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Asymmetric Quantum Rabi Models for Light-Matter Interaction

Zimin Li

Published Jan 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.25911/ezta-qb77

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非対称量子ラビ模型の固有値に関する研究

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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.

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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.
Abstract The asymmetric quantum Rabi model (AQRM) exhibits level crossings in the eigenspectrum for the values ϵ ∈ 1 2 Z of the bias parameter ϵ . Such level crossings are expected to be associated with some hidden symmetry of the model. The origin of this hidden symmetry is established by finding the operators which commute with the AQRM Hamiltonian at these special values. The construction is given explicitly for the first several cases ϵ = 0 , 1 2 , 1 and can be applied to other related light–matter interaction models for which similar level crossings have been observed in the presence of a bias term.
The asymmetric quantum Rabi model (AQRM) has a broken Z2{\mathbb{Z}}_{2}Z2​ symmetry, with generally a nondegenerate eigenvalue spectrum. In some special cases where the asymmetric parameter is a multiple of the cavity frequency, stable level crossings typical of the Z2{\mathbb{Z}}_{2}Z2​-symmetric quantum Rabi model are recovered, however, without any obvious paritylike symmetry. This unknown ``symmetry'' has thus been referred to as hidden symmetry in the literature. Here, we show that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models with an asymmetric qubit bias term. Conditions under which the hidden symmetry exists in these models are determined and discussed. By investigating tunneling dynamics in the displaced oscillator basis, a strong connection is found between the hidden symmetry and selective tunneling.
We study the two-qubit asymmetric quantum Rabi model (AQRM) and find its dark-state solution. Such solutions have at most one photon and constant eigenenergy in the whole coupling regime, causing level crossings in the spectrum, although there is no explicit conserved quantity except energy. We find an operator in the eigenenergy basis to label all the degeneracies with its eigenvalues, and compare it with the well-known hidden symmetry which exists when bias parameter εεε is a multiple of half of the resonator frequency ωωω. Extended to the multimode case, we find symmetries related with conserved bosonic number operators, which also cause level crossings. This provides a perspective for symmetry studies on generalized Rabi models.
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.
Abstract We study the two-qubit asymmetric quantum Rabi model (AQRM) and find its dark-state solution. Such solutions have at most one photon and constant eigenenergy in the whole coupling regime, causing level crossings in the spectrum, although there is no explicit conserved quantity except for energy. We find an operator in the eigenenergy basis to label all the degeneracies with its eigenvalues, and compare it with the well-known hidden symmetry which exists when bias parameter ϵ is a multiple of half of the resonator frequency ω . Extended to the multimode case, we find symmetries related with conserved bosonic number operators, which also cause level crossings. This paves the way for further symmetry studies on generalized Rabi models.
We have studied the two-qubit quantum Rabi model in the asymmetric case and its generalizations with dipole and Heisenberg-type qubit-qubit interactions. The solutions are obtained analytically with eigenstates given in terms of the extended coherent states or photon number states. For identical qubit-photon couplings, a novel type of quasi-exact solution which exists for all coupling values with constant eigenenergy is found, leading to level crossings within the same parity subspace even for non-identical qubits. In contrast to the quasi-exact eigenstates of the single-qubit model, these exceptional eigenstates are formed by just a few Fock states (photon number bounded from above at one or three), and the condition for them depends only on a fine-tuning of the qubit level splittings with respect to the photon energy, but not on the coupling to the photon field if the qubit-qubit interaction is not taken into consideration. This makes them excellent candidates for direct experimental observation within circuit quantum electrodynamics (QED) and application in single photon experiments. Besides, exceptional solutions with finite photon numbers N are also found.