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arXiv:1702.07131 · New Journal of Physics

Mean photon number dependent variational method to the Rabi model

Maoxin Liu, Zu-Jian Ying, Jun-Hong An, Hong-Gang Luo

Published Apr 2, 2015• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/17/4/043001

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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.
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.
The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.
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.
We propose an extended transformed rotating-wave approximation (ETRWA) for the two-photon quantum Rabi model. This method retains the spin-dependent squeezing transformation and the qubit-frequency-dependent parameter-selection condition of the existing TRWA, but replaces its excitation-independent vacuum coefficients with exact photon-number-dependent diagonal and nearest-neighbour two-photon matrix elements. We compute the energy spectra, mean photon numbers, and eigenstate fidelities using the ETRWA, TRWA, and S-GRWA methods. Comparison with exact numerical results shows that the ETRWA improves the 2017 TRWA most clearly at intermediate coupling. The comparison with the S-GRWA is observable-, detuning-, and coupling-dependent. Near spectral collapse, the S-GRWA is superior because it retains the vanishing soft-mode scale, while the finite-squeeze ETRWA omits interblock processes needed for the softening of the continuum and higher excited states. The ETRWA retains closed 2×22\times 22×2 diagonalizations, with an excitation-dependent finite-sum cost for generating their coefficients.
The single-mode spin-boson model exhibits behavior not included in the rotating-wave approximation (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave approximation that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave approximation both off-resonance and at deep-strong coupling. The symmetric rotating-wave approximation allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime. Additionally, we symmetrize the generalized form of the rotating-wave approximation to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave approximation. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.