January 01, 1970
We consider the transition from discrete to continuous spectrum which occurs in some quantum Rabi models. We present recent developments of the subordinacy theory and their applications to study the spectral properties of the intensity-dependent Rabi model, the anisotropic two-photon Rabi model, and the two-photon Rabi-Stark model for the whole range of parameters. Our analysis allows us to detect and locate the continuous spectrum. Moreover, we prove the absence of eigenvalues in the interior of the continuous spectrum and the absence of a singular spectrum for all models considered.
The simplest model of interaction between light and matter was proposed in [1], [2] by I. I. Rabi. Its fully quantized version is called the quantum Rabi model (QRM). The QRM couples a two-level system with quantized single-mode radiation and was used to study real atoms in quantum optics, e.g. to experimentally test the field energy quantization of the cavity electrodynamics (see [3]). We refer to the survey [4] for a list of research work and experimental realizations of the QRM and its generalizations that describe various quantum devices.
One of the generalizations is the two-photon QRM, which describes the situation when the change of level is associated with the absorption/emission of two photons instead of one photon. The corresponding two-photon QRM was applied to describe a two-level atom interacting with squeezed light (see [5], [6]), quantum dots inserted in a cavity (see [7]–[9]), trapped ions experiments (see [10], [11]), and superconducting circuits (see [12], [13]). The two-photon QRM gave rise to many research works (see [14]–[19]). The great interest in this model was motivated by the so-called spectral collapse phenomenon, i.e., the transition from discrete to continuous spectrum that occurs when the coupling constant \(g\) reaches a critical value \(g_{\rm cr}\). In particular, the spectrum of the two-photon QRM is discrete if \(0<g<g_{\rm cr}\), but the spacing of the eigenvalues shrinks as \(g\) approaches \(g_{\rm cr}\) (see [14], [16], [20]). A half-line of the continuous spectrum appears if \(g=g_{\rm cr}\) (see [19]) and the spectrum becomes the whole real line if \(g>g_{\rm cr}\) (see [14]). This phenomenon was also studied numerically (see, e.g. [15], [21]–[24]) and experimentally (see, e.g. [10], [25]). Spectral transitions also occur for the anisotropic two-photon Rabi model (see [26]–[29]), Rabi-Stark models (see [30]–[39]) and other generalizations of the two-photon Rabi model (see [40]–[49]). We note that spectral transitions have recently attracted attention in connection with the quantum metrology (see [48], [49]).
In this paper, we consider Rabi models which can be expressed as a direct sum of operators defined by symmetric Jacobi matrices, i.e. symmetric tridiagonal matrices acting in the Hilbert space of square-summable sequences \(\ell^2(\mathbb{N}_0)\) (see Section 2.2). The spectral properties of the corresponding operators have been the subject of numerous mathematical works. Our purpose is to present basic notions of the spectral theory and to apply recent developments of the subordinacy theory for periodically modulated Jacobi matrices. We note that this theory is based on the asymptotic analysis of the transfer matrix and was applied to investigate the spectral transitions in the fundamental work [50]. We have chosen to characterize the spectrum using the mathematical notions introduced in Section 2 and we have decided to avoid using the terms "bound states" and "the spectral collapse", which require additional clarifications. Our main results are presented in Section 3. The intensity-dependent Rabi model (see [51], [52]) is considered in Theorem 1. The two-photon Rabi model is considered in Theorem 2 and the two-photon anisotropic Rabi model in Theorem 3. Finally, in Theorem 4 we consider the two-photon Rabi-Stark model. The general scheme of our approach is described in Section 4. In Section 5 we give the proofs of Theorems 1-4. In Section 6 we describe the most important aspects of our research.
Let \(\mathcal{H}\) be a complex Hilbert space equipped with the scalar product \(\langle \cdot ,\cdot \rangle\) and let \(H: \, \mathcal{D}(H)\to \mathcal{H}\) be a linear map defined on a dense subspace of \(\mathcal{H}\). The spectrum of \(H\) is defined by the formula \(\sigma (H)=\mathbb{C}\setminus \rho (H)\), where \(\rho (H)\) is the set of complex numbers \(\lambda\) such that \((H-\lambda I)^{-1}\) exists and is a bounded operator on \(\mathcal{H}\). We say that \(\lambda \in \mathbb{C}\) is an eigenvalue of \(H\) if the eigenspace \(E_\lambda (H):=\{ x\in \mathcal{D}(\mathcal{H}): Hx=\lambda x\}\) is not equal to \(\{0\}\) and the multiplicity of \(\lambda\) is defined as the dimension of \(E_\lambda (H)\). The point spectrum of \(H\) is \[\sigma_{\rm p}(H)= \{ \lambda \in \mathbb{C}: \lambdais an eigenvalue ofH\},\] the discrete spectrum of \(H\) is \[\sigma_{\rm discr}(H)= \{ \lambda \in \sigma_{\rm p}(H) : \dim E_{\lambda}(H)<\inftyand\lambdais isolated from\sigma (H)\setminus \{\lambda \} \}\] and the essential spectrum, \(\sigma_{\mathrm{ess}}(H)= \sigma (H)\setminus \sigma_{\rm discr}(H)\).
We say that \(H\) closed if and only if its graph \(\mathcal{G}(H)=\{ (x,Hx)\in \mathcal{H}\times \mathcal{H}: x\in \mathcal{D}(H) \}\) is closed in \(\mathcal{H}\times \mathcal{H}\). If \(\hat{H}: \, \mathcal{D}(\hat{H})\to \mathcal{H}\) is a linear map defined on a dense subspace of \(\mathcal{H}\) and \(\hat{H}\) is symmetric (i.e. \(\langle \hat{H}x,y\rangle =\langle x,\hat{H} y\rangle\) for \(x,y\in \mathcal{D}(\hat{H})\)), then the closure of its graph in \(\mathcal{H}\times \mathcal{H}\) is the graph of the symmetric operator called the closure of \(\hat{H}\) (see, e.g. [53], Section 3.1).
In what follows, \(\mathbb{N}_0\) is the set of non-negative integers, \(\ell^2(\mathbb{N}_0)\) is the complex Hilbert space of square-summable complex sequences \((x_n)_{n\in \mathbb{N}_0 }\) equipped with the scalar product \[\label{not1a} \langle {x,y} \rangle_{\ell^2(\mathbb{N}_0)} =\sum_{n=0}^\infty \, x_n {\overline{y_n}}\tag{1}\] and \(\ell^2_{\rm{fin}}(\mathbb{N}_0)\) is the vector subspace of \(\ell^2(\mathbb{N}_0)\) composed of finite linear combinations of vectors from the canonical basis \(\{ e_n\}_{ n\in \mathbb{N}_0}\). We identify the Fock space with \(\ell^2(\mathbb{N}_0)\) and define the Hamiltonian of single-mode radiation, \(\hat{N}\), as the closed linear operator in \(\ell^2(\mathbb{N}_0)\) satisfying \[\label{not1d} \hat{N} e_n =n e_nforn\in \mathbb{N}_0\tag{2}\] We define the annihilation and creation operators, \(\hat{a}\) and \(\hat{a}^{\dagger}\), as linear operators \(\mathcal{D}(\hat{N}^{1/2})\to \ell^2(\mathbb{N}_0)\) such that \[\label{not1c} \hat{a}^{\dagger}\, e_n = \sqrt{n+1} \, e_{n+1} \, forn\in \mathbb{N}_0\tag{3}\] \[\label{not1c39} \hat{a} \, e_0=0 \, and \,\hat{a} \, e_n = \sqrt{n} \, e_{n-1} \, forn\in \mathbb{N}_0 \setminus \{0\}\tag{4}\] We note that \(\hat{N} =\hat{a}^\dagger \hat{a}\). We also consider the linear operators defined in \(\mathbb{C}^2\) by the matrices \[I_2 :=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} , \sigma_{x} :=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \sigma_{z} :=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} , \sigma_{+} :=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} , \sigma_{-} :=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\] and we denote \[e^{1}:=\begin{pmatrix} 1\\ 0 \end{pmatrix}e^{-1}:=\begin{pmatrix} 0\\ 1 \end{pmatrix}\] If \(\nu =\pm 1\) and \(g_+\), \(g_-\) are real parameters, then we obtain \[\sigma_x e^{\nu} = e^{-\nu},\sigma_z e^\nu =\nu e^\nu,(g_- \sigma_- +g_+ \sigma_+ )e^{\pm 1} = g_\pm e^{\mp 1}\, .\]
In Section 3 we describe Rabi models defined by self-adjoint operators in the Hilbert space \(\mathbb{C}^2\otimes \ell^2(\mathbb{N}_0)\) equipped with the canonical orthonormal basis \(\{ e^\nu_n\}_{(\nu ,n)\in \{-1,1\} \times \mathbb{N}_0}\) of the form \[\label{enun} e^{\nu}_n :=e^\nu \otimes e_n .\tag{5}\]
Assume that \((a_n)_{n\in \mathbb{N}_0}\) and \((b_n)_{n\in \mathbb{N}_0}\) are two sequences of positive and real numbers, respectively. Then the infinite tridiagonal symmetric matrix \[\label{Jacobimatrix} \mathcal{J} ((a_n),(b_n)) = \begin{pmatrix} b_0 & a_0 & 0 & \\ a_0 & b_1 & a_1 & \\ 0 & a_1 & b_2 & \ddots \\ & & \ddots & \ddots \end{pmatrix}\tag{6}\] is called Jacobi matrix. We let \(\hat{J}((a_n),(b_n))\) denote the linear map in \(\ell^2_{\rm fin}(\mathbb{N}_0)\), defined by using complex valued sequences as column vectors, i.e. \[\label{J} \hat{J}((a_n),(b_n))e_n =a_ne_{n+1}+a_{n-1}e_{n-1} +b_ne_n,\tag{7}\] where \(a_je_j:=0\) when \(j<0\). The map \(\hat{J}((a_n),(b_n))\) is symmetric in the Hilbert space \(\ell^2(\mathbb{N}_0)\) and its closure defines the Jacobi operator \(J((a_n),(b_n))\).
Assume that the Jacobi operator \(J=J((a_n),(b_n))\) is self-adjoint \(\ell^2(\mathbb{N}_0)\). It is well known (see Section 2.5 in [54]) that \(e_1\) is cyclic (i.e. span\(\{ J^k e_1: k\in \mathbb{N}_0 \}\) is dense in \(\ell^2(\mathbb{N}_0)\)) and the operator \(J\) is unitary similar to the operator of multiplication by \(\lambda\) in \(L^2(\mathbb{R},d\rho (\lambda ))\) where \(d\rho\) is a Borel measure on \(\mathbb{R}\). The measure \(d\rho\) is determined by the conditions that its support is \(\sigma (J)\) and \[\langle e_1, J^k e_1\rangle = \int_{\mathbb{R}} \lambda^k \, d\rho (\lambda )for everyk\in \mathbb{N}_0.\] Consider the Lebesgue decomposition \(d\rho =d\rho_{\rm {pp}} +d\rho_{\rm{ac}} + d\rho_{\rm{sc}}\), where pp, ac, and sc refer to the pure point, absolutely continuous, and singularly continuous part of the measure \(d\rho\) with respect to Lebesgue measure. Then the pure point spectrum \(\sigma_{\rm {pp}} (J)\), the absolutely continuous spectrum \(\sigma_{\rm {ac}} (J)\) and the singular continuous spectrum \(\sigma_{\rm {sc}} (J)\) are defined as the support of \(d\rho_{\rm {pp}}\), \(d\rho_{\rm{ac}}\) and \(d\rho_{\rm{sc}}\) respectively. We note that \(\sigma_{\rm {pp}}(J)\) is the closure of \(\sigma_{\rm {p}}(J)\).
In what follows, we say that the self-adjoint operators \(H\) and \(H'\) are unitarily similar if and only if \(H'=U^{-1}HU\) where \(U\) is an isometric bijection between two Hilbert spaces. We will investigate Rabi models defined by a self-adjoint operator \(H\) unitarily similar to a direct sum \(J_1 \oplus \dots \oplus J_l\), where \(\{ J_k\}_{1\le k\le l}\) is a finite family of Jacobi operators. In this case \(\sigma (H)=\sigma (J_1)\cup \dots \cup \sigma (J_l)\) and similar equalities remain valid if \(\sigma\) is replaced by \(\sigma_{\rm {pp}}\), \(\sigma_{\rm {ac}}\) or \(\sigma_{\rm {sc}}\) (see [53], Section 9.1).
Assume \(\kappa \ge 0\), \(g>0\), \(\Delta \in \mathbb{R}\), and consider the linear map in \(\mathbb{C}^2 \otimes \ell^2_{\rm fin}(\mathbb{N}_0)\) given by the formula \[\label{21} \hat{H} = I_2 \otimes \hat{N} +\frac{\Delta}{2}\, \sigma_z \otimes I + g \sigma_x \otimes \left( {(\hat{N} +2\kappa )^{1/2} \, \hat{a} +\hat{a}^\dagger \, (\hat{N} +2\kappa )^{1/2} }\right)\tag{8}\] The operator \(\hat{H}\) is symmetric in the Hilbert space \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\) and \(H\), the Hamiltonian of the intensity-dependent QRM, is defined as the closure of \(\hat{H}\) in \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\). It is easy to see that the subspaces \(\mathcal{H}^+\) and \(\mathcal{H}^-\) spanned by \(\{ e^1_0,e^{-1}_1,e^{1}_2,e^{-1}_3, \dots \}\) and \(\{ e^{-1}_0,e^{1}_1,e^{-1}_2,e^{1}_3, \dots \}\) are invariant for \(H\) (see the action of \(H\) on \(e^{\pm 1}_n\) described in Section 5.1) and \(H\) can be written as a direct sum of two operators that can be expressed by Jacobi matrices. This decomposition is usually called the parity decomposition and, it allows us to deduce the the properties of the spectrum of \(H\) from the properties of corresponding Jacobi operators.
Theorem 1. The operator \(H\) is unitarily similar to the direct sum \[\label{decomp1} J^-\oplus J^+ ,\tag{9}\] where \(J^\pm =J((a_n ),(b_n^\pm ))\) is the Jacobi operator with \[\label{thm1} \begin{cases}a_n = g \sqrt{(n+1)(n+ 2\kappa )},\\b_n^{\pm} = n \pm (-1)^n \, \frac{\Delta}{2} \end{cases}\tag{10}\] The operator \(J^\pm\) is self-adjoint in \(\ell^2(\mathbb{N}_0)\) and \[\label{thm1a} \sigma(J^\pm ) =\sigma_{\rm discr}(J^\pm )\, when0<g<\tfrac{1}{2}\tag{11}\] \[\label{thm1c} \mathbb{R}=\sigma(J^\pm )= \sigma_{\mathrm{ac}}(J^\pm ) \, wheng>\tfrac{1}{2}\tag{12}\] \[\label{thm1b} [-\kappa , \infty ) = \sigma_{\mathrm{ac}}(J^\pm ) = \sigma(J^\pm )\setminus \sigma_{\rm discr}(J^\pm ) \, wheng=\tfrac{1}{2}.\tag{13}\] Moreover, in all cases, \(\sigma_{\mathrm{sc}}(J^\pm) = \emptyset\), and \(\sigma_{\mathrm{p}}(J^\pm)\) is disjoint from the interior of \(\sigma_{\mathrm{ac}}(J^\pm)\).
The statement of Theorem 1 ensures the following information on the spectral transition of the model: if \(0<g<\frac{1}{2}\) then the whole spectrum of \(H\) is discrete; if \(g>\frac{1}{2}\) then the spectrum of \(H\) is \(\mathbb{R}\) and \(H\) has no eigenvalues; in the critical case \(g=\frac{1}{2}\) the spectrum of \(H\) consists of the half-line \([-\kappa , \infty )\) with no eigenvalues in \((-\kappa , \infty )\) and a possible discrete spectrum in \((-\infty ,-\kappa )\).
Assume \(g>0\), \(\Delta \in \mathbb{R}\), and consider the linear map in \(\mathbb{C}^2 \otimes \ell^2_{\rm fin}(\mathbb{N}_0)\) given by \[\label{22} \hat{H} = I_2 \otimes \hat{N} + \frac{\Delta}{2}\, \sigma_z \otimes I + g \sigma_x \otimes \left( { \hat{a}^2 + ({\hat{a}}^\dagger )^2 }\right) .\tag{14}\] The operator \(\hat{H}\) is symmetric in the Hilbert space \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\) and \(H\), the Hamiltonian of the two-photon QRM, is defined as the closure of \(\hat{H}\) in \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\). Due to the definition of \(\hat{a}^\dagger\) and \(\hat{a}\) given in 3 and 4 , we get \[\label{320} \big( \hat{a}^{\dagger}\big)^2\, e_n = \sqrt{(n+2)(n+1)} \, e_{n+2},\hat{a}^2 \, e_n = \sqrt{n(n-1)} \, e_{n-2} \, ifn\ge 2,\hat{a}^2 \, e_n =0 \, ifn\ge 1.\tag{15}\] Due to 15 , the subspaces \(\mathcal{H}_0\) and \(\mathcal{H}_1\) spanned by \(\{ e^\nu_{2n}\}_{(n,\nu )\in \mathbb{N}_0 \times \{ -1,1\} }\) and \(\{ e^\nu_{2n+1}\}_{(n,\nu )\in \mathbb{N}_0 \times \{ -1,1\} }\) are invariant for \(H\). Thus \(H\) can be written as a direct sum of two operators usually called the decomposition with respect to the Bargmann index \(q=\frac{1}{4}\) and \(q=\frac{3}{4}\). If \(\mu =0\) or 1, then an additional parity decomposition \(\mathcal{H}_\mu = \mathcal{H}^+_\mu \oplus \mathcal{H}^-_\mu\) with \(\mathcal{H}^+_\mu\) and \(\mathcal{H}^-_\mu\) spanned by \(\{ e_{2n+\mu}^{(-1)^n} \}_{n\in \mathbb{N}_0}\) and \(\{ e_{2n+\mu}^{(-1)^n} \}_{n\in \mathbb{N}_0}\), allows us to write \(H\) as a direct sum of four operators that can be expressed by Jacobi matrices and the properties of the spectrum of \(H\) can be deduced from the properties of the corresponding four Jacobi operators.
