Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Theoretical and Mathematical Physics

Quantum Analogue of Unstable Limit Cycles of a Periodically Perturbed Inverted Oscillator

V. V. Chistyakov

Published Jan 1, 2019• 2 citations
Publisher

Abstract

To study the quantum analogue of classical limit cycles, we consider the behavior of a particle in a negative quadratic potential perturbed by a sinusoidal field. We propose a type of wave function asymptotically satisfying the operator of initial conditions and still admitting analytic integration of the nonstationary Schrödinger equation. The solution demonstrates that for certain perturbation phases determined by the forcing frequency and the initial indeterminacy of the coordinate, the wave-packet center temporarily stabilizes near the potential maximum for approximately two “natural periods” of the oscillator and then moves to infinity with bifurcations in the drift direction. The effect is not masked by packet spreading, because the packet undergoes anomalous narrowing (collapse) to a size of the order of the characteristic length on the above time interval and its unbounded spreading begins only after this.

DOI
10.1134/s0040577919010021

Related papers

On the Evolution of Converging Wave Packet of an Inverted Quantum Oscillator Driven by Homogeneous Harmonic Field

V.V. Chistyakov

The problem investigated refers to periodically driven 1D quantum inverted harmonic oscillator (IHO) with the Hamiltonian of . The model is used widely in huge quantum applications concerned unstable molecular complexes and ions under laser light affection. Non-stationary Schrödinger equation (NSE) was solved analytically and numerically by means of Maple 17 with initial wave function (w.f.) of generalized Gaussian type. This one described the converging 1D probability flux and fitted well the quantum operator of initial conditions (IC). For the IC one can observe, first, the collapse of w.f. packet into extremely narrow 1D space interval of length and, second, its spreading back up to its starting half width, and all that - at dimensionless times. At certain phases j defined by W and s0 the wave packet center displayed nonharmonic oscillating behavior near some slowly drifting space position within this time interval and after that leaved onto infinity while the unlimited packet spreading. And the phases themselves served as bifurcation points separating the NSE solutions with the outgoing to from those with. In “resonant” case of the values obeyed an inverted Fermi-Dirac formula of; for differing the asymptotic of obeyed well classical law.

Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика•Jan 1, 2017•1 citation

On quantum analogue of instable oscillating states of an inverted oscillator in external poly-harmonic field

V. V. Chistyakov

Abstract Nonstationary Schroedinger equation (NSE) is solved analytically and numerically to study a phenomenon of dynamical stabilization of the inverted oscillator driven by polyharmonic in time and spatially uniform force with specially chosen phase shifts. It is shown that for Gaussian wave packet asymptotically fitting the initial condition (IC) it occurs temporary delay of the packet center about top of the parabolic potential for about 2 fundamental time periods followed by the center bifurcation.

Journal of Physics Conference Series•Nov 1, 2021•0 citations

Forced quantum inverted oscillator

Pavel A. Golovinski

New exact and asymptotic results for a quantum inverted oscillator, driven by the variable external force, are presented. To illustrate the advantages of our approach, we applied the obtained propagator to the descriptions of evolution the initial Gaussian wave packet under arbitrary time-dependent force or a δ-pulse action. The alternative problem of tunneling a particle through a parabolic barrier under the influence of a low-frequency harmonic force we solve in the quasistatic approximation. Going to the Heisenberg picture in the Caldeira-Leggett model, we describe the influence of a time-dependent force on an open inverted oscillator. The resulting dynamics is presented in the form of combination the evolution of the average value of a coordinate operator and the stochastic diffusion, being induced the wave packet spreading and the Brownian motion.

arXiv:1905.04591•May 11, 2019•0 citations

Almost-periodic wave packets and wave packets of invariant shape

Detlef Mentrup, Marshall Luban

Wave packets of one-dimensional, single-particle quantum systems exhibit a rich diversity of dynamical behavior. We focus on confining potentials where the entire energy spectrum is discrete so that wave packets do not spread indefinitely with time and instead exhibit characteristics of almost-periodic functions. For the harmonic oscillator, for which all wave packets are necessarily strictly periodic, Senitzky has shown that is possible to construct an infinite class of normalized wave packets that maintain a fixed shape while oscillating sinusoidally with time. We present two alternate and instructive methods for deriving this result using coherent states and the displacement operator.

