Bridging the gap between the Jaynes–Cummings and Rabi models using an intermediate rotating wave approximation

•The intermediate rotating wave approximation (IRWA) is introduced.•The co-rotating and counter-rotating terms in Rabi model are expressed separately.•IRWA is applied to the near resonance case and the large detuning case.•The continuity between the Jaynes–Cummings model and the Rabi model is established.

We present a novel approach called the intermediate rotating wave approximation (IRWA), which employs a time-averaging method to encapsulate the dynamics of light-matter interaction from strong to ultrastrong coupling regime. In contrast to the ordinary rotating wave approximation, this method addresses the co-rotating and counter-rotating terms separately to trace their physical consequences individually, and thus establishes the continuity between the Jaynes–Cummings model and the quantum Rabi model. We investigate IRWA in near resonance and large detuning cases. Our IRWA not only agrees well with both models in their respective coupling strengths, but also offers a good explanation of their differences.