## Dynamics of a Josephson Array in a Resonant Cavity

#### E. Almaas and D. Stroud,
Department of Physics, The Ohio State University, Columbus,
Ohio 43210

We derive dynamical equations for a Josephson array coupled to a
resonant cavity by applying the Heisenberg equations of motion to a
model Hamiltonian described by us earlier. By means of a canonical
transformation, we also show that, in the absence of an applied
current and dissipation, our model reduces to one described by
Shnirman et al [Phys. Rev. Lett. 79, 2371
(1997)] for coupled qubits, and that it corresponds to a
capacitive coupling between the array and the cavity mode. From
extensive numerical solutions of the model in one dimension, we find
that the array locks into a coherent, periodic state above a critical
number of active junctions, that the current-voltage characteristics
of the array have self-induced resonant steps (SIRS's), that when
N_a active junctions are synchronized on a SIRS, the energy emitted
into the resonant cavity is quadratic in N_a, and that when a fixed
number of junctions is biased on a SIRS, the energy is linear in the
input power. All these results are in agreement with recent
experiments. By choosing the initial conditions carefully, we can
drive the array into any of a variety of different integer SIRS's. We
tentatively identify terms in the equations of motion which give rise
to both the SIRS's and the coherence threshold. We also find
higher-order integer SIRS's and fractional SIRS's in some simulations.
We conclude that a resonant cavity can produce threshold behavior and
SIRS's even in a one-dimensional array with appropriate experimental
parameters.