  Long Josephson tunnel junction are non-linear transmission lines that allow
propagation of current vortices (fluxons) and electromagnetic waves and are
used in various applications within superconductive electronics. Recently, the
Josephson vortex has been proposed as a new superconducting qubit. We describe
a simple method to create a double-well potential for an individual fluxon
trapped in a long elliptic annular Josephson tunnel junction characterized by
an intrinsic non-uniform width. The distance between the potential wells and
the height of the inter-well potential barrier are controlled by the strength
of an in-plane magnetic field. The manipulation of the vortex states can be
achieved by applying a proper current ramp across the junction. The read-out of
the state is accomplished by measuring the vortex depinning current in a small
magnetic field. An accurate one-dimensional sine-Gordon model for this strongly
non-linear system is presented, from which we calculate the position-dependent
fluxon rest-mass, its Hamiltonian density and the corresponding trajectories in
the phase space. We examine the dependence of the potential properties on the
annulus eccentricity and its electrical parameters and address the requirements
for observing quantum-mechanical effects, as discrete energy levels and
tunneling, in this two-state system.
