Navigating the Hilbert space of nonseparable elastic states in arrays of periodically coupled one-dimensional waveguides

A planar array of three one-dimensional elastic waveguides mutually coupled periodically along their length and driven externally is shown theoretically and numerically to support nonseparable superpositions of states. These states are the product of Bloch waves describing the elastic displacement along the waveguides and spatial modes representing the displacement across the array of waveguides. For a system composed of finite length waveguides, the frequency, relative amplitude, and phase of the external drivers can be employed to selectively excite specific groups of discrete product modes. The periodicity of the coupling is used to fold bands enabling superpositions of states that span the complete Hilbert space of product states. We show that we can realize a transformation from one type of nonseparable superposition to another one that is analogous to a nontrivial quantum gate. This transformation is also interpreted as the complex conjugation operator in the space of the complex amplitudes of individual waveguides.