  Plasmons are fundamental excitations of metals which can be described in
terms of electron dynamics, or in terms of the electromagnetic fields
associated with them. In this work we develop a quantum description of plasmons
in a double layer structure, treating them as confined electromagnetic modes of
the structure. The structure of the resulting bosonic Hamiltonian indicates the
presence of virtual plasmons of the individual layers which appear as quantum
fluctuations in the ground state. For momenta smaller than the inverse
separation between layers, these modes are in the ultrastrong coupling regime.
Coherence terms in the Hamiltonian indicate that modes with equal and opposite
momenta are entangled. We consider how in principle these entangled modes might
be accessed, by analyzing a situation in which the conductivity of one of the
two layers suddenly drops to zero. The resulting density matrix has a large
entanglement entropy at small momenta, and modes at $\pm \mathbf{q}$ that are
inseparable. More practical routes to releasing and detecting entangled
plasmons from this system are considered.
