# Macroscopic Quantum Tunneling & Coherence

##### 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 correspond to the local minima of the energy:

Since is an axial vector (aka pseudvector) any minimum of the energy is at least twice degenerate with respect to two opposite directions of .

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

The projection of onto one of the easy directions in general does not commute with the energy , this means that the eigenvalues of the projection in general are not conserved quantum numbers even at (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, 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 and . Angular brackets denote quantum average.

In presence of a magnetic field, the potential energy 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 separated by the time interval one should obtain the effect of macroscopic quantum coherence:

at , and neglecting dissipation. is the tunneling matrix element.

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