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Physical Review A

Using nonclassical states to explore the superradiant phase transition in the quantum Rabi model

Junpeng Liu, Miaomiao Zhao, Yun-Tong Yang, Hong‐Gang Luo

Published Feb 27, 2024• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.109.023721

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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.
The ground state of quantum Rabi model (QRM) exhibits rich nonclassical states including squeezed state, cat state, and entangled state in different parameter regimes. In this paper, we firstly use the polaron picture to figure out the process of superradiant phase transition (SPT) in view of the nonclassical states. According to the polaron wave functions, the squeezed state extends to whole parameter regimes, and a small but non-zero weighted antipolaron is necessary to form novel semi-cat states. Moreover, the behavior of superradiance in the QRM can be viewed as the same displacement of the cat states from zero to a finite value, while the ground state becomes entangled state resulting from the emergence of spin-up state. On the other hand, the nonclassical states can be intuitively characterized by the Wigner functions with analytical expressions in the polaron picture, and the Wigner negativity is also evaluated to measure the nonclassicality. Based on the squeezing and superradiance, a classification of the coupling strength for the nonclassical states containing in the ground state is presented, and the process of the SPT is also revealed by the photon number distribution in Fock space, which is important to understand the fundamental quantum physics in the QRM. Our work provides a guidance for preparing the nonclassical states in experiments based on the QRM.
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.
Superradiant phase transitions (SPTs) are important for understanding light-matter interactions at the quantum level, and play a central role in criticality-enhanced quantum sensing. So far, SPTs have been observed in driven-dissipative systems, but the emergent light fields did not show any nonclassical characteristic due to the presence of strong dissipation. Here we report an experimental demonstration of the SPT featuring the emergence of a highly nonclassical photonic field, realized with a resonator coupled to a superconducting qubit, implementing the quantum Rabi model. We fully characterize the light-matter state by Wigner matrix tomography. The measured matrix elements exhibit quantum interference intrinsic of a photonic mesoscopic superposition, and reveal light-matter entanglement.
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.
We theoretically study the superradiant phase transition (SPT) in an indirect Rabi model, where both a two-level system and a single-mode bosonic field couple to an auxiliary bosonic field. We find that the indirect spin-field coupling induced by the virtual excitation of the auxiliary field can allow the occurrence of a SPT at a critical point, and the influence of the so-called A2{A}^{2}A2 term in the normal Rabi model is naturally avoided. In the large detuning regime, we present the analytical expression of quantum critical point in terms of the original system parameters. The critical atom-field coupling strength is tunable, which will loosen the conditions on realizing the SPT. Considering a hybrid magnon-cavity-qubit system, we predict the squeezed cat state of magnon generated with feasible experimental parameters, which has potential applications in quantum metrology and quantum information processing.
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.