Second day of Lux Out-of-Equilibrium, after the pre-workshop. Unfortunately, for some reasons, I only have comments on one seminar.
– Quantal and classical shortcuts to adiabacity (C. Jarzynski, NESM): The classical adiabatic theorem states that, under a slow modification of an Hamiltonian, an arbitrary set of initial conditions along an energy shell evolves into a set of initial conditions into the new energy shell. Faster transformations do not respect this condition. By “shortcut to adiabacity” one means how to perturb an Hamiltonian achieving the benefits of adiabatic driving in finite time, by suitably adding “counter-adiabatic” terms. A strong shortcut is one that drags the initial states along the perturbed eigenstate all along the transformation, a weak shortcut is one where only the final states finally coincide with the deisred ones. Solutions tp the first problem are very awkward. Strong shortcuts for transitionless quantum driving have been given by Demirplak & Rice (JPC 2003), Berry (J Phys A 2009) and Jarzynski (PRL 1995), and they involve some sort of projection along the direction where eigenstates are mapped. This extra term in the Hamiltonian generates a transformation of the phase of the wave function that turns out to be the Berry phase. The classical analog can be found by replacing commutators by Poisson brackets and eigenvalues of the Hamiltonian by microcanonical averages (Jarzynski, PRA 2013). For the weak shortcut, Chris displayed some funny videos showing that the initial conditions eventually come back to the energy shell, and for fast enough drivings the perturbation needed to adjust the Hamiltonian can be quite dramatic. The advantage of the weak shortcut is that the extra-term is local, while the strong shortcut leads to nonlocal corrections even for the simplest system (e.g. particle in a square well with varying width). Apparently there is a stochastic analogue of these techniques.