Unified generation and fast emission of arbitrary single-photon multimode $W$ states

We propose a unified and deterministic scheme to generate arbitrary single-photon multimode $W$ states in circuit QED. A three-level system (qutrit) is driven by a pump-laser pulse and coupled to $N$ spatially separated resonators. The coupling strength for each spatial mode $g_i$ totally decide the generated single-photon N-mode $W$ state $\vert W_N \rangle=\frac{1}{A}\sum_{i=1}^N g_i|0_1 0_2 \cdots 1_i 0_{i+1}\cdots 0_N\rangle$, so arbitrary $\vert W_N \rangle$ can be generated just by tuning $g_i$. We could not only generate $W$ states inside resonators but also release them into transmission lines on demand. The time and fidelity for generating (or emitting) $\vert W_N \rangle$ can both be the same for arbitrary $N$. Remarkably, $\vert W_N\rangle$ can be emitted with probability reaching $98.9\%$ in $20-50$ ns depending on parameters, comparable to the recently reported fastest two-qubit gate ($30-45$ ns). Finally, the time evolution process is convenient to control since only the pump pulse is time-dependent.