Abstract The Thomas–Reiche–Kuhn (TRK) sum rule is a fundamental consequence of the position–momentum commutation relation for an atomic electron, and it provides an important constraint on the transition matrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light–matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light–matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

中文翻译：

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Thomas–Reiche–Kuhn (TRK) 相互作用光子的求和规则

摘要 Thomas-Reiche-Kuhn (TRK) 求和规则是原子电子位置-动量交换关系的基本推论，它为原子的跃迁矩阵元素提供了重要约束。在这里，我们提出了电磁场的 TRK 和规则，即使在存在非常强的光-物质相互作用和/或光学非线性的情况下也是有效的。虽然标准 TRK 求和规则涉及在原子能级之间计算的偶极矩阵矩（在没有与场相互作用的情况下），但这里提出的求和规则涉及在相互作用的光-物质系统的一般本征态之间计算的场算子的期望值。