Parameter-dependent unitary transformation approach for quantum Rabi model

Quantum Rabi model has been exactly solved by employing the parameter-dependent unitary transformation method in both the occupation number representation and the Bargmann space. The analytical expressions for the complete energy spectrum consisting of two double-fold degenerate sub-energy spectra are presented in the whole range of all the physical parameters. Each energy level is determined by a parameter in the unitary transformation, which obeys a highly nonlinear equation. The corresponding eigenfunction is a convergent infinite series in terms of the physical parameters. Due to the level crossings between the neighboring eigenstates at certain physical parameter values, such the degeneracies could lead to novel physical phenomena in the two-level system with the light-matter interaction.

Quantum Rabi model usually has the Hamiltonian H = ωa † a + g(a † + a)σ x + λσ z + ǫσ x , where σ x and σ z are the Pauli matrices for the two-level system with level splitting 2λ, a † and a are the creation and annihilation operators for the single bosonic mode with frequency ω, respectively, the light-matter interaction is controlled by the coupling parameter g, and the last term ǫσ x is the driving term which leads to tunnelling between the two levels. We note that the competition between g and ω produces the different experimental regimes. When g/ω is small, by applying the rotatingwave approximation, the Rabi model (1) with ǫ = 0 is equivalent to the so-called Jaynes-Cummings model [15], which is relevant to most experimental regimes. Because the Jaynes-Cummings model is integrable, it is easy to derive its analytical solution. With increasing g/ω, the ultrastrong coupling regime (∼ 0.1 < g/ω <∼ 1.0) [12] or the deep strong coupling regime (g/ω >∼ 1.0) [9] is reached, where the Jaynes-Cummings model is invalid and cannot be used to investigate the interaction between light and matter. Recently these regimes have rapidly growing interesting due to their fundamental characteristics and the potential applications in quantum devices [11][12][13][14].
Although the Hamiltonian (1) has a simple form, it has not been possible to obtain its correct analytical solution, which is considerably important for exploring accurately the light-matter interaction from weak to extreme strong coupling. In Ref. [16], Braak presented an analytical solution of the Rabi model (1) by using the representation of bosonic operators in the Bargmann space of analytical functions. The energy spectrum consists of two parts, i.e. the regular and the exceptional spectrum. However, such a spectrum structure is incorrect due to the derivation error in solving the time-independent Schrodinger equation in the positive and negative parity parts (see APPENDIX).
In this article, we exactly diagonalize the Hamiltonian (1) by using the parameter-dependent unitary transformation technique in both the occupation number representation and the Bargmann space. Such a direct and powerful approach has been used to solve successfully the complex two-dimensional electron gas in the presence of both Rashba and Dresselhaus spin-orbit interactions under a perpendicular magnetic field [17,18].

II. OCCUPATION NUMBER REPRESENTATION
The two-component eigenstate of the Hamiltonian (1) for the nth energy level with quantum number s has the general form where the 2 × 2 matrix is a unitary one, s = ±1 are associated with the two components under the level quantum number n, respectively, A ns is the normalized factor, ∆ ns is a real parameter to be determined below by requiring the coefficients α ns m and β ns m to be nonzero, φ m is the eigenstate of the mth energy level in the occupation number representation, i.e. a + φ m = √ m + 1φ m+1 , aφ m = √ mφ m−1 and < φ m ′ |φ m >= δ mm ′ . When m → +∞, α ns m = β ns m = 0. Substituting |n, s > into the eigen-equation H|n, s >= E ns |n, s > and letting the coefficients of φ m to be zero, we obtain a coupled system of infinite homogeneous linear equations for α ns m and β ns m 2g∆ns 1+∆ 2 where m = 0, 1, 2, · · · , ∞, and α ns m = β ns m ≡ 0 for m < 0.
A. Sub-energy spectrum I In order to obtain the analytical solution of the Hamiltonian (1) in the whole parameter space, we first choose which come from the vanishing of the two terms about α ns n+1 and β ns n in Eq. (3) with m = n + 1 and Eq. (4) with m = n, respectively. Such a choice is based on the observation of exact solution of the Hamiltonian (1) for the nth energy level with quantum number s when g = 0. We find that the non-zero eigenfunction associated with the eigenvalue E ns is solely fixed by letting [2λ∆ ns + ǫ(1 − ∆ 2 ns )]β ns n − 2g∆ ns √ n + 1α ns n+1 = 0, (7) or [2λ∆ ns + ǫ(1 − ∆ 2 ns )]α ns n+1 + 2g∆ ns √ n + 1β ns n = 0. The low-lying energy levels of the energy spectrum (13) in unit of ω as a function of the coupling parameter g at different λ under ǫ = 0. The solid lines denote n = 0, 1, · · · , 5 and s = 1 while the dash lines mean n = 1, 2, · · · , 5 and s = −1.
We solve the homogenous linear equations (5) and (6) about α ns n+1 and β ns n by vanishing of the coefficient determinant. Then the eigenvalue for the nth eigenstate with s has the analytical expression (9) Note that the quasiparticle energy E ns must be larger than zero. From Eqs. (6) and (7) or Eqs. (5) and (8), the parameter ∆ ns is determined by the highly nonlinear equation or After analysing carefully, we discover that Eq. (10) with s = −1(1) coincides with Eq. (11) with s = 1(−1). In other words, ∆ ns is independent of quantum number s, i.e. ∆ n,1 ≡ ∆ n,−1 , which leads to Ξ n,1 ≡ Ξ n,−1 . So we have where σ = ±1. It is easy to see from Eq. (12) that the analytical solution (9) is physical if and only if ∆ ns → 0 when ǫ → 0. Otherwise, Ξ ns ≡ σω/2, which is not true for arbitrary λ and g. When ǫ = 0, then ∆ ns = 0 according to Eq. (12). Therefore, the eigenvalue (9) has a simple formula in the absence of the driving term ǫ. Obviously, the eigenvalue (13)  crossings between the neighboring eigenstates. With increasing λ, the energy levels with s = 1(−1) become higher (lower), and these crossing points move toward the origin.
For the eigenstate associated with the sub-energy spectrum I, from Eq. (6), we have where β ns n is an arbitrary constant and can be set to 1, and the coefficients α ns m and β ns m are uniquely determined by the recursion relations for m = 0, 1, 2, · · · , n, and (16) for m = n + 1, n + 2, · · · , +∞. Here we have defined where I is the 2 × 2 unit matrix. From the recursion equation (15), we can see that α ns m−1 and β ns m−1 (m = 1, 2, · · · , n) are linear functions of α ns n and β ns n+1 , which are obtained by solving Eq. (15) with m = 0.
For the nth eigenstate with s in the sub-energy spectrum II, we have where α ns n is an arbitrary constant and is set to 1. The other coefficients α ns i and β ns i also obey the same recursion relations (15) and (16) in the sub-energy spectrum I.

