Photon counting beyond the rotating-wave approximation

TL;DR

The paper addresses photon counting for open quantum systems beyond the rotating-wave approximation by deriving a photon current operator from the quantum Langevin equation for linear environments. It provides a concrete formula I(t) = (1/ħ) ∫∫ dω dν/(2π)^2 Re(f_ω) A_{ω+ν/2}^ag A_{ω−ν/2} e^{iν t}, enabling direct access to radiation statistics without relying on the Lindblad structure. Applying this to a damped harmonic oscillator, the authors obtain finite photon currents and first-order coherence beyond the RWA, analyze the pole structure, and reveal how an effective Lindblad description can still be constructed via a Lamb-shifted frequency tilde ω and a quasiparticle weight r. This work broadens the applicability of Lindblad-type analyses to regimes of stronger damping and finite temperatures, with potential impact on cavity/circuit QED, optomechanics, and quantum thermodynamics beyond the RWA.

Abstract

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.

Figures (5)

• Figure 1: Sketch of the Caldeira-Leggett model. A system with Hamiltonian $H_0$ is coupled to an environment that consists of a bath of harmonic oscillators. The coupling of the system operator $X$ to the bath modes allows for an exchange of energy. When tracing out the environment, the dynamics of the system can be described by a QLE, see Eq. (\ref{['eq:QLE']}).
• Figure 2: Two setups from cQED which are described by the QLE of a damped harmonic oscillator. The oscillators consist of an inductance $L$, capacitance $C$, and Ohmic resistor $R$ depicted as a transmission line. For the in-series circuit (left), the loop charge $Q$ takes the place of the position, while in the parallel case (right), this role is played by the node flux $\phi$.
• Figure 3: Pole structure of the integrand in Eq. (\ref{['eq:G1']}) in the complex $\omega$-plane. The poles $\omega_\pm$ (green) start on the real axis in the rotating-wave limit $\gamma \to 0^+$ and approach the imaginary axis close to the critical damping. At the critical point $\gamma=2\omega_0$ (red) they coalesce at an exceptional point. For larger damping, the poles move apart along the imaginary axis. Additionally, there are the Matsubara poles (blue) that gain relevance outside the rotating-wave limit, see also the main text.
• Figure 4: Average photon current $\langle I \rangle$ as a function of damping rate $\gamma$. (a) Average photon current in the high temperature regime with $k_B T \approx 5.5 \hbar\omega_0$ such that $n_0 = 5$. The full line displays the exact result while the dashed and dotted line represent the approximate results given by Eq. (\ref{['eq:high-T']}) and (\ref{['eq:n_overdamped']}) respectively. (b) Low-temperature regime with $k_BT=0.1\hbar\omega_0$ ($n_0 \approx 5\times 10^{-5}$). While the photon current is exponentially small in the rotating-wave limit, it increases for stronger damping, Eq. (\ref{['eq:low-T']}), until the ultra-strong damping regime of Eq. (\ref{['eq:n_overdamped']}) is reached. For both, the high- and low-temperature regimes, the transition to the case of ultra-strong damping is indicated by the gray line.
• Figure 5: Comparison of the exact solution of $G^{(1)}(\tau)$ from Eq. (\ref{['eq:G1']}) (real part in blue, imaginary part in orange), to the approximate solution given by the effective Lindblad equation (\ref{['eq:Lindblad_eff']}), in black (real part as full line, imaginary part dashed), outside the rotating-wave limit with $\gamma = \omega_0$ and $k_BT \approx 5.5\hbar \omega_0$ ($n_0=5$). Note that the imaginary part is well approximated for all times, while the real part provides a good approximation for $\gamma \tau \gtrsim 1$, as expected.