Extended transformed rotating-wave approximation for the two-photon quantum Rabi model

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× 2 diagonalizations, with an excitation-dependent finite-sum cost for generating their coefficients.