  We make a detailed theoretical description of the two-dimensional nature of a
dc-SQUID, analyzing the coupling between its two orthogonal phase oscillation
modes. While it has been shown that the mode defined as "longitudinal" can be
initialized, manipulated and measured, so as to encode a quantum bit of
information, the mode defined as "transverse" is usually repelled at high
frequency and does not interfere in the dynamics. We show that, using typical
parameters of existing devices, the transverse mode energy can be made of the
order of the longitudinal one. In this regime, we can observe a strong coupling
between these modes, described by an Hamiltonian providing a wide range of
interesting effects, such as conditional quantum operations and entanglement.
This coupling also creates an atomic-like structure for the combined two mode
states, with a V-like scheme.