American Journal of Physics•May 12, 2003•3 citations

Time behavior of a Gaussian wave packet accompanying the generalized coherent state for the inverted oscillator

Mustapha Maamache, Yacine Bouguerra, Jeong Ryeol Choi

A Gaussian wave packet of the inverted oscillator is investigated using the invariant operator method together with the unitary transformation method. A simple wave packet directly derived from the eigenstates of the invariant operator of the system corresponds to a plane wave that is fully delocalized. However, we can construct a weighted wave packet in terms of such plane waves, which corresponds to a Gaussian wave. This wave packet is associated with the generalized coherent state, which can be crucially utilized for investigating the classical limit of quantum wave mechanics. Various quantum properties of the system, such as fluctuations of the canonical variables, the uncertainty product, and the motion of the wave packet or quantum particle, are analyzed by means of this wave packet. We have confirmed that the time behavior of such a wave packet is very similar to the counterpart classical state. The wave packet runs away from the origin in the positive or negative direction in the 1D coordinate depending on the condition of the initial state. We have confirmed that this wave packet not only moves acceleratively but also spreads out during its propagation.

Progress of Theoretical and Experimental Physics•Jun 1, 2016•8 citations

Nonlinear Schrödinger equation and dissipative quantum dynamics in periodic fields

Youhong Huang, Shih‐I Chu, Joseph O. Hirschfelder

The nonlinear dynamics of dissipative quantum systems in periodic fields is studied in the framework of a Gisin-like nonlinear Schr\"odinger equation with deterministic nonunitary quantum friction terms describing the system-bath couplings. The virtue of this nonunitary evolution is that it is compatible with Dirac's superposition principle and the Hilbert-space structure of quantum kinematics. Floquet theory and the generalized Van Vleck nearly degenerate perturbation method are used to facilitate both analytical and numerical solutions. Closed-form analytic solutions can be obtained in the long-time average approximation or within the rotating-wave approximation. The methods are applied to the study of dissipative quantum dynamics of two-level systems driven by intense periodic fields. It is found that the system asymptotically approaches a limit cycle (whose orientation is subject to the quantum friction constraint), regardless of the strength of the perturbed fields and the nonlinearity constant, indicating quantum suppression of classical chaos. Further, each point of the limit cycle is found to be an attractor and \ensuremath{\psi}(t) exhibits a fractal-like evolution pattern in the course of time. The structure of the limit cycle depends strongly upon field intensity and frequency as well as the order of nonlinear multiphoton transitions. The power spectrum of the Bloch vector trajectory exhibits a dynamical symmetry inherent in the dissipative system and in the asymptotic limit cycle. A theoretical analysis is presented for the understanding of the origin and the role of the dynamical symmetry.

