A continued fraction based approach for the Two-photon Quantum Rabi Model

Lupo, Elena; Napoli, A.; Messina, A.; Solano, E.; Egusquiza, I. L.

doi:10.1038/s41598-019-39281-4

Cited by 20 publications

(21 citation statements)

References 47 publications

(107 reference statements)

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“…Recently a number of theoretical studies on the spectral collapse [28][29][30][31][32][33][34][35][36][37][38] of the model have indeed confirmed the results and observations of Ng et al 27 . Nevertheless, our understanding of the quantum two-photon Rabi model at the critical coupling ǫ c is still very limited because current theoretical approaches (both analytical and numerical) fail in dealing with the collapse point rigorously.…”

Section: Demystifying the Spectral Collapse In Two-photon Rabi Model supporting

confidence: 58%

“…Nevertheless, our understanding of the quantum two-photon Rabi model at the critical coupling is still very limited because current theoretical approaches (both analytical and numerical) fail in dealing with the collapse point rigorously. While numerical methods (such as numerical exact diagonalization 27 , 35 and the ones based upon spectral function and continued fraction 34 ) suffer from the demand of a huge amount of computational power and unstable convergence, analytical analyses (like variational approximation 24 and Braak’s G-function method 15 , 30 – 32 ) are unable to approach the collapse point satisfactorily. For instance, as pointed out by Duan et al 33 , the G-function method seems to suggest that the eigenenergy spectrum at the critical coupling consists of a discrete part in addition to a continuum: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense, whereas the perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, as pointed out by Duan et al 33 , the G-function method seems to suggest that the eigenenergy spectrum at the critical coupling consists of a discrete part in addition to a continuum: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense, whereas the perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature. In addition, performing a numerical study of both the spectral functions and survival probabilities based upon a continued fraction approach, Lupo et al 34 identifies a signal suggesting that there is a remaining relevant discrete point in the spectrum.…”

Section: Introductionmentioning

confidence: 99%

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Demystifying the spectral collapse in two-photon Rabi model

2020

Sci Rep

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We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature “spectral collapse”, by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a “Lorentzian function” potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the “spectral collapse” can be monitored in a straightforward manner.

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Section: Demystifying the Spectral Collapse In Two-photon Rabi Model supporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Demystifying the spectral collapse in two-photon Rabi model

2020

Sci Rep

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“…It is because the wavefunction is widely spread and even no longer bounded, so it cannot be represented by number states. Fake converging states have been reported for > ω/2 if truncation number is not large enough [22,26]. The truncated wavefunction is stabilized by the local potential but globally it is unstable [25].…”

Section: Introductionmentioning

confidence: 97%

Bound states of two-photon Rabi model at the collapse point

Chan¹

2020

J. Phys. A: Math. Theor.

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This paper presents a proof of the existence of novel bound states of the two-photon quantum Rabi model at the collapse point. The two-photon Rabi model is interesting not only for its important role on non-linear light-matter interaction, but also for the exhibition of many-energy-levels degenerating process called the "spectral collapse". The squeezing property of the two-photon annihilation and creation operators is the origin for this phenomenon which is well studied without the energy-slitting term ω 0. However, many numerical studies have pointed out that with the presence of ω 0 , some low-level isolated states exist while other high energy states collapse to E = − ω 2 , which known as incomplete spectral collapse. From the eigenvalue equation in real space, pair of second order differential equations, which are similarly to the Schrodinger equation, are derived at the collapse point. These differential equations provide explanation to the existence of isolated bound states below E = − ω 2 with the presence of the spin slitting ω 0 and better numerical method to generate those bound states.

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“…20 using Bogoliubov transformations. These solutions have stimulated extensive research interests in the exact solutions to the unmixed QRMs and their variants with either one-photon 21–26 or two-photon term 27–34 . In the literature, many analytical approximate but still very accurate results have also been given 35–47 .…”

Section: Introductionmentioning

confidence: 99%

The mixed quantum Rabi model

Duan¹,

Xie²,

Chen³

2019

Sci Rep

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The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4 × 4 determinants responsible for the exact solutions are derived. These so-called G-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of G-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point. All energy levels only collapse to the lower one, which diverges negatively. The level crossings in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM due to absence of parity symmetry. In the weak two-photon coupling regime, the mixed QRM is equivalent to an one-photon QRM with an effective positive bias, suppressed photon frequency and enhanced one-photon coupling, which may pave a highly efficient and economic way to access the deep-strong one-photon coupling regime.

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A continued fraction based approach for the Two-photon Quantum Rabi Model

Cited by 20 publications

References 47 publications

Demystifying the spectral collapse in two-photon Rabi model

Demystifying the spectral collapse in two-photon Rabi model

Bound states of two-photon Rabi model at the collapse point

The mixed quantum Rabi model

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