Generalised fractional Rabi problem

TL;DR

This work investigates fractional-order quantum dynamics in a two-level system using Caputo time derivatives to model memory effects. It develops a Green's-function–based fractional Dyson perturbation framework, yielding an effective evolution operator $U_\alpha(t)$ and a leading-order state $|\Psi_\alpha(t)\rangle$. Applied to a generalized Rabi problem, the static term induces damping through memory captured by Mittag-Leffler functions, while periodic driving creates a competition between energy injection and memory, producing rich dynamics in spin polarization, fidelity, and autocorrelation. The results provide analytic observables and resonant insights, with potential experimental tests and relevance for graphene-like materials and SSH chains where non-integer temporal evolution could reveal novel relaxation phenomena.

Abstract

Fractional quantum dynamics provides a natural framework to capture nonlocal temporal behavior and memory effects in quantum systems. In this work, we analyze the physical consequences of fractional-order quantum evolution using a Green's function formulation based on the Caputo fractional derivative. Explicit iterative expressions for the evolved state are derived and applied to an extended two-level Rabi model, a paradigmatic setting for coherent quantum control. We find that even in the absence of external driving, the static Hamiltonian term induces non-trivial spin dynamics with damping features directly linked to the fractional temporal nonlocality. When a periodically varying driving field is introduced, the competition between energy injection and memory effects gives rise to a richer dynamical behavior, manifest in the evolution of spin polarization, autocorrelation function, and fidelity. Unlike the standard Rabi oscillations characterized by a fixed frequency, the fractional regime introduces controllable damping and dephasing governed by the degree of fractionality. These distinctive signatures could be observable through the Loschmidt echo and autocorrelation function, and would offer potential routes to probe fractional quantum dynamics experimentally. Our findings open pathways toward exploring memory-induced dynamical phenomena in other systems effectively described by a two-level approximation, such as graphene-like materials and topological SSH chains, where non-integer order evolution may reveal novel topological or relaxation effects.

Figures (4)

• Figure 1: Contour plots for the time evolution of spin polarization components $\langle \sigma_x(t) \rangle$, and $\langle \sigma_y(t) \rangle$, under a static fractional Hamiltonian. As $\alpha$ decreases from $1$ to lower values, the system exhibits slower relaxation and enhanced memory effects, with separated open and closed contours. These results illustrate how fractional dynamics alter the coherence and population evolution in comparison to the standard ($\alpha = 1$) unitary case.
• Figure 2: Contour plots of the time evolution for the spin polarization components $\langle \sigma_x(t) \rangle$ and $\langle \sigma_y(t) \rangle$ for a two-level system governed by a fractional time evolution with a Mittag-Leffler memory kernel. The fractional order is $\alpha \in [0.2, 1.0]$. The system is subjected to a harmonic external perturbation with frequency $\Omega = \Delta/\hbar$, and the coupling strength is fixed at $\lambda = 0.1$.
• Figure 3: Time evolution of fidelity as given in equation (68), for $\Omega=\Delta/\hbar$ and $\lambda=0.1$. For the initially well separated evolutions generated by $\alpha=0.6$ and $0.8$, the memory effects decay at longer times which is a desirable property of non Markovian systems. As expected, the approximate leading order solution breaks down for $\alpha= 1$.
• Figure 4: Time evolution of the autocorrelation, for different values of the fractional parameter $\alpha$, for $\lambda=0.1$ and $\Omega=\Delta/\hbar$. Partial quantum revivals persists up to $\alpha=0.8$ Interestingly, the closer $\alpha$ is to one, the fading of memory effects is faster. For $\alpha=0.6$ it is seen that quantum revivals are rapidly washed out by the stronger memory effects.