Theorem 2. The operator \(H\) is unitarily similar to the direct sum \[\label{decomp2} J_0^-\oplus J_0^+ \oplus J_1^- \oplus J_1^+\tag{16}\] where \(J_\mu^\pm =J((a_{\mu ,n} ),(b_{\mu ,n}^\pm ))\) are the Jacobi operators with \[\label{thm2} \begin{cases}a_{\mu ,n} = g\sqrt{(2n+1+\mu )(2n+2+\mu )},\\b_{\mu ,n}^{\pm} = 2n+\mu \pm (-1)^n \, \frac{\Delta}{2} \end{cases}\tag{17}\] The operator \(J^\pm_\mu\) is self-adjoint in \(\ell^2(\mathbb{N}_0)\) and \[\label{thm2a} \sigma(J_\mu^\pm ) =\sigma_{\rm discr}(J_\mu^\pm )\, when0<g<\tfrac{1}{2}\tag{18}\] \[\label{thm2b} \mathbb{R}=\sigma(J_\mu^\pm )= \sigma_{\mathrm{ac}}(J_\mu^\pm ) \, wheng>\tfrac{1}{2}\tag{19}\] \[\label{thm2c} [-\tfrac 12 , \infty ) = \sigma_{\mathrm{ac}}(J_\mu^\pm ) = \sigma(J_\mu^\pm )\setminus \sigma_{\rm discr}(J_\mu^\pm ) \, wheng=\tfrac{1}{2}\tag{20}\] Moreover, in all cases, \(\sigma_{\mathrm{sc}}(J_{\mu}^\pm) = \emptyset\), and \(\sigma_{\mathrm{p}}(J_{\mu}^\pm)\) is disjoint from the interior of \(\sigma_{\mathrm{ac}}(J_{\mu}^\pm)\).
The statement of Theorem 2 ensures the following information on the spectral transition of the model: if \(0<g<\frac{1}{2}\) then the whole spectrum of \(H\) is discrete; if \(g>\frac{1}{2}\) then the spectrum of \(H\) is \(\mathbb{R}\) and \(H\) has no eigenvalues; in the critical case \(g=\frac{1}{2}\) the spectrum of \(H\) consists of the half-line \([-\tfrac 12 , \infty )\) with no eigenvalues in \((-\tfrac 12 , \infty )\) and a possible discrete spectrum in \((-\infty ,-\tfrac 12 )\).
Assume \(\Delta \in \mathbb{R}\), \(g_->0\), \(g_+>0\), \(g_- \ne g_+\), and consider the linear map in \(\mathbb{C}^2 \otimes \ell^2_{\rm fin}(\mathbb{N}_0)\) given by the formula \[\label{23} \hat{H} = I_2 \otimes \hat{N} +\frac{\Delta}{2}\, \sigma_z \otimes I + (g_-\sigma_- +g_+\sigma_+) \otimes (\hat{a}^\dagger )^2 +(g_+\sigma_- +g_-\sigma_+) \otimes \hat{a}^2\tag{21}\] Then \(\hat{H}\) is symmetric in the Hilbert space \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\) and \(H\), the Hamiltonian of the two-photon anisotropic Rabi model, is defined as the closure of \(\hat{H}\) in \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\). Similarly as in Section 3.2, the operator \(H\) can be written as a direct sum of four operators that can be expressed by Jacobi matrices and the properties of the spectrum of \(H\) can be deduced from the properties of the corresponding four Jacobi operators.
Theorem 3. The operator \(H\) is unitarily similar to the direct sum 16 , where \(J_\mu^\pm =J((a_{\mu ,n}^\pm ),(b_{\mu ,n}^\pm ))\) are Jacobi operators with \[\label{thm3} \begin{cases}a_{\mu ,n}^\pm = (g\mp (-1)^n\, g') \sqrt{(2n+1+\mu )(2n+2+\mu )},\\b_{\mu ,n}^{\pm} = 2n+\mu \pm (-1)^n \, \frac{\Delta}{2} \end{cases}\tag{22}\] where \[\label{thm3g} g:=\frac{g_+ +g_-}{2} , \quad g':=\frac{g_+ -g_-}{2} .\tag{23}\] The operators \(J^\pm_\mu\) are self-adjoint in \(\ell^2(\mathbb{N}_0)\) and \[\label{thm3c} \sigma(J_\mu^\pm ) =\sigma_{\rm discr}(J_\mu^\pm )\, wheng< \tfrac{1}{2}\tag{24}\] \[\label{thm3d} \sigma(J_\mu^\pm ) =\sigma_{\rm discr}(J_\mu^\pm ) \, when|g'|> \tfrac{1}{2}\tag{25}\] \[\label{thm3a} \mathbb{R}=\sigma(J_\mu^\pm )= \sigma_{\mathrm{ac}}(J_\mu^\pm )\, when|g'|<\tfrac{1}{2}<g\tag{26}\] \[\label{thm3b} [-\tfrac{1}{2}, \infty ) = \sigma_{\mathrm{ac}}(J_\mu^\pm ) = \sigma (J_\mu^\pm ) \setminus \sigma_{\mathrm{discr}}(J_\mu^\pm ) \, wheng= \tfrac{1}{2}\tag{27}\] \[\label{thm3e} (-\infty, -\tfrac{1}{2}] = \sigma_{\mathrm{ac}}(J_\mu^\pm ) = \sigma (J_\mu^\pm ) \setminus \sigma_{\mathrm{discr}}(J_\mu^\pm ) \, when|g'|= \tfrac{1}{2}\tag{28}\] Moreover, in all cases, \(\sigma_{\mathrm{sc}}(J_{\mu}^\pm) = \emptyset\), and \(\sigma_{\mathrm{p}}(J_{\mu}^\pm)\) is disjoint from the interior of \(\sigma_{\mathrm{ac}}(J_{\mu}^\pm)\).
We discuss the above statement, following the notation of [27]. Assume \(0<g_+<g_-\) and denote \(g:=g_-\), \(r:={g_+}/{g_-} \in (0,1)\). Then Theorem 3 ensures the following:
If \(g<\frac{1}{1+r}\), then the whole spectrum of \(H\) is discrete
If \(g=:g_{\rm{cr}}=\frac{1}{1+r}\), then the spectrum of \(H\) consists of the half-line \([-\frac{1}{2},\infty )\) with no eigenvalues in \((-\frac{1}{2},\infty )\) and a possible discrete spectrum in \((-\infty ,-\frac{1}{2})\)
If \(\frac{1}{1+r} < g < \frac{1}{1-r}\), then the spectrum of \(H\) is \(\mathbb{R}\) and there is no eigenvalue
If \(g=g'_{\rm{cr}}=\frac{1}{1-r}\), then the spectrum of \(H\) consists of the half-line \((-\infty ,-\frac{1}{2}]\) with no eigenvalues in \((-\infty ,-\frac{1}{2} )\) and a possible discrete spectrum in \((-\frac{1}{2},\infty )\)
If \(g>\frac{1}{1-r}\), then the whole spectrum of \(H\) is discrete
We note that the cases (4), (5) have not been investigated in [26]–[29]. Only the case (1) was investigated numerically.
Assume \(g>0\), \(\Delta \in \mathbb{R}\), \(\kappa \in \mathbb{R}\) and consider the linear map in \(\mathbb{C}^2 \otimes \ell^2_{\rm fin}(\mathbb{N}_0)\) given by the formula \[\label{24} \hat{H} = I_2 \otimes \hat{N} +\, \sigma_z \otimes \Big( {\kappa \hat{N} +\frac{\Delta}{2}}\Big) + g \sigma_x \otimes \left( { \hat{a}^2 + ({\hat{a}}^\dagger )^2 }\right)\tag{29}\] The operator \(\hat{H}\) is symmetric in the Hilbert space \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\) and \(H\), the Hamiltonian of the two-photon Rabi-Stark quantum model, is defined as the closure of \(\hat{H}\) in \(\mathbb{C}^2 \otimes \ell^2(\mathbb{N}_0)\). Similarly as in Sections 3.2 and 3.3, the operator \(H\) can be written as a direct sum of four operators that can be expressed by Jacobi matrices and the properties of the spectrum of \(H\) can be deduced from the properties of the corresponding four Jacobi operators.
