Preparation and Detection of Magnetic Quantum Phases in Optical SuperlatticesA. M. Rey,1V. Gritsev,2I. Bloch,3E. Demler,1,2and M. D. Lukin1,21Institute for Theoretical Atomic, Molecular and Optical Physics, Harvard-Smithsonian Center of Astrophysics,Cambridge, Massachusetts 02138, USA2Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA3Johannes Gutenberg-Universita¨t, Institut fu¨r Physik, Staudingerweg 7, 55099 Mainz, Germany(Received 11 April 2007; revised manuscript received 31 July 2007; published 5 October 2007)We describe a novel approach to prepare, detect, and characterize magnetic quantum phases in ultracoldspinor atoms loaded in optical superlattices. Our technique makes use of singlet-triplet spin manipulationsin an array of isolated double-well potentials in analogy to recently demonstrated control in quantum dots.We also discuss the many-body singlet-triplet spin dynamics arising from coherent coupling betweennearest neighbor double wells and derive an effective description for such systems. We use it to study thegeneration of complex magnetic states by adiabatic and nonequilibrium dynamics.DOI: 10.1103/PhysRevLett.99.140601 PACS numbers: 05.50.+q, 03.67.Mn, 05.30.Fk, 05.30.JpRecent advances in the manipulations of ultracold atomsin optical lattices have opened new possibilities for ex-ploring many-body systems [1]. A particular topic of con-tinuous interest is the study of quantum magnetism in spinsystems [2– 4]. By loading spinor atoms in optical latticesit is now possible to ‘‘simulate’’ spin models in controlledenvironments and to explore novel spin orders.In this Letter we describe a new approach for prepara-tion and probing of many-body magnetic quantum statesthat makes use of coherent manipulation of singlet-tripletpairs of ultracold atoms loaded in deep period-two opticalsuperlattices. Our approach makes use of a spin dependentenergy offset between the double-well minima to com-pletely control and measure the spin state of two-atompairs, in a way analogous to the recently demonstratedmanipulations of coupled electrons in quantum doubledots [5]. As an example, we show how this techniqueallows one to detect and analyze antiferromagnetic spinstates in optical lattices. We further study the many-bodydynamics that emerge when tunneling between nearestneighbor double wells is allowed. As two specific ex-amples, we show how a set of singlet atomic states canbe evolved into singlet-triplet cluster-type states and into amaximally entangled superposition of two antiferromag-netic states. Finally, we discuss the use of our projectiontechnique to probe the density of spin defects (kinks) instates prepared via equilibrium and nonequilibriumdynamics.The key idea of this work is illustrated by considering apair of ultracold atoms with two relevant internal states,which we identify with spin up and down  " , # in anisolated double-well (DW) potential as shown in Fig. 1.Bydynamically changing the optical lattice parameters, it ispossible to completely control this system and measure itin an arbitrary two-spin basis. For concreteness, we firstfocus on the fermionic case. The physics of this system isgoverned by three sets of energy scales: (i) the on siteinteraction energy U  U"#between the atoms, (ii) thetunneling energy of the  species Jand (iii) the energydifference between the two DW minima 2for each ofthe two species. The  index in J and  is due to the factthat the lattice that the " and # atoms feel can be engineeredto be different by choosing laser beams of appropriatepolarizations, frequencies, phases, and intensities. In thefollowing we assume that the atoms are strongly interact-ing, U  J, and that effective vibrational energy of eachwell @!0is the largest energy scale in the system @!0U, , J, i.e., deep wells.Singlet jsi and triplet jti states form the natural basis forthe two-atom system. The relative energies of these statescan be manipulated by controlling the energy bias between the two wells. In the unbiased case (U  2)0.0 0.2 0.4 0.6 0.8 1.0U0-U-U-4J /U2UJ∆∆(b)∆∆(c)ζY= -Energy∆∆/U(a)(0,2)s + (2,0)s(1,1)s(1,1)t(0,2)s(1,1)s,(1,1)tJ∆∆FIG. 1 (color online). (a) Energy levels of fermionic atoms in aspin independent double well as =U is varied: While in theregime 2 U, 1; 1jsi is the lowest energy state, when 2 *U, 0; 2jsi becomes the state with lowest energy. (b) In spindependent potentials the two species feel different lattice pa-rameters. (c) Restricted to the (1,1) subspace  acts as aneffective magnetic field gradient and couples jsi and tzi.PRL 99, 140601 (2007)PHYSICAL REVIEW LETTERSweek ending5 OCTOBER 20070031-9007=07=99(14)=140601(4) 140601-1only states with one atom per site (1,1) are populated, asthe large atomic repulsion energetically suppresses doubleoccupancy [here, labels (m, n) indicate the integer numberof atoms in the left and right sites of the DW]. For weaktunneling and spin independent lattices (J" J# J, "# ) the states 1; 1jsi and 1; 1jti are nearly degen-erated. The small energy splitting between them is4J2=U, with the singlet being the low energy state[Fig. 1(a)]. As  is increased the relative energy of doublyoccupied states (0, 2) decreases. Therefore, states 1; 1jsiand 0; 2jsi will hybridize. When 2 * U the atomicrepulsion is overwhelmed and consequently the 0; 2jsibecomes the ground state. At the same time, the Pauliexclusion results in a large energy splitting @!0betweendoubly occupied singlet and triplet states as the latter musthave an antisymmetric orbital wave function. Hence,1; 1jti does not hybridize with its doubly occupied coun-terpart, and its relative energy becomes large as comparedto the singlet state. Thus the energy difference betweensinglet and triplet states can be controlled using .Further control is provided by changing Jand inspin dependent lattices [see Fig. 1(b)]. Specifically, let usnow consider the regime 2 U in which only (1,1)subspace is populated. Within this manifold we define [6]jsi^syj0i12pj"#i  j#"i, jtzi^tyzj0i12pj"#ij#"i,jtxi^tyxj0i12pj""i  j##i, jtyi^tyyj0ii2pj""ij##i.Here^tyand^s are operators that create triplet and singletstates from the vacuum j0i (state with no atoms). Theysatisfy