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arXiv:1910.13043 · Journal of Physics A Mathematical and Theoretical

Quantum phase transitions and critical behaviors in the two-mode three-level quantum Rabi model

Yan Zhang, Bin-Bin Mao, Dazhi Xu, Yu-Yu Zhang, Wen-Long You, Maoxin Liu, Hong-Gang Luo

Published May 13, 2020• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ab92be

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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.
We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio η of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer–Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete Z 2 symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1) symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu–Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.
The second-order quantum phase transition and quench dynamics in the two-mode Rabi model are investigated. We propose a diagonalization approach of jointly using a beam-splitter operator and a squeezed operator for each effective low-energy Hamiltonian when the ratio of the qubit transition frequency to the oscillator frequency approximates infinity. Eigenenergy and eigenstate of the normal and superradiant phases in the system are analytically derived, demonstrating the second-order quantum phase transition at a critical point. This critical point means that the requirement of coupling strength between the qubit and two oscillators is largely loosened by joint effect of two modes when compared with the standard Rabi model. The universal scaling between residual energy and quench time from ${\ensuremath{\tau}}^{\ensuremath{-}\frac{1}{3}}$ to ${\ensuremath{\tau}}^{\ensuremath{-}2}$ is confirmed.
We consider the Rabi Hamiltonian, which exhibits a quantum phase transition (QPT) despite consisting only of a single-mode cavity field and a two-level atom. We prove QPT by deriving an exact solution in the limit where the atomic transition frequency in the unit of the cavity frequency tends to infinity. The effect of a finite transition frequency is studied by analytically calculating finite-frequency scaling exponents as well as performing a numerically exact diagonalization. Going beyond this equilibrium QPT setting, we prove that the dynamics under slow quenches in the vicinity of the critical point is universal; that is, the dynamics is completely characterized by critical exponents. Our analysis demonstrates that the Kibble-Zurek mechanism can precisely predict the universal scaling of residual energy for a model without spatial degrees of freedom. Moreover, we find that the onset of the universal dynamics can be observed even with a finite transition frequency.
The dipole-coupled two-level atoms (qubits) in a single-mode resonant cavity are studied by extended bosonic coherent states. The numerical solution is presented. The first-order quantum phase transitions may occur in finite systems, in contrast to the original Dicke model. This system also exhibits a second-order quantum phase transition from the normal to the superradiant phase, as in the original Dicke model. Finite-size scaling for several observables---such as average fidelity susceptibility, order parameter, and concurrence---is performed for different interatomic interactions. The scaling exponents obtained suggest that interatomic interactions do not change the universality class.
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.
Quantum criticality usually occurs in many-body systems. Recently it was shown that the quantum Rabi model, which describes a two-level atom coupled to a single model cavity field, presents quantum phase transitions from a normal phase to a superradiate phase when the ratio between the frequency of the two-level atom and the frequency of the cavity field extends to infinity. In this work, we study quantum phase transitions in the quantum Rabi model from the fidelity susceptibility perspective. We found that the fidelity susceptibility and the generalized adiabatic susceptibility present universal finite-size scaling behaviors near the quantum critical point of the Rabi model if the ratio between frequency of the two-level atom and frequency of the cavity field is finite. From the finite-size scaling analysis of the fidelity susceptibility, we found that the adiabatic dimension of the fidelity susceptibility and the generalized adiabatic susceptibility of fourth order in the Rabi model are 4/34/34/3 and 2, respectively. Meanwhile, the correlation length critical exponent and the dynamical critical exponent in the quantum critical point of the Rabi model are found to be 3/23/23/2 and 1/31/31/3, respectively. Since the fidelity susceptibility and the generalized adiabatic susceptibility are the moments of the quantum noise spectrum which are directly measurable by experiments in linear response regime, the scaling behavior of the fidelity susceptibility in the Rabi model could be tested experimentally. The simple structure of the quantum Rabi model paves the way for experimentally observing the universal scaling behavior of the fidelity susceptibility at a quantum phase transition.