  To generate a NOON state with a large photon number $N$, the number of
operational steps could be large and the fidelity will decrease rapidly with
$N$. Here we propose a method to generate a new type of quantum entangled
states, $(|NN00\rangle+|00NN\rangle)/\sqrt{2}$ called "double NOON" states,
with a setup of two superconducting flux qutrits and five circuit cavities.
This scheme operates essentially by employing a two-photon process, i.e., two
photons are simultaneously and separately emitted into two cavities when each
coupler qutrit is initially in a higher-energy excited state. As a consequence,
the "double" NOON state creation needs only $N$+2 operational steps. One
application of double NOON states is to get a phase error of $1/(2N)$ in phase
measurement. In comparison, to achieve the same error, a normal NOON state of
the form $(|2N,0\rangle+|0,2N\rangle)/\sqrt{2}$ is needed, which requires at
least $2N$ operational steps to prepare by using the existing schemes. Our
numerical simulation demonstrates that high-fidelity generation of the double
NOON states with $N\leq 10$ even for the imperfect devices is feasible with the
present circuit QED technique.
