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arXiv:1511.00342 · Physical Review A

Ground-state phase diagram of the quantum Rabi model

Zu‐Jian Ying, Maoxin Liu, Hong‐Gang Luo, Hai‐Qing Lin, J. Q. You

Published Nov 9, 2015• 74 citations
arXivHTMLPDF

Abstract

The Rabi model plays a fundamental role in understanding light-matter interaction. It reduces to the Jaynes-Cummings model via the rotating-wave approximation, which is applicable only to the cases of near resonance and weak coupling. However, recent experimental breakthroughs in upgrading light-matter coupling order require understanding the physics of the full quantum Rabi model (QRM). Despite the fact that its integrability and energy spectra have been exactly obtained, the challenge to formulate an exact wave function in a general case still hinders physical exploration of the QRM. Here we unveil a ground-state phase diagram of the QRM, consisting of a quadpolaron and a bipolaron as well as their changeover in the weak-, strong-, and intermediate-coupling regimes, respectively. An unexpected overweighted antipolaron is revealed in the quadpolaron state, and a hidden scaling behavior relevant to symmetry breaking is found in the bipolaron state. An experimentally accessible parameter is proposed to test these states, which might provide novel insights into the nature of the light-matter interaction for all regimes of the coupling strengths.

DOI
10.1103/physreva.92.053823

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The Rabi model plays a fundamental role in understanding light-matter interaction. It reduces to the Jaynes-Cummings model via the rotating-wave approximation, which is applicable only to the cases of near resonance and weak coupling. However, recent experimental breakthroughs in upgrading light-matter coupling order require understanding the physics of the full quantum Rabi model (QRM). Despite the fact that its integrability and energy spectra have been exactly obtained, the challenge to formulate an exact wave function in a general case still hinders physical exploration of the QRM. Here we unveil a ground-state phase diagram of the QRM, consisting of a quadpolaron and a bipolaron as well as their changeover in the weak-, strong-, and intermediate-coupling regimes, respectively. An unexpected overweighted antipolaron is revealed in the quadpolaron state, and a hidden scaling behavior relevant to symmetry breaking is found in the bipolaron state. An experimentally accessible parameter is proposed to test these states, which might provide novel insights into the nature of the light-matter interaction for all regimes of the coupling strengths.
Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalizations to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus, one is able to see how the phase transition occurs and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition takes place and further exhibits the statistics of Gaussian unitary ensemble with increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.
Abstract The light–matter interaction not only is ubiquitous in nature but also can be simulated in various artificial systems. Besides the tunability of artificial quantum systems, the experimental access to the ultra‐strong and even deep‐strong couplings has brought a regime with novel phenomenology. Theoretically the finding of the integrability for the quantum Rabi model (QRM) has attracted tremendous attention to the study of the QRM and its extensions which are fundamental models of the light–matter interaction. With the continuing enhancements of coupling strengths, a most fascinating phenomenon of these light–matter models is that they can exhibit few‐body quantum phase transitions (QPTs), while a full understanding of these QPTs is a challenge. The polaron picture is a method capable of analyzing these fundamental models in the entire coupling regime and extracting the essential physics behind the QPTs. A review of its extensive applications on the fundamental light–matter models is presented, with discussions on the phase diagrams and QPT‐related features. The universality of critical exponent of the anisotropic QRM as well as the possibility of QPTs in A 2 terms are also addressed.
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.
The light-matter interaction described by the Rabi model and Jaynes-Cummings (JC) model under local bias fields is investigated by parity breaking as well as the scaling behavior of the ground state. We show that the parity breaking leads to different scaling behaviors in the two models, where the biased Rabi model demonstrates scaling invariance, but the biased JC model behaves in a cusp-like way. Our study helps further understand the rotating-wave approximation and could present more subtle physics than other characteristic parameters for the difference between the two models. More importantly, the scaling behavior is observable using currently available techniques in light-matter interaction.
The ground state of the quantum Rabi model (QRM) exhibits rich nonclassical states including the squeezed state, cat state, and entangled state in different parameter regimes. In this paper, we first use the polaron picture with high precision and fidelity to figure out the process of the superradiant phase transition (SPT) in view of the nonclassical states. The squeezing effect is considered to be a result of the higher-order excitations and important in order to understand the fundamental quantum physics of the SPT in the QRM. According to the polaron wave functions, the squeezed state extends across the entire range of parameters. Based on the squeezing and superradiance, a classification of the coupling strength for the nonclassical states containing the ground state is presented; the system sequentially goes through the vacuum states, the squeezed vacuum states, the squeezed cat states, and the cat states as the coupling strength increases. The process of the SPT is also demonstrated by the photon number distribution in Fock space, and then the novel behavior of parity chains for the SPT is revealed. Additionally, the nonclassical states can be intuitively characterized by the Wigner functions with analytical expressions in the polaron picture, and the Wigner negativity is evaluated to measure the nonclassicality. We find that a small but nonzero weighted antipolaron is necessary to form novel semicat states. Our paper provides guidance for preparing nonclassical states in experiments based on the light-matter interaction.
We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two bare polarons. Importantly, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional induced polarons. According to this picture, the variational ground-state wave function of the two-photon QRM can be correctly constructed, with the ground-state energy and other physical observables in good agreement with the exact numerics in all the coupling regimes. Furthermore, generalization to multiple induced polarons involving higher orders in the tunneling effect provides a systematic way to yield a rapid convergence in accuracy even around the difficult spectral collapse point. In addition, the polaron picture provides a distinctive understanding of an incomplete spectral collapse behavior that is about the existence of discrete energy levels apart from the collapsed spectrum at the spectral collapse point. This work illustrates that the polaron picture is helpful to capture the key physics in this nonlinear light-matter interaction model and indicates that this method can be applicable to more complicated QRM-related models.