Geometric signature of non-Markovian dynamics

TL;DR

The paper addresses detecting non-Markovian dynamics in open quantum systems by linking memory effects to geometric properties of the TLS evolution in a hierarchical environment (TLS–cavity–bath). Using a quantum-state-diffusion framework with dual-noise projection, it derives a complex geometric phase where the imaginary part $\beta_I$ diverges when the auxiliary function $g(t)$ vanishes, signaling information backflow from the bath. The divergence of $\beta_I$ serves as a strong witness for non-Markovianity, with phase-diagram boundaries showing regimes where the divergence is a sufficient or (for certain memory strengths) necessary-and-sufficient indicator. The approach provides a computationally efficient, trajectory-based method for identifying non-Markovianity and suggests potential applications in quantum noise spectroscopy and metrology.

Abstract

Non-Markovian effects in the dynamics of an open system are typically characterized by non-monotonic information flows from the system to its environment or by information backflows from the environment to the system. Using a two-level system (TLS) coupled to a dissipative single-mode cavity, we demonstrate that the geometric decoherence of the open quantum system can serve as a reliable indicator of non-Markovian dynamics. This geometric approach also reveals finer details of the dynamics, such as the specific time points when non-Markovian behavior emerges. In particular, we show that the divergence of the geometric decoherence factor of the TLS can serve as a sufficient condition for non-Markovian dynamics, and in certain cases, it can even be both a necessary and sufficient condition.

Figures (4)

• Figure 1: (Color online) Panel (a): The imaginary part of the complex-value geometric phase $\beta_I(t)$ as a function of time, in the parameter region where non-Markovian dynamics may be found ($\gamma_w=0.9$ and $\kappa = 0.43$). The dashed pink lines denote the time when $F_{z,R}(t)$ becomes negative and there is a back-flow of information from the bath to the system, which coincides with the time when the geometric decoherence divergence takes place at $t \approx 5.19, 8.85, 14.87$. Panel (b): Non-Markovianity as integrated up to time $t$, where it can be observed that the non-Markovianity measure increases in the region where the dynamics is non-contractive. Blue dashed lines denotes when $F_{z,R}(t)$ becomes positive and the dynamics becomes contractive. Panel (c): The expectation values $\langle\sigma_{x,y,z}\rangle$ as a function of time, which are smooth in time and do not display singular behaviors.
• Figure 2: (Color online) Panel (a) Phase diagram shows when $\beta_I$ diverges, in the parameter domain ($\kappa,\gamma_w$). The color bar indicates the first divergent time point $t_0$ (log-scale). The black dashed line shows the boundary between the Markov and non-Markovian dynamics, where the region above the black dashed line is the non-Markovian region. However, when $\kappa, \gamma_\omega$ fall into the region enclosed by the black dashed line and the green line, the system is still non-Markovian, but the divergence is not observed. Panel (b) Plot of $\beta_I(t)$ against time with the parameters $\gamma_w=0.3$ and $\kappa=0.23$, marked by a blue pentagon in the inset. Inset shows a zoom-in view of the area boxed by the dashed orange line in Panel (a). Shaded area are time ranges when the information back-flow occurs.
• Figure 3: (Color online) The auxiliary function $g(t)$ as a function of time. The blue dashed line is from the non-Markovian region where the geometric phase can show a sudden jump, with the same parameters as Fig. \ref{['fig_gpdiv']}. The orange dots on the line signifies times when $g(t)=0$ and $\partial_t|g(t)|$ becomes positive, such that the geometric phase diverges and the information flow between the system and bath changes direction. The green solid line is from the "exceptional case" where the system dynamics can be non-Markovian but the geometric phase does not diverge, with the same parameter as Fig. \ref{['fig_pd']}(b), and the red dots shows the times $\partial_t g(t)$ flips signs. The orange dot-dashed line is from the Markovian region, with $\gamma_w=0.9$ and $\kappa=0.1$.
• Figure 4: (Color online) Panel (a) $g'(t)$ in the non-Markovian region $(\gamma_w =0.5, \kappa=0.4)$ in the upper plot, and on the boundary $(\gamma_w =0.5, \kappa \approx 0.27475)$ in the lower plot. It can be seen that $g'(t)$ intersects with the x-axis in the non-Markovian region but is tangent to the x-axis on the boundary. Panel (b) Analytically derived Markov to non-Markovian boundary for $\gamma_w \in [0.08, 1.5]$ shown as pink dots, where the black dashed line is the numerically obtained Markov to non-Markovian boundary, and the green line is the boundary between divergent and non-divergence geometric decoherence.