Deterministic multi-mode photonic device for quantum information processing Anne E. B. Nielsen 1 and Klaus Mølmer 1 $^{1}$Lundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, Aarhus University, DK-8000 ˚ Arhus C, Denmark (Dated: December 3, 2018) We propose the implementation of a light source, which can deterministically generate a rich variety of multi-mode quantum states. The desired states are encoded in the collective population of different ground hyperfine states of an atomic ensemble and converted to multi-mode photonic states by excitation to optically excited levels followed by cooperative spontaneous emission. Among our examples of applications, we demonstrate how two-photon entangled states can be prepared and implemented in a protocol for reference frame free quantum key distribution and how one- dimensional as well as higher-dimensional cluster states can be produced. I. INTRODUCTION arXiv:1001.1429v2 [quant-ph] 17 Mar 2010 Quantum states of light are efficient carriers of quan- tum information with applications in quantum com- puting, quantum cryptography, and quantum networks. Their most serious drawback is the absence of suitable non-linear interactions which can produce and transform the desired states. Measurement back action presents one means to accommodate the required non-linearities and paves the way for universal quantum computing with the tools of linear optics, but certain resource states such as single-photon states [1] or superpositions of coherent states [2] are required, while even more involved states, such as cluster states [3], are needed to perform suc- cessfully with high probability. An efficient method to prepare general quantum states of light deterministically would constitute a significant step forward for optical quantum information processing. There are a number of different strategies to produce non-classical states of light. Squeezed states and multi- mode states with quantum correlated amplitudes can be produced directly by optical parametric oscillators and beam splitters, while these states can be further trans- formed into states with non-Gaussian amplitude distri- butions by conditional measurements [4–6]. The prob- abilistic nature of the schemes, however, limits the va- riety of states that can be produced by this method. Another successful approach uses laser excitation of an atomic ensemble with all atoms in the same single atom state and detection of a spontaneously emitted photon to conditionally prepare a superposition state with different atoms populating the final state of the spontaneous emis- sion process. The atomic state is generated at a random instant and the ensemble is then ready to determinis- tically emit a single photon at the moment when the experimentalist couples the singly occupied state to an optically excited state [7–9]. In the present paper, we propose a source, which is capable of producing a large variety of quantum states of light. We focus on coding of quantum bits for quan- tum computing and communication, and we use multiple spatial and temporal modes to provide multi-bit registers with different entangled states. The basic idea is to pre- pare the equivalent state in an ensemble of cold atoms and, in a step-wise process, convert the atomic state to a quantum state of light, Fig. 1(a), as proposed for two- mode states in [10]. To avoid the requirement of individ- ual addressing of the atoms and to facilitate the efficient conversion to light, we encode the quantum state in the collective populations of different internal states of the atoms as proposed in [11]. The transfer of precisely one atom in the ensemble to an initially unoccupied internal FIG. 1: (Color online) (a) Step-wise transfer of a quantum state encoded in a cold atomic ensemble to light. The initial state of the ensemble is an arbitrary superposition of states with zero or one atom in the single-atom levels | 1 〉 to | 5 〉 , and during the conversion process, occupation by one atom trans- lates into a single photon in a corresponding spatio-temporal light mode. For the case illustrated, the states | 1 〉 , | 2 〉 , | 4 〉 , and | 5 〉 are occupied initially. (b) Preparation of the atomic state. The excitation from | 0 〉 to a Rydberg state | r 〉 is off-resonant if more than a single atom is excited, and a collectively dis- tributed single excitation can be subsequently transferred to a unit atomic population of the single-atom ground state level | j 〉 . (c) The feeding of state | j 〉 by the attempted transfer 0 → | r 〉 → | j 〉 can also be made conditional on the popula- tion of another state | j $^{′}$〉 if that state is excited to a Rydberg state | r $^{′}$〉 interacting with | r 〉 . | j 〉 | 0 〉 | r 〉 1 2 | j 〉 | 0 〉 | j $^{′}$〉 | r 〉 | r $^{′}$〉 1 2 3 4 (a) (b) (c) 4 3 2 1 | 1 〉 | 2 〉 | 3 〉 | 4 〉 | 5 〉 | 0 〉 | e 〉