The Time-Fractional Schrödinger Equation (TFSE) is an effective tool for simulating the dynamics of non-Markovian quantum systems. The Quantum Speed Limit (QSL) time characterizes the minimum time required for the evolution of a non-Markovian quantum system. In this paper, Wei's TFSE is employed to simulate the non-Markovian quantum accelerated evolution process in the Resonant Dissipative Jaynes-Cummings (RDJC) model. By solving the QSL time of a time-fractional single-qubit open system, the enhancement mechanism of the system evolution speed induced by the non-Markovian memory effects of the environment is revealed. Further studies show that the optimized acceleration of the system evolution can be achieved by jointly regulating the fractional order, coupling strength, and photon number. Comparative analyses indicate that Wei's TFSE can accurately capture the non-Markovian accelerated dynamical features of the system over the entire fractional order range, whereas Naber's TFSE is applicable only within a limited fractional order interval. In addition, the comparisons of the average simulation time for calculating the dynamical trajectory of the excited-state probability demonstrate that Wei's TFSE has a significant simulation advantage in computational efficiency. Therefore, Wei's TFSE is more accurate and efficient for simulating the accelerated dynamics of non-Markovian quantum systems.
The Time-Fractional Schrödinger Equation (TFSE) is an effective tool for simulating the dynamics of non-Markovian quantum systems. The Quantum Speed Limit (QSL) time characterizes the minimum time required for the evolution of a non-Markovian quantum system. In this paper, Wei's TFSE is employed to simulate the non-Markovian quantum accelerated evolution process in the Resonant Dissipative Jaynes-Cummings (RDJC) model. By solving the QSL time of a time-fractional single-qubit open system, the enhancement mechanism of the system evolution speed induced by the non-Markovian memory effects of the environment is revealed. Further studies show that the optimized acceleration of the system evolution can be achieved by jointly regulating the fractional order, coupling strength, and photon number. Comparative analyses indicate that Wei's TFSE can accurately capture the non-Markovian accelerated dynamical features of the system over the entire fractional order range, whereas Naber's TFSE is applicable only within a limited fractional order interval. In addition, the comparisons of the average simulation time for calculating the dynamical trajectory of the excited-state probability demonstrate that Wei's TFSE has a significant simulation advantage in computational efficiency. Therefore, Wei's TFSE is more accurate and efficient for simulating the accelerated dynamics of non-Markovian quantum systems.
Applications of Time-Fractional Schrodinger Equations (TFSEs) to quantum processes are instructive for understanding and describing the time behavior of real physical systems. By applying three popular TFSEs, namely Naber's TFSE I, Naber's TFSE II, and XGF's TFSE, to a basic open system model of a two-level system (qubit) coupled resonantly to a dissipative environment, we solve exactly for Time-Fractional Single Qubit Open Systems (TFSQOSs). However, the three TFSEs perform badly for the following reasons. On the other hand, in the respective frameworks of the three TFSEs, the total probability for obtaining the system in a single-qubit state is not equal to one with time at fractional order, implying that time-fractional quantum mechanics violates quantum mechanical probability conservation. On the other hand, the latter two TFSEs are not capable of describing the non-Markovian dynamics of the system at all fractional order, only at some fractional order. To address this, we introduce a well-performed TFSE by constructing a new analytic continuation of time combined with the conformable fractional derivative, in which for all fractional order, not only does the total probability for the system equal one at all times but also the non-Markovian features can be observed throughout the time evolution of the system. Furthermore, we study the performances of the four TFSEs applying to an open system model of two isolated qubits each locally interacting with its dissipative environment. By deriving the exact solutions for time-fractional two qubits open systems, we show that our TFSE still possesses the above two advantages compared with the other three TFSEs.
• A generalized time-dependent Jaynes–Cummings model is investigated within the framework of the time fractional Schr¨odinger equation. • Two distinct fractional formulations are compared, encompassing oscillatory and dissipative non- Markovian dynamics. • Our results shows that fractional order introduces memory effects leading to damped oscillations and asymptotic decay in population inversion and entanglement. • Time-dependent couplings (linear, exponential, and sinusoidal) strongly modulate entanglement generation and decay. • Fractional order acts as a control parameter, preserving non-periodic dynamics under sinusoidal coupling within specific regimes. We investigate the fractional time description of a generalized quantum light-matter system modeled by a time-dependent Jaynes-Cummings (JC) interaction, with different coupling types: constant, linear, exponential, and sinusoidal. Two formulations of the time fractional Schrödinger equation (TFSE) are examined, with a focus on their impact on population inversion and entanglement. Our findings highlight that the introduction of fractional order introduces memory effects, associated with damped oscillations and asymptotic decay. Furthermore, we find that the time-dependent couplings, combined with distinct fractional formulations, influence how these effects occur, ultimately resulting in high or low entanglement. A key finding of our work is that, under sinusoidal coupling, non-periodic dynamics is preserved for both formulations of the TFSE; however, within a certain range, the fractional order can act as a control mechanism for the non-periodic evolution.
In this paper, we examine the time-fractional Schrödinger equation from the perspective of non-Markovian dynamics in dissipative systems. First, we determine the range of the fractional derivative's order by examining the memory properties of the time-fractional Schrödinger equation. Next, we employ the Jaynes-Cummings model to identify the appropriate mathematical form of the imaginary unit. Finally, we use the refined equation to study quantum teleportation under amplitude damping noise. It was found that the time-fractional Schrödinger equation without fractional operations on the imaginary unit i might be more suitable for describing non-Markovian dynamics in dissipative systems. Our research may provide a new perspective on the time-fractional Schrödinger equation, contributing to a deeper understanding and further development of time-fractional quantum mechanics.
This study investigates the time-fractional Schrödinger equation (TFSE) for a free particle using the Caputo fractional derivative. The model extends the classical Schrödinger equation by incorporating memory effects through a non-integer order time derivative. By applying the Fourier transform method, an explicit solution is derived in terms of the Mittag–Leffler function, which generalizes the classical exponential evolution. The analysis reveals that the resulting dynamics are non-unitary, leading to a time-dependent total probability that deviates from the conservation property of standard quantum mechanics. Numerical illustrations demonstrate the influence of the fractional order ν on wave packet evolution and probability behavior. The results highlight the role of fractional calculus in modeling non-Markovian quantum systems and provide insight into the transition from fractional to classical dynamics as v→1. .
Quantum speed limit (QSL) time under noise has drawn considerable attention in real quantum computational processes. Though non-Markovian noise is found to be able to accelerate quantum evolution for a damped Jaynes—Cummings model, in this work we show that non-Markovianity will slow down the quantum evolution of an experimentally controllable photon system. As an application, QSL time of a photon can be controlled by regulating the relevant environment parameter properly, which nearly reaches the currently available photonic experimental technology.
A significant obstacle impeding the advancement of the time fractional Schrödinger equation lies in the challenge of determining its precise mathematical formulation. In order to address this, we undertake an exploration of the time fractional Schrödinger equation within the context of a non-Markovian environment. By leveraging a two-level atom as an illustrative case, we find that the choice to raise i to the order of the time derivative is inappropriate. In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment, the time fractional Schrödinger equation, when devoid of fractional-order operations on the imaginary unit i, emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects. Meanwhile, we also prove that it is meaningless to study the memory of time fractional Schrödinger equation with time derivative 1 < α ≤ 2. It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrödinger equation. This will be the focus of future research. Our study might provide a new perspective on the role of time fractional Schrödinger equation.