state is carried out via intermediate excitation of a Ryd- berg excited state and makes use of the so-called Rydberg blockade mechanism [12, 13]. It is thus possible to ini- tialize the ensemble with all atoms populating a definite internal ”reservoir” state, denoted | 0 〉 in the following, and to subsequently encode, in principle, any quantum superposition states of an N -bit register based on N dif- ferent internal states, labeled | 1 〉 , | 2 〉 , . . . , | N 〉 , with each qubit value set to zero if the state is unpopulated, and to unity, if the corresponding state is populated by pre- cisely one atom. This is ensured by the universality of the collective quantum computing proposal with Ryd- berg blockade [11]. Here, however, we also have the free- dom to process the state of the atomic ensemble between light emission events. We can thus reuse the atomic lev- els and generate entangled light states with more than N qubits. Our current task with the Rydberg blocked ensemble is a more dedicated one than quantum com- puting, and rather than appeal to universality, we will describe explicit schemes which in few, efficient steps use the Rydberg interaction, the physical properties of the in- ternal atomic level structure, and the collective emission properties of the ensemble to provide the optical states of interest. laser traveling wave fields are taken into account. The state | 10 . . . 0 〉 , for instance, with one atom in | 1 〉 and zero atoms in | 2 〉 to | N 〉 is where K ≫ N is the number of atoms in the ensemble, | j 〉 k means that the k th atom is in the internal state | j 〉 , r j is the position of the j th atom, and k 0 is the resulting wave vector of the light pulses used to bring the ensemble to the above state, i.e., k 0 = ∑ $_{i}$δn$_{i}$ k $_{i}$, where k i is the wave vector of the i th laser pulse and δn$_{i}$ is 1 if a photon is absorbed and − 1 if a photon is emitted during the i th pulse. If we apply a π -pulse with wave vector k e to transfer the atom in the state | 1 〉 to the optically excited state | e 〉 , the amplitude for emission of a photon with wave vector k and polarization e k when the atom decays to the state | 0 〉 is proportional to K − 1 / $^{2}$∑ K j =1 〈 0 | e k · d | e 〉 e i ( k ′ $_{0}$− k ) · $^{r$_{j}$}$, where d is the dipole operator and k ′ 0 = k 0 + k $_{e}$. The transition probability P ( k ) is thus proportional to The Rydberg blockade mechanism [12] utilizes the large dipole-dipole interaction between atoms in certain Rydberg states that are not separated by more than about 10 µ m. Due to this interaction, states with two or more atoms in Rydberg excited states have energies that are significantly different from the sum of the single atom excitation energies, and therefore resonant excitation of a few µ m sized ensemble from the reservoir ground atomic state will coherently drive the system towards the state with a single coherently distributed Rydberg excitation in the ensemble. The Rabi frequency for this transition is collectively enhanced by the factor √$_{K}$$_{, where}$ K is the number of atoms. This enhancement and the block- ing of further excitation has been seen for a pair of atoms ( K = 2) [14, 15], and blocking and enhanced coupling has also been observed in larger samples of atoms [16]. Af- ter a π pulse on the coherent transition, there is a single ensemble Rydberg excitation and one can subsequently drive this collectively shared population into a low ly- ing state, Fig. 1(b). To provide the coupling in a fully selective manner between the desired internal states, we suggest to use polarization selection rules and the optical frequency resolution of different Zeeman ground states of the atoms in a homogeneous magnetic field switched on during state selective processes. In the following, the ket | n$_{1}$n$_{2}$ . . . n$_{N}$ 〉 denotes a state with n$_{j}$ ∈ { 0 , 1 } atoms in the single-atom state | j 〉 , j = 1 , 2 , . . . , N , and the remaining atoms in the reser- voir state | 0 〉 . In writing this state, phase factors arising due to the different positions of the atoms within the We now turn to a description of explicit schemes to generate particular quantum states of light. We start with the M -mode GHZ state ( | + 〉 1 | + 〉 $_{2}$. . . | + 〉 M + |−〉 1 |−〉 $_{2}$. . . |−〉 $_{M}$) $_{/}$√$_{2,}$ where | + / −〉 m denotes a single/zero-photon state in the m th mode. To prepare this state, we first apply a π/ 2-pulse between | 0 〉 and a Rydberg level | r $^{′}$〉 followed by a π -pulse between | r $^{′}$〉 and the ground state level | j $^{′}$〉 , which leads to a superposition of having zero and one The factor 1 /K suppresses emission in all directions ex- cept k ≈ k ′ $_{0}$, for which the sum is approximately K and P ( k ) ∝ K . c | k | must also equal the atomic excitation frequency. The factor 1 /K is replaced by 1 /K 2 if the atom decays to a different hyperfine level, and such tran- sitions are thus also suppressed. With an ensemble of about 1000 atoms it is possible both to satisfy the re- quirement of all atoms being at most 10 µ m apart, while the ensemble is still large enough to ensure directional emission of the photon [17, 18]. A detailed analysis of light emission from various arrangements of atoms can be found in [10, 19]. To map a general superposition of the states | n$_{1}$n$_{2}$ . . . n$_{N}$ 〉 to N temporally distinct light pulses, we simply transfer one internal state population after the other to the optically excited state as illustrated schematically for N = 5 in Fig. 1(a), and we can even renew the populations of the atomic states during the conversion process if desired. P ( k ) ∝ 1 K ∣ ∣ K ∑ j =1 e i ( k ′ $_{0}$− k ) · $_{r$_{j}$}$∣ ∣ ∣ ∣$^{2}$$^{.}$ (2) | 10 . . . 0 〉 = 1 √$_{K}$ K ∑ j =1 e i k 0 · $^{r$_{j}$}$| 0 〉 1 | 0 〉 2 · · · | 1 〉 j · · · | 0 〉 $_{K}$, (1) 2 II. STATE PREPARATION AND EMISSION III. GHZ AND BELL STATES