Symmetric rotating-wave approximation for the generalized single-mode spin-boson system

Abstract

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 twophoton spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.which have not been thoroughly explored in the past and have shown that the RWA breaks down in those regions [13,19]. Specifically, a number of recent theoretical studies have shown that contributions of counter-rotating terms, which are ignored in the RWA, prove important in these parameter regions and exhibit dynamical behavior different from the weak-coupling case [12,[20][21][22][23]. Additionally, counter-rotating terms dominate the short-time dynamical behavior for some parameter regions, leading to important Zeno and anti-Zeno effects that are not reproduced by the RWA [24][25][26].The Hamiltonian of a two-level system coupled quadratically to a quantum harmonic oscillator, the twophoton Rabi Hamiltonian, has also been studied within the RWA [27]. Limitations of the RWA have been outlined for this system [28], but so far limited effort has been directed to studying it outside of the RWA [29][30][31][32]. This Hamiltonian arose in quantum optics as a phenomenological model for a three-level system interacting with two photons [28,30,33] and is also relevant in modeling pure dephasing in crystals [34]. As opposed to a displacement in position in the case of the Rabi Hamiltonian, the coupling in the two-photon Rabi Hamiltonian is through frequency displacement or "squeezing" [35,36]. With bi-exciton effects and two-photon processes occurring in experimental systems [37], more work is needed to determine whether this Hamiltonian can successfully model these effects.The two Hamiltonians (m = 1,...

Solvability of the two-photon Rabi Hamiltonian

“…The values ω 0 = 1 and ω = 2 are chosen so that the spectrum at g = 0 becomes equidistant, see eq. eters was already published [2], where the authors plot also numerical results from larger matrices diagonalization.…”

“…The same model is known under several pseudonyms, e. g. single-mode spin-boson system, Jaynes-Cummings model without rotating wave approximation and others; a brief survey can be found in [2]. Let us introduce a more general form of Rabi Hamiltonians [3], describing the interaction between a bosonic mode with energy ω and a two-level atom with level spacing ω 0…”

Exact solvability of the quantum Rabi model using Bogoliubov operators

“…When the intermediate transition frequencies are strongly detuned from cavity frequency, after adiabatically eliminating the intermediate levels, one arrives at the effective Hamiltonian. This two-photon QRM has also been studied for a long time with the RWA [26] and beyond the RWA [17,18,27]. It may describe the twophoton processes occurring in Rubidium atoms [28] and quantum dots [29].…”

“…then we have (17) with f ′ 0 = 1 and f ′ 1 = Ω ′ (0). If both wavefunctions (5) and (14) are the true eigenfunction for the same eigenvalue E, they should be in principle only different by a complex constant r if this eigenvalue is not degenerate…”

“…The key procedures can be described with a simple tutorial, and therefore it is straightforward to extend to study the exact solvability or integrability of various spin-boson systems. The extension to the twophoton QRM [17,18] is performed in this paper as the first example.…”