The sum rules of quantum field theory should not be confused with the sum rules in quantum chromodynamics or quantum mechanics. The relationships in (4) and (7) are very general. They also apply to systems with multiple dipoles (see e.g. , [, 78]) and modes (see e.g. ). These relationships provide a very useful check of the consistency of approximation models in quantum optics. Rough Hamiltonians and efficient models can hurt any of them. Such a violation suggests that the model may neglect some relevant physics . For example, we have shown that the JC model, a widely used description for the dipole coupling between a two-stage atom and a quantized electromagnetic field, violates the relation (4).
Another example of a model that violates this relationship is the well-known and widely used Hamiltonian optomechanical cavity interaction ħgaˆ†aˆ(bˆ+bˆ†) (here bˆ is the destruction operator of the mechanical oscillator) . On the contrary, the interaction obtained by Hamiltonian through a microscopic model  ħg(aˆ†+aˆ)2(bˆ+bˆ†) satisfies these two relations [Eqs (7), (14)]. It turns out that such an interaction describes Hamiltonian not only the standard optomechanical effects, but also the dynamic Casimir effect , [, 64]. Thomas-Reiche-Kuhn (TRK) Summation rule for interacting photons in the three-component system described by Hamilton in (11). (a) subsum rules ∑j=1NF0ja with respect to the first resonator and (b) ∑j=1NF1jb with respect to the second resonator, both for values different from the N levels. The black segmented line describes the case of zero distuning δ = 0, while the dotted blue segmented lines refer to the case δ=(ω0−ω ̅0)/ω̅0=−6×10−3. The parameters are specified in the text. The eigenstates of these systems, including the ground state, can have a complex structure that involves a superposition of several eigenstates of the subsystems without interaction , ,  and are difficult to calculate.
As a result, a number of approximation methods have been developed , [, 61]. In addition, the correlation functions of the initial field associated with the measures depend on these eigenstates (see for example , [, 62]). Therefore, summation rules that contain general guidelines and constraints can be very useful for testing the validity of approximations. The general sum rule proposed in this paper can also be used to test the validity of effective Hamiltonians, commonly used in quantum optics and cavity optomechanics , , . Moreover, this generalized TRK summation rule applies to the broad emerging domain of non-perturbative light-matter interactions, including multiple parameters and subdomains, such as quantum electrodynamics of cavities and circuits (QED) , collective excitations in solids , optomechanics , photochemistry, and QED chemistry , [, 66]. To understand how the sum rule in (7) applies to Rabi`s quantum model, we calculate subtotals with an increasing number of states. In particular, we calculate  A. Settineri, O. Di Stefano, D. Zueco, S. Hughes, S. Savasta and F.
Nori, “Gauge freedom, quantum measurements, and time-dependent interactions in the cavity and QED circuit,” arXiv: 1912.08548, 2019.Search Google Scholar The proposed TRK summation rule for interacting photons may be useful for the study of general nonlinear quantum optical effects and multi-body physics in photonic systems (see, For example, , , , , ), such as the corresponding summation rules for interacting electronic systems, which has played a fundamental role in understanding the multi-body physics of interacting electronic systems , , . The Thomas-Reiche-Kuhn (TRK) sum rule is a fundamental consequence of the position-moment switching principle for an atomic electron and represents an important constraint for the elements of an atom`s transition matrix. Here we propose a TRK summation rule for electromagnetic fields, which is also valid for very strong light-matter interactions and/or optical nonlinearities. While the standard TRK sum rule includes calculated dipole matrix moments between atomic energy levels (in the absence of field interaction), the sum rule proposed here includes the expected values of field operators calculated between the general eigenstates of the interacting light-matter system. This summation rule provides constraints and guidance for the analysis of strongly interacting light-matter systems and can be used to test the validity of approximately efficient Hamiltonians commonly used in quantum optics. The Rabi-Hamilton quantum describes the dipole coupling between a two-stage atom and a single mode of the quantized electromagnetic field.