A hidden analytic structure of the Rabi model

The Rabi model describes the simplest interaction between a cavity mode with a frequency ω_c${\omega }_c$ and a two-level system with a resonance frequency ω₀${\omega }_0$. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to Δ=ω₀/(2ω_c)=0$\Delta ={\omega }_0/\left(2{\omega }_c\right)=0$, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy ?$?$, which are orthogonal on an equidistant lattice. A non-zero value of Δ$\Delta$ leads to non-classical discrete orthogonal polynomials ?_k(?)${?}_k\left(?\right)$ and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak’s solution. Any first n$n$ eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ?_N(?)${?}_N\left(?\right)$ of at least the degree N=n+n_t$N=n+n_t$. The value of n_t>0$n_t>0$, which is slowly increasing with n$n$, depends on the required precision. For instance, n_t?26$n_t?26$ for n=1000$n=1000$ and dimensionless interaction constant κ=0.2$\kappa =0.2$, if double precision is required. Given that the sequence of the l$l$th zeros x_nl$x_{nl}$’s of ?_n(?)${?}_n\left(?\right)$’s defines a monotonically decreasing discrete flow with increasing n$n$, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant κ<1$\kappa <1$, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1$\kappa \ge 1$.     

Geometric curvature and phase of the Rabi model

We study the geometric curvature and phase of the Rabi model. Under the rotating-wave approximation (RWA), we apply the gauge independent Berry curvature over a surface integral to calculate the Berry phase of the eigenstates for both single and two-qubit systems, which is found to be identical with the system of spin-1/2 particle in a magnetic field. We extend the idea to define a vacuum-induced geometric curvature when the system starts from an initial state with pure vacuum bosonic field. The induced geometric phase is related to the average photon number in a period which is possible to measure in the qubit–cavity system. We also calculate the geometric phase beyond the RWA and find an anomalous sudden change, which implies the breakdown of the adiabatic theorem and the Berry phases in an adiabatic cyclic evolution are ill-defined near the anti-crossing point in the spectrum.     

Rabi模型的光场压缩

电路量子电动力学的实验实现了光与人造原子的超强耦合相互作用, 相互作用强度与光场频率在同一个数量级.在超强耦合区域, 著名的旋波近似失效, 因此系统的动力学必须用含有反旋波项的Rabi模型描述.本文研究Rabi模型中的光场压缩.数值模拟结果发现, 光场压缩不是随耦合强度线性增加, 而是在合适的超强区域获得最大值.同时, 我们还发现, 较小的反旋波项有助于提高光场压缩.所得结果有利于实验上在超强区域中制备所需的压缩态. 关键词： Rabi模型 超强耦合 光场压缩     

Full spectrum of the Rabi model

It is shown that in the Rabi model, for an integer value of the spectral parameter x, in addition to the finite number of the classical Judd states there exist infinitely many possible eigenstates. These eigenstates exist if the parameters of the problem are zeros of a certain transcendental function; in other words, there are infinitely many possible choices of parameters for which integer x belongs to the spectrum. Moreover, it is shown that the classical Judd eigenstates appear as degenerate cases of the confluent Heun function.     

Exact solution of the two-mode quantum Rabi model

An analytical method is developed to study the two-mode quantum Rabi model. For certain specific parameter conditions, especially for the resonant conditions, we obtain an infinite number of the exact solutions of the eigenfunctions and associated energies. It is shown that there exist new types of the exact energies which do not correspond to the level-crossings. Our analytical method may find applications in some related models.     

Haydock’s recursive solution of self-adjoint problems. Discrete spectrum

Haydock’s recursive solution is shown to underline a number of different concepts such as (i) quasi-exactly solvable models, (ii) exactly solvable models, (iii) three-term recurrence solutions based on Schweber’s quantization criterion in Hilbert spaces of entire analytic functions, and (iv) a discrete quantum mechanics of Odake and Sasaki. A recurrent theme of Haydock’s recursive solution is that the spectral properties of any self-adjoint problem can be mapped onto a corresponding sequence of polynomials {p_n(E)}$\left\{p_n\left(E\right)\right\}$ in energy variable E$E$. The polynomials {p_n(E)}$\left\{p_n\left(E\right)\right\}$ are orthonormal   with respect to the density of states n₀(E)$n_0\left(E\right)$ and energy eigenstate |E〉$|E〉$ is the generating function   of {p_n(E)}$\left\{p_n\left(E\right)\right\}$. The generality of Haydock’s recursive solution enables one to see the different concepts from a unified perspective and mutually benefiting from each other. Some results obtained within the particular framework of any of (i) to (iv) may have much broader significance.     