Physical Review A•Oct 1, 1989•18 citations
To study the quantum analogue of classical limit cycles, we consider the behavior of a particle in a negative quadratic potential perturbed by a sinusoidal field. We propose a type of wave function asymptotically satisfying the operator of initial conditions and still admitting analytic integration of the nonstationary Schrödinger equation. The solution demonstrates that for certain perturbation phases determined by the forcing frequency and the initial indeterminacy of the coordinate, the wave-packet center temporarily stabilizes near the potential maximum for approximately two “natural periods” of the oscillator and then moves to infinity with bifurcations in the drift direction. The effect is not masked by packet spreading, because the packet undergoes anomalous narrowing (collapse) to a size of the order of the characteristic length on the above time interval and its unbounded spreading begins only after this.
The problem investigated refers to periodically driven 1D quantum inverted harmonic oscillator (IHO) with the Hamiltonian of . The model is used widely in huge quantum applications concerned unstable molecular complexes and ions under laser light affection. Non-stationary Schrödinger equation (NSE) was solved analytically and numerically by means of Maple 17 with initial wave function (w.f.) of generalized Gaussian type. This one described the converging 1D probability flux and fitted well the quantum operator of initial conditions (IC). For the IC one can observe, first, the collapse of w.f. packet into extremely narrow 1D space interval of length and, second, its spreading back up to its starting half width, and all that - at dimensionless times. At certain phases j defined by W and s0 the wave packet center displayed nonharmonic oscillating behavior near some slowly drifting space position within this time interval and after that leaved onto infinity while the unlimited packet spreading. And the phases themselves served as bifurcation points separating the NSE solutions with the outgoing to from those with. In “resonant” case of the values obeyed an inverted Fermi-Dirac formula of; for differing the asymptotic of obeyed well classical law.
Abstract Nonstationary Schroedinger equation (NSE) is solved analytically and numerically to study a phenomenon of dynamical stabilization of the inverted oscillator driven by polyharmonic in time and spatially uniform force with specially chosen phase shifts. It is shown that for Gaussian wave packet asymptotically fitting the initial condition (IC) it occurs temporary delay of the packet center about top of the parabolic potential for about 2 fundamental time periods followed by the center bifurcation.
New exact and asymptotic results for a quantum inverted oscillator, driven by the variable external force, are presented. To illustrate the advantages of our approach, we applied the obtained propagator to the descriptions of evolution the initial Gaussian wave packet under arbitrary time-dependent force or a δ-pulse action. The alternative problem of tunneling a particle through a parabolic barrier under the influence of a low-frequency harmonic force we solve in the quasistatic approximation. Going to the Heisenberg picture in the Caldeira-Leggett model, we describe the influence of a time-dependent force on an open inverted oscillator. The resulting dynamics is presented in the form of combination the evolution of the average value of a coordinate operator and the stochastic diffusion, being induced the wave packet spreading and the Brownian motion.
Wave packets of one-dimensional, single-particle quantum systems exhibit a rich diversity of dynamical behavior. We focus on confining potentials where the entire energy spectrum is discrete so that wave packets do not spread indefinitely with time and instead exhibit characteristics of almost-periodic functions. For the harmonic oscillator, for which all wave packets are necessarily strictly periodic, Senitzky has shown that is possible to construct an infinite class of normalized wave packets that maintain a fixed shape while oscillating sinusoidally with time. We present two alternate and instructive methods for deriving this result using coherent states and the displacement operator.
A Gaussian wave packet of the inverted oscillator is investigated using the invariant operator method together with the unitary transformation method. A simple wave packet directly derived from the eigenstates of the invariant operator of the system corresponds to a plane wave that is fully delocalized. However, we can construct a weighted wave packet in terms of such plane waves, which corresponds to a Gaussian wave. This wave packet is associated with the generalized coherent state, which can be crucially utilized for investigating the classical limit of quantum wave mechanics. Various quantum properties of the system, such as fluctuations of the canonical variables, the uncertainty product, and the motion of the wave packet or quantum particle, are analyzed by means of this wave packet. We have confirmed that the time behavior of such a wave packet is very similar to the counterpart classical state. The wave packet runs away from the origin in the positive or negative direction in the 1D coordinate depending on the condition of the initial state. We have confirmed that this wave packet not only moves acceleratively but also spreads out during its propagation.
The nonlinear dynamics of dissipative quantum systems in periodic fields is studied in the framework of a Gisin-like nonlinear Schr\"odinger equation with deterministic nonunitary quantum friction terms describing the system-bath couplings. The virtue of this nonunitary evolution is that it is compatible with Dirac's superposition principle and the Hilbert-space structure of quantum kinematics. Floquet theory and the generalized Van Vleck nearly degenerate perturbation method are used to facilitate both analytical and numerical solutions. Closed-form analytic solutions can be obtained in the long-time average approximation or within the rotating-wave approximation. The methods are applied to the study of dissipative quantum dynamics of two-level systems driven by intense periodic fields. It is found that the system asymptotically approaches a limit cycle (whose orientation is subject to the quantum friction constraint), regardless of the strength of the perturbed fields and the nonlinearity constant, indicating quantum suppression of classical chaos. Further, each point of the limit cycle is found to be an attractor and \ensuremath{\psi}(t) exhibits a fractal-like evolution pattern in the course of time. The structure of the limit cycle depends strongly upon field intensity and frequency as well as the order of nonlinear multiphoton transitions. The power spectrum of the Bloch vector trajectory exhibits a dynamical symmetry inherent in the dissipative system and in the asymptotic limit cycle. A theoretical analysis is presented for the understanding of the origin and the role of the dynamical symmetry.