Mean-Correlation Reconstruction of Forward Dissipation and Non-Markovian Reverse Flow in a Trajectory-Resolved Thermo-Field Two-Spin Model

Abstract

We develop a trajectory-resolved extension of the thermo-field entanglement description for a minimal dissipative two-spin system. For the isolated exchange-coupled model, the intrinsic thermo-field entanglement coefficient is $b_0(t)=\frac{1}{4}{\sin }^2(\omega t)$. We promote the corresponding open-system coefficient to a stochastic trajectory observable, $b_{qe}^{(\xi )}(t)=b_0(t)X_t$, where the binary variable $X_t$ specifies whether the trajectory occupies the one-excitation entangling sector or the decayed sector. For a bidirectional time-local jump process $1\stackrel{a(t)}{\to }0$ and $0\stackrel{b(t)}{\to }1$, we derive an exact two-time connected correlation function, \[ C_{qe}(t,s)=b_0(t)b_0(s)S(s)[1-S(s)] \exp\!\left[-\int_s^t(a(u)+b(u))\,du\right], \qquad t\ge s, \] where $S(t)=⟨X_t⟩$. The one-time mean determines the net probability current, $\dot{S}=J_R-J_F$, whereas the normalized two-time correlation determines the rate sum, $a+b$. Combining the two quantities yields an exact reconstruction of the forward and reverse currents, \[ J_F=S(1-S)q-S\dot S,\qquad J_R=S(1-S)q+(1-S)\dot S, \] with $q=-{\partial }_t\ln [C_{qe}(t,s)/b_0(t)]$. This leads to a three-current decomposition of the mean thermo-field entanglement dynamics into coherent generation, dissipative loss, and memory-induced return. For Markovian amplitude damping the reconstruction gives $J_R=0$, while in a pure non-Markovian revival interval it gives $J_F=0$ and $J_R=\dot{S}>0$. The result shows that the mean alone measures only net backflow, whereas mean plus two-time fluctuations resolves hidden bidirectional traffic between system and environment.

13 pages, 4 figures