Lindblad Master Equation Capable of Describing Hybrid Quantum Systems in the Ultrastrong Coupling Regime

Despite significant theoretical efforts devoted to studying the interaction between quantized light modes and matter, the so-called ultrastrong coupling regime still presents significant challenges for theoretical treatments and prevents the use of many common approximations. Here we demonstrate an approach that can describe the dynamics of hybrid quantum systems in any regime of interaction for an arbitrary electromagnetic (EM) environment. We extend a previous method developed for few-mode quantization of arbitrary systems to the case of ultrastrong light-matter coupling, and show that even such systems can be treated using a Lindblad master equation where decay operators act only on the photonic modes by ensuring that the effective spectral density of the EM environment is sufficiently suppressed at negative frequencies. We demonstrate the validity of our framework and show that it outperforms current state-of-the-art master equations for a simple model system, and then study a realistic nanoplasmonic setup where existing approaches cannot be applied.

Figure 1

A realistic spectral density at zero temperature and its Lorentzian approximation, corresponding to a single-mode Lindblad master equation. The tail at negative frequencies causes artificial pumping of energy into the emitter, leading to incorrect dynamics.

Figure 2

(a) The spectral density of the system under study. Black dashed line represents the preset threshold value for model spectral density at negative frequencies. Gray lines show the real part of the complex resonances of the model spectral density, and the brown dotted line indicates the emitter transition frequency. (b) The population of the two-level emitter interacting with photonic environment in the USC regime for different methods.

Figure 3

Average relative error of the emitter population dynamics obtained with our approach for a different number of modes involved into fitting. Red and green arrows show the error of GME and BR master equations. In inset: time dependence of relative error. Orange, red, and green lines represent results for our approach (with 10 modes at the threshold $\mathrm{value}={10}^{-8}\mathrm{meV}$), GME, and BR master equation, respectively.

Figure 4

(a) The spectral density of the considered system. Black dashed line represents the preset threshold value for the model spectral density at negative frequencies and the brown dotted line indicates the emitter transition frequency. (b) Emitter population as a function of time for the numerically exact and our approach applied for narrow [(0.7; 4) eV] and broad [$(-5;5)\mathrm{eV}$] spectral windows. In inset: The total number of photons accumulated in the system and bath modes.