Quantum coherence and classical yet quantum materials

Because I haven't seen this explicitly discussed anywhere, I think it's worth pointing out that everyday materials around us demonstrate some features of coherence and decoherence in quantum mechanics.

Quantum mechanics allows superposition states to exist - an electron can be in a state with a well-defined momentum, but that is a superposition of all possible position states along some wavefront.   As I mentioned here, empirically a strong measurement means coupling the system being measured to some large number of degrees of freedom, such that we don't keep track of the detailed evolution of quantum entanglement.  In my example, that electron hits a CCD detector and interacts locally with the silicon atoms in one particular pixel, depositing its charge and energy there and maybe creating additional excitations.  That "collapses" the state of the electron into a definite position.  This kind of measurement is a two-way street - a quantum system leaves its imprint on the state of the measuring apparatus, and the measurement changes the quantum system's state.

One fascinating aspect of the emergence of materials properties is that we can have systems that act both very classically (as I'll explain in a minute) and also very quantum mechanically at the same time, for different aspects of the material.  

If I have a piece of aluminum sitting in front of me (like the case of my laptop) that hunk of metal does not show up in a superposition of positions or orientations.  It surely seems to have a definite position and orientation, and if I looked closely at a given moment I would find the aluminum atoms arranged in crystal lattices, with clear atomic positions.  Somehow, the interactions of the aluminum with the broader environment have washed out the quantumness of the atomic positions.  (Volumes have been written about interpretations of quantum mechanics and "the measurement problem", as I touched on here.  In the many-worlds view, we live in a particular branch of reality, while there are other branches that correspond to other possible positions and orientations of the aluminum piece, one for each possible outcome of a positional or orientational measurement.  I'm not going to touch on the metaphysics behind how to think about this here, except to say that somehow the position of the aluminum empirically acts classically.)

What about the electrons in the piece of crystalline aluminum?  Well, we've learned about band structure.  The allowed quantum states of electrons in a periodic potential consists of bands of states.  Each of these states has an associated crystal momentum \(\hbar \mathbf{k}\), and there is some relationship between energy and crystal momentum, \(E(\mathbf{k})\).  There are values of energy between the bands that do not correspond to any allowed electronic quantum states in that periodic lattice.  In aluminum, the electronic states are filled up to states in the middle of a band.  (One can be more rigorous that this, but it's beside the point I'm trying to make.)  Interestingly, the electrons in those filled states energetically far away from the highest occupied states are coherent - they are wavelike and extended, and indeed the Bloch waves themselves are a direct consequence of quantum interference throughout the periodic lattice.  Why haven't these electrons somehow decohered into some classical situation?   If you imagine some dynamic interaction that would "measure" the location, say, of one of those electrons, you have to consider some final state in which the electron would end up.  Because all of the states at nearby energies are already occupied, and the electrons obey the Pauli Principle, there is no low-energy (on the scale of, say, the thermal energy available, \(k_{\mathrm{B}}T\)) path to decoherence.  You'd need much larger energy/higher momentum/shorter wavelength processes to reach those electrons and scatter them to empty final states (as in ARPES).

By that argument, though, the electrons that are energetically close to the Fermi level in metals should be vulnerable to decoherence - they have energetically nearby states into which they can be scattered, and a variety of comparatively low energy scattering processes (electron-electron scattering, electron-phonon scattering).   Is  that true?  Yes.  This is exactly why you can't see quantum interference effects in electrical conduction in metals at room temperature, but at low temperatures you can see interference effects like universal conductance fluctuations and understand the effects of decoherence on those effects quantitatively.

I find it remarkable that a piece of aluminum can show both the emergence of classical physics (the piece of aluminum is not spatially delocalized) while having quantum coherent degrees of within.  Understanding how to engineer robust quantum coherent systems despite the tendency toward environmental decoherence is key to future quantum information science and technology.