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Fractional time approach to a generalized quantum light-matter system

This paper investigates a generalized quantum light-matter system with time-dependent Jaynes-Cummings interactions using a fractional time approach, revealing how distinct fractional orders and various coupling modulations uniquely influence population inversion, entanglement, and the preservation or suppression of non-periodic dynamics.

Original authors: Enrique C. Gabrick, Thiago T. Tsutsui, Danilo Cius, Ervin K. Lenzi, Antonio S. M. de Castro, Fabiano M. Andrade

Published 2026-04-10

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CC BY 4.0

Original authors: Enrique C. Gabrick, Thiago T. Tsutsui, Danilo Cius, Ervin K. Lenzi, Antonio S. M. de Castro, Fabiano M. Andrade

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a dance between two partners: an atom (the dancer) and a light beam (the music). In the world of quantum physics, this dance is called the Jaynes-Cummings model. Usually, they dance in a perfect, predictable rhythm, swapping energy back and forth like a perfectly timed tango. If the atom gets excited, it gives energy to the light; if the light gets strong, it excites the atom. This is a "Markovian" dance, meaning the partners only care about what is happening right now.

This paper explores what happens if we change the rules of the dance to include memory.

The Big Idea: The "Memory" Dance

The authors ask: What if the atom remembers not just the current beat, but also the beats from the past?

In physics, this is called Non-Markovian dynamics. To study this, they use a mathematical tool called Fractional Calculus. Think of standard calculus as counting steps in whole numbers (1, 2, 3). Fractional calculus allows for "half-steps" or "quarter-steps." It's like saying the dancer doesn't just move from one spot to the next instantly, but glides through a blur of past positions.

They tested two different ways to write the "memory rules" (mathematically speaking, two ways to handle the imaginary number $i$ in their equations):

The "Wick Rotation" Style ( $\beta = \alpha$ ): This creates a dance where the partners keep spinning, but they get a little tired and wobbly over time. The rhythm slows down, and the energy fades away slowly, but they never stop dancing completely.
The "Dissipative" Style ( $\beta = 1$ ): This is like the dancer getting stuck in mud. The energy drains away quickly, and the dance stops abruptly, settling into a calm, quiet state.

The Four Different "Songs" (Couplings)

The researchers didn't just watch the dance; they changed the music (the coupling strength) to see how the memory affected different scenarios. They tried four types of songs:

Constant Beat (The Metronome): The music stays the same volume.
- Result: With memory, the dance becomes a bit chaotic. The partners don't swap energy perfectly anymore; they get "stuck" in a state where they are half-excited, half-calm, and the rhythm fades out.
Linear Ramp (The Volume Knob): The music starts quiet and gets louder steadily.
- Result: In a normal world, the dance would speed up as the music gets louder. With memory, the dance starts slow, then accelerates, but the "memory" makes the partners lag behind the beat, creating a delayed, stretched-out reaction.
Exponential Surge (The Explosion): The music gets louder very, very fast.
- Result: The dance becomes frantic. The partners swap energy incredibly quickly at first, but the memory effect causes the energy to drain away even faster than usual, leaving them exhausted sooner.
Sinusoidal Wave (The Rollercoaster): The music goes up and down in a wave (like an atom flying through a standing wave of light).
- Result: This is the most surprising finding. In a normal world, this dance is perfectly periodic (it repeats the same pattern forever). But with the "memory" rules, the dance stops repeating. It becomes a unique, non-repeating pattern.
- The Twist: The authors found that by adjusting the "fractional order" (how much memory the system has), they could actually control this chaos. If they lowered the memory parameter enough, they could turn a wild, unpredictable dance into a smoother, more regular one. It's like a DJ using a memory effect to stabilize a broken record.

Why Does This Matter?

You might wonder, "Why do we care about a math-heavy dance?"

Quantum Computers: These "dancers" are the basis of quantum computers. If we can control how they remember and interact, we can build better, more stable quantum machines.
Experimental Verification: The paper suggests that scientists can actually test this in real labs (using things like superconducting circuits or trapped ions). By watching how the "dance" fades or changes rhythm, they can prove that "fractional time" (memory) is a real physical phenomenon, not just a math trick.
New Control Mechanisms: The biggest takeaway is that memory can be a tool. Instead of just being a nuisance that causes errors, the "fractional order" can be used as a knob to tune the system, turning chaotic, non-repeating behavior into something stable and useful.

The Bottom Line

This paper is about teaching a quantum system to remember its past. By doing so, the authors discovered that the system's dance changes from a perfect, repeating loop into something more complex and interesting. They found that by tweaking the "memory settings," they could control whether the system stays chaotic or settles down, offering a new way to manipulate light and matter for future technologies.

1. Problem Statement

The paper addresses the dynamics of a two-level atom interacting with a quantized electromagnetic field (the Jaynes–Cummings or JC model) within the framework of fractional quantum mechanics. Specifically, the authors investigate the Time-Dependent Jaynes–Cummings (TDJC) model where the light-matter coupling strength, $\lambda(t)$ , varies with time.

The core problem involves understanding how non-Markovian memory effects, introduced via the Time Fractional Schrödinger Equation (TFSE), alter the coherent dynamics of this system. Standard quantum mechanics assumes unitary evolution and Markovian processes (no memory). The authors seek to determine how fractional time derivatives (order $\alpha < 1$ ) and different formulations of the imaginary unit in the TFSE affect:

Atomic population inversion ( $W$ ).
Atom-field entanglement (measured by von Neumann entropy, $S_A$ ).
The conservation of probability.

2. Methodology

Theoretical Framework

The authors employ the Caputo fractional derivative to formulate the TFSE. They explore two distinct formulations of the equation, unified as:
$i^\beta \frac{\partial^\alpha}{\partial t^\alpha} |\Psi_{\alpha,\beta}(t)\rangle = \hat{V}(t) |\Psi_{\alpha,\beta}(t)\rangle$
Where:

$\alpha \in (0, 1]$ is the fractional order of the time derivative, representing the degree of memory.
$\beta$ $β$ determines the power of the imaginary unit $i$ $i$ .
- Case 1 ( $\beta = \alpha$ ): Corresponds to a Wick rotation, leading to oscillatory dynamics.
- Case 2 ( $\beta = 1$ ): Retains the standard imaginary unit, associated with dissipative dynamics.
$\hat{V}(t)$ is the effective time-dependent Hamiltonian including detuning $\Delta$ .

Coupling Modulations

The study analyzes four distinct time-dependent coupling functions $\lambda(t)$ :

Constant: $\lambda(t) = \lambda_0$ .
Linear: $\lambda(t) = \lambda_0 (\zeta t)$ .
Exponential: $\lambda(t) = \lambda_0 e^{\zeta t}$ .
Sinusoidal: $\lambda(t) = \lambda_0 \sin(p\zeta t)$ (modeling an atom traversing a standing-wave cavity).

Numerical and Analytical Approach

Analytical Solutions: Derived for the constant coupling case using the Mittag-Leffler function ( $E_\alpha$ ), which generalizes the exponential function.
Numerical Integration: For time-dependent couplings, the authors use the Adams-Bashforth-Moulton method to solve the coupled fractional differential equations.
Normalization: Since fractional evolution is non-unitary (probability is not conserved), the authors normalize the density matrix and observables ( $W$ and $S_A$ ) by the total probability $P_{\alpha,\beta}(t) = \langle \Psi | \Psi \rangle$ to ensure a valid statistical interpretation.

