NATURE|VOL 415|3 JANUARY 2002|www.nature.com 39articlesQuantum phase transition from asuper¯uid to a Mott insulator ina gas of ultracold atomsMarkus Greiner*, Olaf Mandel*, Tilman Esslinger², Theodor W. HaÈnsch* & Immanuel Bloch** Sektion Physik, Ludwig-Maximilians-UniversitaÈt, Schellingstrasse 4/III, D-80799 Munich, Germany, and Max-Planck-Institut fuÈr Quantenoptik, D-85748 Garching,Germany²Quantenelektronik, ETH ZuÈrich, 8093 Zurich, Switzerland............................................................................................................................................................................................................................................................................For a system at a temperature of absolute zero, all thermal ¯uctuations are frozen out, while quantum ¯uctuations prevail. Thesemicroscopic quantum ¯uctuations can induce a macroscopic phase transition in the ground state of a many-body system when therelative strength of two competing energy terms is varied across a critical value. Here we observe such a quantum phase transitionin a Bose±Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potentialdepth of the lattice is increased, a transition is observed from a super¯uid to a Mott insulator phase. In the super¯uid phase, eachatom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atomsare localized at individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in theexcitation spectrum. We can induce reversible changes between the two ground states of the system.A physical system that crosses the boundary between two phaseschanges its properties in a fundamental way. It may, for example,melt or freeze. This macroscopic change is driven by microscopic¯uctuations. When the temperature of the system approaches zero,all thermal ¯uctuations die out. This prohibits phase transitions inclassical systems at zero temperature, as their opportunity to changehas vanished. However, their quantum mechanical counterparts canshow fundamentally different behaviour. In a quantum system,¯uctuations are present even at zero temperature, due to Heisen-berg's uncertainty relation. These quantum ¯uctuations may bestrong enough to drive a transition from one phase to another,bringing about a macroscopic change.A prominent example of such a quantum phase transition is thechange from the super¯uid phase to the Mott insulator phase in asystem consisting of bosonic particles with repulsive interactionshopping through a lattice potential. This system was ®rst studiedtheoretically in the context of super¯uid-to-insulator transitions inliquid helium1. Recently, Jaksch et al.2have proposed that such atransition might be observable when an ultracold gas of atoms withrepulsive interactions is trapped in a periodic potential. To illustratethis idea, we consider an atomic gas of bosons at low enoughtemperatures that a Bose±Einstein condensate is formed. Thecondensate is a super¯uid, and is described by a wavefunctionthat exhibits long-range phase coherence3. An intriguing situationappears when the condensate is subjected to a lattice potential inwhich the bosons can move from one lattice site to the next only bytunnel coupling. If the lattice potential is turned on smoothly, thesystem remains in the super¯uid phase as long as the atom±atominteractions are small compared to the tunnel coupling. In thisregime a delocalized wavefunction minimizes the dominant kineticenergy, and therefore also minimizes the total energy of the many-body system. In the opposite limit, when the repulsive atom±atominteractions are large compared to the tunnel coupling, the totalenergy is minimized when each lattice site is ®lled with the samenumber of atoms. The reduction of ¯uctuations in the atomnumber on each site leads to increased ¯uctuations in the phase.Thus in the state with a ®xed atom number per site phase coherenceis lost. In addition, a gap in the excitation spectrum appears. Thecompetition between two terms in the underlying hamiltonian(here between kinetic and interaction energy) is fundamental toquantum phase transitions4and inherently different from normalphase transitions, which are usually driven by the competitionbetween inner energy and entropy.The physics of the above-described system is captured by theBose±Hubbard model1, which describes an interacting boson gas ina lattice potential. The hamiltonian in second quantized form reads:H  2 J^hi;jiÃa²iÃaj^ieiÃni12U^iÃniÃni2 11HereÃaiandÃa²icorrespond to the bosonic annihilation andcreation operators of atoms on the ith lattice site,ÃniÃa²iÃaiisthe atomic number operator counting the number of atoms onthe ith lattice site, and eidenotes the energy offset of the ithlattice site due to an external harmonic con®nement of theatoms2. The strength of the tunnelling term in the hamiltonianis characterized by the hopping matrix element between adja-cent sites i,j J  2 ed3xwx 2 xi 2~2=2=2m  Vlatxwx 2 xj,where wx 2 xi is a single particle Wannier function localized tothe ith lattice site (as long as ni< O1), Vlat(x) indicates the opticallattice potential and m is the mass of a single atom. The repulsionbetween two atoms on a single lattice site is quanti®ed by the on-siteinteraction matrix element U 4p~2a=mejwxj4d3x, with abeing the scattering length of an atom. In our case the interactionenergy is very well described by the single parameter U, due to theshort range of the interactions, which is much smaller than thelattice spacing.In the limit where the tunnelling term dominates the hamilto-nian, the ground-state energy is minimized if the single-particlewavefunctions of N atoms are spread out over the entire lattice withM lattice sites. The many-body ground state for a homogeneoussystem (ei const:) is then given by:jªSFiU0~^Mi1Ãa²i !Nj0i 2Here all atoms occupy the identical extended Bloch state. Animportant feature of this state is that the probability distribution© 2002 Macmillan Magazines Ltdfor the local occupation niof atoms on a single lattice site ispoissonian, that is, its variance is given by VarnihÃnii. Further-more, this state is well described by a macroscopic wavefunctionwith