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arXiv:1702.06641 · Physical Review Letters

Universal Scaling and Critical Exponents of the Anisotropic Quantum Rabi Model

Maoxin Liu, Stefano Chesi, Zu‐Jian Ying, Xiaosong Chen, Hong‐Gang Luo, Hai‐Qing Lin

Published Nov 28, 2017• 166 citations
arXivHTMLPDF

Abstract

We investigate the quantum phase transition of the anisotropic quantum Rabi model, in which the rotating and counterrotating terms are allowed to have different coupling strengths. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches, we extract the phase diagram, scaling functions, and critical exponents, which determine the universality class at finite anisotropy (identical to the isotropic limit). We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are extended to the few-body quantum phase transitions with N>1 spins, where we expose the same effective parameters, scaling properties, and phase diagram. Thus, a stronger form of universality is established, valid from N=1 up to the thermodynamic limit.

DOI
10.1103/physrevlett.119.220601

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We investigate the quantum phase transition of the anisotropic quantum Rabi model, in which the rotating and counterrotating terms are allowed to have different coupling strengths. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches, we extract the phase diagram, scaling functions, and critical exponents, which determine the universality class at finite anisotropy (identical to the isotropic limit). We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are extended to the few-body quantum phase transitions with N>1 spins, where we expose the same effective parameters, scaling properties, and phase diagram. Thus, a stronger form of universality is established, valid from N=1 up to the thermodynamic limit.
We present our recent studies on the quantum Rabi model (QRM). Firstly, by using a variational wave function, which facilitates to extract physics in entire parameter regime with high accuracy, we unveil a ground-state phase diagram of the QRM and argue that the main constituents are polaron and anti-polaron. Secondly, introducing an anisotropy into the QRM, in which the rotating- and counter-rotating terms are allowed to have different coupling strength, we explore that how the criticality of the model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches we compute phase diagram, scaling functions and critical exponents, and establish that the universality class at the finite anisotropy is the same as that of the isotropic limit.
We investigate the universal nonequilibrium dynamics of the superradiant phase transition in the anisotropic quantum Rabi model. By introducing position and momentum operators, we find the generators and the displaced operators to diagonalize the anisotropic Hamiltonian, and then obtain the ground states and their excitation gaps. According to the Kibble-Zurek mechanism, we find that the excitation gap near the critical point vanishes as $\ensuremath{\omega}\ensuremath{\sim}{|\ensuremath{\epsilon}|}^{\ensuremath{\nu}z}$ with $\ensuremath{\nu}z=1/2$, and the length scale diverges according to $\ensuremath{\zeta}\ensuremath{\sim}{\ensuremath{\epsilon}}^{\ensuremath{-}\ensuremath{\nu}}$, which determines the universality class $(z=2,\ensuremath{\nu}=1/4)$ of the anisotropic quantum Rabi model. Moreover, by simulating the real-time dynamics across the critical point, we numerically extract the critical exponents from the phase transition delay ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{d}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{-}1/(1+\ensuremath{\nu}z)}$ and the diverging length scale $\mathrm{\ensuremath{\Delta}}x$, $\mathrm{\ensuremath{\Delta}}p\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+\ensuremath{\nu}z)}$ (where ${\ensuremath{\tau}}_{Q}$ is the quench time). The numerical critical exponents are well consistent with the analytical ones. Our study provides a dynamic way to explore universal critical behaviors in the anisotropic quantum Rabi model.
We study the nonequilibrium steady states of an open anisotropic quantum Rabi trimer. Combining semiclassical and quantum mechanical treatments, we map out the system's phase diagram and elucidate the critical phenomena induced by anisotropic light-matter coupling, cavity dissipation, and complex photon hopping, which give rise to abundant multistable phases. Counterintuitively, the frustrated superradiant phase (FSP) can be stabilized by both positive and negative photon hopping, a feature absent in closed Rabi and Dicke trimers. We further uncover two independent dissipation thresholds for the normal superradiant phase and the FSP. Unlike isotropic open quantum Rabi trimers, both phases are suppressed once the dissipation exceeds their respective critical values. The rotating and counter-rotating interaction terms exhibit mutually reciprocal effects on the quantum phase transitions. By analyzing universal scaling behaviors with quantum fluctuations incorporated, we reveal an unprecedented finite-fluctuation critical exponent for the FSP. Moreover, the critical exponents of individual phases are coupling-strength dependent. The multistability and FSP can be tuned by anisotropy and dissipation. Our results broaden the scope of nonequilibrium quantum phase physics in few-body systems and hold promising applications for quantum state engineering and cavity/circuit quantum electrodynamics.
Abstract We explore an extended quantum Rabi model describing the interaction between a two-mode bosonic field and a three-level atom. Quantum phase transitions of this few degree of freedom model is found when the ratio η of the atom energy scale to the bosonic field frequency approaches infinity. An analytical solution is provided when the two lowest-energy levels are degenerate. According to it, we recognize that the phase diagram of the model consists of three regions: one normal phase and two superradiant phases. The quantum phase transitions between the normal phase and the two superradiant phases are of second order relating to the spontaneous breaking of the discrete Z 2 symmetry. On the other hand, the quantum phase transition between the two different superradiant phases is discontinuous with a phase boundary line relating to the continuous U (1) symmetry. For a large enough but finite η , the scaling function and critical exponents are derived analytically and verified numerically, from which the universality class of the model is identified.
Although the superradiant phase transition (SRPT) is prohibited in the paradigmatic quantum Rabi model due to the no-go theorem caused by the A2{\mathbf{A}}^{2}A2 term, we demonstrate two distinct types of SRPTs emerging from the normal phase in the anisotropic quantum Rabi model. A discontinuous phase transition between the two types of superradiant phases also emerges in the presence of a strong A2{\mathbf{A}}^{2}A2 term. Additionally, a rich phase diagram featuring a triple point, which connects first- and second-order phase transitions, is derived analytically and confirmed through numerical diagonalization at large effective system sizes. Finally, distinct critical behavior at the triple point is revealed and contrasted with that of a single continuous SRPT. This work may open a new avenue for observing SRPTs in their intrinsic form without altering the A2{\mathbf{A}}^{2}A2 term, while also offering a practical platform for exploring rich quantum phenomena.
We investigate the nonequilibrium steady state of the anisotropic open quantum Rabi model, which exhibits first-order and second-order dissipative phase transitions upon varying the degree of anisotropy between the coupling strengths of rotating and counter-rotating terms. Using both semiclassical and quantum approaches, we find a rich phase diagram resulting from the interplay between the anisotropy and the dissipation. First, there exists a bistable phase where both the normal and superradiant phases are stable. Second, there are multicritical points where the phase boundaries for the first- and second-order phase transitions meet. We show that a new set of critical exponents governs the scaling of the multicritical points. Finally, we discuss the feasibility of observing the multicritical transitions and bistability using a pair of trapped ions where the anisotropy can be tuned by controlling the intensity of the Raman transitions. Our study enlarges the scope of critical phenomena that may occur in finite-component quantum systems, which could be useful for applications in critical quantum sensing. Published by the American Physical Society 2024