Figure 1

(Color online) A photonic module [14] at the center of a photonic chip, implemented here in a photonic band-gap structure. Photons are adiabatically loaded from the first $Q$-switch cavity into the module cavity which contains the atomic system with a differential coupling between photon polarization. Once the interaction is complete photons are out coupled from the second $Q$-switch cavity into the right waveguide mode. The atomic qubit has laser control for initialization and measurement and each $Q$-switch has Stark shift controls.Reuse & Permissions

Figure 2

A three-level atomic system in the central cavity of the photonic module provides the nondemolition measurements at the heart of the scheme. The system is initialized in the state $\mid 1⟩$. With the resonant rf field, the system is pumped to the state $(\mid 1⟩+\mid 2⟩)∕\sqrt{2}$. The cavity mode is coupled to the $\mid 1⟩\to \mid 3⟩$ transition via the Hamiltonian $H=\beta a^{†}a{\sigma }_z$, and we assume no dipole moment for the coupling from ∣2⟩ to ∣3⟩. Introducing a single photon into the cavity mode will induce a phase shift on the atomic state ∣1⟩. The photon is controllably out-coupled from the cavity mode (via $Q$-switched cavities) once the accumulated phase shift reaches $\pi$, hence the atomic system will oscillate between the $(\pm \mid 1⟩+\mid 2⟩)∕\sqrt{2}$ states with each sequential photon. After all photons have passed through the system the rf pulse is applied again and the system measured in the {∣1⟩, ∣2⟩} basis.Reuse & Permissions

Figure 3

(Color online) Preparation of a two-dimensional cluster state in five time steps. Each vertex of the lattice represents a photon in the cluster and vertices are numbered to indicate the time step during which its associated stabilizer operator [see Eq. (4)] is measured. Green vertices represent stabilizer operators that have been measured, orange vertices represent stabilizer operators that are being measured, and red vertices represent stabilizer operators that are yet to be measured. By extending this pattern vertically and horizontally a larger cluster can be prepared.Reuse & Permissions

Figure 4

(Color online) Continuous preparation of a two-dimensional cluster state by the synchronous network. Each cross represents a photonic chip that will measure a stabilizer of the form of Eq. (4) whenever a photon occupies the central vertex. Other symbolic conventions are as in Fig. 3. In every time step after the first four a new column of the two-dimensional cluster state is prepared. Unentangled photons can enter from the left indefinitely and the network can be extended vertically to prepare a larger cluster.Reuse & Permissions

Figure 5

Photonic chip for the synchronous network. Each shaded circle represents a buffer that effects a delay of ${\delta }_t$ and each shaded triangle represents a buffer that effects a delay of ${\delta }_t^{\prime }$. The input and output switches are operated according to Table . Note that line $d$ is a feedback loop that can return photons to the input switch. Before they exit the module photons that entered along lines $a$ and $c$ are delayed by $5{\delta }_t+{\delta }_t^{\prime }$ and photons that entered along line $b$ are delayed by $15{\delta }_t+3{\delta }_t^{\prime }$.Reuse & Permissions

Figure 6

Asynchronous network for the continuous preparation of a two-dimensional cluster state. Every column is an identical set of photonic chips as illustrated in Fig. 7. The time interval between input photons on each row is $2{\delta }_t$ and adjacent rows are staggered by ${\delta }_t$.Reuse & Permissions

Figure 7

One of five identical columns in the asynchronous network of Fig. 6. Each box represents a photonic chip as shown in Fig. 9, all of which are the same. The timing of their operation can be found in Table .Reuse & Permissions

Figure 8

The six switching settings required for the photonic chip in the asynchronous network. Solid and broken lines indicate the presence and absence of a photon, respectively. Parentheses indicate that the photon should be switched to the cavity of the photonic module, no parentheses indicate that the photon should be switched to the bypass line, and $H$ indicates that a Hadamard wave plate is required. These six settings can be achieved using the photonic chip in Fig. 9.Reuse & Permissions

Figure 9

Photonic chip for the asynchronous network, also illustrated in Fig. 1. See Table for details of the routing of photons through the photonic module.Reuse & Permissions

Figure 10

(Color online) Continuous preparation of a two-dimensional cluster state by the asynchronous network. As described in the caption of Table , the label over each column indicates how photons are switched in the photonic modules at that point in time. Red circles indicate photons that have not yet entered any modules and so are unentangled. Orange circles indicate photons that have passed through at least one module but not the entire network. Filled orange circles represent photons that have been used as the center photon in a stabilizer measurement. Lines indicate the creation of entanglement between photons. Green lines and green photons indicate cluster photons exciting the network and their complete stabilizers.Reuse & Permissions