaz P Lamata L Rico E Kono J Solano E 2019 Rev. Mod. Phys. 91 025005
24
Wu Y Yang X Xiao Y 2001 Phys. Rev. Lett. 86 2200
25
Wu Y Yang X 2003 Phys. Rev. A 68 013608
26
Olver J W F Lozier W D Boisvert F R Clark W C 2010 NIST Handbook of Mathematical Functions Cambridge Cambridge University Press
27
Ishkhanyan A M 2016 Mod. Phys. Lett. A 31 1650177
28
Ishkhanyan A M 2016 Phys. Lett. A 380 3786
29
Ishkhanyan A M 2016 Phys. Lett. A 380 640
30
Ishkhanyan A M 2015 Eur. Phys. Lett. 112 10006

RIGHTS & PERMISSIONS

© 2020 Chinese Physical Society and IOP Publishing Ltd
PDF(355 KB)

327

Accesses

0

Citation

Detail

Sections
Recommended

/