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Journal of Physics A Mathematical and Theoretical

Squeezing based analytical variational method for the biased quantum Rabi model in the ultrastrong coupling regime

Weijie Xie, Bin-Bin Mao, Guangzhuo Li, Weihang Wang, Chen Sun, Yimin Wang, Wen‐Long You, Maoxin Liu

Published Oct 7, 2019• 5 citations
Publisher

Abstract

Abstract In order to obtain the ground state of the biased quantum Rabi model in the ultrastrong coupling regime, a generalized squeezing rotating-wave approximation method including the displacement and the squeezing transformations is proposed in this work. While considering the deformation of the oscillator state, our method evidently improves the generalized variational method. The analyses of a few physical quantities show that our method works well especially in the ultrastrong coupling and large atomic frequency regime.

DOI
10.1088/1751-8121/ab4b7a

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Abstract In order to obtain the ground state of the biased quantum Rabi model in the ultrastrong coupling regime, a generalized squeezing rotating-wave approximation method including the displacement and the squeezing transformations is proposed in this work. While considering the deformation of the oscillator state, our method evidently improves the generalized variational method. The analyses of a few physical quantities show that our method works well especially in the ultrastrong coupling and large atomic frequency regime.
Generalized squeezing rotating-wave approximation (GSRWA) is proposed by employing both the displacement and the squeezing transformations. A solvable Hamiltonian is reformulated in the same form as the ordinary RWA ones. For a qubit coupled to oscillators experiment, a well-defined Schr\"odinger-cat-like entangled state is given by the displaced-squeezed oscillator state instead of the original displaced state. For the isotropic Rabi case, the mean photon number and the ground-state energy are expressed analytically with additional squeezing terms, exhibiting a substantial improvement of the GSRWA. And the ground-state energy in the anisotropic Rabi model confirms the effectiveness of the GSRWA. Due to the squeezing effect, the GSRWA improves the previous methods only with the displacement transformation in a wide range of coupling strengths even for large atom frequency.
An analytical variational method for the ground state of the biased quantum Rabi model in the ultra-strong coupling regime is presented. This analytical variational method can be obtained by a unitary transformation or alternatively by assuming the form of the ground state wave function. The key of the method is to introduce a variational parameter λ , which can be determined by minimizing the energy functional. Using this method, we calculate the physical observables with high accuracy in comparison with the numerical exact ones. Our method evidently improves over the widely used general rotating-wave approximation (GRWA) in both qualitative and quantitative aspects.
We present an analytical method for the two-qubit quantum Rabi model. While still operating in the frame of the generalized rotating-wave approximation (GRWA), our method further embraces the idea of introducing variational parameters. The optimal value of the variational parameter is determined by minimizing the energy function of the ground state. Comparing with numerically exact results, we show that our method evidently improves the accuracy of the conventional GRWA in calculating fundamental physical quantities, such as energy spectra, mean photon number, and dynamics. Interestingly, the accuracy of our method allows us to reproduce the asymptotic behavior of a mean photon number in a large frequency ratio for the ground state and investigate the quasiperiodical structure of the time evolution, which the GRWA is incapable of predicting. The applicable parameter ranges cover the ultrastrong-coupling regime, which will be helpful to recent experiments.
Recent experiments about the circuit cavity quantum electrodynamics have realized the ultrastrong couplings between the artificial atom and the photon, in which the coupling strengths have the same order of the photon frequency. In such a regime, the well-known rotating wave approximation is invalid, and the system dynamics is thus governed by the Rabi model. In this paper, we investigate the photon squeezing of the Rabi model. We find numerically that with the increase of the atom-photon coupling strength, the photon squeezing does not increase linearly, but displays a maximum in the ultrastrong coupling regime. In addition, we also reveal that the photonsqueezing can be enhanced by the counter-rotating terms of the Rabi model. Our results are of benefit to preparing the required squeezing state of the photon in experiment.
We present a mean photon number dependent variational method, which works well in the whole coupling regime if the photon energy is dominant over the spin-flipping, to evaluate the properties of the Rabi model for both the ground state and excited states. For the ground state, it is shown that the previous approximate methods, the generalized rotating-wave approximation (only working well in the strong coupling limit) and the generalized variational method (only working well in the weak coupling limit), can be recovered in the corresponding coupling limits. The key point of our method is to tailor the merits of these two existing methods by introducing a mean photon number dependent variational parameter. For the excited states, our method yields considerable improvements over the generalized rotating-wave approximation. The variational method proposed could be readily applied to more complex models, for which it is difficult to formulate an analytic formula.
An unexplored nonperturbative deep strong coupling (npDSC) achieved in superconducting circuits has been studied in the anisotropic Rabi model by the generalized squeezing rotating-wave approximation. Energy levels are evaluated analytically from the reformulated Hamiltonian and agree well with numerical ones in a wide range of coupling strength. Such improvement ascribes to deformation effects in the displaced-squeezed state presented by the squeezed momentum variance, which are omitted in previous displaced states. The atom population dynamics confirms the validity of our approach for the npDSC strength. Our approach offers the possibility to explore interesting phenomena analytically in the npDSC regime in qubit-oscillator experiments.