Controllability in tunable chains of coupled harmonic oscillators

We prove that temporal control of the strengths of springs connecting N harmonic oscillators in a chain provides complete access to all Gaussian states of N-1 collective modes. The proof relies on the construction of a suitable basis of cradle modes for the system. An iterative algorithm to reach any desired Gaussian state requires at most 3N(N-1)/2 operations. We illustrate this capability by engineering squeezed pseudo-phonon states - highly non-local, strongly correlated states that may result from various nonlinear processes. Tunable chains of coupled harmonic oscillators can be implemented by a number of current state-of-the-art experimental platforms, including cold atoms in lattice potentials, arrays of mechanical micro-oscillators, and coupled optical waveguides.