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Physical Review A

Quantum phase transition and quench dynamics in the anisotropic Rabi model

Li-Tuo Shen, Zhen‐Biao Yang, Huai‐Zhi Wu, Shi‐Biao Zheng

Published Jan 11, 2017• 54 citations
Publisher

Abstract

We investigate the quantum phase transition (QPT) and quench dynamics in the anisotropic Rabi model when the ratio of the qubit transition frequency to the oscillator frequency approaches infinity. Based on the Schrieffer-Wolff transformation, we find an anti-Hermitian operator that maps the original Hamiltonian into a one-dimensional oscillator Hamiltonian within the spin-down subspace. We analytically derive the eigenenergy and eigenstate of the normal and superradiant phases and demonstrate that the system undergoes a second-order quantum phase transition at a critical border. The critical border is a straight line in a two-dimensional parameter space which essentially extends the dimensionality of QPT in the Rabi model. By combining the Kibble-Zurek mechanism and the adiabatic dynamics method, we find that the residual energy vanishes as the quench time tends to zero, which is a sharp contrast to the universal scaling where the residual energy diverges in the same limit.

DOI
10.1103/physreva.95.013819

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We investigate the quantum phase transition (QPT) and quench dynamics in the anisotropic Rabi model when the ratio of the qubit transition frequency to the oscillator frequency approaches infinity. Based on the Schrieffer-Wolff transformation, we find an anti-Hermitian operator that maps the original Hamiltonian into a one-dimensional oscillator Hamiltonian within the spin-down subspace. We analytically derive the eigenenergy and eigenstate of the normal and superradiant phases and demonstrate that the system undergoes a second-order quantum phase transition at a critical border. The critical border is a straight line in a two-dimensional parameter space which essentially extends the dimensionality of QPT in the Rabi model. By combining the Kibble-Zurek mechanism and the adiabatic dynamics method, we find that the residual energy vanishes as the quench time tends to zero, which is a sharp contrast to the universal scaling where the residual energy diverges in the same limit.
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.
In this manuscript, we analyze the mechanism of the superradiant phase transition in the anisotropic Rabi model under the classical oscillator limit using the pattern picture. By expanding the anisotropic Rabi model Hamiltonian in operator space, we obtained three patterns, and we find that the phase transition arises from the competition between patterns. The difficulty in achieving the classical oscillator limit motivates our investigation into the quantum phase transition within a parametrically-driven Jaynes-Cummings model. This parametrically-driven Jaynes-Cummings model can reproduce the dynamics of an ultrastrong-coupling anisotropic Rabi model in a squeezed-light frame. According to the eigenenergies and eigenstates of the normal and superradiant phases of this equivalent anisotropic Rabi model, we find that the excitation energy of the normal phase and the superradiant phase vanishes at the critical point. The photon number becomes infinite beyond the critical point. These results indicate that the system undergoes a superradiant phase transition at the critical point.
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.
Dynamics of a system exhibits non-adiabaticity even for slow quenches near critical points. We analyze the response to disorder in quenches on a non-adiabaticity quantifier for the quantum Rabi model, which possesses a phase transition between normal and superradiant phases. We consider a disordered version of a quench in the Rabi model, in which the system residing in the ground state of an initial Hamiltonian of the normal phase is quenched to the final Hamiltonian corresponding to the critical point. The disorder is inserted either in the total time the quench or in the quench parameter itself. We solve the corresponding quantum dynamics numerically, and find that the non-adiabatic effects are unaffected by the presence of disorder in the total time of the quench. This result is then independently confirmed by the application of adiabatic perturbation theory and the Kibble- Zurek mechanism. For the disorder in the quench parameter, we report a monotonic increase in the adiabaticity with the strength of the disorder. Lastly, we consider a quench where the final Hamiltonian is chosen as the average over the disordered final Hamiltonians, and show that this quench is more adiabatic than the average of the quenches with the disorder in final Hamiltonian.
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.