Dynamics of a Quantum System Interacting with White Non-Gaussian Baths: Poisson Noise Master Equation

Quantum systems are unavoidably open to their surrounding degrees of freedom. The theory of open quantum systems is thus crucial to understanding the fluctuations, dissipation, and decoherence of a quantum system of interest. Typically, the bath is modeled as an ensemble of harmonic oscillators, which yields Gaussian statistics of the bath influence on the quantum systems. However, there are also phenomena in which the bath consists of two-state systems, spins, or anharmonic oscillators; therefore, the non-Gaussian properties of the bath become important. Nevertheless, a theoretical framework to describe quantum systems under the influence of such non-Gaussian baths is not well established. Here, we develop a theory to describe quantum dissipative systems affected by Poisson noise properties of the bath, because the Lévi-Itô decomposition theorem asserts that Poisson noise is fundamental in describing arbitrary white noise beyond Gaussian properties. We introduce a quantum bath model that allows for the consistent description of dissipative quantum systems. The obtained master equation reveals non-Gaussian bath effects in the white noise regime, and provides an essential step toward describing open quantum dynamics under the influence of generic baths.

Figure 1

Plot of the ground state probability as a function of time for the collective system-bath coupling model. The black dotted curve is obtained from the Poisson noise master equation, Eq. (12). Dashed gray line is calculated by using the Gibbs state with inverse temperature $\beta$, i.e., ${\rho }_{\mathrm{S}}^{\mathrm{eq}}=e^{-\beta H_{\mathrm{S}}}/\mathrm{Tr}[e^{-\beta H_{\mathrm{S}}}]$. The solid curves are obtained by directly solving Eq. (1) for different values of ${\mathrm{\Gamma }}_1^{-}={\mathrm{\Gamma }}_2^{-}$ and $\lambda =\mu {\mathrm{\Gamma }}_1^{-}$. The figure shows that as ${\mathrm{\Gamma }}_1^{-}$ increases, the condition ${\mathrm{\Gamma }}_i^{+}\ll {\mathrm{\Gamma }}_1^{-},\lambda$ is better satisfied and converges to the result calculated by Eq. (12). The initial state is given by the ground state, and the parameters are $N=6$, $\mu =2.0$, $\beta =1.5$, $\omega ={\omega }_1-{\omega }_2=1$, ${\mathrm{\Gamma }}_2^{+}=1$.

Figure 2

Plot of the effective decay rate ${\mathrm{\Gamma }}_{\mathrm{eff}}/{\mathrm{\Gamma }}_2^{+}$ [Eq. (14)], as a function of ${\mu }^{2}N$, where $N$ is the number of two-level systems and $\mu$ is the effective coupling strength. The red curve is obtained using the Poisson noise master equation (12), which converges to the rate ${\mathrm{\Gamma }}_{\mathrm{eff}}={\mathrm{\Gamma }}_2^{+}/2=O(1)$ in large $N$, where ${\mathrm{\Gamma }}_2^{+}$ is the average rate of the Poisson noise. Gaussian noise limit corresponds to $\mu \ll 1$ and agrees with the $O(N)$ scaling of the effective decay rate for the conventional Gaussian noise master equation (13) analysis, plotted as the black dashed line.