Photon counting beyond the rotating-wave approximation

Open quantum systems are often described by a Lindblad master equation, which relies on a set of approximations, most importantly the rotating-wave approximation which is only valid for weak damping. In the Lindblad setting, dissipative processes are described through jump operators, distinguishing between absorption and emission of photons. This enables the simple identification of emitted photons which provides a straightforward way to obtain the radiation statistics. Outside the rotating-wave limit, the Lindblad approach does not work. Open quantum systems can then be described by, e.g., the quantum Langevin equation. However, in this framework the number of emitted photons is not easily accessible. In this work, we point out how to obtain the photon counting statistics from a quantum Langevin equation and provide an expression for the photon current operator, for arbitrary systems coupled to linear environments. As an example, we employ the method to study the radiation statistics of a damped harmonic oscillator at finite temperature beyond the rotating-wave approximation. We show that even outside the rotating-wave limit, the most important contribution to the radiation statistics can be captured by an effective Lindblad equation, thus extending the range of possible applications of the Lindblad framework.

💡 Research Summary

The paper addresses a fundamental limitation of the widely used Lindblad master‑equation description of open quantum systems: its reliance on the rotating‑wave approximation (RWA) and weak damping. When the system’s natural frequency becomes comparable to the dissipation rate—as in many microwave and circuit‑QED platforms—the RWA breaks down and the Lindblad formalism no longer yields a faithful description of photon emission and absorption processes. To overcome this, the authors return to the Caldeira‑Leggett model of a system linearly coupled to a bath of harmonic oscillators and derive the quantum Langevin equation (QLE) for an arbitrary system operator (Y): \