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Recent measurement on an LC resonator magnetically coupled to a superconducting qubit [arXiv:1005[arXiv: .1559] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes-Cummings model exactly without the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch-Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch-Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.

The light-matter interaction described by Rabi model and Jaynes-Cummings (JC) model is investigated by parity breaking as well as the scaling behavior of ground-state population-inversion expectation. We show that the parity breaking leads to different scaling behaviors in the two models, where the Rabi model demonstrates scaling invariance, but the JC model behaves in cusp-like way. Our study helps further understanding rotating-wave approximation and could present more subtle physics than any other characteristic parameter for the difference between the two models. More importantly, our results could be straightforwardly applied to the understanding of quantum phase transitions in spin-boson model. Furthermore, the scaling behavior is observable using currently available techniques in light-matter interaction.Comment: 3 figure

We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition (QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of no-degeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations. A two-level system coupled to an environment provides a unique test-bed for fundamental interests of quantum physics. Denoting the environment by a multimode harmonic oscillator, the spin-boson model (SBM) [1, 2] presents a phenomenological description of the open quantum system, which plays an important role in quantum information science and condensed matter physics. Particularly, for the SBM at zero temperature, it has attracted intensive interests for the quantum phase transition (QPT) happening between localized and delocalized phases regarding the spin.The standard SBM Hamiltonian in units of = 1 is given by [1],where σ z and σ x are usual Pauli operators, ǫ and ∆ are, respectively, the local field (also called c-number bias [1]) and tunneling regarding the two levels of the spin. a † k and a k are creation and annihilation operators of the bath modes with frequencies ω k , and λ k is the coupling between the spin and the bath modes. The effect of the harmonic oscillator environment is reflected by the spectral function J(ω) = π k λ 2 k δ(ω − ω k ) for 0 < ω < ω c with the cutoff energy ω c . In the infrared limit, i.e., ω →0, the power laws regarding J(ω) are of particular importance. Considering the low-energy details of the spectrum, we have J(ω) = 2παω 1−s c ω s with 0 < ω < ω c and the dissipation strength α. The exponent s is responsible for different bath with super-Ohmic bath s >1, Ohmic bath s =1 and sub-Ohmic bath s <1. * Electronic address: liutao849@163.com † Electronic address: mangfeng@wipm.ac.cnThe local field ǫ introduces asymmetry in the model, which was considered to be less important than the tunneling ∆ and thereby sometimes neglected for convenience of treatments. For the Ohmic dissipation, it was mentioned [3,4] that the SBM has a delocalized and a localized zero temperature phase, separated by a Kosterlitz-Thouless (KT) transition in the case of ǫ = 0. In the delocalized phase, realized at small dissipation strength α, the non-degenerate ground state behaves like a damped tunneling particle. In contrast, for large α, the dissipation leads to a localization of the particle ...

We study the breaking of parity in the spin-boson model and demonstrate unique scaling behavior of the magnetization and entanglement around the critical points for the parity breaking after suppressing the infrared divergence existing inherently in the spectral functions for Ohmic and sub-Ohmic dissipations. Our treatment is basically analytical and of generality for all types of the bath. We argue that the conventionally employed spectral function is not fully reasonable and the previous justification of quantum phase transition for localization needs to be more seriously reexamined.