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arXiv:0706.2087 · Physical Review Letters

Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling

E. K. Twyeffort Irish

Published Oct 22, 2007• 317 citations
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Abstract

A generalized version of the rotating-wave approximation for the single-mode spin-boson Hamiltonian is presented. It is shown that performing a simple change of basis prior to eliminating the off-resonant terms results in a significantly more accurate expression for the energy levels of the system. The generalized approximation works for all values of the coupling strength and for a wide range of detuning values, and may find applications in solid-state experiments.

DOI
10.1103/physrevlett.99.173601

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A generalized version of the rotating-wave approximation for the single-mode spin-boson Hamiltonian is presented. It is shown that performing a simple change of basis prior to eliminating the off-resonant terms results in a significantly more accurate expression for the energy levels of the system. The generalized approximation works for all values of the coupling strength and for a wide range of detuning values, and may find applications in solid-state experiments.
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The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.
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An efficient and stable numerical scheme based on Hill’s determinant is proposed for computing the whole spectrum of the single-mode spin-boson model. Its shown that for the low-lying levels a simple version of Hill’s determinant method works well, while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels. It is very convenient to use this entire function to compute all the energy levels numerically. Some high levels of the model are computed and compared with Irish’s general rotating-wave approximation. A scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated.