Generalised fractional Rabi problem

This paper investigates a generalized fractional Rabi model using Caputo derivatives to demonstrate how fractional temporal nonlocality induces controllable damping and dephasing in two-level quantum systems, offering new experimental signatures and pathways for exploring memory effects in materials like graphene and topological chains.

The Gist

Imagine you are watching a spinning top. In the world of standard physics, if you give it a gentle, rhythmic push (like a periodic driving field), it spins in a perfect, predictable rhythm forever. It's like a dancer who never gets tired, moving in perfect sync with the music. This is the classic "Rabi problem," a fundamental way physicists understand how tiny quantum particles, like electrons or atoms, behave when nudged by energy.

But what if the universe had a "memory"? What if the top didn't just react to the push you are giving it right now, but also remembered every push it received in the past? And what if that memory made it feel a little "sticky" or sluggish, causing it to slow down or wobble differently than expected?

This paper explores exactly that scenario. The authors ask: What happens to a quantum system if we replace the standard rules of motion with "fractional" rules?

The "Fractional" Twist: A Sticky Memory

In standard physics, time flows smoothly, and a system's future depends only on its present state. In this paper, the authors use a mathematical tool called fractional calculus. Think of this as giving the system a "sticky memory."

Instead of moving like a fresh, clean dancer, the quantum particle moves like a dancer in a room filled with thick honey. Every time it tries to spin, it drags the past with it. This "honey" is the memory effect. The authors found that even without any external music (driving field), just having this sticky memory changes how the particle spins. It doesn't just spin; it slowly loses energy and dampens, a behavior that wouldn't happen in a normal, non-sticky world.

The Experiment: The Two-Level System

To test this, the authors looked at a "two-level system." Imagine a light switch that can be either ON or OFF, or a coin that is either Heads or Tails. In quantum mechanics, this particle can be in a mix of both states at once.

1. The Static Case (No Music): When they just let the particle sit with its "sticky memory" (no external pushing), they found the particle's spin didn't just stay still or oscillate perfectly. It showed a unique kind of damping. The "memory" of its past positions caused it to lose its rhythm over time, creating a pattern that looked like a fading echo rather than a steady beat.

2. The Driven Case (With Music): Then, they started pushing the particle rhythmically (like a periodic driving field). In a normal world, the particle would lock into a perfect dance with the push. In this "fractional" world, a tug-of-war began:

   • The push tried to inject energy and keep the dance going.
   • The memory (honey) tried to drag it back and dampen the motion.

The result was a complex, rich dance. The particle didn't just follow the music; it showed a mix of rhythmic steps and fading echoes. The authors discovered that by changing the "stickiness" of the memory (a number called $\alpha$), they could control how much the particle slowed down or how quickly it lost its rhythm.

How They Measured It: The "Echo" and The "Snapshot"

How do you see this invisible sticky memory? The authors used two clever tools:

• The Autocorrelation Function (The "Snapshot"): This measures how much the particle looks like its original self after a while. In a normal world, it would look exactly the same at specific times (like a perfect loop). In this fractional world, the "snapshots" started to blur. The particle would return to its starting shape, but less perfectly each time, like a photo that gets slightly fuzzier with every replay.
• The Fidelity or Loschmidt Echo (The "Rewind"): Imagine playing a movie forward, then hitting "rewind" to see if the particle goes back to exactly where it started. In a normal world, it would return perfectly. In this sticky world, the "rewind" wasn't perfect. The memory of the past pushes made it hard for the particle to retrace its steps exactly.

The Big Picture

The paper concludes that this "fractional" behavior creates a unique signature. If you were to observe a quantum system that behaves like this, you wouldn't see the perfect, endless oscillations of standard physics. Instead, you would see controllable damping—a slowing down and a loss of rhythm that is directly linked to how much "memory" the system has.

The authors suggest that these specific patterns (the way the "echo" fades or the "snapshots" blur) could be the key to spotting this strange, memory-filled physics in real experiments. They mention that this could help us understand complex materials like graphene or special topological chains (materials with unique electrical properties), where these "sticky" memory effects might be hiding in plain sight, waiting to be discovered.

In short: The paper shows that if you give a quantum particle a memory, it stops dancing perfectly and starts moving like it's wading through water, creating a new kind of rhythm that we can now predict and potentially measure.

Technical Summary

Technical Summary: Generalised Fractional Rabi Problem

Problem Statement
This work addresses the physical consequences of fractional-order quantum evolution in two-level systems (TLS), specifically extending the paradigmatic Rabi problem to a fractional regime. While standard quantum dynamics are governed by the first-order time-dependent Schrödinger equation, recent interest has focused on fractional derivatives to model non-Markovian, memory-dependent phenomena in open quantum systems and optics. The authors investigate the "fractional Rabi problem," where the standard time derivative is replaced by the Caputo fractional derivative. A primary theoretical challenge identified is that fractional dynamics may be non-unitary, complicating the probabilistic interpretation of the wavefunction and rendering standard perturbative approaches (relying on the Leibniz rule and interaction pictures) inapplicable. Furthermore, the static Hamiltonian term in fractional systems induces non-trivial dynamics even without external driving, a feature not captured by naive perturbation expansions that neglect the static component's influence on the memory kernel.

Methodology
The authors formulate the fractional time-dependent Schrödinger equation (FTSE) using the Caputo derivative:
$$i\hbar^\alpha D_t^\alpha \Psi(t) = H(t)\Psi(t)$$
where $0 < \alpha < 1$.

To solve this, the paper develops a fractional Green's function expansion rather than relying on standard Dyson series or interaction pictures, which fail due to the lack of the Leibniz property in fractional calculus. The methodology proceeds as follows:

1. Static Solution: The authors first solve the eigenvalue problem for a static Hamiltonian $H_s$, yielding solutions expressed in terms of the one-parameter Mittag-Leffler function, $E_\alpha(z)$.
2. Green's Function Formulation: The total Hamiltonian is split into a static part ($H_s$) and a time-dependent perturbation ($V(t)$). A fractional Green's function $G_\alpha(t, t')$ is defined as the solution to $(i\hbar^\alpha D_t^\alpha - H_s)G_\alpha(t, t') = \delta(t-t')$.
3. Iterative Expansion: The evolved state is expressed as a self-consistent integral equation involving the static solution and the Green's function convolved with the perturbation. This leads to a perturbative series where the leading-order correction depends explicitly on the Green's function of the static system and the perturbation potential.
4. Application to Rabi Model: The formalism is applied to a generalized Rabi Hamiltonian $H_{R,\alpha}(t) = \Delta_\alpha \sigma_z + \xi_\alpha (\sigma_x \cos \Omega t + \sigma_y \sin \Omega t)$. The authors derive explicit expressions for the time evolution of spin polarization, the autocorrelation function, and the fidelity (Loschmidt echo).

Key Contributions

• Fractional Green's Function Expansion: The paper introduces a specific expansion scheme for the fractional time-dependent Schrödinger equation that isolates the static Hamiltonian's influence within the Green's function. This overcomes the limitations of standard perturbation theory in fractional calculus where the static term cannot be simply factored out via a phase transformation.
• Analytical Expressions for TLS Dynamics: The authors derive explicit iterative expressions for the evolved state of a two-level system under fractional evolution, providing leading-order formulas for spin polarization components ($\langle \sigma_x \rangle, \langle \sigma_y \rangle, \langle \sigma_z \rangle$), autocorrelation, and fidelity.
• Characterization of Memory Effects: The work quantifies how the fractional order parameter $\alpha$ governs the transition between coherent oscillations and damped, memory-induced dynamics.

Results

• Static Dynamics: Even in the absence of external driving, the static Hamiltonian term induces non-trivial spin dynamics. The spin polarization components exhibit damping features directly linked to the fractional temporal nonlocality. For $\alpha < 1$, the system shows slower relaxation and enhanced memory effects compared to the standard unitary case ($\alpha=1$). The contours of spin polarization in the $x-y$ plane transition from closed (periodic) to open (decaying) as $\alpha$ decreases, reflecting the power-law asymptotic regime of the Mittag-Leffler function.
• Driven Dynamics: When a periodic driving field is introduced, a competition arises between energy injection and memory effects.
  • Spin Polarization: The driving field converts the open contours of the static fractional case into closed contours, indicating a stabilization of the dynamics. The minima and maxima of the spin components show an alternating pattern.
  • Fidelity (Loschmidt Echo): The fidelity exhibits transient memory effects for intermediate $\alpha$ values (e.g., 0.6, 0.8) at early times. However, at longer times, the memory effects fade, and the curves synchronize, suggesting the driving protocol stabilizes the dynamics. The leading-order approximation breaks down for $\alpha=1$, as expected, but also shows deviations for small interaction strengths in the fractional regime.
  • Autocorrelation Function: This quantity displays behavior complementary to the fidelity. While fidelity stabilizes, the autocorrelation function shows that lower fractional orders (stronger memory) dominate over light-matter coupling, leading to rapid decay of partial quantum revivals. As $\alpha$ approaches 1, the fading of memory effects accelerates.
• Distinctive Signatures: The paper identifies that the fractional regime introduces controllable damping and dephasing governed by $\alpha$, distinct from the fixed-frequency Rabi oscillations of the standard model.

Significance and Claims
The authors claim that their findings provide a framework to probe fractional quantum dynamics experimentally. The distinctive signatures—specifically the complementary time evolution of the fidelity (Loschmidt echo) and the autocorrelation function—offer potential routes for experimental observation. The work suggests that these memory-induced dynamical phenomena could be relevant in other systems effectively described by a two-level approximation, such as graphene-like materials and topological SSH chains, where non-integer order evolution might reveal novel topological or relaxation effects. The paper positions the fractional Rabi problem as a rich playground for exploring the interplay between memory, non-locality, and quantum coherence, while acknowledging theoretical challenges regarding non-unitarity and the formulation of appropriate fractional perturbation theories.