Emergence of non-Markovian Decoherent Histories in Integrable Environment: A "Tape Recorder" Model for Local Quantum Observables

TL;DR

The paper tackles how classical multi-time histories emerge in quantum systems coupled to non-Markovian, integrable environments. It introduces a physically grounded tape-recorder mechanism in which the environment sequentially and irreversibly stores information about the system, enabling explicit construction of decoherent histories via irreversibly decoupled environmental modes. The authors develop an OTOC-based computational framework to identify the minimal set of light-cone interior modes and prove that the decoherence functional becomes approximately diagonal as a function of a significance threshold, allowing a stochastic unraveling of dynamics through Monte Carlo sampling. The approach provides a practical and general method to study history-based descriptions in settings where traditional Markovian or non-integrable assumptions fail, with potential experimental tests and links to quantum Darwinism.

Abstract

We propose a new approach to coarse-grained description of quantum evolution that provides an explicit recipe to construct and evaluate multi-time decoherent histories in a controlled way, applicable to non-Markovian and integrable systems. Specifically, we study local interaction quench of a local degree of freedom (an open quantum system) within a noninteracting integrable environment. This setting allows us to identify the environmental degrees of freedom that irreversibly store records of the system's past. These modes emerge sequentially in time and define the projectors required for decoherent histories. We show numerically that the off-diagonal elements of the decoherence functional are exponentially suppressed relative to a significance threshold.

Figures (11)

• Figure 1: Tape recorder scheme. i --- non-interacting edges; ii --- future interaction regions; iii --- active recording region; iv --- irreversibly decoupled domains. In analogy with such a model, we coarse-grain the total system comprising the open system and the environment.
• Figure 2: When local open system is coupled to the environment at $t=0$, the disturbance propagates in a light-cone-like way. In (a) $\hat{n}_{p}\left(t\right)=\bra{\Psi\left(t\right)}\hat{a}_{p}^{\dagger}\hat{a}_{p}\ket{\Psi\left(t\right)}$ is plotted depending on chain site over time. In (b) the light-cone-like spread of the coupling site $\hat{a}_{0}^{\dagger}\left(t\right)$, we plot $\left|\alpha_{k}\left(t\right)\right|$, eq. (\ref{['eq:spread_equation']}), with respect to the chain site index $k$ and time $t$.
• Figure 3: Mapping of the environmental modes onto a "moving tape". At time $t$, the tape is divided into four regions: edge, incoming ($m_+(t)$ modes), relevant ($r(t)$ modes), and outgoing domains ($m_-(t)$ modes), where $r(t) = m_+(t) - m_-(t)$.
• Figure 4: The number of arrived modes $m_{+}\left(t\right)$ vs time $t$ on different scales. $m_{+}\left(t\right)$ is defined as the number of modes $\kappa$ such that $t_{\rm in}\left(\kappa\right)\leq t$. The parameters used are: $a_{\rm cut}=10^{-5},\,\varepsilon_{j}\equiv1$, $h_{j}\equiv0.05$. The chain sites couple one-by-one at an asymptotically constant rate (blue curve). At the same time, the light cone interior normal modes couple almost instantly at $t=0$ (green curve).
• Figure 5: Square-root infidelity between the state $\left|\Psi\left(T\right)\right\rangle ^{\prime}$ obtained for $\hat{H}_{+ \rm min}\left(t\right)$ eq. (\ref{['eq:H_min']}), and the state $\left|\Psi\left(T\right)\right\rangle$ containing all modes $\hat{\kappa}_{k}^{\rm in}$. The nonstationary Schrodinger equation was solved numerically in a Fock space which was truncated at a maximal number of environment's quanta $n_{\rm cut}=4$ for $T=100$. The driven qubit $\hat{H}_{s}\left(t\right)=\hat{\sigma}_{+}\hat{\sigma}_{-}+\hat{\sigma}_{x}0.1\cos t$ was taken as an open system, $g=0.1$, $\hat{V}_{s}=\hat{\sigma}_{-}$, $\varepsilon_{j}\equiv1$, $h_{j}\equiv0.05$. The semiinfinite chain eq. (\ref{['eq:semiinf_chain']}) was truncated at $30$ sites.