Purified pseudomode model for nonlinear system-bath interactions

TL;DR

This work extends the purified pseudomode framework to general nonlinear system-bath interactions by combining spectral decompositions of the bath correlation function with a purification protocol that preserves pure pseudomode states. A zero-frequency pseudomode is introduced to capture the second moment C(0), and the nonlinear coupling Q(X) = \sum_n α_n X^n is incorporated, yielding a Lindblad-like generator in the a→∞ limit and a HEOM-compatible formulation. The authors validate the approach with two nonperturbative demonstrations: spontaneous decay of a two-level atom in a lossy cavity and the resonance fluorescence of a quantum dot coupled to a phonon bath, showing exact agreement with traditional master-equation results and revealing how nonlinear couplings modify hybridization and spectra. The method holds promise for accurate, scalable simulations of nonlinear light- and phonon-matter interactions across quantum optics, nanophotonics, and related fields.

Abstract

The theory of purified pseudomodes [arXiv:2412.04264 (2024)] was recently developed to provide a numerical tool for the analysis of the properties of a quantum system and the environment it couples to via linear system-bath interactions. Here we extend this theory to allow for the description of general nonlinear system-bath interactions. We demonstrate the validity of our method by considering the spontaneous decay of a two-level atom placed inside a single-mode lossy cavity and furthermore, its potential application to nanophotonics by calculating the resonance fluorescence spectrum of a quantum dot in the presence of a phonon environment. Our method provides a useful tool for the study of phonon-assisted emission in quantum dots and holds the the promise for broad applications in fields like quantum biology, nonlinear phononics, and nanophotonics.

Figures (3)

• Figure 1: Spontaneous decay of a two-level atom placed inside a lossy cavity. Panel (a) and (b) show the evolution of the excited state population $\langle \sigma_+\sigma_-\rangle$ and the von Neumann entropy $S_\text{vN}$, respectively. Colored solid curves are obtained using the corresponding purified pseudomode model (abbreviated as "PPM") for four different combinations of the coefficients $(\alpha_1,\alpha_2,\alpha_3)$ in the coupling operator $Q_\text{ac} = \alpha_1X_\text{c} + \alpha_2X_\text{c}^2 + \alpha_3X_\text{c}^3$ with $X_\text{c}=\lambda(b+b^\dagger)$. Black dashed lines are the solution of the master equation in Eq. (\ref{['eq:lossymodeexample']}). Simulation parameters are $\omega_S=0.5$, $\nu=0.5$, $\gamma=0.1$, and $\lambda=0.3$.
• Figure 2: Excited state population $\langle\sigma_+\sigma_-\rangle$ in panel (a) and von Neumann entropy $S_\text{vN}$ in panel (b) for the steady-state of the two-level atom placed in a lossy cavity. Here $\alpha_1=1$ is fixed in the coupling operator $Q_\text{ac} = \alpha_1X_\text{c} + \alpha_2X_\text{c}^2 + \alpha_3X_\text{c}^3$. Other simulation parameters are the same as those in Fig. \ref{['fig:singlemode']}.
• Figure 3: (a) Power spectrum $S(\omega)$ (blue solid) of the phonon environment and its approximation (red dashed) generated with the AAA algorithm. The positive (negative) frequency domain of $S(\omega)$ corresponds to emission (absorption) of phonons, respectively. The upper inset shows the fitting error $\epsilon_S = \int d\omega\lvert S(\omega) - S_\text{AAA}(\omega)\rvert$ for different $N$, while the lower inset shows the fitting error $\epsilon_C = \lvert C(t) - C_\text{AAA}(t)\rvert$ as a function of time $t$. (b) Resonance fluorescence spectra in the absence (blue line) and presence (orange line for the pure linear coupling and green line for the linear+quadratic coupling) of the phonon environment. Other simulation parameters are $\kappa=0.1meV$ and $\Omega=1meV$.