Open-loop linear control of quadratic Hamiltonians with applications

The quantum harmonic oscillator is one of the most fundamental objects in physics. We consider the case where it is extended to an arbitrary number modes and includes all possible terms that are bilinear in the annihilation and creation operators, and we assume that we also have an arbitrary time-dependent drive term that is linear in those operators. Such a Hamiltonian is very general, covering a broad range of systems, including quantum optics, superconducting circuit QED, quantum error correcting codes, Bose-Einstein condensates, atomic wave packet transport beyond the adiabatic limit, and many others. We examine this situation from the point of view of quantum control, making use of optimal control theory to determine what can be accomplished, both when the controls are arbitrary and when they must minimize some cost function. In particular we develop a class of analytical pulses. We then apply our theory to a number of specific topical physical systems to illustrate its use and provide explicit control functions, including the case of the continuously driven conditional displacement gate.