Exact treatment of the memory kernel under time-dependent system-environment coupling via a train of delta distributions

TL;DR

This work tackles the challenge of nonstationary memory kernels in open quantum systems by introducing a nonperturbative train-of-Dirac-delta switching framework that renders the integro-differential dynamics solvable in closed form. The method discretizes the time-dependent coupling into a finite sequence of delta interactions, yielding a compact solution that separates the free evolution, noise impulses, and memory-propagation contributions, and it converges to the standard continuum result as the number of deltas grows. Applied to the damped Jaynes–Cummings model and the Caldeira–Leggett model, the approach not only reproduces known exact solutions in the continuum limit but also provides a clear, diagrammatic view of how memory effects propagate through the environment. The framework enables visualization of Markovian versus non-Markovian memory via simple memory-arc diagrams and is extendable to a range of environments through the discrete-time spectral-density formalism, with potential relevance to finite-time quantum thermodynamics and relativistic detector-field setups.

Abstract

Memory effects in a quantum system coupled to an environment are one of the central features in the theory of open quantum systems. The dynamics of such quantum systems are typically governed by an equation of motion with a time-convolution integral of the memory kernel. However, solving such integro-differential equations is challenging, especially when the memory kernel is nonstationary (not time-translation invariant). In this paper, we analytically and nonperturbatively solve such integro-differential equations with a nonstationary memory kernel by employing a train of Dirac-delta switchings. We then apply this method to the damped Jaynes-Cummings model and the damped harmonic oscillator model to demonstrate that (i) our solution asymptotes to the well-known exact solution in the continuum limit, and that (ii) our method also enables us to visualize the memory effect in the environment.

Figures (5)

• Figure 1: The basic diagrams that constitute \ref{['eq:Gft']}. Each $t_i$, $i\in \{ 1,2, \ldots, N \}$ represents the time at which the instantaneous interaction occurs. Here, we use a simplified notation $\Sigma_{ki}\equiv \Sigma(t_k, t_i)$.
• Figure 2: The diagrams illustrating $\mathcal{O}(t)$ in \ref{['eq:Gft']} for (a) $N=2$, (b) $N=3$, and (c) $N=4$. An arc connecting $t_i$ and $t_j$ ($j>i$) corresponds to $\Sigma(t_j, t_i) \equiv \Sigma_{ji}$.
• Figure 3: Comparison between the solutions to the integro-differential equation \ref{['eq:Schrodinger eq JC model']} with a constant switching (dashed line) and our train of delta switchings (solid curve). The horizontal axis is taken to be the number of delta switchings $N$. Here, we chose $\kappa/\Lambda=0.1$. The function \ref{['eq:JC pure delta sol']} at fixed time $\Lambda T=1$ asymptotes to \ref{['eq:JC exact sol']} at $\Lambda t=1$ as the number of delta switchings $N$ increases.
• Figure 4: A measure of non-Markovianty of quantum processes. Non-Markovian behavior arises during the time intervals with $\gamma_{k}/\Lambda<0$, while the dynamics is Markovian otherwise. Here, we chose $\kappa/\Lambda=5/2$, $N=40$ and $\Lambda T=30$. The integer $j$ denotes the extent of memory propagation $t_l \xrightarrow{\Sigma} t_{l+j}$ allowed in the dynamics. $j=1$ in (a) allows only nearest-neighbor propagation, whereas $j\geq 2$ in (b)-(d) permits propagation up to $j$ steps away. Note that the time evolution is Markovian for $t\geq t_{40}$, where $k=40$ denotes the time at which the interaction ends. Therefore, we exclude the point $k=40$ from our analysis.
• Figure 5: Comparison between the solutions to the QLE with a constant switching (dashed line) and our train of delta switchings (solid curve). The horizontal axis is taken to be the number of delta switchings $N$. Here, we choose $\kappa/\Omega=0.1, \Lambda/\Omega=2$. (a) The function $G_{f_P}$ at fixed time $T=1$ (in units of $\Omega$) asymptotes to $G(t=1)$ as the number of delta switchings $N$ increases. (b) The correlation function evaluated with the train of delta switchings also asymptotes to the result for constant switching.