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

The Rhythm of Quantum Revivals

Quantum systems interacting with their environments rarely behave in a simple, predictable way. Instead of losing information to the surroundings in a smooth, one-way street, they often exhibit "memory" effects. In these cases, information leaks out and then flows back in. This causes the system to spontaneously recover its lost state. Researchers have sought ways to mathematically model these complex movements. They focus on systems that transition from steady decay to sudden, rhythmic "kicks" or revivals.

The Geometry of Decay and Revival

At its core, this research addresses the problem of non-Markovianity. In physics, a Markovian process is one where the future state depends only on the present. It is "memoryless," like a coin flip where the previous toss does not influence the next. Most simple models of quantum decay assume this memoryless behavior. This leads to a smooth, exponential fade-out of a state's probability.

However, real quantum systems are often non-Markovian. This happens when the environment—the collection of other particles or fields surrounding a system—retains a record of the interaction. The Bixon-Jortner (BJ) model is a specialized mathematical sandbox designed to study this. It consists of a single discrete state (a specific energy level) coupled to an infinite "ladder" of other states. Rather than behaving like a simple sink that swallows energy, the ladder acts like a reservoir. It can reflect information back to the discrete state. This creates a tug-of-war between irreversible decay and periodic revivals.

The Ladder and the Memory Effect

To understand the math, one must first grasp the architecture of the Bixon-Jortner system. As illustrated in, the system is built from a single discrete state $|\phi\rangle$ and an infinite ladder of states $|n\rangle$. The unperturbed energies of the ladder states are equally spaced by an amount $\Delta$. This creates a regular, repeating structure. The perturbation $V$ provides a constant coupling $v$ between the discrete state and every rung of the ladder.

When a researcher attempts to track the probability of the system remaining in the discrete state, they encounter a difficult integro-differential equation. This is an equation where the rate of change at the current moment is tied to an integral of all previous moments. This is the mathematical signature of memory. The system’s current behavior is weighted by its entire history.

Standard approximations usually fall into two camps. The first is the Markovian approximation. This assumes the environment responds instantly. This results in the Fermi golden rule, which calculates a simple, smooth exponential decay rate. The second is a perturbative approach that only looks at the very beginning of the decay. Neither captures the "kicks" seen in reality. As shown in, the actual dynamics of the BJ model involve an initial exponential decay.

This is followed by non-smooth, sudden increases in probability at integer time intervals. These are the revivals. They appear almost "out of the blue" in traditional mathematical treatments.

From History to Delays

The authors, Osman Cevheroğlu and Arkaş Özakın, propose a shift in perspective to make these kicks transparent. Instead of struggling with an integral that sums up the entire past, they use the Poisson summation formula to transform the integro-differential equation. They turn it into a delay differential equation.

A delay differential equation is a specific class of equation where the derivative (the rate of change) at time $\tau$ depends on the value of the function at a specific previous time. For example, it might depend on $\tau - 1$ or $\tau - 2$. This is a profound conceptual simplification. In this new framework, the non-Markovianity is not hidden in a continuous smear of historical data. It is manifested as discrete, periodic updates. Every time the dimensionless time $\tau$ hits an integer, the equation receives a "kick" from a previous version of itself.

To solve this, the authors employ an intuitive ansatz (a mathematical starting guess). They treat the solution as a sum of additive pieces. Each piece "kicks in" at successive integer intervals. Through a sophisticated use of generating functions—tools used to encode an entire sequence of numbers into a single function—they relate the system's behavior to Laguerre polynomials. These are a special class of orthogonal polynomials used in quantum mechanics.

The resulting analytical solution provides a complete description of the state's evolution. As shown in, the researchers demonstrate how the system moves from near-exponential decay to highly structured, oscillatory behavior.

They achieve this by varying the parameter $\beta$ (a dimensionless constant involving the coupling strength and ladder spacing).

Decoding the Quantum Pulse

This work changes how we visualize the "memory" of a quantum system. Instead of viewing non-Markovianity as a complex, continuous convolution of past events, it can be viewed as a series of discrete, timed echoes. The ability to write down an exact analytical solution using Laguerre polynomials is significant. Physicists no longer have to rely solely on heavy numerical simulations to predict when these revivals will occur.

Specifically, the paper shows that the "kicks" in the probability density are not mere mathematical artifacts. They are the direct consequence of the delay terms in the differential equation. This provides a clear bridge between the abstract concept of a "non-Markovian environment" and the tangible, periodic pulses observed in the system's dynamics.

Limits of the Model

While the solution is exact for the Bixon-Jortner model, its applicability is bounded by specific constraints. The researchers note that the system is a direct sum of two parts. It is not a standard bipartite system (where the total space is a tensor product of the system and environment). Furthermore, the model relies on several idealizations. The coupling $v$ between the state and the ladder is assumed to be constant. Additionally, the energy spacing $\Delta$ of the ladder rungs is assumed to be perfectly uniform.

Future research may seek to extend this "delay equation" approach to more realistic scenarios. These could include ladders with non-constant coupling or true bipartite systems. Such an extension would test if the clarity of the Bixon-Jortner model holds when the environmental structure becomes more chaotic.

Figures from the paper