Theorem 4. The operator \(H\) is unitarily similar to the direct sum 16 , where \(J_\mu^\pm =J((a_{\mu ,n} ),(b_{\mu ,n}^\pm ))\) are Jacobi operators with \[\label{thm4} \begin{cases}a_{\mu ,n} = g \sqrt{(2n+1+\mu )(2n+2+\mu )},\\b_{\mu ,n}^{\pm} = (2n+\mu ) \left({ 1 \pm (-1)^n \kappa }\right) \pm (-1)^n \frac{\Delta}{2} \end{cases}\tag{30}\] The operators \(J^\pm_\mu\) are self-adjoint in \(\ell^2(\mathbb{N}_0)\) and \[\label{thm4a} \sigma(J_\mu^\pm ) =\sigma_{\rm discr}(J_\mu^\pm )\, when|\kappa |>1\tag{31}\] \[\label{thm4e} \sigma(J_\mu^\pm ) =\sigma_{\rm discr}(J_\mu^\pm )\, when\kappa^2 +4g^2<1\tag{32}\] \[\label{thm4c} \mathbb{R}=\sigma(J_\mu^\pm )= \sigma_{\mathrm{ac}}(J_\mu^\pm )\, when|\kappa |<1and\kappa^2 +4g^2>1\tag{33}\] \[\label{thm4d} [ \tfrac {\kappa^2 -1 - \kappa \Delta}{2} , \infty ) = \sigma_{\mathrm{ac}}(J_\mu^\pm ) = \sigma (J_\mu^\pm ) \setminus \sigma_{\mathrm{discr}}(J_\mu^\pm )\, when\kappa^2 +4g^2=1\tag{34}\] \[\label{thm4b} (-\infty, -\tfrac { \kappa \Delta}{2} ] = \sigma_{\mathrm{ac}}(J_\mu^\pm ) = \sigma (J_\mu^\pm ) \setminus \sigma_{\mathrm{discr}}(J_\mu^\pm )\, when|\kappa |=1\tag{35}\] Moreover, in all cases, \(\sigma_{\mathrm{sc}}(J_{\mu}^\pm) = \emptyset\), and \(\sigma_{\mathrm{p}}(J_{\mu}^\pm)\) is disjoint from the interior of \(\sigma_{\mathrm{ac}}(J_{\mu}^\pm)\).
The statement of Theorem 4 ensures the following:
the whole spectrum of \(H\) is discrete in the case \(\kappa^2+4g^2<1\) and in the case \(|\kappa |>1\)
if \(|\kappa |<1\) and \(\kappa^2+4g^2>1\) then the spectrum of \(H\) is \(\mathbb{R}\) and there is no eigenvalue
in the critical case \(\kappa^2+4g^2=1\) the spectrum of \(H\) consists of the half-line \([ \tfrac {\kappa^2 -1 - \kappa \Delta}{2}, \infty )\) with no eigenvalues in \((\tfrac {\kappa^2 -1 - \kappa \Delta}{2}, \infty )\) and a possible discrete spectrum in \((-\infty ,\tfrac {\kappa^2 -1 - \kappa \Delta}{2})\)
in the critical case \(|\kappa |=1\) the spectrum of \(H\) consists of the half-line \((-\infty ,-\tfrac { \kappa \Delta}{2}]\) with no eigenvalues in \((-\infty ,-\tfrac { \kappa \Delta}{2})\) and a possible discrete spectrum in \((-\tfrac { \kappa \Delta}{2},\infty )\)
We note that this model was investigated theoretically and numerically by J. Li, Q.-H. Chen [33], [34], assuming \(|\kappa |<1\). It seems that other cases have not been investigated by now.
We begin by the following well-known result (see, e.g. [55])
Theorem 5 (Carleman). Let \((a_n)\) and \((b_n)\) be two sequences of real numbers. If \(J((a_n),(b_n))\) is the closure of the linear operator defined in \(\ell^2_{\rm fin}(\mathbb{N}_0)\) by 7 and \[\label{eq:Carleman} \sum_{n=0}^\infty \frac{1}{|a_n|} =\infty\tag{36}\] then \(J((a_n),(b_n))\) is self-adjoint in \(\ell^2(\mathbb{N}_0)\).
The assertion of Theorem 5 ensures the self-adjointness of the operators \(H\) introduced in Sections 3.1-3.4. This fact results from the decompositions 9 , 16 , and the following
Corollary 6. The operators \(J^\pm\) (respectively \(J^\pm_\mu\)) introduced in Theorem 1 (respectively Theorem 2, 3 or Theorem 4) are self-adjoint.
Proof. Each operator in question is defined as the closure of the operator defined in \(\ell^2_{\rm fin}(\mathbb{N}_0)\) by 7 and \[|a_n |\le c(n+1)\] holds with a certain \(c>0\), hence \((a_n)\) satisfies the Carleman’s condition 36 . ◻
Let \(N\) be a positive integer. We say that a sequence \((x_n : n \geq 1)\) belongs to \(\mathcal{D}_1^N\) if \[\sum_{n=1}^\infty |x_{n+N} - x_n| < \infty.\] Notice that \((x_n : n \geq 1) \in \mathcal{D}_1^N\) if and only if for any \(i \in \{0,1,\ldots,N-1\}\) the sequence \((x_{nN+i} : n \geq 1)\) belongs to \(\mathcal{D}_1^1\), so, in particular, it is convergent. Moreover, we have the following
Proposition 7. Suppose that \((x_n),(y_n) \in \mathcal{D}_1^N\). Then
for any \(\alpha \in \mathbb{C}\) we have \((\alpha \cdot x_n) \in \mathcal{D}_1^N\),
\((x_n + y_n) \in \mathcal{D}_1^N\),
\((x_n \cdot y_n) \in \mathcal{D}_1^N\),
suppose that \(f\) is a Lipschitz continuous function on a compact interval \([a,b]\). If for some \(M\) we have \(\{x_n : n \geq M\} \subset [a,b]\), then \((f(x_n)) \in \mathcal{D}_1^N\).
The proof of Proposition 7 is straightforward.
Let \(N\) be a positive integer. We say that Jacobi parameters \((a_n),(b_n)\) are \(N\)-periodically modulated if there exist \(N\)-periodic sequences \((\alpha_n : n \in \mathbb{Z}),(\beta_n : n \in \mathbb{Z})\) of positive and real numbers, respectively, such that \[\label{eq:5} \lim_{n \to \infty} \bigg| \frac{a_{n-1}}{a_n} - \frac{\alpha_{n-1}}{\alpha_n} \bigg| = 0, \quad \lim_{n \to \infty} \bigg| \frac{b_n}{a_n} - \frac{\beta_n}{\alpha_n} \bigg| = 0, \quad \lim_{n \to \infty} a_n = \infty.\tag{37}\]
Proposition 8. If Jacobi parameters satisfy \[\label{eq:1} \bigg( \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} \bigg), \bigg( \frac{\beta_{n}}{\alpha_n} a_n - b_n \bigg), \bigg( \frac{1}{\sqrt{a_n}} \bigg) \in \mathcal{D}_1^N,\qquad{(1)}\] then \[\label{eq:139} \bigg( \frac{a_{n-1}}{a_n} \bigg), \bigg( \frac{b_n}{a_n} \bigg), \bigg( \frac{1}{a_n} \bigg) \in \mathcal{D}_1^N.\qquad{(2)}\]
Proof. We shall repeatedly use Proposition 7. Suppose that ?? is satisfied. Thus \[\frac{1}{a_n} = \frac{1}{\sqrt{a}_n} \frac{1}{\sqrt{a}_n}\] also belongs to \(\mathcal{D}_1^N\). Next, since \[\begin{align} \frac{a_{n-1}}{a_n} &= -\frac{1}{a_n} \Big( \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} \Big) + \frac{\alpha_{n-1}}{\alpha_n} \\ \frac{b_n}{a_n} &= -\frac{1}{a_n} \Big( \frac{\beta_{n}}{\alpha_n} a_n - b_{n} \Big) + \frac{\beta_{n}}{\alpha_n} \end{align}\] these sequences also belong to \(\mathcal{D}_1^N\), which ends the proof. ◻
For any \(n \in \mathbb{Z}\) let us define \[\mathfrak{X}_n(x) = \mathfrak{B}_{n+N-1}(x) \ldots \mathfrak{B}_{n+1}(x) \mathfrak{B}_n(x), \quad \text{where} \quad \mathfrak{B}_n(x) = \begin{pmatrix} 0 & 1 \\ -\frac{\alpha_{n-1}}{\alpha_n} & \frac{x-\beta_n}{\alpha_n} \end{pmatrix}.\] Spectral properties of Jacobi matrices with \(N\)-periodically modulated parameters depend on \(\operatorname{tr}\mathfrak{X}_0(0)\). The following theorem follows from [56] (see also [57]).
