The memory fractional effects of a one-axis twisting model on the dynamics of two-qubit entanglement and non-locality are discussed. It consists of solving the time-dependent fractional Schrödinger equation by extending any integration into non-integer orders using Riemann–Liouville integration. The obtained results present the possibility of controlling the fractional order of memory, varying the parameters to significantly generate concurrency and Bell’s non-locality. Under the current investigation setup, it is noticeable that the behaviors of the proposed quantifiers are similar to each other, but with a small difference in the amplitude of non-locality with respect to entanglement. Importantly, we show that the most intriguing aspect of this paper is to detect that pair-qubit entanglement and non-locality can be preserved for an indefinite time, which still holds significance in quantum information processing.
The memory fractional effects of a one-axis twisting model on the dynamics of two-qubit entanglement and non-locality are discussed. It consists of solving the time-dependent fractional Schrödinger equation by extending any integration into non-integer orders using Riemann–Liouville integration. The obtained results present the possibility of controlling the fractional order of memory, varying the parameters to significantly generate concurrency and Bell’s non-locality. Under the current investigation setup, it is noticeable that the behaviors of the proposed quantifiers are similar to each other, but with a small difference in the amplitude of non-locality with respect to entanglement. Importantly, we show that the most intriguing aspect of this paper is to detect that pair-qubit entanglement and non-locality can be preserved for an indefinite time, which still holds significance in quantum information processing.
Abstract Our work explores the time evolution of entanglement, local quantum uncertainty, and correlated coherence within a system modeled by two coupled double quantum dots. The dynamics is described using a time-fractional Schrödinger equation, which is treated here as a phenomenological effective framework introducing temporal nonlocality and memory-like effects, rather than as a first-principles open-system reduction. We vary the fractional parameter τ , the tunneling amplitudes δ A and δ B , and the inter-dot Coulomb interaction strength V , in order to investigate how these key parameters govern the generation, stabilization, and decay of quantum resources within the system. The obtained results reveal that, for both initial states (separable and partially entangled), fractional dynamics with a small τ rapidly generates entanglement reaching maximal values LN ≈ 1 together with strong non-classical correlations quantified by the local quantum uncertainty. Conversely, when τ is closer to unity, entanglement builds up more slowly but the memory-like effects allow quantum resources to remain significant for a longer time, with the negativity remaining above ≈ 0.6 . We also find that higher Coulomb-interaction frequencies V accelerate the build-up of correlations and stabilize coherence, while a strong tunneling asymmetry degrades entanglement and coherence despite the initial benefits of increasing quantum resources. We find that the standard unitary Schrödinger evolution is recovered analytically at τ = 1 , where the Mittag–Leffler functions reduce to ordinary exponentials, while decreasing τ progressively introduces fractional-time nonlocal (memory-like) effects.
Our work explore the time evolution of entanglement, local quantum uncertainty, and correlated coherence, within a system modeled by two double quantum dots. The dynamics is represented using a time-fractional Schrödinger equation, which includes memory effects in a non-Markovian regime. We vary the fractional parameter τ, the tunneling amplitudes δA and δB, as well as the inter-dot interaction strength V, to investigate how these key parameters govern the generation, stabilization, and decay of quantum resources within the system. The obtained results reveal that, for both initial states, fractional dynamics with a low τ rapidly generates entanglement expecting maximal values LN≈1 and non-classical correlations quantified by local quantum uncertainty. Conversely, higher values of τ lead to slower entanglement but memory effects allow quantum resources to remain significant for a longer time, with the negativity remaining above (≈0.6). We also find that higher interaction frequencies V accelerate correlations and stabilize coherence, while a strong tunneling asymmetry degrades entanglement and coherence despite the initial benefits of increasing quantum resources.
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.
• 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.
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.