大光子数下二次谐波哈密顿量的能级
Energy levels of the second-harmonic Hamiltonian at large photon numbers
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中文总结 AI 辅助
该研究针对总激发数很大的简并χ⁽²⁾哈密顿量，通过映射为双势阱的一维薛定谔方程推导隐式量子化条件，得到互补谱区的渐近公式，为常规近似失效的谱中心附近能级提供精确解析描述。
中文摘要 AI 辅助
我们研究描述二次谐波产生和参量下转换的简并χ⁽²⁾哈密顿量在总激发数很大时的能谱。由于总激发数守恒，该谱问题可简化为有限维三对角矩阵的对角化。遵循Alvarez和Alvarez-Estrada的方法，我们将该问题映射为具有双势阱的有效一维薛定谔方程。随后，我们推导势垒顶部附近能级的隐式量子化条件，并得到适用于互补谱区的两个显式渐近公式。确定了所得公式的适用范围，并将其精度与精确矩阵对角化及常规JWKB近似进行比较。所提方法为谱中心附近的能级提供了精确的解析描述，而常规近似在该区域会丧失精度。
英文摘要
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.
发表机构
- Russian Quantum Center（俄罗斯量子中心）
- Moscow Institute of Physics and Technology（莫斯科物理技术学院）
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