Control of Quantum SystemsThere's much interest among certain control engineers, physicists and physical chemists in trying to integrate ideas from control design with those of quantum dynamics. The applications range from laser chemistry (molecular vibrations control with femtosecond laser pulses) and NMR (nuclear spin control) to quantum computing. As a control problem, the dynamics are governed by the Schroedinger equation where u(t) is an external (control) signal (e.g. an electromagnetic field strength in the laser chemistry case, or a magnetic field in the NMR case), is the nominal (internal) Hamiltonian, and is the “control” Hamiltonian representing the coupling of the external control to the internal dynamics (e.g. dipole moment coupling). Notice that this is a socalled “bilinear system”. These equations look deceptively simple. However, this hides the fact that infinitedimensional bilinear systems are a universal class representing effectively all dynamical systems with an input (see “Myths, ….” in the talks page). Optimal State Transfer
Timescale Separation in Optimal ControlIn fact one can be much more precise. We can show that in the limit of large transfer times , the optimal control has the following form where the exponential term represents the Bohr frequency between the and states of the system. Thus, up to first order in , the optimal control is of the form of Bohr frequencies modulated by slowly varying (realtive to the time scale ) envelopes. This makes precise the notion of time scale separation for such problems, and reduces the optimal control problem (for terms) to that of designing the envelopes. The slow envelopes obey optimalcontroltype differential equations (which are derived via an averaging procedure) that are orders of magnitude less stiff than the original ones, and thus much easier to obtain numerically.
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