Theorem 9. Suppose that Jacobi parameters \((a_n),(b_n)\) are \(N\)-periodically modulated and \(\operatorname{tr}\mathfrak{X}_0(0) \in (-2,2)\). Assume further that \[\bigg( \frac{a_{n-1}}{a_n} \bigg), \bigg( \frac{b_n}{a_n} \bigg), \bigg( \frac{1}{a_n} \bigg) \in \mathcal{D}_1^N.\] Then \(J\) is self-adjoint if and only if the Carleman’s condition 36 is satisfied. If that is the case, then \[\sigma_{\mathrm{ac}}(J) = \mathbb{R}, \quad{and} \quad \sigma_{\mathrm{sc}}(J) = \emptyset, \quad{and} \quad \sigma_{\rm {p}}(J) = \emptyset.\]
The following theorem is a consequence of [58], but its hypotheses are taken from a less general [59].
Theorem 10. Suppose that Jacobi parameters \((a_n),(b_n)\) are \(N\)-periodically modulated and \(\mathfrak{X}_0(0)\) is not diagonalizable1. Assume further that \[\label{eq:4} \bigg( \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} \bigg), \bigg( \frac{\beta_{n}}{\alpha_n} a_n - b_n \bigg), \bigg( \frac{1}{\sqrt{a_n}} \bigg) \in \mathcal{D}_1^N.\tag{38}\] Then \(J\) is self-adjoint. Define \(N\)-periodic sequences \((s_n),(r_n)\) by \[\lim_{n \to \infty} \bigg| \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} - s_n \bigg| = 0, \quad \text{and} \quad \lim_{n \to \infty} \bigg| \frac{\beta_{n}}{\alpha_n} a_n - b_n - r_n \bigg| = 0.\] Let \(\varepsilon = \operatorname{sign}({\operatorname{tr}\mathfrak{X}_0(0)})\) and define a polynomial \[\label{eq:3} \tau(x) = \sum_{i=0}^{N-1} \bigg( \frac{s_i}{\alpha_{i-1}} \big( 1-\varepsilon [\mathfrak{X}_i(0)]_{1,1} \big) - \frac{x+r_i}{\alpha_{i-1}} \varepsilon [\mathfrak{X}_i(0)]_{2,1} \bigg).\tag{39}\] Then \[\sigma_{\mathrm{ac}}(J) = \sigma(J) \setminus \sigma_{\mathrm{discr}}(J) = \operatorname{cl}({\tau^{-1} \big( (-\infty,0) \big)}), \quad \text{and} \quad \sigma_{\mathrm{sc}}(J) = \emptyset, \quad \text{and} \quad \sigma_{\mathrm{p}}(J) \cap \tau^{-1}\big( (-\infty,0) \big) = \emptyset.\]
Proof. We are going to show that the hypotheses of [58] are satisfied for \(\gamma_n=a_n\). Notice that \[\begin{align} \tag{40} \sqrt{a_n} \bigg( \frac{\alpha_{n-1}}{\alpha_n} - \frac{a_{n-1}}{a_n} \bigg) &= \frac{1}{\sqrt{a_n}} \bigg( \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} \bigg) \\ \tag{41} \sqrt{a_n} \bigg( \frac{\beta_{n}}{\alpha_n} - \frac{b_{n}}{a_n} \bigg) &= \frac{1}{\sqrt{a_n}} \bigg( \frac{\beta_{n}}{\alpha_n} a_n - b_{n} \bigg). \end{align}\] Thus, in view of 38 the sequences on the left-hand sides belong to \(\mathcal{D}_1^N\). Next, by Proposition 8 we have that ?? is satisfied. In view of 37 and Proposition 7 we have \[\bigg( \sqrt{\frac{a_{n-1}}{a_n}} \bigg) \in \mathcal{D}_1^N.\] Since \[\sqrt{a_n} \bigg( \sqrt{\frac{\alpha_{n-1}}{\alpha_n}} - \sqrt{\frac{a_{n-1}}{a_n}} \bigg) = \sqrt{a_n} \bigg( \frac{\alpha_{n-1}}{\alpha_n} - \frac{a_{n-1}}{a_n} \bigg) \bigg( \sqrt{\frac{\alpha_{n-1}}{\alpha_n}} + \sqrt{\frac{a_{n-1}}{a_n}} \bigg)^{-1}\] the sequence on the left-hand side belongs to \(\mathcal{D}_1^N\). Consequently, we have shown [58]. According to [59] the sequence \((a_{n+N} - a_n)\) is bounded. Therefore, \[\lim_{n \to \infty} (\sqrt{a_{n+N}} - \sqrt{a_{n}}) = \lim_{n \to \infty} \frac{a_{n+N} - a_n}{\sqrt{a_{n+N}} + \sqrt{a_n}} = 0\] and we have verified that [58] holds true. Finally, the condition 38 easily implies [58]. Therefore, we have shown that the hypotheses of [58] are satisfied.
It remains to compare the formula 39 with [58]. By 40 we have \[\lim_{n \to \infty} \sqrt{a_n} \bigg( \frac{\alpha_{n-1}}{\alpha_n} - \frac{a_{n-1}}{a_n} \bigg) = 0,\] which leads to \(\mathfrak{S}= 0\) (cf. [58]). Next, \(\mathfrak{t}=1\) (cf. [58]). Finally, we have \[\mathfrak{u}_n = s_n \big( 1-\varepsilon [\mathfrak{X}_n(0)]_{1,1} \big) - r_n \varepsilon [\mathfrak{X}_n(0)]_{2,1}\] (cf. [58]), which leads to the equality of 39 and [58]. ◻
The following theorem follows from [60].
Theorem 11. Suppose that Jacobi parameters \((a_n),(b_n)\) are \(N\)-periodically modulated and \(\operatorname{tr}\mathfrak{X}_0(0) \in \mathbb{R}\setminus [-2,2]\). Assume further that \[\bigg( \frac{a_{n-1}}{a_n} \bigg), \bigg( \frac{b_n}{a_n} \bigg), \bigg( \frac{1}{a_n} \bigg) \in \mathcal{D}_1^N.\] Then \(J\) is self-adjoint and \(\sigma_{\mathrm{ess}}(J) =\sigma (J)\setminus \sigma_{\mathrm{discr}}(J)=\emptyset\).
The following proposition will be instrumental for our studies of Rabi models.
Proposition 12. Let \(N\) be a positive integer. Consider \[a_n = \alpha_n \sqrt{(n+t)(n+s)}, \quad b_n = \beta_n n + \gamma_n,\] where \(t,s>0\), \((\alpha_n),(\beta_n),(\gamma_n)\) are \(N\)-periodic sequences with \((\alpha_n)\) positive and \((\beta_n),(\gamma_n)\) real. Then ?? holds true. Moreover, \[\label{eq:2} \lim_{n \to \infty} \bigg| \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} - s_n \bigg| = 0, \quad \text{and} \quad \lim_{n \to \infty} \bigg| \frac{\beta_{n}}{\alpha_n} a_n - b_n - r_n \bigg| = 0\qquad{(3)}\] holds for \[s_n = \alpha_{n-1}, \quad r_n = \frac{\beta_n}{2} (t+s) - \gamma_n.\]
Proof. Consider a sequence \[\tilde{a}_n = \sqrt{(n+t)(n+s)}, \quad n \geq 0.\] It is immediate that \[\label{eq:6} \lim_{n \to \infty} (\tilde{a}_n - \tilde{a}_{n-1}) = 1.\tag{42}\] Let us define \[f(x) = \sqrt{(x+t)(x+s)}, \quad x > \max(-t,-s).\] Notice that \[\label{eq:7} f'(x) = \frac{2x+s+t}{2\sqrt{(x+t)(x+s)}}, \quad f''(x) = - \frac{(t-s)^2}{((x+t)(x+s))^{3/2}}.\tag{43}\] Thus the function \(f\) is increasing and concave (the case \(t=s\) corresponds to a linear function). Since \(\tilde{a}_n = f(n)\) by [59] we have \[(\tilde{a}_n - \tilde{a}_{n-1}), \Big( \frac{1}{\sqrt{\tilde{a}_n}} \Big) \in \mathcal{D}_1^1.\] Thus, these sequences belong also to \(\mathcal{D}_1^N\). Notice \[a_n = \alpha_n \tilde{a}_n, \quad \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} = \alpha_{n-1}(\tilde{a}_n - \tilde{a}_{n-1}).\] Since \(\mathcal{D}_1^N\) is an algebra we get \[\Big( \frac{\alpha_{n-1}}{\alpha_n} a_n - a_{n-1} \Big), \Big( \frac{1}{\sqrt{a_n}} \Big) \in \mathcal{D}_1^N.\] Moreover, in view of 42 we get the first equality in ?? . Next, define \(g(x) = f(x) - x\). In view of 43 we have \[g'(x) = \frac{2x+t+s}{2\sqrt{(x+t)(x+s)}} -1.\] By direct computations we get that \(g'\) is strictly positive if and only if \(t \neq s\) (if \(t=s\), then \(g\) is a constant function). Therefore, \(g\) is non-decreasing. Moreover, \[\label{eq:8} \lim_{x \to \infty} g(x) = \frac{t+s}{2}.\tag{44}\] Notice \(g(n) = \tilde{a}_n - n\). Therefore, \((\tilde{a}_n - n)\) is non-decreasing and bounded, thus it belongs to \(\mathcal{D}_1^1\), so also to \(\mathcal{D}_1^N\). Since \[\frac{\beta_n}{\alpha_n} a_n - b_n = \beta_n (\tilde{a}_n - n) - \gamma_n\] we have that this sequence also belongs to \(\mathcal{D}_1^N\). In view of 44 the second equality in ?? follows. The proof of ?? is complete. ◻
In this section, \(H\) is the Hamiltonian of the intensity-dependent Rabi model defined as the closure of \(\hat{H}\) given in 8 .
