We show that a quantum system with long-range interparticle interaction can be described by invoking fractional calculus tools. The system under consideration is nonlocal exciton-phonon quantum dynamics on a 1D lattice. It has been shown that long-range power-law exciton-exciton interaction leads to a nonlocal integral term in the motion equation of an exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction with noninteger power, the nonlocal integral term can be expressed through a spatial derivative of fractional order. Considering exciton-phonon dynamics with long-range exciton-exciton interaction, we have obtained the system of two coupled equations, where one is the quantum fractional differential equation for the exciton subsystem while the other is a standard differential equation for the phonon subsystem. It has been found that the system of two coupled equations can be further simplified to come up with the following fractional nonlinear equations of motion: nonlinear fractional Schrödinger equation, nonlinear Hilbert-Schrödinger equation, fractional generalization of Zakharov system, and fractional Ginzburg-Landau equation. The appearance of fractional differential equations in the continuum limit of lattice dynamics allows us to apply powerful tools of fractional calculus to study nonlocal quantum phenomena.
We show that a quantum system with long-range interparticle interaction can be described by invoking fractional calculus tools. The system under consideration is nonlocal exciton-phonon quantum dynamics on a 1D lattice. It has been shown that long-range power-law exciton-exciton interaction leads to a nonlocal integral term in the motion equation of an exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction with noninteger power, the nonlocal integral term can be expressed through a spatial derivative of fractional order. Considering exciton-phonon dynamics with long-range exciton-exciton interaction, we have obtained the system of two coupled equations, where one is the quantum fractional differential equation for the exciton subsystem while the other is a standard differential equation for the phonon subsystem. It has been found that the system of two coupled equations can be further simplified to come up with the following fractional nonlinear equations of motion: nonlinear fractional Schrödinger equation, nonlinear Hilbert-Schrödinger equation, fractional generalization of Zakharov system, and fractional Ginzburg-Landau equation. The appearance of fractional differential equations in the continuum limit of lattice dynamics allows us to apply powerful tools of fractional calculus to study nonlocal quantum phenomena.
Fractional dynamics of open quantum systems with power-law memory is described. We show that the quantum processes can demonstrate a universal dynamic behavior caused by the effects of power-law fading memory. We propose generalizations of time-ordered exponential and time-ordered product for processes with powerlaw memory. As examples, we consider equations of the N-level open quantum system with memory, quantum oscillator with friction and memory. The solutions of the corresponding fractional differential equations with derivatives of noninteger orders are obtained.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.
We propose and investigate a generalized Schrödinger equation by introducing a fractional Riesz derivative to account for anomalous transport and a memory kernel to describe temporal nonlocal effects. Additionally, we include a long-range interactions term, modeled by an integral operator, which captures spatially extended interactions. Using the Green function approach, we derive analytical solutions and explore their implications in the time-space domain. Our findings reveal novel quantum phenomena arising from the interplay of fractional dynamics, nonlocal potentials, and memory effects, including the emergence of new local maxima in the evolution of Green's functions and distinct localization behaviors.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.
A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.
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.