The role of singular control in frictionless atom cooling in a harmonic trapping potential

Carl Wieman, David E. Pritchard, David J. Wineland

Dionisis Stefanatos, Shin Li

In this article we study the frictionless cooling of atoms trapped in a harmonic potential, while minimizing the transient energy of the system. We show that in the case of unbounded control, this goal is achieved by a singular control, which is also the time-minimal solution for a “dual” problem, where the energy is held fixed. In addition, we examine briefly how the solution is modified when there are bounds on the control. The results presented here have a broad range of applications, from the cooling of a Bose-Einstein condensate confined in a harmonic trap to adiabatic quantum computing and finite time thermodynamic processes.

Dionisis Stefanatos, Heinz Schäettler, Jr-Shin Li

Frictionless atom cooling in harmonic traps is formulated as a time-optimal control problem, and a synthesis of optimal controlled trajectories is obtained.

Harold J. Metcalf, P. van der Straten

Dionisis Stefanatos, Justin Ruths, Jr-Shin Li

In this article we formulate frictionless atom cooling in harmonic traps as a time-optimal control problem, permitting imaginary values of the trap frequency for transient time intervals during which the trap becomes an expulsive parabolic potential. We show that the minimum time solution has a ``bang-bang'' form, where the frequency jumps suddenly at certain instants and then remains constant, and calculate estimates of the minimum cooling time for various numbers of such jumps. A numerical optimization method based on pseudospectral approximations is used to obtain suboptimal realistic solutions without discontinuities, which may be implemented experimentally.

Quan Long, Yuzhu Wang