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American Journal of Mathematics

Long-time dynamics of coherent states in strong magnetic fields

Grégory Boil, San Vũ Ngọc

Published Jan 1, 2021• 6 citations
Publisher

Abstract

We consider the Schr\"odinger evolution of strongly localized wave packets under the magnetic Laplacian in the plane $\Bbb{R}^2$. When the initial energy is low, we obtain a precise control, in Schwartz seminorms, of the propagated states for times of order $1/\hbar$, where $\hbar$ is Planck's constant. In this semiclassical regime, we prove that the initial particle will always split into multiple coherent states, each one following the average dynamics of the guiding center motion but at its own speed, demonstrating a purely quantum ``ubiquity'' phenomenon.

DOI
10.1353/ajm.2021.0045

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We consider the Schr"odinger evolution of strongly localized wave packets under the magnetic Laplacian in the plane R2\Bbb{R}^2R2. When the initial energy is low, we obtain a precise control, in Schwartz seminorms, of the propagated states for times of order 1/ℏ1/\hbar1/ℏ, where ℏ\hbarℏ is Planck's constant. In this semiclassical regime, we prove that the initial particle will always split into multiple coherent states, each one following the average dynamics of the guiding center motion but at its own speed, demonstrating a purely quantum ``ubiquity'' phenomenon.
Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.
We present the quantal dynamics of electronic wave packets in the hydrogen atom in the presence of circularly polarized microwave and magnetic fields in regions of classical stability. Whereas wave packets may disperse without the stabilizing influence of a magnetic field, in its presence stable motion can be maintained for wave packets that are localized either at global equilibria that are either a maximum or minimum in a zero-velocity surface. Because these extrema are locally harmonic their vacuum states are truly coherent states in the original sense of Schr\"odinger.
We study the time propagation of an initially localized wave packet for a generic one-dimensional time-independent system, using the ``nonlinear wave-packet dynamics'' [S. Tomsovic and E. J. Heller, Phys. Rev. Lett. 67, 664 (1991)], a semiclassical approximation using a local linearization of the wave packet in the vicinity of classical reference trajectories. Several reference trajectories are needed to describe the behavior of the full wave packet. The introduction of action-angle variables allows us to obtain a simple analytic expression for the autocorrelation function, and to show that a universal behavior (quantum collapses, quantum revivals, etc.) is obtained via interferences between the reference trajectories. A connection with the standard WKB approach is established. Finally, we apply the nonlinear wave-packet dynamics to the case of the hydrogen atom in a weak magnetic field, and show that the semiclassical expressions obtained by nonlinear wave-packet dynamics are extremely accurate. \textcopyright{} 1996 The American Physical Society.
In this article, we consider fixed spin-1/2 particles interacting through the quantized electromagnetic field in a constant magnetic field. We give approximate evolutions of coherent states. This uses spins-photon classical Hamiltonian mechanics. These approximations enable to derive that the approximate average fields and spins follow Maxwell-Bloch equations with a current density coming from spins. In addition, we obain a law concerning the evolution of the approximate average number of photons. Next, we provide stationary points of the spins-photon Hamiltonian when the spin particles belong to an orthogonal plane to the constant magnetic field. This allows for the construction of quasimodes with spins colinear to the constant magnetic field. Finally, a quasimode with an arbitrary high accuracy is built up and its first order radiative correction is computed.
We study numerically the one-dimensional quantum evolution of a coherent state when the particle is bound by a nonlinear force. Snapshots of the Husimi function show that there exist times where the initial Gaussian distribution in phase space splits into chains of 2,3,…2,3,\dots{}2,3,… distributions of similar form. During their finite lifetime the members of such a string are equally distributed along the classical orbit which passes through the center of the initial distribution, and they move along this curve with the corresponding velocity. This effect, which is generic for integrable bounded motion, is explained in terms of the quantum analogs of action and angle variables.
A Gaussian wave packet confined to move on a plane perpendicular to a magnetic field remains a Gaussian wave packet in its time evolution. The average position and momentum follow the Ehrenfest equations which are identical to the classical Hamilton equations. A set of nonlinear equations decoupled from the Ehrenfest equation is derived for the parameters describing the time evolution of the density distribution and phases of a wave packet. Explicit solutions are then obtained when the “internal” angular momentum of the wave packet vanishes. In this case it is shown that the motion of the wave packet is a superposition of a translational motion, a rotation and a vibration.