The quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum.
The quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum.
The quantum Rabi model is a model in quantum optics used to describe the interaction between a two-level atom and a single-level photon field.This model, described in 1963 by Jaynes and Cummings, is the fully quantized version of the original semi-classical model proposed in 1936 by Rabi.In previous years, the study of the properties of Hamiltonian of the quantum Rabi model and its energy levels (eigenvalues of the Hamiltonian) has been relegated mainly for two reasons.The first one was the belief that the quantum Rabi model was not an exactly solvable model, that is, that its energy levels could not be described analytically.The second reason is that in the parameter regime achievable at the time in experiments and applications, the quantum Rabi model could be approximated successfully by the Jaynes-Cummings model, a simpler model which is known to be exactly solvable.Both situations changed in recent years.In 2011, Daniel Braak proved the exact-solvability of the quantum Rabi model by constructing analytical solutions and defining a transcendental function, the G-function, whose zeros determine all the spectrum (with the exception of a (possible finite) set of exceptional eigenvalues).This pioneering technique has since then been successfully applied to show the exact-solvability of several models in quantum optics.On the other hand, due to the advances in experimental physics, starting from the first decade of this century, experiments have been steadily reaching parameter regimes where the Jaynes-Cummings model becomes unsuitable for approximation and in 2014, the experiments by Maissen et al reached regimes where it is imperative to consider the full Hamiltonian of the quantum Rabi model.Adding to this, recently there has also been proposals for applications of the quantum Rabi model in quantum information theory and quantum computing.For these reasons, there has been a large amount of research done in theoretical and experimental physics, and, more recently, mathematics on the subject of the quantum Rabi model, its generalizations and applications.In this thesis, we study the asymmetric quantum Rabi model, one of the generalizations of the quantum Rabi model.In this generalization, an additional non-trivial interaction term is introduced in the Hamiltonian, breaking the Z 2Z-symmetry of the quantum Rabi Hamiltonian.The presence of this symmetry in the quantum Rabi model explains the presence of crossings in the energy levels (degeneracies in the spectrum), so a priori there was no reason to expect degeneracies in the spectrum of the asymmetric version.In this work, we show that when the coefficient of the symmetry-breaking term is halfinteger, degeneracies appear in the spectrum of the asymmetric quantum Rabi model.This is done by studying the properties of the constraint polynomials, polynomials appearing in certain conditions that the parameters of the model must satisfy in order to have certain eigenvalues, called Juddian eigenvalues.The spectrum of the asymmetric quantum Rabi model is divided into different sets, an infinite set of regular eigenvalues, the zeros of the G-function and the exceptional eigenvalues, not captured as zeros of the G-function.The exceptional spectrum is further classified into Juddian and non-Juddian, according to the type of solution in the Segal-Bargmann space realization.These solutions are also described in terms of the sl 2 picture of the Hamiltonian in terms of irreducible representations of sl 2 .The case for Juddian and regular eigenvalues was already known, and in this thesis we complete the picture for the non-Juddian eigenvalues.Another main result of this work is to characterize the degeneracies of the asymmetric quantum Rabi model in terms of the types of eigenvalues and the parameters.Furthermore, by carefully studying the poles of the G-function, we define a new G-function whose zeros give the full spectrum of the asymmetric quantum Rabi model.Finally, we present a study on continued fractions expansions of integer powers of the Napier constant e with a nice representation and good convergence properties.This study is a byproduct of the study of certain orthogonal polynomial families related to the constraint polynomials of the AQRM.
The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.
The analytically exact solutions to the mixed quantum Rabi model including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4×4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can be also obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point 1/2. Surprisingly, all energy levels only collapse to the lower one, which diverges negatively. All the level crossings in the unmixed models are relaxed to avoided crossings in the present mixed model due to absence of parity symmetry. Very interestingly, the effective one-photon coupling is enhanced in the presence of the two-photon coupling, which paves a highly efficient and economic way to access the deep-strong coupling regime even though the original one-photon coupling is weak.
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.
The Rabi model is a paradigm for interacting quantum systems.It couples a bosonic mode to the smallest possible quantum model, a two-level system.I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit.A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry.Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.
The well-known quantum Rabi model strictly describes the simplest interaction between matter and the quantum light.We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced.We consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction.The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom.