THE precise manner in which quantum-mechanical behaviour at the microscopic level underlies classical behaviour at the macroscopic level remains unclear, despite seventy years of theoretical investigation. Experimentally, the crossover between these regimes can be explored by looking for signatures of quantum-mechanical behaviour—such as tunneling—in macroscopic systems1. Magnetic systems (such as small grains, spin glasses and thin films) are often investigated in this way2–12 because transitions between different magnetic states can be closely monitored. But transitions between states can be induced by thermal fluctuations, as well as by tunnelling, and definitive identification of macroscopic tunnelling events in these complex systems is therefore difficult13. Here we report the results of low-temperature experiments on a single crystal composed of super-paramagnetic manganese clusters (Mn12-ac), which clearly demonstrate the existence of quantum-mechanical tunnelling of the bulk magnetization. In an applied magnetic field, the magnetization shows hysteresis loops with a distinct 'staircase' structure: the steps occur at values of the applied field where the energies of different collective spin states of the manganese clusters coincide. At these special values of the field, relaxation from one spin state to another is enhanced above the thermally activated rate by the action of resonant quantum-mechanical tunnelling. These observations corroborate the results of similar experiments performed recently on a system of oriented crystallites made from a powdered sample4.

Macroscopic quantum tunnelling of magnetization in a single crystal of nanomagnets

Magnetic Relaxation in Fine‐Particle Systems

The quantum dynamics of mesoscopic or macroscopic systems is always complicated by their coupling to many `environmental' modes. At low T these environmental effects are dominated by localized modes, such as nuclear and paramagnetic spins, and defects (which also dominate the entropy and specific heat). This environment, at low energies, maps onto a `spin bath' model. This contrasts with `oscillator bath' models (originated by Feynman and Vernon) which describe delocalized environmental modes such as electrons, phonons, photons, magnons, etc. The couplings to N spin bath modes are independent of N (rather than the ~O(1/(N )1/2 ) dependence typical of oscillator baths), and often strong. One cannot in general map a spin bath to an oscillator bath (or vice versa); they constitute distinct `universality classes' of quantum environment. We show how the mapping to spin bath models is made, and then discuss several examples in detail, including moving particles, magnetic solitons, nanomagnets, and SQUIDs, coupled to nuclear and paramagnetic spin environments. We then focus on the `central spin' model, which couples a central two-level system to a background spin bath. It is the spin bath analogue of the famous `spin-boson' oscillator model, and describes, e.g., the tunnelling dynamics of nanoscopic and mesoscopic magnets and superconductors. We show how to average over (or `integrate out') spin bath modes, using an operator instanton technique, to find the central spin dynamics. The formal manouevres involve four separate averages - each average corresponds physically to a different `decoherence' mechanism acting on the central spin dynamics. Each environmental spin has its own topological `spin phase', which by interacting with the phase of the central system, decoheres it - this can happen even without dissipation. We give analytic results for the central spin correlation functions, under various conditions. We then describe the application of this theory to magnetic and superconducting systems. Particular attention is given to recent work on tunnelling magnetic macromolecules, where the role of the nuclear spin bath in controlling the tunnelling is very clear; we also discuss other magnetic systems in the quantum regime, and the influence of nuclear and paramagnetic spins on flux dynamics in SQUIDs. Finally, we discuss decoherence mechanisms and coherence experiments in superconductors and magnets. We show that a spin bath environment causes decoherence even in the T 0 limit. Control of this decoherence will be essential in the effort to construct `qubits' for quantum computers.

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Theory of the spin bath

Alternating current (ac) magnetic susceptibility data are presented for six distorted cubane complexes of the composition [MnIVMnIII3O3X] Each of these complexes has a well isolated S = 9/2 ground state There is zero-field splitting (ZFS) in the ground states where D, the axial ZFS parameter, is found to be in the range of −027 to −038 cm-1 As a result of the big spin ground state and appreciable magnetic anisotropy, an out-of-phase ac magnetic susceptibility signal is seen for each of the six Mn4 complexes The out-of-phase ac susceptibility signal reflects slow magnetization relaxation which is taken to indicate that individual molecules are acting as magnets Alternating current susceptibility data are presented for a frozen glass of one of the Mn4 complexes to confirm that the out-of-phase ac signal is associated with isolated molecules The factors that influence whether a given complex can function as a single-molecule magnet are described The above Mn4 complexes represent only the second type

Distorted MnIVMnIII3 Cubane Complexes as Single-Molecule Magnets

For many years physicists thought small structures would be nearly ideal systems in which to explore and manipulate magnetic interactions. On a small enough length scale the interactions between individual atomic spins cause their magnetic moments to align in the ordered pattern of a single domain, without the complication of domain walls separating regions of varying orientation. For particle sizes at or below that of a single domain, many theoretical models of dynamical behavior predict simple, stable magnets with controllable classical properties. However, as with advances in semiconductor physics, the process of miniaturizing magnetic materials has unexpectedly revealed fascinating new classical and quantum mechanical phenomena. Even the simplest magnetic system, the isolated single‐domain particle, exhibits a wealth of exotic behavior that pushes us to the limits of our present understanding of the fundamentals of magnetism.

Complex dynamics of mesoscopic magnets

Magnetic relaxation experiments constitute a unique method of determining the nature of fluctuations in dissipative magnetic systems. At high temperatures these fluctuations are thermal and strongly temperature dependent. At low temperatures, where quantum fluctuations dominate, magnetic relaxation becomes independent of temperature. Such behavior has been observed in many systems. In this review we emphasize the study of low temperature relaxation in ferromagnetic nanoparticles, layers, and multilayers (including ‘‘domain wall junctions’’), and large single crystals. The results of magnetic relaxation experiments are shown to agree with theoretical predictions of quantum tunneling of the magnetization. When dissipation becomes important, in large and complex systems, a time dependent WKB exponent needs to be introduced.

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Quantum tunneling in magnetic systems of various sizes (invited)

The magnetic relaxation of a set of small ferromagnetic particles of Tb0.5Ce0.5Fe2 of mean diameter 150 A has been measured at low temperature (down to 50 mK) and with a field up to 8 Tesla. A change between thermally activated and temperature-independent relaxation has been observed near Tc = 0.6 K. All the results obtained at low temperature are consistent with theoretical predictions of Macroscopic Quantum Tunnelling with weak dissipation in a ferromagnetic system.

https://iopscience.iop.org/article/10.1209/0295-5075/19/7/013/pdf

Macroscopic Quantum Tunnelling Effects of Bloch Walls in Small Ferromagnetic Particles

Mesoscopic quantum tunneling in small ferromagnetic particles

The time delayed reversal of the magnetization of magnetic system with distributions of energy barriers, is in general strongly dependent on temperature at high temperature. At low temperature it becomes independent of the temperature. This result is interpreted in terms of a crossover between a thermal regime, in which the irreversible rotation of the magnetization is thermally activated, and a quantum regime in which it proceeds by quantum tunnelling. In this paper we show some results obtained in nanoparticles, layers and multilayers. In particular an analogy with Josephson junction will be made for the case of the sandwiches (domain wall junction). The role of dissipation will be discussed.

M. Uehara

Papers

Quantum tunneling in magnetic systems of various sizes (invited)

Macroscopic Quantum Tunnelling Effects of Bloch Walls in Small Ferromagnetic Particles

Mesoscopic quantum tunneling in small ferromagnetic particles

Quantum tunnelling in magnetic particles, layers and multilayers

Mesoscopic quantum tunneling in small ferromagnetic particles