Lewis-Riesenfeld invariants and transitionless tracking algorithm

Different methods have been recently put forward and implemented experimentally to inverse engineer the time dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via non-adiabatic shortcuts. In the "transitionless tracking algorithm" proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on designing first a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian connecting these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are in fact strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study as explicit examples the expansions of time-dependent harmonic traps and state preparation of two level systems.Comment: 9 pages, 2 figure

Engineering fast and stable splitting of matter waves

When attempting to split coherent cold atom clouds or a Bose-Einstein condensate (BEC) by bifurcation of the trap into a double well, slow adiabatic following is unstable with respect to any slight asymmetry, and the wave "collapses" to the lower well, whereas a generic fast chopping splits the wave but it also excites it. Shortcuts to adiabaticity engineered to speed up the adiabatic process through non-adiabatic transients, provide instead quiet and robust fast splitting. The non-linearity of the BEC makes the proposed shortcut even more stable

Optimal trajectories for efficient atomic transport without final excitation

We design optimal harmonic-trap trajectories to transport cold atoms without final excitation, combining an inverse engineering techniqe based on Lewis-Riesenfeld invariants with optimal control theory. Since actual traps are not really harmonic, we keep the relative displacement between the center of mass and the trap center bounded. Under this constraint, optimal protocols are found according to different physical criteria. The minimum time solution has a "bang-bang" form, and the minimum displacement solution is of "bang-off-bang" form. The optimal trajectories for minimizing the transient energy are also discussed.Comment: 10 pages, 7 figure