##### E. M.Chudnovsky, L. Gunther – Quantum Tunneling of Magnetization in Small Ferromagnetic Particles. Phys. Rev. Lett. 60, 661 (1988)

Macroscopic Quantum Tunneling (MQT) correspond to the tunneling of a macroscopic variable trough the barrier between two minima of the effective potential of a macroscopic system.

We consider a small ferromagnetic particle enough small to form a single magnetic domain (→ Stoner-Wohlfarth model). Equilibrium easy directions of the magnetic moment $\mathbf{M}$ correspond to the local minima of the energy:

$E=-\mathbf{M}\cdot \mathbf{H}+A_{ik}M_iM_k+B_{iklm}M_iM_kM_lM_m+\ldots$

Since $\mathbf{M}$ is an axial vector (aka pseudvector) any minimum of the energy $H_0=0$ is at least twice degenerate with respect to two opposite directions of $\mathbf{M}$.

If the exchange interaction is so much strong to suppress the dynamics of the individual spins of the particle, $\mathbf{M}$ can be regarded as a single quantum variable.

The projection of $\mathbf{M}$ onto one of the easy directions in general does not commute with the energy $E$, this means that the eigenvalues of the projection in general are not conserved quantum numbers even at $H_0=0$ (this is not surprising because the magnetic anisotropy appears as a result of  relativistic interactions → Landau, Lifshitz – Electrodynamics of Continuous Media).

As consequence of the last statement, $\mathbf{M}$ can tunnel between the energy minima.

Tunneling removes the degeneracy of the ground state and put the particle into a state of lower energy wherein $\langle \mathbf{M} \rangle = 0$ and $\langle \mathbf{M}^2 \rangle = M_0^2$. Angular brackets denote quantum average.

In presence of a magnetic field, the potential energy $E$ has, in general, one absolute minimum and several local minima so that the problem of MQT from metastable states arises.

The first reference to quantum tunneling of the magnetic moment was proposed as explanation of the experimental data indicating that transitions between different orientations of the magnetic moment in single domain nickel particles do not disappear completely with a decrease in temperature at absolute zero.

Macroscopic Quantum Coherence (experimental point of view)
For two successive measurements of $\mathbf{M}$ separated by the time interval $\Delta t$ one should obtain the effect of macroscopic quantum coherence:

$\langle \mathbf{M}(t)\mathbf{M}(t+\Delta t) \rangle = M_0^2\cos(2P\Delta t)$

at $T=0$, $H=0$ and neglecting dissipation. $\hbar P$ is the tunneling matrix element.

For both macroscopic quantum coherence and MQT the key quantity is the tunneling rate $P$, which should be calculated in terms of the macroscopic parameters describing single-domain particles.