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arXiv:2012.04568

Scaling of non-adiabaticity in disordered quench of quantum Rabi model close to phase transition

Chirag Srivastava, Ujjwal Sen

Published Dec 8, 2020• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.2012.04568

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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.
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 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 investigate the quench dynamics of an open quantum system involving a quantum phase transition. In the isolated case, the quench dynamics involving the phase transition exhibits a number of scaling relations with the quench rate as predicted by the celebrated Kibble-Zurek mechanism. In contact with an environment however, these scaling laws break down and one may observe an anti-Kibble-Zurek behavior: slower ramps lead to less adiabatic dynamics, increasing thus nonadiabatic effects with the quench time. In contrast to previous works, we show here that such anti-Kibble-Zurek scaling can acquire a universal form in the sense that it is determined by the equilibrium critical exponents of the phase transition, provided the excited states of the system exhibit singular behavior, as observed in fully connected models. This demonstrates novel universal scaling laws granted by a system-environment interaction in a critical system. We illustrate these findings in two fully connected models, namely, the quantum Rabi and the Lipkin-Meshkov-Glick models. In addition, we discuss the impact of nonlinear ramps and finite-size systems.
We investigate the statistics of the work performed during a quench across a quantum phase transition using the adiabatic perturbation theory when the system is characterized by independent quasiparticles and the "single-excitation" approximation is assumed. It is shown that all the cumulants of work exhibit universal scaling behavior analogous to the Kibble-Zurek scaling for the average density of defects. Two kinds of transformations are considered: quenches between two gapped phases in which a critical point is traversed, and quenches that end near the critical point. In contrast to the scaling behavior of the density of defects, the scaling behavior of the cumulants of work are shown to be qualitatively different for these two kinds of quenches. However, in both cases the corresponding exponents are fully determined by the dimension of the system and the critical exponents of the transition, as in the traditional Kibble-Zurek mechanism (KZM). Thus, our study deepens our understanding about the nonequilibrium dynamics of a quantum phase transition by revealing the imprint of the KZM on the work statistics.
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.