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arXiv:1708.04382 · Physical Review A

Analytical solutions by squeezing to the anisotropic Rabi model in the nonperturbative deep-strong-coupling regime

Yuyu Zhang, Xiang-You Chen

Published Dec 15, 2017• 29 citations
arXivHTMLPDF

Abstract

An unexplored nonperturbative deep strong coupling (npDSC) achieved in superconducting circuits has been studied in the anisotropic Rabi model by the generalized squeezing rotating-wave approximation. Energy levels are evaluated analytically from the reformulated Hamiltonian and agree well with numerical ones in a wide range of coupling strength. Such improvement ascribes to deformation effects in the displaced-squeezed state presented by the squeezed momentum variance, which are omitted in previous displaced states. The atom population dynamics confirms the validity of our approach for the npDSC strength. Our approach offers the possibility to explore interesting phenomena analytically in the npDSC regime in qubit-oscillator experiments.

DOI
10.1103/physreva.96.063821

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An unexplored nonperturbative deep strong coupling (npDSC) achieved in superconducting circuits has been studied in the anisotropic Rabi model by the generalized squeezing rotating-wave approximation. Energy levels are evaluated analytically from the reformulated Hamiltonian and agree well with numerical ones in a wide range of coupling strength. Such improvement ascribes to deformation effects in the displaced-squeezed state presented by the squeezed momentum variance, which are omitted in previous displaced states. The atom population dynamics confirms the validity of our approach for the npDSC strength. Our approach offers the possibility to explore interesting phenomena analytically in the npDSC regime in qubit-oscillator experiments.
Generalized squeezing rotating-wave approximation (GSRWA) is proposed by employing both the displacement and the squeezing transformations. A solvable Hamiltonian is reformulated in the same form as the ordinary RWA ones. For a qubit coupled to oscillators experiment, a well-defined Schr\"odinger-cat-like entangled state is given by the displaced-squeezed oscillator state instead of the original displaced state. For the isotropic Rabi case, the mean photon number and the ground-state energy are expressed analytically with additional squeezing terms, exhibiting a substantial improvement of the GSRWA. And the ground-state energy in the anisotropic Rabi model confirms the effectiveness of the GSRWA. Due to the squeezing effect, the GSRWA improves the previous methods only with the displacement transformation in a wide range of coupling strengths even for large atom frequency.
Abstract In order to obtain the ground state of the biased quantum Rabi model in the ultrastrong coupling regime, a generalized squeezing rotating-wave approximation method including the displacement and the squeezing transformations is proposed in this work. While considering the deformation of the oscillator state, our method evidently improves the generalized variational method. The analyses of a few physical quantities show that our method works well especially in the ultrastrong coupling and large atomic frequency regime.
Anisotropic quantum Rabi model is a generalization of quantum Rabi model, which allows its rotating and counter-rotating terms to have two different coupling constants. It provides us with a fundamental model to understand various physical features concerning quantum optics, solid-state physics, and mesoscopic physics. In this paper, we propose an experimental feasible scheme to implement anisotropic quantum Rabi model in a circuit quantum electrodynamics system via periodic frequency modulation. An effective Hamiltonian describing the tunable anisotropic quantum Rabi model can be derived from a qubit-resonator coupling system modulated by two periodic driving fields. All effective parameters of the simulated system can be adjusted by tuning the initial phases, the frequencies and the amplitudes of the driving fields. We show that the periodic driving is able to drive a coupled system in dispersive regime to ultrastrong coupling regime, and even deep-strong coupling regime. The derived effective Hamiltonian allows us to obtain pure rotating term and counter-rotating term. Numerical simulation shows that such effective Hamiltonian is valid in ultrastrong coupling regime, and stronger coupling regime. Moreover, our scheme can be generalized to the multi-qubit case. We also give some applications of the simulated system to the Schrödinger cat states and quantum gate generalization. The presented proposal will pave a way to further study the stronger anisotropic Rabi model whose coupling strength is far away from ultrastrong coupling and deep-strong coupling regimes in quantum optics.
The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.
In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called GGG-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.
The quantum Rabi model describes the fundamental mechanism of light-matter interaction. It consists of a two-level atom or qubit coupled to a quantized harmonic mode via a transversal interaction. In the weak coupling regime, it reduces to the well-known Jaynes-Cummings model by applying a rotating wave approximation. The rotating wave approximation breaks down in the ultra-strong coupling regime, where the effective coupling strength g is comparable to the energy ω of the bosonic mode, and remarkable features in the system dynamics are revealed. Here we demonstrate an analog quantum simulation of an effective quantum Rabi model in the ultra-strong coupling regime, achieving a relative coupling ratio of g/ω ~ 0.6. The quantum hardware of the simulator is a superconducting circuit embedded in a cQED setup. We observe fast and periodic quantum state collapses and revivals of the initial qubit state, being the most distinct signature of the synthesized model.An analog quantum simulation scheme has been explored with a quantum hardware based on a superconducting circuit. Here the authors investigate the time evolution of the quantum Rabi model at ultra-strong coupling conditions, which is synthesized by slowing down the system dynamics in an effective frame.