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arXiv:1601.00803 · New Journal of Physics

Quantum revivals and magnetization tunneling in effective spin systems

M Krizanac, D. Altwein, E. Y. Vedmedenko, R. Wiesendanger

Published Mar 18, 2016• 8 citations
arXivHTMLPDF

Abstract

Quantum mechanical objects or nano-objects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump–probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetization tunneling. The quantum revival time has been initially defined as the recurrence time of a total wave-function. Here we show that the quantum revivals of wave-functions and expectation values in spin systems may be quite different which gives rise to a more sophisticated definition of the quantum revival within the realm of experimental research. Particularly, the revival times for integer spins coincide which is not the case for half-integer spins. Furthermore, the quantum revival is found to be shortest for integer ratios between the on-site anisotropy and an external magnetic field paving the way to novel methods of anisotropy measurements. We show that the quantum tunneling of magnetization at avoided level crossing is coherent to the quantum revival time of expectation values, leading to a connection between these two fundamental properties of quantum mechanical spins.

DOI
10.1088/1367-2630/18/3/033029

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Quantum mechanical objects or nano-objects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump–probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetization tunneling. The quantum revival time has been initially defined as the recurrence time of a total wave-function. Here we show that the quantum revivals of wave-functions and expectation values in spin systems may be quite different which gives rise to a more sophisticated definition of the quantum revival within the realm of experimental research. Particularly, the revival times for integer spins coincide which is not the case for half-integer spins. Furthermore, the quantum revival is found to be shortest for integer ratios between the on-site anisotropy and an external magnetic field paving the way to novel methods of anisotropy measurements. We show that the quantum tunneling of magnetization at avoided level crossing is coherent to the quantum revival time of expectation values, leading to a connection between these two fundamental properties of quantum mechanical spins.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
We investigate the reversal of magnetization and the coherence of tunneling when an external magnetic field is rotated instantaneously in systems of a few (N) spin 1/2 particles described by an anisotropic Heisenberg Hamiltonian at T=0. The temporal evolution is calculated by a numerically exact solution of the time-dependent Schrödinger equation, and the mean value in time of each spin component is computed as a function of the magnetic field. The correlation function and the spectrum are analyzed in terms of the macroscopic quantum coherence. Our calculations demonstrate that this model for small (N<11) magnetic particles exhibit collective tunneling of the magnetization only for some specific resonant values of the applied magnetic field, at variance with the Stoner–Wohlfarth model.
The theory for the quantum tunneling of nano-magnets is developed within the intermediate spin framework. Periodic magnetic effects are seen to reflect that associated with a change of flux by a single flux quantum Φ0Φ_{0}Φ0​. Essential are Schrödinger cat wave functions which involve superpositions of different magnitudes of the applied magnetic field. For systems in which the tunneling paths are not co-planar the theory leads to essentially complex magnetic fields although the expectation values of the fields remain real. In general the ground states correspond to minimal uncertainty squeezed states. The degree of squeezing depends on the anisotropy parameters.
Motivated by the experimental observation of resonant tunneling in systems of half-integer spin, such as V15 and Mn4, we study the mechanism of the adiabatic change of the magnetization in systems with time-reversal symmetry. Although tunneling between the degenerate ground states is suppressed in a simple Kramers doublet, some types of perturbational interactions cause an avoided level crossing structure in the energy level structures when the system possesses additional degeneracy of the ground state. We show that a nonadiabatic transition governed by the Landau-Zener-Stückelberg mechanism occurs in many cases, and also find more general cases of adiabatic change of the magnetization in which LZS mechanism cannot be applied directly.
Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.
In the usual quantum tunneling, a low-energy quantum particle penetrates across a physical barrier of higher potential energy, by traversing a classically forbidden region, and finally escapes into another region. In an analogous scenario, a classical particle inside a closed regular region in the phase space is dynamically bound from escaping to other regions of the phase space. Here, the physical potential barrier is replaced by dynamical barriers which separate different regions of the phase space. However, in the quantum regime, the system can overcome such dynamical barriers and escape through them, giving rise to dynamical tunneling. In chaotic Hamiltonian systems, dynamical tunneling refers to quantum tunneling between states whose classical limits correspond to symmetry-related regular regions separated by a chaotic zone between which any classical transport is prohibited. Here, an experimental realization of dynamical tunneling in spin systems is reported using nuclear magnetic resonance (NMR) architecture. In particular, dynamical tunneling in quantum kicked tops of spin-1 and spin-3/2 systems using two- and three-qubit NMR registers is investigated. By extracting time-dependent expectation values of the angular momentum operator components, size-dependent tunneling behavior for various initial states is systematically investigated. Further, by monitoring the adverse effects of dephasing noise on the tunneling oscillations, we assert the importance of quantum coherence in enabling dynamical tunneling.