Perfect pattern formation of neutral atoms in an addressable optical lattice J. Vala, 1,3 A. V. Thapliyal, 2,3 S. Myrgren, 1 U. Vazirani, 2,3 D. S. Weiss, 4 and K. B. Whaley 1,3 1 Department of Chemistry and Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA 2 Department of Computer Science, University of California, Berkeley, California 94720, USA 3 Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720-5070, USA 4 Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802-6300, USA sReceived 10 July 2003; revised manuscript received 15 June 2004; published 18 March 2005d We propose a physical scheme for formation of an arbitrary pattern of neutral atoms in an addressable optical lattice. We focus specifically on the generation of a perfect optical lattice of simple orthorhombic structure with unit occupancy, as required for initialization of a neutral atom quantum computer. The scheme employs a compacting process that is accomplished by sequential application of two types of operations: a flip operator that changes the internal state of the atoms, and a shift operator that selectively moves the atoms in one internal state along the lattice principal axis. Realizations of these elementary operations and their physical limitations are analyzed. The complexity of the compacting scheme is analyzed and we show that this scales linearly with the number of lattice sites per row of the lattice. DOI: 10.1103/PhysRevA.71.032324 PACS numberssd: 03.67.Lx, 32.80.Pj, 42.50.Vk I. INTRODUCTION Neutral atoms trapped in an optical lattice constitute an attractive system for implementation of scalable quantum computation f1–3g, simulation of many-body systems f4g, and implementation of topological quantum computing f5–9g. In standard optical lattices, small lattice constants present a serious obstacle to implementing quantum compu- tation, since it is difficult to address individual qubits with an external field. An optical lattice with a large lattice constant is in principle addressable, and can allow for the quantum state manipulation of individual atoms by an optical field. High addressability and controllability and low decoherence make addressable lattices promising candidates for large- scale quantum computer implementation. The present work focuses on preparation of the initializa- tion of an addressable optical lattice for the purposes of quantum computing. The objective is a perfectly filled, regu- lar optical lattice, with each site occupied by a single atom in its motional ground state and in a specific internal state. We consider one-dimensional s1Dd, orthorhombic two- dimensional s2Dd, and three-dimensional s3Dd lattices. After loading and laser-cooling atoms in the optical lattice, half the sites have a single atom and half are vacant. In order to use this system for scalable quantum computation, a perfect lat- tice with each site occupied by a single atom is required. We propose here an efficient, feasible scheme for compacting the optical lattice, i.e., for removing vacant sites to the edge of the lattice, thus creating a smaller lattice, but one more suit- able for quantum computation. The scheme presented here can as well be used to make arbitrary patterns of neutral atoms in an addressable optical lattice. These include lattices with fractional occupation, a specific translational and rotational lattice symmetry, a bro- ken symmetry, and heteroatomic patterns. Another important property of the scheme is that it can be applied recursively to reach any desired accuracy of the pattern formation. After a large number of elementary operations, the lattice can be cooled and imaged again. The remaining defects can be eliminated by repeating the compacting scheme. This recur- sion increases the total pattern formation time by the total duration of additional cooling and imaging cycles, but does not result in any increase in the scaling of the pattern forma- tion, i.e., the algorithmic complexity of the scheme is un- changed. The possibility of preparing any homoatomic or heteroatomic pattern of neutral atoms to an arbitrarily high degree of perfection makes this scheme attractive for initial- ization of quantum simulations of condensed phase systems, in addition to initialization of quantum computation. Before proceeding, we briefly discuss sid the possibility of quantum computation using an imperfect pattern of atoms with vacant sites, and siid other possible approaches to prepa- ration of an optical lattice with single occupancy at each site. One can imagine starting with a known imperfect lattice pat- tern, and then instead of simplifying the distribution, devis- ing a quantum algorithm that accounts for the known loca- tions of the vacancies. Our analysis of this procedure suggests that even if the vacancy locations are known, they will cause bottlenecks in quantum information flow. These bottlenecks eventually occur when the computer size, i.e., the number of atomic qubits, or equivalently, the number of occupied lattice sites, is scaled up. In fact, we maintain that the probability of finding a “good” sublattice, where “good” means that each filled site is connected to another filled site, is exponentially small for any constant filling factor f . This is because the probability that there will be “insurmountable” blocks of defects sgapsd in any chosen sublattice increases rapidly with its size. The site percolation threshold has the following values for lattices of various dimensions: 1 s1Dd, 0.59 s2Dd, and 0.31 s3Ddf10,11g. So an initial filling factor f < 0.5 does exceed the percolation threshold in a three- dimensional lattice. While this implies a nonzero probability that distant qubits are connected, it does not guarantee that they are connected via independent routes, nor even that they are connected at all. Perhaps more importantly, the mapping of a quantum algorithm onto an imperfectly filled lattice may be a hard classical computational problem. Based on these considerations, an imperfect lattice structure does not appear PHYSICAL REVIEW A 71, 032324 s2005d 1050-2947/2005/71s3d/032324s13d/$23.00 ©2005 The American Physical Society 032324-1