An analytical solution of a quantum system with non-Markovian behavior: The Bixon-Jortner system in time domain

Technical Summary: An Analytical Solution of the Bixon-Jortner System in the Time Domain

Problem Statement
The paper addresses the challenge of obtaining an exact, analytical description of non-Markovian dynamics in quantum systems. While non-Markovian behavior is typically studied in bipartite systems (system plus environment) where tracing out the environment leads to memory effects, standard approaches often rely on approximations like the Lindblad master equation (Markovian limit) or perturbative corrections (e.g., Nakajima-Zwanzig formalism). The authors focus on the Bixon-Jortner (BJ) model, a system consisting of a single discrete state $|\phi\rangle$ coupled to an infinite "ladder" of states $|n\rangle$. Unlike standard system-environment models, the BJ Hilbert space is a direct sum rather than a tensor product. Although the BJ model is known to exhibit exponential decay (Wigner-Weisskopf limit) and subsequent decay-revival dynamics, previous analytical treatments relied on Laplace transforms that obscured the physical origin of the "kicks" (non-smooth transitions) observed at integer time intervals. The goal is to derive the exact time-domain dynamics and explicitly demonstrate the non-Markovian nature of the system without resorting to approximations.

Methodology
The authors attack the time evolution problem directly in the time domain through the following steps:

1. Formulation: Starting from the time-dependent Schrödinger equation in the interaction picture, they derive an integro-differential equation for the coefficient $b_\phi(t)$ of the discrete state. This equation eliminates the ladder state coefficients in favor of an integral over the past history of $b_\phi$.
2. Transformation to Delay Differential Equation (DDE): By specializing to the BJ model parameters (uniform ladder spacing $\Delta$ and $E_\phi=0$) and applying the Poisson summation formula, the integro-differential equation is transformed into a delay differential equation. In this form, the time derivative of $b_\phi$ at time $\tau$ depends on the values of $b_\phi$ at previous times $\tau - m$ (where $m$ are integers), making the non-Markovian memory effects explicit.
3. Analytical Solution via Ansatz: To solve the resulting DDE, the authors employ an additive decomposition ansatz. They define a transformed function $f(\tau) = b_\phi(\tau)e^{\beta\tau}$ and assume the solution can be written as a sum of additive pieces $f_n(\tau-n)$ that "kick in" at each integer time $\tau = n$.
4. Recurrence and Generating Functions: By analyzing the equation within specific time intervals $(k, k+1)$, they derive a recurrence relation for the additive components. This relation is solved using a generating function approach, which links the solution to Laguerre polynomials $L_n(x)$.

Key Results

• Exact Non-Markovian Equation: The paper derives an exact delay differential equation (Eq. 19) for the discrete state amplitude. The equation explicitly shows that the system's evolution is governed by a Markovian exponential decay term plus a sum of delayed terms. These delayed terms introduce "kicks" at every integer time unit $\tau = n$, causing the dynamics to deviate from simple exponential decay and leading to partial revivals.
• Closed-Form Solution: The authors obtain an exact analytical solution for the probability amplitude $b_\phi(\tau)$ (Eq. 56). The solution is expressed as a sum involving Laguerre polynomials:
  $$b_\phi(\tau) = e^{-\beta\tau} \left( 1 + \sum_{n=1}^{\infty} H(\tau-n) e^{\beta n} [L_n(2\beta(\tau-n)) - L_{n-1}(2\beta(\tau-n))] \right)$$
  where $H$ is the Heaviside step function and $\beta$ is a dimensionless coupling parameter.
• Decay and Revival Dynamics: The solution confirms that for $0 \le \tau < 1$, the system undergoes pure exponential decay with a rate $\Gamma = 2\pi v^2 / \Delta$ (consistent with the Fermi golden rule). At $\tau = 1, 2, 3, \dots$, the delayed terms activate, modifying the decay rate and causing the probability to exhibit non-smooth transitions and partial revivals, as visualized in the provided plots for various $\beta$ values.

Significance and Claims
The paper claims that its primary contribution is providing an explicit, accessible, and exact case study of non-Markovianity in a quantum system.

• Transparency: Unlike previous Laplace transform methods which yield analytical formulas but obscure the physical mechanism of the "kicks," the time-domain delay differential equation approach makes the non-Markovian nature of the system completely transparent. The memory effects are directly visible as dependencies on past values at integer intervals.
• Accessibility: The solution techniques (integro-differential to DDE transformation, simple ansatz, and generating functions) are presented as intuitive and accessible, offering a clear pedagogical example of how non-Markovian dynamics arise and can be solved analytically.
• Model Utility: The authors position the BJ system as a valuable "pristine" model for understanding decay and revival phenomena, stripping away the mathematical complications of more complex light-matter interactions while retaining the essential physics.

The paper concludes modestly, suggesting that while the exact solution is derived for this specific direct-sum system, the approach could potentially be generalized to other systems, such as ladders with non-constant coupling or traditional tensor-product bipartite systems, to compare with standard master equation formulations.