Multiphonon-resonance quantum Rabi model and adiabatic passage in a cavity-optomechanical system

In this paper,we propose a scheme to achieve a multiphonon-resonance quantum Rabi model and adiabatic passage in a strong-coupling cavity optomechanical system.In the scheme,when the driving bichromatic laser beam is adjusted to the off-resonant j-order red-and blue-sideband,the interaction between the cavity and mechanical oscillator leads to a j-phonon resonance quantum Rabi model.Moreover,we show that there exists a resonant multi-phonon coupling via intermediate states connected by counter-rotating processes when the frequency of the simulated bosonic mode is near a fraction of the transition frequency of the simulated two-level system.As a typical example,we theoretically analyze the two-phonon resonance quantum Rabi model,and derive an effective Hamiltonian of the six-phonon coupling.Finally,we present a method of six-phonon generation based on adiabatic passage across the resonance.Numerical simulations confirm the validity of the proposed scheme.Theoretically,the proposed scheme can be extended to the realization of 3j-phonon state.     

Concise analytic solutions to the quantum Rabi model with two arbitrary qubits

Using extended coherent states, an analytical exact study has been carried out for the quantum Rabi model (QRM) with two arbitrary qubits in a very concise way. The G$G$-functions with 2×2$2\times 2$ determinants are generally derived. For the same coupling constants, the simplest G$G$-function, resembling that in the one-qubit QRM, can be obtained. Zeros of the G$G$-function yield the whole regular spectrum. The exceptional eigenvalues, which do not belong to the zeros of the G$G$ function, are obtained in the closed form. The Dark states in the case of the same coupling can be detected clearly in a continued-fraction technique. The present concise solution is conceptually clear and practically feasible to the general two-qubit QRM and therefore has many applications.     

囚禁离子非线性Rabi模型量子场熵演化特性

本文研究了基于单个囚禁离子与两束激光场相互作用的非线性Rabi模型中的纠缠动力学.利用Von-Neumann熵来表征纠缠度.通过数值模拟,详细讨论了Lamb-Dicke参数对量子场熵的影响.在单光子非线性Rabi模型中,随着Lamb-Dicke参数和初始平均光子数的增加,场熵的平均值并不单调增加.在双光子非线性Rabi模型中,场熵的时间演化是周期性振荡的.随着Lamb-Dicke参数的增加,场熵的周期振荡逐渐消失.对于较小的Lamb-Dicke参数,纠缠和退纠缠周期性地出现.     

A simple,analytic solution of the semiclassical Rabi model in the red-detuned regime

We consider the semiclassical Rabi model in the large, red-detuned regime. Using the method of multiple-scales we obtain a simple, analytic, and approximate solution that describes the evolution of the system accurately for long times and for arbitrary values of the qubit-field coupling. It is used to characterize the probability to find the qubit in the excited state and the trajectory of the associated Bloch vector. Finally, we present physical situations where the results can be applied.     

Out-of-Time-Order Correlators and Quantum Phase Transitions in the Rabi and Dicke Models

The out-of-time-order correlators (OTOCs) is used to study the quantum phase transitions (QPTs) between the normal phase and the superradiant phase in the Rabi and few-body Dicke models with large frequency ratio of the atomic level splitting to the single-mode electromagnetic radiation field frequency. The focus is on the OTOC thermally averaged with infinite temperature, which is an experimentally feasible quantity. It is shown that the critical points can be identified by long-time averaging of the OTOC via observing its local minimum behavior. More importantly, the scaling laws of the OTOC for QPTs are revealed by studying the experimentally accessible conditions with finite frequency ratio and finite number of atoms in the studied models. The critical exponents extracted from the scaling laws of OTOC indicate that the QPTs in the Rabi and Dicke models belong to the same universality class.     

The effect of the counter-rotating terms on the Rabi oscillations and squeezing in the Dicke model with cavity losses

The effect of the counter-rotating terms in the Dicke model with cavity losses is examined. Numerical results for the time evolutions of the atomic population inversion, dipole moment, mean-photon number and atomic and field squeezing parameters for an initial coherent field are presented for various numbers of atoms and different cavity dampings. As a consequence of the counter-rotating terms, the appearance of new steady states for atomic population inversion and the mean-photon number is pointed out.     

