Energy levels of the second-harmonic Hamiltonian at large photon numbers
We study the energy spectrum of the degenerate $χ^{(2)}$ Hamiltonian describing second-harmonic generation and parametric down-conversion at large total excitation numbers. Owing to the conservation of the total excitation number, the spectral problem reduces to the diagonalization of finite-dimensional tridiagonal matrices. Following the approach of Alvarez and Alvarez-Estrada, we map this problem onto an effective one-dimensional Schrödinger equation with a double-well potential. We then derive an implicit quantization condition for the energy levels near the top of the potential barrier and obtain two explicit asymptotic formulas applicable in complementary spectral regions. The ranges of applicability of the resulting formulas are established, and their accuracy is compared with exact matrix diagonalization and with the conventional JWKB approximation. The proposed approach provides an accurate analytical description of the energy levels near the center of the spectrum, where the conventional approximation loses accuracy.