Periodic and discrete Zak bases

Abstract

Weyl's unitary operators for displacement in position and momentum commute with one another if the product of the elementary displacements equals Planck's constant. Then, their common eigenstates constitute the Zak basis, with each state specified by two phase parameters. Accordingly, the transformation function from the position basis to the Zak basis maps the Hilbert space on the line onto the Hilbert space on the torus. This mapping is one to one provided that the Zak basis states are periodic functions of their phase parameters, but then the mapping cannot be continuous on the whole torus. With the periodicity of the Zak basis enforced, the basis has a double Fourier series. The Fourier coefficients identify a discrete basis which complements the periodic Zak basis to form a pair of mutually unbiased bases. The discrete basis states are the common eigenstates of the two complementary partners to the two unitary displacement operators. These partner operators are of angular-momentum type, with integer eigenvalues, and generate the fundamental rotations of the torus. Conversely, the displacement operators are the ladder operators for their partners. For each consistent phase convention for the periodic Zak basis, and thus for the line-onto-torus mapping, there is a corresponding discrete Zak basis and a corresponding pair of partner operators. Examples of particular interest are the conventions that give a continuous mapping in one phase parameter or are symmetric in both phase parameters. The latter emphasizes the Heisenberg-Weyl symmetry between position and momentum. We discuss briefly the relation between the Zak bases and Aharonov's modular operators. Finally, as an application of the Zak operators for the torus, we mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits that are potentially entangled. © 2006 IOP Publishing Ltd.