III. THE BARGMANN SPACE
In this section, we reinvestigate the eigenvalue problem for the Hamiltonian (1) in the Bargmann space [16], where the bosonic creation and anihilation operators in terms of a complex variable z can be transformed as In this representation, the state Ψ(z) can be normalized according to We assume that the two-component eigenstate of the Hamiltonian (28) for the nth energy level with quantum number s possesses the general form where s = ±1, ∆ ns is a real parameter in the unitary matrix to be determined below by requiring the coefficients A ns i and B ns i to be nonzero. When i → +∞, A ns i → 0 and B ns i → 0, so that Ψ ns is finite at any z in the Bargmann space. Substituting the eigenfunction (30) into the eigen-equation HΨ ns = E ns Ψ ns and requiring the coefficients of z i to be zero, we obtain the infinite system of homogeneous linear equations with the variables A ns i and B ns i 2g∆ns where i = 0, 1, 2, · · · , ∞, A ns m = B ns m ≡ 0 for m < 0. Eqs. (31) and (32) can be also solved exactly by employing the same procedure in the occupation number representation in section II.
A. Sub-energy spectrum I Following the trick presented in the occupation number representation, we let which come from the vanishing of the two terms about A ns n+1 and B ns n in Eq. (31) with i = n + 1 and Eq.(32) with i = n, respectively. Then the non-zero eigenfunction associated with the eigenvalue E ns is solely fixed by requiring [2λ∆ ns + ǫ(1 − ∆ 2 ns )]B ns n − 2g∆ ns (n + 1)A ns n+1 = 0, (35) or 2g∆ ns B ns n + [2λ∆ ns + ǫ(1 − ∆ 2 ns )]A ns n+1 = 0.
Solving the homogenous linear equations (33) and (34) about A ns n+1 and B ns n , we have For the eigenstate for the sub-energy spectrum I, from Eq. (34), we have where B ns n is a constant to be determined by the normalized condition (29). The coefficients α ns i and β ns i , proportional to B ns n , are obtained by the recursion relations (41) for i = 0, 1, 2, · · · , n, and [ω(n + 1) − The equations above originate in the vanishing of the two terms about A ns n and B ns n+1 in Eq. (31) with i = n and Eq. (32) with i = n + 1, respectively. The corresponding eigenfunction is uniquely determined by the condition Solving Eqs. (43) and (44), we have which is consistent with the eigenvalue (22) in the occupation number representation. Here ∆ ns satisfies the nonlinear equation or (1 + ∆ 2 ns )(E ns − nω) − λ(1 − ∆ 2 ns ) + 2ǫ∆ ns g(1 − ∆ 2 ns )(n + 1) A ns n , (50) where A ns n is a constant to be determined by the normalized condition (29). The other coefficients A ns i and B ns i , proportional to A ns n , also satisfy the same recursion relations (41) and (42) in the sub-energy spectrum I.
In order to compare with the energy spectrum of the Rabi model presented by Braak, here we employ the physical parameters in Ref. [16]. Figs. 5 and 6 exhibit the low-lying energy levels of the sub-energy spectrum I and II as a function of g at λ = 0.4ω and ǫ = 0 and at λ = 0.7ω and ǫ = 0.2ω, respectively. We can see that the energy spectrum possesses the level crossings between the neighboring eigenstates, which is dramatically different from that in Ref. [16]. It is expected that such the degeneracies at certain physical parameter values could produce novel physical phenomena in the two-level system with the light-matter interaction, similar to the two-dimensional electron gas with spin-orbit interaction under a perpendicular magnetic field [19][20][21].

IV. SUMMARY
We have exactly solved the quantum Rabi model (1) in both the occupation number representation and the Bargmann space. The complete energy spectrum is comprised of two double-fold degenerate sub-energy spectrum I and II. Such the exact solution can help us to deeply understand the light-matter interaction, especially in strong coupling regimes. Because the analytical expressions of the eigenvalue E ns in the occupation number representation are completely identical to those in the Bargmann space, this exact solution for quantum Rabi model is definitely correct.