Let \(\{ e^\nu_n \}_{(\nu ,n)\in \{ -1,1\} \times \mathbb{N}_0}\) be the basis of \(\mathbb{C}^2\otimes \ell^2(\mathbb{N}_0)\) given by 5 . Since \(e^\nu_n=e^\nu \otimes e_n\) and \(\sigma_x e^\nu =e^{-\nu}\), \(\sigma_z e^\nu =\nu e^\nu\), we have \[g\big(\sigma_x \otimes \hat{a}^\dagger (\hat{N}+2\kappa )^{1/2} \big) e_n^{\nu} =g\big(\sigma_x e^\nu \otimes \hat{a}^\dagger (\hat{N}+2\kappa )^{1/2} e_n \big)= a_n\, e^{-\nu}_{n+1},\] \[g\big(\sigma_x \otimes (\hat{N}+2\kappa )^{1/2} \hat{a} \big) e_n^{\nu} =g\big(\sigma_x e^\nu \otimes (\hat{N}+2\kappa )^{1/2} \hat{a} e_n \big)= a_{n-1}\, e^{-\nu}_{n-1}\] with \((a_n)\) given by 10 . Moreover, \[\Big( I_2 \otimes \hat{N} +\frac{\Delta}{2}\, \sigma_z \otimes I\Big) e_n^{\nu} =\left( {n+\nu \frac{\Delta}{2} }\right) e_n^{\nu}\] and, introducing \(\mathfrak{F}^\pm :=\{ f^\pm_n\}_{n\in \mathbb{N}_0}\) with \[\label{41B} f^\pm_n:=e_n^{\pm (-1)^n},\tag{45}\] we obtain \[\label{41H} \hat{H} f^\pm_n =a_nf^\pm_{n+1} +a_{n-1}f^\pm_{n-1} + b^{\pm}_n f^\pm_n,\tag{46}\] where \((b_n^\pm )\) is given by 10 and \(a_{j}f^\pm_{j}:=0\) when \(j<0\). In order to obtain 9 , we consider the orthogonal decomposition \[\label{12orth} \mathbb{C}^2\otimes \ell^2(\mathbb{N}_0) =\mathcal{H}^- \oplus \mathcal{H}^+\tag{47}\] with \(\mathcal{H}^\pm\) defined as the closure of \({\rm span}(\mathfrak F^\pm )\). Due to 46 , \(\mathcal{H}^\pm\) are invariant subspaces for \(\hat{H}\) and \(J((a_{n}),(b_{n}^\pm ))\) is the matrix of \(\hat{H}|_{\mathcal{H}^\pm}\) in the basis \(\mathfrak F^\pm\).
To begin the analysis of \(J^\pm\) defined by \(J((a_n),(b_n^\pm ))\), we write \[a_n = \alpha_n \sqrt{(n+1) (n+2\kappa )}\, with \,\alpha_n \equiv g,\] \[b_n^\pm =\beta_n n +\gamma_n^\pm\, with \,\beta_n \equiv 1,\gamma_n^\pm = \pm (-1)^n \frac{\Delta}{2}\] Then ?? and ?? are satisfied with \(N=2\) due to Proposition 12 and 8. Moreover, \[\label{eq:100} {\mathfrak B}_n(0) = \begin{pmatrix} 0 & 1 \\ -1 & -\frac{1}{g} \end{pmatrix} , \quad \mathfrak{X}_n(0) = {\mathfrak B}_n(0)^2 = \begin{pmatrix} -1 & -\frac{1}{g} \\ \frac{1}{g} & -1 + \frac{1}{g^2} \end{pmatrix}\tag{48}\] and \[\label{1tr} \operatorname{tr}\mathfrak{X}_0(0) = -2 + \frac{1}{g^2}.\tag{49}\]
In this case, 49 gives \(\operatorname{tr}\mathfrak{X}_0(0) > 2\), and Theorem 11 ensures \(\sigma(J^\pm )=\sigma_{\mathrm{discr}}(J^\pm )\).
In this case, 49 gives \(-2<\operatorname{tr}\mathfrak{X}_0(0) < 2\), and Theorem 9 ensures \(\sigma_{\mathrm{ac}}(J^\pm )=\mathbb{R}\).
In this case \(\operatorname{tr}\mathfrak{X}_0(0)=2\), \(\mathfrak{X}_0(0)\) is not diagonalizable and, using Proposition 12 with \(s+t=2\kappa +1\), we obtain ?? with \[s_n =\alpha_{n-1} \equiv g=\frac{1}{2}\] and \[r_n^\pm =\frac{\beta_n}{2}(s+t) -\gamma_n^\pm = \kappa + \frac{1}{2} \mp (-1)^n \frac{\Delta}{2}.\] Since \(1-\varepsilon [\mathfrak{X}_0(0)]_{1,1}=2\) and \([\mathfrak{X}_0(0)]_{2,1}=\frac{1}{g} =2\), we find that 39 gives \[\tau^\pm (x) =4-4(2x+r_0^\pm +r_1^\pm )= -8(x+ \kappa ).\] Therefore \(\tau^\pm (x) < 0 \Leftrightarrow x> -\kappa\) and the conclusion follows from Theorem 10.
In this section, \(H\) is the Hamiltonian of the two-photon Rabi model defined as the closure of \(\hat{H}\) given in 14 .
If \(e^\nu_n :=e^\nu \otimes e_n\) as before and \(\mu \in \{0,1\}\), then \[g\big(\sigma_x \otimes (\hat{a}^\dagger )^2 \big) e_{2n+\mu}^{\nu} =a_{\mu ,n}\, e^{-\nu}_{2(n+1)+\mu},\] \[g\big(\sigma_x \otimes \hat{a}^2 \big) e_{2n+\mu}^{\nu} =a_{\mu ,n-1} \, e^{-\nu}_{2(n-1)+\mu}\] with \((a_{\mu ,n} )\) given by 17 . Moreover, \[\Big( I_2 \otimes \hat{N} +\frac{\Delta}{2}\, \sigma_z \otimes I\Big) e_{2n+\mu}^{\nu} = \left( {2n+\mu +\nu \frac{\Delta}{2} }\right) e_{2n+\mu}^{\nu}\] and, introducing \(\mathfrak{F}_\mu^\pm :=\{ f_{\mu ,n}^\pm \}_{n\in \mathbb{N}_0}\) with \[\label{42f} f_{\mu ,n}^\pm :=e_{2n+\mu}^{\pm (-1)^n},\tag{50}\] we obtain \[\label{42H} \hat{H} f_{\mu ,n}^\pm =a_{\mu ,n} f_{\mu ,n+1}^\pm +a_{\mu ,n-1}f^\pm_{\mu ,n-1} + b^{\pm}_{\mu ,n} f^\pm_{\mu ,n} ,\tag{51}\] where \((b^\pm_{\mu ,n} )\) is given by 17 and \(a_{\mu ,j}f^\pm_{\mu ,j}:=0\) when \(j<0\). In order to obtain 16 , we consider the orthogonal decomposition \[\label{42orth} \mathbb{C}^2\otimes \ell^2(\mathbb{N}_0) =\mathcal{H}_0^- \oplus \mathcal{H}_0^+ \oplus \mathcal{H}_1^- \oplus \mathcal{H}_1^+\tag{52}\] with \(\mathcal{H}_\mu^\pm\) defined as the closure of \({\rm span}(\mathfrak F_\mu^\pm )\). Due to 51 , \(\mathcal{H}_\mu^\pm\) are invariant subspaces for \(\hat{H}\) and \(J((a_{\mu ,n}),(b_{\mu ,n}^\pm ))\) is the matrix of \(\hat{H}|_{\mathcal{H}_\mu^\pm}\) in the basis \(\mathfrak F_\mu^\pm\).
As before, Proposition 12 and 8 ensure that ?? , ?? are satisfied with \(N=2\) and \[\label{42alpha} a_{\mu ,n} = \alpha_n \sqrt{(n+\tfrac{1}{2} + \tfrac{\mu}{2}) (n+1+\tfrac{\mu}{2})}\, with \,\alpha_n \equiv 2g,\tag{53}\] \[\label{beta} b_{\mu ,n}^\pm =\beta_n n +\gamma_{\mu ,n}^\pm\, with \,\beta_n \equiv 2,\gamma_{\mu ,n}^\pm = \mu \pm (-1)^n \frac{\Delta}{2} .\tag{54}\] Moreover, \({\mathfrak B}_n(0)\), \(\mathfrak{X}_n(0)\) and \({\rm tr} \mathfrak{X}_n(0)\) are given by 48 and 49 , respectively. This gives the following three cases similar to before.
We have \(\operatorname{tr}\mathfrak{X}_0(0) > 2\) and Theorem 11 ensures \(\sigma(J_\mu^\pm )=\sigma_{\mathrm{discr}}(J_\mu^\pm )\).
We have \(-2<\operatorname{tr}\mathfrak{X}_0(0) < 2\) and Theorem 9 ensures \(\sigma_{\mathrm{ac}}(J_\mu^\pm )=\mathbb{R}\).
We have \(\operatorname{tr}\mathfrak{X}_0(0)=2\) and, using Proposition 12 with \(s+t=\mu +\frac{3}{2}\), we obtain ?? with \[s_n =\alpha_{n-1} \equiv 2g=1\] and \[r_n^\pm = \frac{\beta_n}{2}\Big( \, \mu +\frac{3}{2} \, \Big) -\gamma_{\mu ,n}^\pm =\frac{3}{2} \mp (-1)^n \frac{\Delta}{2} .\] Therefore, \[\tau^\pm (x)=4-2(2x+r_0^\pm +r_1^\pm )=-4x-2.\] It is clear that \(\tau^\pm (x) < 0 \Leftrightarrow x> -\frac{1}{2}\) and the conclusion follows from Theorem 10.
In this section, \(H\) is the Hamiltonian of the anisotropic two-photon Rabi model defined as the closure of \(\hat{H}\) given in 21 .
Using \(g\) and \(g'\) given by 23 , we find \[(g_- \sigma_- +g_+ \sigma_+ )e^{\pm 1} = g_\mp e^{\mp 1} =(g \mp g') e^{\mp 1}\] and, using \(f_{\mu ,n}^\pm =e^{\pm (-1)^n}_{2n+\mu}\), we can express \[\big( (g_-\sigma_- +g_+\sigma_+) \otimes (\hat{a}^\dagger )^2 \big) f_{\mu ,n}^\pm = a_{\mu ,n}^\pm f_{\mu ,n+1}^\pm ,\] \[\big( (g_+\sigma_- +g_-\sigma_+) \otimes \hat{a}^2 \big) f_{\mu ,n}^\pm = a_{\mu ,n-1}^\pm f_{\mu ,n-1}^\pm\] with \(a_{\mu ,n}^\pm\) given by 22 . Moreover, \[\label{43H} \hat{H} f_{\mu ,n}^\pm =a_{\mu ,n}^\pm f_{\mu ,n+1}^\pm +a_{\mu ,n-1}^\pm f^\pm_{\mu ,n-1} + b^{\pm}_{\mu ,n} f^\pm_{\mu ,n}\tag{55}\] with \(b_{\mu ,n}^\pm\) given by 22 and \(a^\pm_{\mu ,j}f^\pm_{\mu ,j}:=0\) when \(j<0\). It is clear that we obtain 16 , using 52 similarly as in Section 3.2. Then Proposition 12 and 8 ensure that ?? , ?? are satisfied with \(N=2\), \[a_{\mu ,n}^\pm = \alpha_n^\pm \sqrt{(n+\tfrac{1}{2} + \tfrac{\mu}{2}) (n+1+\tfrac{\mu}{2})}\, with \,\alpha_n^\pm = 2(g \mp (-1)^n g'),\] and \(\beta_n\), \(\gamma_{\mu ,n}^\pm\) as in 54 . Moreover, \[\mathfrak{B}_n^\pm (0) = \begin{pmatrix} 0 & 1 \\ -\frac{\alpha^\pm_{n+1}}{\alpha^\pm_n} & -\frac{2}{\alpha^\pm_n} \end{pmatrix}, \quad \mathfrak{X}_n^\pm (0) = \begin{pmatrix} \frac{-\alpha^\pm_{n+1}}{\alpha^\pm_n} & -\frac{2}{\alpha^\pm_n} \\ \frac{2}{\alpha^\pm_n} & \frac{4-(\alpha^\pm_n)^2}{\alpha^\pm_n \alpha^\pm_{n+1}} \end{pmatrix}\] If \(\sigma \in \{ -1, 1\}\), then \[\operatorname{tr}\mathfrak{X}^\pm_0(0) -2\sigma = \frac{4-(\alpha^\pm_0)^2-(\alpha^\pm_1)^2}{\alpha^\pm_0 \alpha^\pm_1} - 2\sigma = \frac{4-(\alpha^\pm_0 +\sigma \alpha^\pm_1)^2}{\alpha^\pm_0 \alpha^\pm_1}.\]
In this case \(4-(\alpha^\pm_0 +\alpha^\pm_1)^2=4-16g^2>0\), therefore \({\rm tr\,} \mathfrak{X}^\pm_0(0) -2>0\), i.e. Theorem 11 ensures \(\sigma(J_\mu^\pm )=\sigma_{\mathrm{discr}}(J_\mu^\pm )\).
In this case \(4-(\alpha^\pm_0 -\alpha^\pm_1)^2=4-16g'^2<0\), therefore \({\rm tr\,} \mathfrak{X}^\pm_0(0) +2<0\), i.e. Theorem 11 ensures \(\sigma(J_\mu^\pm )=\sigma_{\mathrm{discr}}(J_\mu^\pm )\).
In this case \(-2<{\rm tr\,} \mathfrak{X}^\pm_0(0)<2\), i.e. Theorem 9 ensures \(\sigma_{\mathrm{ac}}(J_\mu^\pm )=\mathbb{R}\).
In this case \({\rm tr\,} \mathfrak{X}^\pm_0(0)=2\), \(\mathfrak{X}^\pm_0(0)\) is not diagonalizable and, using Proposition 12 with \(s+t=\mu +\frac{3}{2}\), we obtain ?? with \[s_n^\pm =\alpha^\pm_{n-1} = 2 \big( g \pm (-1)^n g' \big) , \quad \beta_n \equiv 2 ,\gamma_{\mu ,n}^\pm = \mu \pm (-1)^n \frac{\Delta}{2}\] and \[\frac{\beta_n}{2}\Big( \, \mu +\frac{3}{2} \, \Big) -\gamma_{\mu ,n}^\pm = \frac{3}{2} \mp (-1)^n \frac{\Delta}{2} =r_n^\pm .\] Since \(\varepsilon =1\) and \(\alpha^\pm_0+\alpha^\pm_1 =4g=2\), we find \[2-\varepsilon [\mathfrak{X}^\pm_0(0)+\mathfrak{X}^\pm_1(0)]_{1,1} =2+\frac{\alpha^\pm_1}{\alpha^\pm_0}+ \frac{\alpha^\pm_0}{\alpha^\pm_1} = \frac{(\alpha^\pm_0+\alpha^\pm_1)^2}{\alpha^\pm_0\alpha^\pm_1} = \frac{4}{\alpha^\pm_0\alpha^\pm_1} .\] Moreover, \[\frac{[\mathfrak{X}^\pm_n(0)]_{2,1}}{\alpha^\pm_{n-1}} = \frac{2}{\alpha^\pm_0\alpha^\pm_1}\] and \[\tau^\pm (x) = \frac{4}{\alpha^\pm_0 \alpha^\pm_1} -\frac{2}{\alpha^\pm_0 \alpha^\pm_1} (2x+r_0^\pm +r_1^\pm ) =-\frac{4x+2}{\alpha^\pm_0 \alpha^\pm_1} .\] Therefore \(\tau^\pm (x) < 0 \Leftrightarrow x> -\frac{1}{2}\) and the conclusion follows from Theorem 10.
