(a) | 1 〉 | 2 〉 | 0 〉 | 3 〉 | 4 〉 | 5 〉 | 6 〉 | 7 〉 | 8 〉 | e L 〉 | e 〉 | e R 〉 | r 〉 | r $^{′}$〉 1 1 2 2 3 6 4 5 (b) (c) FIG. 4: (Color online) Generation of a two-dimensional clus- ter state. Initially, one atom is moved to | 3 〉 , and one atom is moved to | 7 〉 . (a) In each iteration step, a π/ 2-pulse is ap- plied to the transition between | 2 〉 and | 3 〉 (pulse 1) and to the transition between | 6 〉 and | 7 〉 (pulse 2). The π -pulses 3-6 then change the phase of the state by π unless | 3 〉 and | 7 〉 are both unoccupied. Atoms are moved from | 0 〉 to | 1 〉 , | 4 〉 , | 5 〉 , and | 8 〉 conditioned on | 3 〉 , | 2 〉 , | 7 〉 , and | 6 〉 being unoccupied, respectively, and photons are emitted by transferring first the populations in | 1 〉 and | 4 〉 and subsequently the populations in | 5 〉 and | 8 〉 to e L 〉 and | e R 〉 . (b) The first 8 emitted photons and (c) the corresponding graph. 4 by weaving more one-dimensional cluster states together, and one can also form tubes and three-dimensional struc- tures. In reality, the finite number of internal states of the atoms restricts the possibilities, and to comply with this, we suggest to emit the light pulses from a one- or two-dimensional array of individually addressable ensem- bles with separations that are small enough to allow long distance Rydberg phase gates [24] between nearest neigh- bor ensembles. In conclusion, we have proposed a multi-purpose, multi-mode photonic source and constructed explicit schemes to prepare GHZ states, double trine states, and cluster states. We have only considered states with at most one photon per mode, but we anticipate that the same mechanism allows emission of more photons into the same mode [10]. The collective encoding general- izes naturally to this case by allowing atomic state pop- ulations exceeding one. Experiments on Rydberg me- diated one- and two-bit operations are still in their in- fancy, but according to theoretical estimates, their fideli- ties may be high enough that entangled photonic states with tens of modes may be prepared. A Rydberg state mediated quantum gate may typically be accomplished in about 10 − 6 s, and recent experimental results with pulsed laser fields even demonstrate Rabioscillations to Rydberg states taking place on a time scale of about 1 ns [25]. The time required to emit a single photon can be as short as 10 − 7 s [18]. These time scales are much faster than Rydberg state lifetimes on the order of milliseconds. The shape of the laser pulses used to excite the atoms leaves some freedom to tailor the shape and duration of the emitted pulses, and the emission direction is control- lable and may vary from mode to mode. The number of photons that can be emitted by a single ensemble is limited in practice, because the interactions with the en- semble are not perfectly symmetric. The collective state of the ensemble may, however, be brought back to the symmetric state by optical pumping of all atoms into the reservoir state or by a transfer of the atomic internal state populations via the Rydberg gate mechanism to a new ensemble from which the photon emission can proceed. quently transfer the populations in | 1 〉 and | 4 〉 to | e$_{L}$ 〉 and | e$_{R}$ 〉 , respectively. After emission of a photon thus entangled with the atomic population remaining in states | 2 〉 and | 3 〉 , one applies again a π/ 2-pulse between | 2 〉 and 3 followed by a new Rydberg controlled feeding of the populations in | 1 〉 and | 4 〉 and emission of the next pho- ton. This procedure is iterated and one gradually builds up a one-dimensional cluster state of light pulses. After the desired number of iterations, the cluster is decoupled from the atomic state by measuring the state of the last emitted photon in the basis {| L 〉 , | R 〉} . The behavior of one-dimensional cluster states can be efficiently simulated on a classical computer, and they are hence not useful resource states for cluster state quantum computing [23]. Our method can, however, be generalized to higher-dimensional cluster states by in- volving more internal levels of the atoms. To generate, for instance, the two-dimensional cluster state shown in Fig. 4(c), we can use eight hyperfine states to prepare two one-dimensional cluster states in parallel and then add a controlled phase gate in each iteration step to obtain the bonds between them. The details of the processing of the state of the ensemble between each emission of pho- tons are shown in Fig. 4(a), and the emission follows the same procedure as for the one-dimensional cluster state. The cluster state is decoupled from the atomic ensemble by measuring the state of the last two emitted photons. Larger two-dimensional cluster states can be obtained This work was supported by ARO-IARPA. Acknowledgments V. CONCLUSION [1] E. Knill, R. Laflamme, and G. Milburn, Nature (London) 409 , 46 (2001). [2] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Phys. Rev. A 68 , 042319 (2003). [3] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86 , 5188 (2001). [4] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Opt. Express 15 , 3568 (2007).