Ground-state phase diagram,symmetries,excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

We investigate the two-mode quantum Rabi model(QRM)describing the interaction between a two-level atom and a two-mode cavity field.The quantum phase transitions are found when the ratioηof transition frequency of atom to frequency of cavity field approaches infinity.We apply the Schrieffer–Wolff(SW)transformation to derive the low-energy effective Hamiltonian of the two-mode QRM,thus yielding the critical point and rich phase diagram of quantum phase transitions.The phase diagram consists of four regions:a normal phase,an electric superradiant phase,a magnetic superradiant phase and an electromagnetic superradiant phase.The quantum phase transition between the normal phase and the electric(magnetic)superradiant phase is of second order and associates with the breaking of the discrete Z_(2) symmetry.On the other hand,the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1)symmetry.Several important physical quantities,for example the excitation energy and average photon number in the four phases,are derived.We find that the excitation spectra exhibit the Nambu–Goldstone mode.We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities.To confirm the validity of the low-energy effective Hamiltonians analytically derived by us,the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.

Hidden symmetry operators for asymmetric generalized quantum Rabi models

The hidden Z2 symmetry of the asymmetric quantum Rabi model(AQRM)has recently been revealed via a systematic construction of the underlying symmetry operator.Based on the AQRM result,we propose an ansatz for the general form of the symmetry operators for AQRM-related models.Applying this ansatz we obtain the symmetry operator for three models:the anisotropic AQRM,the asymmetric Rabi–Stark model(ARSM),and the anisotropic ARSM.

An analytical variational method for the biased quantum Rabi model in the ultra-strong coupling regime

An analytical variational method for the ground state of the biased quantum Rabi model in the ultra-strong coupling regime is presented. This analytical variational method can be obtained by a unitary transformation or alternatively by assuming the form of the ground state wave function. The key of the method is to introduce a variational parameter λ,which can be determined by minimizing the energy functional. Using this method, we calculate the physical observables with high accuracy in comparison with the numerical exact ones. Our method evidently improves over the widely used general rotating-wave approximation（GRWA） in both qualitative and quantitative aspects.

Quantum exceptional points of non-Hermitian Hamiltonian and Liouvillian in dissipative quantum Rabi model

The open quantum system can be described by either a Lindblad master equation or a non-Hermitian Hamiltonian(NHH).However,these two descriptions usually have different exceptional points(EPs),associated with the degeneracies in the open quantum system.Here,considering a dissipative quantum Rabi model,we study the spectral features of EPs in these two descriptions and explore their connections.We find that,although the EPs in these two descriptions are usually different,the EPs of NHH will be consistent with the EPs of master equation in the weak coupling regime.Further,we find that the quantum Fisher information(QFI),which measures the statistical distance between quantum states,can be used as a signature for the appearance of EPs.Our study may give a theoretical guidance for exploring the properties of EPs in open quantum systems.

Dynamics of Rabi model under second-order Born Oppenheimer approximation

We apply the second-order Born-Oppenheimer （BO） approximation to investigate the dynamics of the Rabi model, which describes the interaction between a two-level system and a single bosonic mode beyond the rotating wave approxi- mation. By comparing with the numerical results, we find that our approach works well when the frequency of the two-level system is much smaller than that of the bosonic mode.

Dynamics of a stochastic rabies epidemic model with Markovian switching

In this paper,we analyze a stochastic rabies epidemic model which is perturbed by both white noise and telegraph noise.First,we prove the existence of the unique global positive solution.Second,by constructing an appropriate Lyapunov function,we establish a sufficient condition for the existence of a unique ergodic stationary distribution of the positive solutions to the model.Then we establish sufficient conditions for the extinction of diseases.Finally,numerical simulations are introduced to illustrate our theoretical results.

Analytical Solution of Nonlinear System of Fractional Differential Equations

In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear system of fractional differential equations (FDEs). The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are discussed. Some applications are solved such as fractional-order rabies model.

Dynamics of two arbitrary qubits strongly coupled to a quantum oscillator

Using adiabatic approximation, a two arbitrary qubits Rabi model has been studied in ultra-strong coupling. The analytical expressions of the eigenvalues and the eigenvalues are obtained. They are in accordance with the numerical determined results. The dynamical behavior of the system and the evolution of entanglement have also been discussed. The collapse and revival phenomena has garnered particular attention. The influence of inconsistent coupling strength on them is studied. These results will be applied in quantum information processing.

Criticality-enhanced quantum sensing in the anisotropic quantum Rabi model

Quantum systems that undergo quantum phase transitions exhibit divergent susceptibility and can be exploited as probes to estimate physical parameters.We generalize the dynamic framework for criticality-enhanced quantum sensing by the quantum Rabi model(QRM)to its anisotropic counterpart and derive the correspondingly analytical expressions for the quantum Fisher information(QFI).We find that the contributions of the rotating-wave and counterrotating-wave interaction terms are symmetric at the limit of the infinite ratio of qubit frequency to field frequency,with the QFI reaching a maximum for the isotropic QRM.At finite frequency scaling,we analytically derive the inverted variance of higher-order correction and find that it is more affected by the rotating-wave coupling than by the counterrotating-wave coupling.

Experimental observation of spontaneous symmetry breaking in a quantum phase transition

Spontaneous symmetry breaking(SSB)plays a central role in understanding a large variety of phenomena associated with phase transitions,such as superfluid and superconductivity.So far,the transition from a symmetric vacuum to a macroscopically ordered phase has been substantially explored.The process bridging these two distinct phases is critical to understanding how a classical world emerges from a quantum phase transition,but so far remains unexplored in experiment.We here report an experimental demonstration of such a process with a quantum Rabi model engineered with a superconducting circuit.We move the system from the normal phase to the superradiant phase featuring two symmetry-breaking field components,one of which is observed to emerge as the classical reality.The results demonstrate that the environment-induced decoherence plays a critical role in the SSB.

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.

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.

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.

Teleportation of unknown states of a qubit and a single-mode field in strong coupling regime without Bell-state measurement

In this paper,we develop the teleportation scheme in[Zheng in Phys Rev A 69,064302,2004],in the sense that,we work in the strong atom-field coupling regime wherein the rotating wave approximation(RWA)is no longer valid.To achieve the purpose,a scheme consisting of a qubit interacting with a single-mode quantized field is described via the Rabi model(counter rotation terms are taken into account).Our first aim is to teleport an unknown atomic state of a qubit(which interacts with the quantized field in a cavity)to a second qubit(exists in another distant cavity field),beyond the RWA and without the Bell-state measurement method.In the continuation,in a similar way,we teleport an unknown state of a single-mode field too.In fact,it is shown that,in this regime,after applying some particular conditions,containing the interaction time of atom-field in the cavities,adjusting the involved frequencies,as well as the atom-field coupling in the model,if a proper measurement is performed on the state of the first qubit(the related field in the cavity),the unknown states of the qubit(field)can be teleported from the first qubit(cavity field)to the second qubit(cavity field),appropriately.We show that in both considered cases,the teleportation protocol is successfully performed with the maximum possible fidelity,1,and the acceptable success probability,0.25.