Quantum Thermalization and Thermal Entanglement in the Open Quantum Rabi Model

Quantum thermalization and thermal entanglement in the open quantum Rabi model (QRM), in which a two-level system and a single-mode bosonic field are coupled to either two individual heat baths or a common heat bath, are studied. By treating the QRM as an effective multilevel system and deriving global quantum master equations in the eigenstate representation of the QRM, the physical conditions for quantum thermalization of the QRM is studied. It is found that, in the individual heat-bath case, the QRM can only be thermalized when either the two heat baths have the same temperature or the QRM is only coupled to one of the two baths. In the common heat-bath case, differently, the QRM can always be thermalized. Thermal entanglement of the QRM in both the resonant- and non-resonant coupling cases is also studied. The logarithmic negativity for the thermal state of the QRM is obtained in a wide parameter space, ranging from the low- to high-temperature limits, and from the weak- to deep-strong-coupling regimes. This work paves the way toward the study of quantum effects in nonequilibrium ultrastrongly-coupled light-matter systems.     

Parity Symmetry and Parity Breaking in the Quantum Rabi Model with Addition of Ising Interaction

We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced. In contrast to a mathematical treatment in a previous publication [J. Phys. A 46(2013) 265302], 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, and then the parity symmetry is broken as well as the scaling behavior of the ground state by introducing a bias. The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom. Experimental feasibility of realizing the models under discussion is investigated.     

Mean field theory for the quantum Rabi model,inconsistency to the rotating wave approximation

Time evolution of pertinent operators in the Rabi Hamiltonian and its rotating wave approximation (RWA) version, the Jaynes-Cummings model (JCM), in the Heisenberg picture, gives systems of nonlinear differential equations (NDEs). Considering well localized atom, the mean field theory (MFT) was applied to replace the operators by equivalent expectation values. The Rabi model was reduced to a fourth orders NDE describing atoms position. Solution by the harmonic balance method (HBM) showed good accuracy and consistency to the numerical results, which introduces it as a useful tool in the quantum dynamics studies. The NDEs describing the JCM in the Heisenberg picture structurally prevent applying the MFT and shows inconsistency to the Ehrenfest's theorem, contrary to the Rabi model.     

U (2) algebraic model applied to vibrational spectra of Nickel Metalloporphyrins

In this paper the highly excited stretching and bending vibrational energy levels of some Nickel Metalloporphyrin molecules are studied by U (2) algebraic model. Its application to Nickel Octaethyl Porphyrin, Nickel Tetra Phenyl Porphyrin and Nickel Porphyrin are presented with fewer algebraic fitting parameters. The highly excited stretching and bending vibrational energy levels of these molecules are showing the energy clustering spectrum.     

On the nature of the Lee-Yang measure for Ising ferromagnets

For the two- and three-dimensional nearest neighbors Ising model in the presence of a magnetic field, we study numerically asymptotic properties of the set of orthogonal polynomials associated with the Lee-Yang measure. This provides an insight into the nature of this measure near its end points, on the Lee-Yang circle. We introduce a smoothness index which analyzes the structure of the measure. Its value is found to be equal to 2 within 10^–3 for all the models tested in two and three dimensions, at any temperatures. The results strongly suggest the absence of any singular part (continuous or pure point) in the measure, even in dimension 3. We also confirm, using a different method, known results on the behavior of the measure near its end points.Research Assistant of the Belgian National Fund for Scientific Research.On leave of absence from CEN-Saclay France     

On the connection between the one-dimensionalS=1/2 Heisenberg chain and Haldane-Shastry model

Extra integrals of motion and the Lax representation are found for interacting spin systems with the HamiltonianH = (J/2) _{j, k=1,j k} ^N P(j – k) _{j k} , where one of the periods of the WeierstrassP function is equal toN. The Heisenberg and Haldane-Shastry chains appear as limiting cases of these systems at some values of the second period. The simplest eigenvectors and eigenvalues ofH corresponding to the scattering of two spin waves are presented explicitly for these finite-dimensional systems and for their infinite-dimensional version.     

Decorated star-triangle relations and exact integrability of the one-dimensional Hubbard model

The exact integrability of the one-dimensional Hubbard model is demonstrated with the help of a novel set of triangle relations, the decorated star-triangle relations. The covering two-dimensional statistical mechanical model obeys the star-triangle or Yang-Baxter relation. A conjecture is presented for the eigenvalues of the transfer matrix.