Abstract The quantum Rabi model (QRM) is identified as the simplest model to study the interaction between light and matter. In recent years, Braak proposed an alternative more efficient method to obtain the exact solvability and spectrum of the QRM in the Bargmann space, which is appropriate frame for extended squeezed coherent states. Based on this feasible method, some study for the two-photon QRM is concentrated. In this paper we diagonalized the Hamiltonian and obtain the Spectrum of the system of two-photon QRM in the special case that the level splitting of the atom was equal to 0, and compared with Travĕnec’s results without diagonalizing the Hamiltonian. Finally, we confirm the validity of diagonalization for the Hamiltonian in the two-photon QRM.
Abstract The quantum Rabi model (QRM) is identified as the simplest model to study the interaction between light and matter. In recent years, Braak proposed an alternative more efficient method to obtain the exact solvability and spectrum of the QRM in the Bargmann space, which is appropriate frame for extended squeezed coherent states. Based on this feasible method, some study for the two-photon QRM is concentrated. In this paper we diagonalized the Hamiltonian and obtain the Spectrum of the system of two-photon QRM in the special case that the level splitting of the atom was equal to 0, and compared with Travĕnec’s results without diagonalizing the Hamiltonian. Finally, we confirm the validity of diagonalization for the Hamiltonian in the two-photon QRM.
Within extended coherent states, a recent exact solution to the quantum Rabi model (QRM) [D. Braak, Phys. Rev. Lett. 107, 100401 (2011)] can be recovered in an alternative simpler and more physical way, without use of any extra conditions. In the same framework, the two-photon QRM is solved exactly by treating extended squeezed states on an equal footing. Concise transcendental functions responsible for the exact solutions are derived. The isolated Juddian solutions are also analytically obtained in terms of degeneracy. Both the extended coherent states and squeezed states employed here are essentially Fock states in the space of the corresponding Bogoliubov operators, which result in free-particle number operators. The present approach can be summarized concisely in a unified way and easily extended to various spin-boson systems with multiple levels, even multiple modes.
The spectrum of the two-photon Rabi Hamiltonian is found, proceeding in full analogy with the solution of the standard (one-photon) Rabi Hamiltonian, published by D. Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The Hamiltonian is rewritten as a set of two differential equations. Symmetries that get hidden after further treatment are found. One can plainly see how the Hilbert space splits into four disjunct subspaces, categorized by four values of the symmetry parameter $c=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}i$. There were only two values, $\ifmmode\pm\else\textpm\fi{}1$, for the standard Rabi model. The values of c are in fact the eigenvalues of the respective parity operators. Four analytic functions are introduced by a recurrence scheme for the coefficients of their series expansion. All their roots yield the complete spectrum of the Hamiltonian. Eigenstates in Bargmann space can also be found.
The quantum Rabi model can be solved exactly by the Bargmann transformation from real coordinate to complex variable recently [Phys. Rev. Lett. \textbf{107}, 100401 (2011)]. By the extended coherent states, we recover this solution in an alternative simpler and perhaps more physical way without uses of any extra conditions, like Bargmann conditions. In the same framework, the two-photon Rabi model are solved exactly by extended squeeze states. Transcendental functions have been derived with the similar form as those in one-photon model. Both extended coherent states and squeeze states are essentially Fock states in the space of the corresponding Bogoliubov operators. The present approach could be easily extended to study the exact solvability or integrability of various spin-boson systems with multi-level, even multi-mode.
The quantum Rabi model is essential for understanding interacting quantum systems. It serves as the simplest non-integrable yet solvable model describing the interaction between a two-level system and a single mode of a bosonic field. In this study, we delve into the exploration of the generalized quantum Rabi model, wherein the bosonic mode of the field undergoes squeezing. Utilizing the Segal-Bargmann representation of the infinite-dimensional Hilbert space, we demonstrate that the energy spectrum of the generalized quantum Rabi model, when both the Rabi coupling strength and the squeezing strength are not significantly large compared to the field mode frequency, can be analytically determined by a bi-confluent Fuchsian equation with two regular singularities at 0 and 1 and an irregular singularity of rank two at infinity.
The quantum Rabi model (QRM), describing a two-level system coupled to a quantum harmonic oscillator, is one of the simplest and most ubiquitous interaction models in quantum mechanics. Despite its simplicity, the QRM has found applications in various fields of physics. The most exciting realization of the QRM nowadays is arguably the circuit quantum electrodynamics (QED) system, where an on-chip artificial atom (or a qubit) is coupled to a transmission line or a waveguide. The qubits in circuit QED systems contain a bias term, which breaks the Z_2 symmetry in the QRM. Therefore, the precise model realized in circuit QED systems is referred to as the asymmetric quantum Rabi model (AQRM). This thesis explores theoretical methods to treat the AQRM and develops a complete approach to describe its energy landscape. The hidden symmetry of the AQRM is investigated in detail, and seen to be a rather general phenomenon in light-matter interaction models. The equivalence between the generalised Poschl-Teller potential and the spectrum of the AQRM is also established. Chapter 1 is dedicated to reviewing the analytic solutions and conventional approximations to the QRM/AQRM. Regarding the analytic solutions, the analytic results in terms of the transcendental functions determining the eigenspectrum of the AQRM are collected. Special emphasis is placed on the isolated closed-form exceptional solutions to the AQRM. Due to the lack of simple closed-form expressions for the general eigenvalues, some well-known approximations are considered. In Chapter 2, a generalised adiabatic approximation (GAA) is developed to calculate the eigenvalues of the excited states of the QRM and apply it to the AQRM. The GAA is based on the similarity between the exact exceptional solutions and the Laguerre polynomials. By construction, the GAA always has exact exceptional energy and approximates the regular spectrum with rather good agreement in a large parameter regime. Furthermore, the GAA is applied to the AQRM and shown to correctly recover the conical intersections in the energy landscape. The geometric phases around these conical intersections are calculated analytically as an illustrative example. In Chapter 3, a physically motivated variational wave function is presented for the ground state of the AQRM. The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states and relies only on three variational parameters in the regimes of interest. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. The results show that the variational expansion is a significant improvement over the existing approximations for the AQRM. In Chapter 4, the hidden symmetry and tunnelling dynamics in the AQRM and several related models are analysed in detail. The AQRM has a broken Z_2 symmetry, with generally a non-degenerate eigenvalue spectrum. In some special cases of the related models, stable level crossings typical of the Z_2-symmetric quantum Rabi model are recovered. It is thus shown that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models. In Chapter 5, starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the AQRM, a spectral equivalence is established with QES hyperbolic Schrodinger potentials. This leads to particular QES Poschl-Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Poschl-Teller potentials. Conclusions and future directions are presented in the final Chapter 6.
We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two bare polarons. Importantly, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional induced polarons. According to this picture, the variational ground-state wave function of the two-photon QRM can be correctly constructed, with the ground-state energy and other physical observables in good agreement with the exact numerics in all the coupling regimes. Furthermore, generalization to multiple induced polarons involving higher orders in the tunneling effect provides a systematic way to yield a rapid convergence in accuracy even around the difficult spectral collapse point. In addition, the polaron picture provides a distinctive understanding of an incomplete spectral collapse behavior that is about the existence of discrete energy levels apart from the collapsed spectrum at the spectral collapse point. This work illustrates that the polaron picture is helpful to capture the key physics in this nonlinear light-matter interaction model and indicates that this method can be applicable to more complicated QRM-related models.