In this case \({\rm tr\,} \mathfrak{X}^\pm_0(0)=-2\), \(\mathfrak{X}^\pm_0(0)\) is not diagonalizable, \(\varepsilon =-1\) and \[2-\varepsilon [\mathfrak{X}^\pm_0(0)+\mathfrak{X}^\pm_1(0)]_{1,1} =2-\frac{\alpha^\pm_1}{\alpha^\pm_0}- \frac{\alpha^\pm_0}{\alpha^\pm_1} = -\frac{(\alpha^\pm_0-\alpha^\pm_1)^2}{\alpha^\pm_0\alpha^\pm_1} = -\frac{4}{\alpha^\pm_0\alpha^\pm_1} ,\] hence \[\tau^\pm (x) = -\frac{4}{\alpha^\pm_0 \alpha^\pm_1} +\frac{2}{\alpha^\pm_0 \alpha^\pm_1} (2x+r_0^\pm +r_1^\pm ) =\frac{4x+2}{\alpha^\pm_0 \alpha^\pm_1}\] Therefore \(\tau^\pm (x) < 0 \Leftrightarrow x< -\frac{1}{2}\) and the conclusion follows from Theorem 10.
In this section, \(H\) is the Hamiltonian of the two-photon Rabi–Stark model defined as the closure of \(\hat{H}\) given in 29 .
Consider 47 with \(\mathcal{H}_\mu^\pm\) generated by \(\mathfrak{F}_\mu^\pm :=\{ f_{\mu ,n}^\pm \}_{n\in \mathbb{N}_0}\) defined in 50 . Then it is easy to check that 51 holds true with \((a_{\mu ,n})\), \((b_{\mu ,n}^\pm )\) given by 30 . Thus, \(\mathcal{H}_\mu^\pm\) are invariant subspaces for \(\hat{H}\) and \(J((a_{\mu ,n}),(b_{\mu ,n}^\pm ))\) is the matrix of \(\hat{H}|_{\mathcal{H}_\mu^\pm}\) in the basis \(\mathfrak F_\mu^\pm\). Then Proposition 12 and 8 ?? , ?? are satisfied with \(N=2\), \(\alpha_n\) as in 53 and \[b_{\mu ,n}^\pm =\beta_n^\pm n +\gamma_{\mu ,n}^\pm\, with \,\beta_n^\pm = 2(1\pm (-1)^n \kappa ),\gamma_{\mu ,n}^\pm = (1\pm (-1)^n \kappa ) \mu \pm (-1)^n \frac{\Delta}{2}\] Moreover, we have \[\mathfrak{B}_{n}^\pm (0) = \begin{pmatrix} 0 & 1 \\ -1 & -\frac{1\pm (-1)^n\kappa }{g} \end{pmatrix} , \quad \mathfrak{X}_{n}^\pm (0) = \begin{pmatrix} -1 & \frac{-1\mp (-1)^n\kappa }{g} \\ \frac{1\mp (-1)^n\kappa }{g} & -1+\frac{1-\kappa^2 }{g^2} \end{pmatrix},\] hence \[\label{4tr} {\rm tr}\mathfrak{X}_{n}^\pm (0) =-2+ \frac{1-\kappa^2 }{g^2}.\tag{56}\]
In this case, 56 gives \(\operatorname{tr}\mathfrak{X}_0^\pm (0) < -2\) and Theorem 11 ensures \(\sigma(J_\mu^\pm )=\sigma_{\mathrm{discr}}(J_\mu^\pm )\).
In this case, 56 gives \(\operatorname{tr}\mathfrak{X}_0^\pm (0)>2\) and Theorem 9 ensures \(\sigma(J_\mu^\pm )=\sigma_{\mathrm{discr}}(J_\mu^\pm )\).
In this case, 56 gives \(|\operatorname{tr}\mathfrak{X}_0^\pm (0)|< 2\) and Theorem 11 ensures \(\sigma_{\mathrm{ac}}(J_\mu^\pm )=\mathbb{R}\).
In this case, 56 gives \(\operatorname{tr}\mathfrak{X}^\pm_0(0)=2\), \(\mathfrak{X}^\pm_0\) is not diagonalizable, and using Proposition 12 with \(s+t=\mu +\frac{3}{2}\), we obtain ?? with \[s_n =\alpha_{n-1} \equiv 2g\] and \[\frac{\beta^\pm_n}{2}\Big( \, \mu +\frac{3}{2} \, \Big) -\gamma_{\mu ,n}^\pm = \frac{3}{2} (1 \pm (-1)^n \kappa) \mp (-1)^n \frac{\Delta}{2} =r_n^\pm .\] Then \(1-\varepsilon [\mathfrak{X}^\pm_n(0)]_{1,1}=2\) and we obtain \[\tau^\pm (x) =-\frac{x}{g^2} + 4 - \frac{3(1-\kappa^2)}{2g^2} - \frac{\kappa \Delta}{2 g^2}\] and \(\tau^\pm (x) < 0 \Leftrightarrow x> \frac{\kappa^2 -1- \kappa \Delta}{2}\) and the conclusion follows from Theorem 10.
In this case, 56 gives \(\operatorname{tr}\mathfrak{X}^\pm_0(0)=-2\) and \(1-\varepsilon [\mathfrak{X}^\pm_n(0)]_{1,1}=0\), hence \[\tau^\pm (x) = \frac{x}{g^2} + \frac{\kappa \Delta}{2g^2}\] Therefore \(\tau^\pm (x) < 0 \Leftrightarrow x < - \frac{\kappa \Delta}{2}\) and the conclusion follows from Theorem 10.
In this paper, we have investigated spectral transitions for Rabi models which are unitarily similar to direct sums of Jacobi operators. We have presented an analysis of the spectrum of Jacobi operators based on the subordinacy theory developed in [56]–[60]. In particular, new results have been given for the intensity-dependent Rabi model. We have proved that the spectral transition in the intensity-dependent Rabi model is similar to that in the two-photon model; the only difference is that the critical coupling gives the essential spectrum, which is a half-line depending on the additional parameter \(\kappa\) of the model. Concerning the two-photon anisotropic Rabi model, we have proved that there are two cases of critical coupling: the well-known case of the essential spectrum \(\sigma_{\mathrm{ess}}(H)=[-\frac{1}{2}, \infty )\) and the case of the essential spectrum \(\sigma_{\mathrm{ess}}(H)=(-\infty ,-\frac{1}{2} ]\), which seems to be a new result. Concerning the two-photon Rabi-Stark model, we have proved that there are two cases of critical coupling: the case \(4g^2+\kappa^2=1\) with \(\kappa \in (-1,1)\) and the case \(\kappa =\pm 1\). The essential spectrum is formed by a half-line bounded below in the first case and by a half-line bounded above in the second case. We note that \(\inf \sigma_{\mathrm{ess}}(H)\) in the first case and \(\sup \sigma_{\mathrm{ess}}(H)\) in the second case depend on the parameters of the model. Our result in the case \(\kappa =\pm 1\) seems to be new for the two-photon Rabi-Stark, but an analogical case was considered for the one-photon Rabi-Stark.
We have also proved that there is no singular spectrum. Moreover, we have proved that there is no eigenvalue in the interior of the continuous spectrum for all models considered. This fact is particularly interesting for the two-photon Rabi-Stark model in the context of the work [31], investigating the question of existence of "bound states embedded in the continuous spectrum" for the one-photon Rabi-Stark model.
The subordinacy theory is a powerful tool to locate the continuous spectrum and to ensure absence of eigenvalues in the interior of the continuous spectrum. On the other hand, it gives little information on the discrete spectrum. For this reason, our descriptions of spectral transitions are essentially limited to the statements about absence or presence of the continuous spectrum. In particular, this approach cannot treat superradiant problems or phase transitions connected with the behaviour of the ground state. We note that numerical analysis based on a truncated Hilbert space does not work well if the model approaches the spectral transition. An analysis of the associated \(G\)-functions is the only method that can detect the first-order phase transition in the anisotropic two-photon Rabi model (see [38]) and the second-order phase transition in the two-photon Rabi-Stark model (see [33]). However, there are perspectives to develop a subordinacy theory of block Jacobi matrices that could be applied to the mixed Rabi model in order to describe phase transitions considered in quantum metrology (see [48]).
rabi
In particular, \(\operatorname{tr}\mathfrak{X}_0(0) \in \{-2,2\}\) and \(\mathfrak{X}_0(0)\) is not a multiple of the identity matrix.↩︎