Quantum Chaos in Non-Markovian Open Quantum Systems: Interferometric OTOC, Loschmidt Echo and Commutator Operator Norm

TL;DR

This work investigates quantum information scrambling in open, non-Markovian quantum systems using interferometric $ ext{F}$-OTOC, Loschmidt echo, and operator growth. By analyzing two open-system models—the Tavis-Cummings atom-field setup and a tilted-field Ising chain—the authors show that dissipation generally suppresses scrambling signatures, but non-Markovian memory effects can revive correlations and partially restore light-cone structure via a corrected $ ext{F}$-OTOC, especially in TFIM. Scrambling behavior depends on model details: TC shows dissipation-dominated decay with limited light-cone visibility, while TFIM exhibits angle-dependent transitions from integrable to chaotic-like dynamics, including robust light-cone propagation for certain $\theta$ values. The results illuminate how non-Markovian environments interplay with chaos indicators, providing insights for diagnosing chaos in realistic, nonunitary quantum systems and suggesting avenues for probing scrambling in larger open quantum platforms.

Abstract

Out-of-time order correlators (OTOCs) are crucial tools for studying quantum chaos as they show distinct scrambling behavior for chaotic Hamiltonians. We calculate OTOC and analyze the quantum information scrambling in atom-field and spin-spin interaction models, which are open-system models and exhibit non-Markovian behavior. We also examine the Loschmidt echo for these models and comment on their chaotic nature. The commutator growth of two local operators, which is upper bounded by the Lieb-Robinson bound, is studied for these models, and the patterns of scrambling are investigated.

Figures (11)

• Figure 1: The plots are of the $\mathcal{F}$-OTOC $\mathcal{F}(t)$ for the Tavis Cummings model for $j_s=0$. The other parameters are, $j_{TC}=\frac{\lambda}{2\sqrt{N}}=0.5$, $\omega_0=2,~\omega_c=2,~T=10$.
• Figure 2: The plots are of the $\mathcal{F}$-OTOC $\mathcal{F}(t)$ in (a) and the corrected $\mathcal{F}$-OTOC $\mathcal{F}_c(t)$ in (b) for the Tavis Cummings model for $j_{TC}=\frac{\lambda}{2\sqrt{N}}=0.5$ and the interaction strength $j_s=0.5$ of the nearest neighbor interaction in the system spin chain. The other parameters are $\omega_0=2,~\omega_c=2,~T=10$.
• Figure 3: The $\mathcal{F}$-OTOC for the TFIM model of four system spins interacting with a four Ising spin bath. The direction of the scrambling shown in the figures by means of an arrow denotes that the information of the action of $B_S(0)$ scrambles up to time $t$ where $A_S$ is applied. In (a) and (b) the $\mathcal{F}$-OTOC is plotted for $\theta=\pi/2$. For (c) and (d), the angle is $\theta=\pi/8$. The other parameters chosen are $\mathcal{B}=0.5$, $j=0.8$.
• Figure 4: The corrected OTOC $\mathcal{F}_c(t)$ without dissipation effects and the $\mathcal{F}(t)$ with dissipation effects are plotted for the TFIM model at $\theta=\pi/2$. The other parameters chosen are $\mathcal{B}=0.5$, $j=0.8$.
• Figure 5: The corrected $\mathcal{F}$-OTOC for the TFIM model of four system spins interacting with a four Ising spin bath. The direction of the scrambling shown in the figures by means of an arrow denotes that the information of the action of $B_S(0)$ scrambles up to time $t$ where $A_S$ is applied. In (a) and (b) the corrected $\mathcal{F}$-OTOC, $\mathcal{F}_c(t)$ is plotted for $\theta=\pi/2$ and $\theta=\pi/8$ respectively. The other parameters chosen are $\mathcal{B}=0.5$, $j=0.8$.