3. Key Contributions

Generalized TFSE Formulation: The paper systematically compares the two established TFSE formulations ( $\beta=\alpha$ vs. $\beta=1$ ) within the context of a time-dependent JC model, a combination not previously explored in this depth.
Control of Non-Periodicity: A novel finding is that the fractional order $\alpha$ acts as a control mechanism for non-periodic dynamics. Specifically, in the sinusoidal coupling regime, reducing $\alpha$ can stabilize irregular, chaotic-like motion into smoother, more regular dynamics.
Comprehensive Coupling Analysis: The work extends beyond constant coupling to include linear, exponential, and sinusoidal modulations, providing a comparative analysis of how different physical scenarios (e.g., atomic motion vs. transient effects) interact with fractional memory.
Normalization Protocol: The authors rigorously apply a normalization procedure to observables, allowing for the extraction of physical insights (entanglement and inversion) despite the inherent non-conservation of probability in fractional time evolution.

4. Key Results

A. Probability and Normalization

Non-Conservation: For $\alpha < 1$ $α < 1$ , the total probability $P_{\alpha,\beta}(t)$ $P_{α, β} (t)$ is not conserved.
- In the $\beta = \alpha$ case, $P(t)$ oscillates around a fixed value with decaying amplitude.
- In the $\beta = 1$ case, $P(t)$ exhibits a rapid initial decay followed by a slow, non-monotonic evolution, indicative of information loss to the environment.
Detuning Influence: The detuning parameter $\Delta$ significantly affects the normalization parameter, particularly at weak coupling strengths.

B. Population Inversion ( $W'$ ) and Rabi Oscillations (RO)

$\beta = \alpha$ (Oscillatory): The system exhibits damped Rabi oscillations that persist around a stationary value. The fractional order $\alpha$ modifies the period and amplitude of these oscillations.
$\beta = 1$ (Dissipative): Oscillations are suppressed, and the population inversion decays to an asymptotic value.
- Constant Coupling: Oscillations cease, and the system settles.
- Linear/Exponential Coupling: The acceleration of RO (due to increasing $\lambda(t)$ ) is observed, but the fractional memory causes faster decay to asymptotic states compared to the standard case.
- Sinusoidal Coupling: This is the most distinct result. The system exhibits non-periodic dynamics for both formulations. However, for lower $\alpha$ (e.g., 0.4), the dynamics become more regular, suggesting fractional order can suppress chaos.

C. Entanglement (Von Neumann Entropy, $S_A'$ )

Entanglement follows the behavior of the population inversion.
$\beta = \alpha$ : High entanglement is maintained as the system oscillates around a mixed state ( $W' \approx 0$ ).
$\beta = 1$ : Entanglement decays to asymptotic values.
Sinusoidal Case: The non-periodic nature of the coupling leads to irregular entanglement dynamics, which are smoothed out by reducing the fractional order $\alpha$ .

5. Significance and Implications

Physical Interpretation: The study demonstrates that fractional calculus provides a viable framework for modeling non-Markovian memory effects in quantum optics without explicitly introducing an external bath. The parameter $\alpha$ effectively tunes the strength of the memory.
Experimental Relevance: The authors propose that these effects could be tested in:
- Cavity QED: Using atoms traversing standing waves (sinusoidal coupling).
- Superconducting Circuit QED: By dynamically modulating coupling parameters.
- Trapped Ions: Using stroboscopic control to emulate fractional evolution.
Control Mechanism: The discovery that fractional order can stabilize non-periodic (potentially chaotic) dynamics offers a new theoretical tool for controlling quantum systems, potentially useful in quantum computing and information processing where stability is crucial.
Theoretical Advancement: By resolving the probability non-conservation issue through normalization, the paper bridges the gap between the mathematical formalism of fractional quantum mechanics and the physical observables required for experimental verification.

In conclusion, the paper establishes that time-fractional generalizations of the Jaynes–Cummings model introduce rich, memory-dependent dynamics that can qualitatively alter entanglement and population inversion, with the fractional order serving as a tunable parameter to control the transition between oscillatory, dissipative, and non-periodic regimes.